6.6. SAMS: Sensitivity Analysis Module for SCALE
K. B. Bekar, J. D. McDonnell, B. T. Rearden, W. A. Wieselquist
ABSTRACT
The Sensitivity Analysis Module for SCALE (SAMS) calculates sensitivity coefficients that predict the expected changes of the calculated value of system responses such as keffand reaction rate ratios due to perturbations in constituent cross-section data. Additionally, the uncertainty in each response due to cross section covariance data can be predicted. For multigroup calculations, SAMS performs sensitivity calculations using linear perturbation theory and requires the calculation of the forward and adjoint flux moments. Multigroup sensitivity coefficients computed by SAMS also include the implicit effect due to resonance self-shielding calculations. In multigroup mode, SAMS works in conjunction with XSDRNPM, NEWT, KENO V.a, or KENO-VI, which are capable of calculating the desired responses as well as associated fluxes, flux moments and/or mesh fluxes. The implicit calculations are performed with data generated by a combination of the BONAMIST and SENLIB codes specially developed to produce resonance self-shielded cross sections and their sensitivities to input data. SAMS is also used to edit the sensitivity coefficients generated by KENO continuous energy sensitivity calculations.
SAMS automatically selects all of the sensitivity parameters that can be calculated or edited for each nuclide in each region of the system based on available cross-section data. Sensitivity parameters for a given nuclide may be generated for a number of parameters, including total, scatter, capture, and fission cross sections, as well as \(\chi\). The uncertainty information is produced for all processes available on the cross section covariance data file.
SAMS has been designed for automated operation with the TSUNAMI-1D, TSUNAMI-3D-K5 and TSUNAMI-3D-K6 SCALE control modules and produces data suitable for use with the TSUNAMI-IP, TSURFER, and TSAR modules.
Version Information
Version 6.2 (2016)
Code Responsible(s): B. T. Rearden, L. M. Petrie, M. A. Jessee, M. L Williams
The following section acknowledgements appeared in the SCALE 6.2 manual.
The author wishes to acknowledge B. L. Broadhead of the Oak Ridge National Laboratory and R. L. Childs, formerly of the Oak Ridge National Laboratory, for their assistance with this work. The support and encouragement of C. V. Parks of the Oak Ridge National Laboratory and C. M. Hopper, formerly of the Oak Ridge National Laboratory is also appreciated. The support of U.S. Department of Energy Nuclear Criticality Safety Program is greatly appreciated.
6.6.1. Introduction
SAMS is the Sensitivity Analysis Module for SCALE, which has been designed for automated operation with the TSUNAMI-1D, TSUNAMI-3D-K5 and TSUNAMI-3D-K6 SCALE control modules. In multigroup mode, SAMS utilizes data from forward and adjoint neutron transport analyses as well as the sensitivity of resonance self-shielded cross-section data to input data to compute sensitivity coefficients that represent the percentage effect on a system response, such as multiplication factor (keff) or reaction rate ratio, to a percentage change in the nuclear reaction probabilities or cross sections. Sensitivity coefficients computed with SAMS can be presented as an energy-integrated value, which represents a uniform change in a given cross section across all energies, or as a “profile” in which a value is calculated for each discrete energy group. When combined with the uncertainties in the cross-section data, the cross-section-covariance data, the uncertainty in keff due to uncertainties in the cross-section data can be assessed. The sensitivity data produced by SAMS are appropriate for use with the TSUNAMI-IP, TSURFER and TSAR functional modules of SCALE and may be plotted in the Fulcrum SCALE user interface.
The methodology for calculating sensitivity and uncertainty parameters in SAMS is an extension of that used in the Fantastic Oak Ridge Sensitivity System (FORSS) [SAMSLWM+81, SAMSWML+76, SAMSWOM+78]. The FORSS system was capable of calculating the sensitivity of the system keff to changes in group-wise cross-section data for any given isotope for a number of reaction types. This data could also be used in conjunction with cross-section covariance data to determine the uncertainty in the calculated keff due to uncertainties in the nuclear data library. The FORSS methodology requires the calculation of the forward and adjoint angular and scalar fluxes and flux moments. Once these flux solutions are determined, the relative change in the system response due to a change in a component cross section can be determined. These sensitivities are calculated for each material region, nuclide, reaction type, and energy group in the system model.
In multigroup transport solution mode, the generation of sensitivity coefficients with SAMS requires forward and adjoint criticality calculations to compute the system responses, required fluxes as well as the volume of each flux region. Additionally, the neutron cross-section data used in the transport solution and the sensitivity of the resonance self-shielded cross-section data to the input material number densities must be available. SAMS currently functions with neutron transport solutions from XSDRNPM, NEWT, KENO V.a and KENO-VI. The sensitivities of the resonance self-shielded cross-section data to the input material number densities is computed by routines from SENLIB and BONAMIST. SAMS reads the required data files and produces sensitivity coefficients for many reactions for all nuclides for all regions in the problem. SAMS offers several output options to provide convenient transfer of the sensitivity data for plotting and additional analysis with other modules, such as TSUNAMI-IP, TSURFER, TSAR and Fulcrum.
As is described in the SCALE TSUNAMI-3D manual section, SAMS is also used in continuous energy mode transport solution sensitivity calculations. In these calculations, sensitivities are generated by TSUNAMI-3D and SAMS is used only to provide edits of sensitivity and uncertainty results. Sensitivity coefficients generated in continuous energy mode are stored in a multigroup sensitivity data file (SDF) format that is consistent with the various SCALE tools that use SDFs. The text provided in the rest of the SAMS manual chapter is oriented toward SAMS use with multigroup transport solutions.
6.6.2. Theory
The methodology used by SAMS to produce sensitivity coefficients is presented in this section. The explicit portion of the sensitivity coefficient represents the sensitivity of the system responses to the problem-dependent resonance self-shielded multigroup cross-section data used in the analysis. The implicit portion of the complete sensitivity coefficient is the sensitivity of the resonance self-shielded multigroup cross-section data to the data input to the resonance self-shielding calculation. Summing the implicit and explicit contributions from a particular cross-section data component produces the complete sensitivity coefficient. Uncertainty analysis involves propagating the uncertainty information from the cross-section-covariance data file to the computed value of keff via the sensitivity coefficients. Problem characterization data and system average quantities are computed, such as the energy of average lethargy causing fission. Each of these quantities is computed by SAMS and is explained in this section. When executed as part of one of the TSUNAMI sequences, the SAMS input is generated automatically by the control module for that sequence.
6.6.2.1. Explicit sensitivity coefficient generation
This methodology used to generate the explicit portion of the sensitivity coefficients is identical to that used in the FORSS [SAMSWML+76] code system for fast reactor applications, with the addition of the sensitivity of keff to the fission energy spectrum (\(\chi\)). The sensitivity coefficients produced with these techniques give the sensitivity of the computed keff to a particular component of the group-wise cross-section data.
The explicit sensitivity coefficients are calculated using the well-established adjoint-based perturbation theory approach [SAMSGan67, SAMSObl76, SAMSSJ72, SAMSUsa64]. The full derivation of the general procedure is not given here; however, the specific theory for the generation of keff sensitivities is presented below. For the full derivation of the general sensitivity equations the reader is referred to [SAMSWML+76].
The Boltzmann transport equation can be written in the form
where
\(\phi\) = neutron flux,
\(\lambda\) = represents the eigenvalues where the largest eigenvalue is 1/keff,
A = operator that represents all of the transport equation except for the fission term,
B = operator that represents the fission term of the transport equation.
Defining perturbed transport operators and the perturbed eigenvalues as
where \(\delta A\) and \(\delta B\) represent small linear perturbations in their corresponding transport operators and \(\delta \lambda\) represents the resulting change in the eigenvalues. The perturbed transport equation can be written in the form
The equation adjoint to Eq. (6.6.1) is
where \(\phi^{\dagger}\) is the adjoint flux and has a special physical significance as the “importance” of the particles within the system, and \(A^{\dagger}\) and \(B^{\dagger}\) are the adjoint operators corresponding to A and B.
Multiplying Eq. (6.6.3) by \(\phi^{\dagger}\), and integrating over all phase space yields
Using the property of adjointness (i.e., \(\langle {\phi }^{\dagger }(A -\lambda B){\phi }'\rangle =\langle {\phi }'({{A}^{\dagger }} -{{\lambda }^{\dagger }}{{B}^{\dagger }})\mathop{\phi }^{\dagger }\rangle\)) and Eq. (6.6.4) to reduce the number of terms yields
Eq. (6.6.6) is further simplified by ignoring the second-order perturbation term (\(\delta\)\(\lambda\)\(\delta\)B) and substituting \(\phi^{\prime}\) with \(\phi\), indicating that the perturbations in the transport operators do not cause significant perturbations in the flux solution. The eigenvalue perturbation becomes
Substituting the perturbation terms with partial derivatives with respect to a macroscopic cross section, \(\Sigma\), of the transport operator at some point in phase space \(\vec{r}\), the relative sensitivity of \(\lambda\), becomes
where \(\xi\) is the phase space vector and the brackets indicate integration over space, direction and energy variables.
Note that since \(\lambda=1 / k\), then \(\partial \lambda / \lambda=-\partial k / k\), where \(k=k_{e f f}\), the sensitivity of k due to a small perturbation in a macroscopic cross section, \(\Sigma\), of the transport operator at some point in phase space \(\vec{r}\) can be expressed as
The k sensitivity for individual cross sections can be obtained from Eq. (6.6.9) using the discrete ordinates form of the transport equation. In doing so, the phase space vector, \(\bar{\xi}\), has been replaced by indices representing discretization in space, energy and angle. It has been demonstrated in Ref. [SAMSWML+76] that sensitivity coefficients for reaction x, isotope i, energy group g, and computational region z can be represented as
where the denominator, D, is expressed as
where
\(\chi _{{g}',z}^{i}\) = average fraction of fission neutrons emitted into energy group from fission of isotope i in region z,
\(\bar{\nu }_{g,z}^{i}\) = average number of fission neutrons emitted from fission of isotope i in region z in energy group g,
\(\Sigma _{f,g,z}^{i}\) = macroscopic cross section for fission of isotope i in region z and energy group g,
I = number of isotopes in the system model,
R = number of computational regions in the system model,
G = number of neutron energy groups in the system model.
Energy-integrated coefficients are obtained by summing the group-wise coefficients over all energy groups. The T terms of Eq. (6.6.10) represent the transport processes for neutron loss, fission production and scattering to the group of interest in T1, T2 and T3, respectively. The first term is expressed as
where
\(\Sigma _{x,g,z}^{i}\) = macroscopic cross section for some reaction x, of isotope i, energy group g, in region z,
\(\ell\) = Legendre order that corresponds to the jth real valued flux moment,
\(\tilde{\phi }_{g,z}^{\dagger j}\) = jth component real valued adjoint flux moment for energy group g, and region z,
NMOM = the total number of real valued flux moments corresponding to the desired Legendre order of expansion.
\(V_z\) = the volume of region z
The second and third terms can be expressed as
where
\(\Sigma _{x,g\to {g}',z}^{\ell ,i}\) = \({{\ell }^{th}}\) moment of the transfer cross section for reaction x of isotope i, from energy group to energy group g in region z.
For specific reactions, not all of the T terms defined above are needed to calculate the sensitivity coefficient. The application of Eq. (6.6.10) for each type of reaction is outlined below. The computational form of each equation is expressed with the volume integrated product of the forward and adjoint flux moments as
For KENO calculations without mesh fluxes, the flux product is computed with Eq. (6.6.12) , where z simply represents each material region. For KENO calculations using the mesh flux generation option, the flux product for each material region is computed as
where m represents flux meshes that occur in region z, and the subscript \(z_m\) denotes fluxes computed in mesh m of region z. Also, \(V_{z_m}\) is the volume of mesh m in region z.
Similarly, for transport calculations using XSDRNPM and NEWT, the flux product is computed over the computational mesh using Eq. (6.6.16) where m represents the computational mesh used in the transport calculation defined for each material region, z.
Once the flux products are computed for each material region or zone, the sensitivity coefficients for each reaction type can be computed as follows.
Capture Reaction Sensitivity (non-fission, non-scattering)
Only the \(T^i_{1,x,g,z}\) term is used for this class of reactions where \(\Sigma^i_{x,g,z}\) is the absorption cross section of interest ((n, \(\gamma\)), (n, \(\alpha\)), (n, p), etc.) and can be expressed as
(6.6.17)\[S_{x,g,z}^{i}=\frac{-\Sigma _{x,g,z}^{i}\sum\limits_{\ell =0}^{ISCT}{(2\ell +1)P_{g,g,z}^{\ell }}}{D}\]where ISCT = the highest Legendre order of scattering used in the sensitivity calculations.
Fission Reaction Sensitivity
The fission reaction requires \(T_{1,x,g,z}^{i}\) and \(T_{2,g,z}^{i}\), where \(\Sigma _{x,g,z}^{i}\) in the definition of \(T_{1,x,g,z}^{i}\) is the fission cross section and can be expressed as
(6.6.18)\[\begin{split}\begin{array}{c} S_{f, g, z}^{i}=\frac{1}{D}\left[\left(\frac{1}{k} \bar{\nu}_{g, z}^{i} \Sigma_{f, g, z}^{i} \chi_{g, z}^{i}-\Sigma_{f, g, z}^{i}\right) P_{g, g, z}^{0}+\sum_{g^{\prime}=1 \atop g \neq g^{\prime}}^{G} \frac{1}{k} \bar{v}_{g, z}^{i} \Sigma_{f, g, z}^{i} \chi_{g^{\prime}, z}^{i} P_{g, g^{\prime}, z}^{0}\right. \\ \left.-\sum_{f, g, z}^{i} \sum_{=1}^{I S C T}(2+1) P_{g, g, z}\right] \end{array}\end{split}\]
\(\bar{\nu}\) Sensitivity
The \(\bar{\nu}\) reaction only requires \(T_{2,g,z}^{i}\) and can be expressed as
(6.6.19)\[S_{\bar{\nu },g,z}^{i}=\frac{\frac{1}{k}\sum\limits_{{g}'=1}^{G}{\bar{\nu }_{g,z}^{i}\Sigma _{f,g,z}^{i}\chi _{{g}',z}^{i}P_{g,{g}',z}^{0}}}{D} .\]\(\chi\) Sensitivity
The computation of the unconstrained \(\chi\) reaction sensitivity only requires \(T_{2,g,z}^{i}\), with the \(\chi\) and \(\nu \Sigma_f\) terms interchanged, and is expressed as
(6.6.20)\[S_{\chi ,g,z}^{i}=\frac{\frac{1}{k}\sum\limits_{{g}'=1}^{G}{\bar{\nu }_{{g}',z}^{i}\Sigma _{f,{g}',z}^{i}\chi _{g,z}^{i}P_{{g}',g,z}^{0}}}{D}\]In Eq. (6.6.20), the sensitivities coefficients sum to 1.0 when added over all energy groups and nuclides. However, since the fission spectrum probability distribution for any nuclide must, by definition, sum to 1.0 over all energy groups, the sensitivity of keff to the fission spectrum should sum to 0.0, as any change in fission spectrum in any group must be compensated by changes in other groups to maintain the constraint that all values sum to 1.0. This methodology was first developed for the SAGEP code [SAMSHTK84] and is implemented in SAMS as
(6.6.21)\[\tilde{S}_{\chi ,g,z}^{i}=S_{\chi ,g,z}^{i}-\chi _{g,z}^{i}\sum\limits_{{g}'=1}^{G}{S_{\chi ,{g}',z}^{i}}\]The constrained \(\chi\) calculation from Eq. (6.6.21) is the default for SAMS.
Scattering Reaction Sensitivity
All scattering reactions (elastic, inelastic, and (n, 2n) reactions) require \(T_{1,x,g,z}^{i}\) and \(T_{3,x,g,z}^{i}\) where \(\Sigma _{x,g,z}^{i}\) in the definition of \(T_{1,x,g,z}^{i}\) is the scattering cross section and \(\Sigma _{x,{g}'\to g,k}^{\ell ,i}\) in the definition of \(T_{3,x,g,z}^{i}\) is a component of the group-to-group scattering matrix for the \({{\ell }^{th}}\) scattering moment of reaction x.
(6.6.22)\[\begin{split}\begin{aligned} S_{x, g, z}^{i}&=\frac{1}{D}\left(\sum_{\ell=0}^{I S C T}\left(\Sigma_{x, g \rightarrow g, z}^{\ell, i}-(2 \ell+1) \Sigma_{x, g, z}^{i}\right) P_{g, g, z}^{\ell}\right.\\ &+\left.\sum_{g^{\prime}=1 \atop g^{\prime} \neq g}^{G} \Sigma_{x, g \rightarrow g^{\prime}, z}^{\ell, i} P_{g, g^{\prime}, z}^{\ell}\right) \end{aligned}\end{split}\]
Total Reaction Sensitivity
The total reaction requires \(T_{1,x,g,z}^{i}\), \(T_{2,g,z}^{i}\), and \(T_{3,x,g,z}^{i}\). Here, \(\Sigma _{x,g,z}^{i}\) in the definition of \(T_{1,x,g,z}^{i}\) is the total cross section and \(\Sigma _{x,{g}'\to g,k}^{\ell ,i}\) in the definition of \(T_{3,x,g,z}^{i}\) is a component of the group-to-group scattering matrix for the \({{\ell }^{th}}\) scattering moment. For non-fissionable isotopes, \(T_{2,g,z}^{i}\) will be zero. The total reaction sensitivity coefficient can be expressed as
(6.6.23)\[\begin{split}\begin{aligned} S_{t, g, z}^{i} &=\frac{1}{D}\left[\left(\sum_{s, g \rightarrow g, z}^{0, i}+\frac{1}{k} \bar{v}_{g, z}^{i} \Sigma_{f, g, z}^{i} \chi_{g, z}^{i}-\Sigma_{t, g, z}^{i}\right) P_{g, g, z}^{0}\right.\\ &+\sum_{g^{\prime}=1 \atop g \neq g^{\prime}}^{G}\left(\sum_{s, g \rightarrow g^{\prime}, z}^{0, i}+\frac{1}{k} \bar{v}_{g, z}^{i} \Sigma_{f, g, z}^{i} \chi_{g^{\prime}, z}^{i}\right) P_{g, g^{\prime}, z}^{0} \\ &+\sum_{=1}^{I S C T}\left(\sum_{s, g \rightarrow g, z}^{, i}-(2+1) \sum_{t, g, z}^{i}\right) P_{g, g, z}^{l}+\sum_{g^{\prime}=1 \atop g^{\prime} \neq g}^{G} \sum_{s, g \rightarrow g^{\prime}, z} P_{g, g^{\prime}, z}^{l} \end{aligned}\end{split}\]
6.6.2.2. Implicit sensitivity coefficient generation
The methodology to calculate the sensitivity coefficients, as presented in the previous section, was developed for fast reactor applications in which the effect of resonance self-shielding in the multigroup cross-section data is minimal. To provide an accurate estimation of the sensitivity coefficients for systems in which resonance self-shielding is important, the sensitivity coefficients as computed in Eq. (6.6.10) require additional terms to account for the first-order implicit effect of perturbations in the material number densities or nuclear data upon the shielded group-wise macroscopic cross-section data [SAMSWBP01].
The sensitivity of the cross-section data to the input data in turn affects the keff sensitivities. The implicit portion of the sensitivity coefficient, the sensitivity of the group-wise data to the input quantities, is defined as
where \(\omega_{i}\) is some input quantity. The \(\omega_{i}\) term could represent the number density of a particular material, a certain nuclear data component or a physical dimension of a system. For the sensitivity coefficients produced by SAMS, which are the sensitivities of keff to the group-wise cross-section data, the effect on keff of perturbing one cross section that affects the resonance-shielded values of other cross sections must be computed. If \(\omega_{i}\) is a certain cross-section data component for process y of nuclide j in energy group h expressed as \(\Sigma _{y,h}^{j}\), which is sensitive to perturbations in process x in energy group g for nuclide i expressed as \(\Sigma _{x,g}^{i}\), the complete sensitivity of keff due to perturbations of \(\Sigma _{x,g}^{i}\) can be defined using the chain rule for derivatives as
where the sensitivity coefficients with respect to keff are the explicit components as computed in Eq. (6.6.10), with the region subscript, z, omitted, and j and h are varied to include all processes that are influenced by the value of \(\Sigma _{x,g}^{i}\).
Subsequent to the computation of the explicit portion of the sensitivity coefficients, data from SENLIB and BONAMIST are used to compute the implicit portion of the sensitivity coefficients. This implementation of the implicit sensitivity coefficients is more general than that presented in the example calculation of [SAMSWBP01], as it allows for the assessment of the implicit components for all reactions due to interaction with all nuclides. Because the sensitivity of a response to a material number density is equivalent to the sensitivity of the same response to the corresponding total macroscopic cross section, the computation of the implicit sensitivity coefficients can be based on the sensitivity to the input material number densities. The implicit effect of the number densities on keff must be accounted for from several sources. One source is the effect of the number densities input to BONAMIST on the shielded cross sections in the unresolved resonance region. For this case, the implicit sensitivity of keff to the total cross section of nuclide i is
where j and y are varied to include all processes that are sensitive to \(N^i\), the number density of the ith nuclide. Additionally, the energy group for the implicit sensitivity, g, is varied over all energies. The sensitivity of the total macroscopic cross section to the group-wise macroscopic total cross section, \({{S}_{\Sigma _{T}^{i},\Sigma _{T,g}^{i}}}\), is simply 1.0. For data computed by SENLIB and input to BONAMIST, an additional term is necessary to account for the sensitivity of the SENLIB parameter, denoted \(\omega_{m}\). The chain rule for derivatives can again be used to propagate this sensitivity to a keff sensitivity. The implicit sensitivity of keff to the input number densities, in this case, is
where m is varied to include all SENLIB computed parameters that use the material number densities in their calculation and are input to BONAMIST.
The calculation of the implicit sensitivity of a total cross-section component in the unresolved resonance region requires the sum of the implicit quantities computed in Eq. (6.6.26) and Eq. (6.6.27) with \(\omega_{m}\) varied to include the Dancoff factor for each zone used in the BONAMIST calculation.
To compute the implicit portion of sensitivity coefficients for reactions, x, other than total, an additional term must be employed. With the implicit sensitivity of keff to the total cross section computed, the chain rule for derivatives is again applied to propagate the sensitivity of keff to the total cross section to the sensitivity of keff to some other process. This is accomplished using the sensitivity of the total cross section to the particular processes, computed from the unshielded cross-section data as
The second term on the right hand side of Eq. (6.6.28) is computed analytically using data from the short master cross-section data library.
6.6.2.2.1. Complete sensitivity coefficient
With the implicit sensitivities properly computed, the complete sensitivity coefficient by group can be computed as the sum of the explicit and implicit terms as
When a Monte Carlo transport solution is used to produce sensitivity coefficients, uncertainties in the forward and adjoint flux solutions and the value of keff are propagated to the final sensitivity results using standard error propagation techniques [SAMSBev69]. The forward and adjoint fluxes are treated as uncorrelated to each other. Also, the group-wise values of each flux solution are treated as uncorrelated. The flux moments within each group are treated as fully correlated. Although this method provides an adequate assessment of the statistical uncertainty in the sensitivity coefficients, a more robust technique may be implemented in subsequent revisions.
6.6.2.3. Summary of sensitivity coefficients calculated by SAMS
Sensitivity coefficients are calculated for the sensitivity of keff to the reactions listed in Table 6.6.1, if appropriate cross-section data is available. The identifier used in the SAMS output for each of these sensitivity types is also given. The MT of 0 assigned to scattering is arbitrary, as a sum of scattering reaction does not exist in the AMPX format.
MT |
Reaction |
SAMS identifier |
0 |
Sum of scattering |
scatter |
1 |
Total |
total |
2 |
Elastic scattering |
elastic |
4 |
Inelastic scattering |
n,n’ |
16 |
n,2n |
n,2n |
18 |
Fission |
fission |
101 |
Neutron disappearance |
capture |
102 |
n,\(\gamma\) |
n,gamma |
103 |
n,p |
n,p |
104 |
n,d |
n,d |
105 |
n,t |
n,t |
106 |
n,3He |
n,he-3 |
107 |
n,\(\alpha\) |
n,alpha |
452 |
\(\bar{\nu}\) |
nubar |
1018 |
\(\chi\) |
chi |
6.6.2.4. Uncertainty analysis
Given uncertainty information for the cross sections for all nuclides and reaction processes that are important to the system of interest, it is possible to estimate the uncertainty in the calculated system multiplication factor due to these data uncertainties.
The nuclear data parameters are represented by the vector \(\mathbf{\alpha}\), the elements of which are (\(\alpha _{x,g}^{i}\)), where i is varied over all isotopes, x is varied over all reactions for each isotope and g is varied over all energy groups. If M is the number of nuclide-reaction pairs \(\times\) the number of energy groups (i.e., the number of elements in \(\mathbf{\alpha}\)), the symmetric M \(\times\) M matrix containing the relative variances (diagonal elements) and relative covariances (off-diagonal elements) in the nuclear data is \(\mathbf{\mathrm{C}_{\alpha \alpha}}\). The elements of \(\mathbf{\mathrm{C}_{\alpha \alpha}}\) are
where i and j are varied over all isotopes, x and y are varied over all reactions for each isotope and g and g’ are varied over all energy groups. Additionally,
where \(\delta \alpha _{x,g}^{i}\) and \(\delta \alpha _{y,{g}'}^{j}\) represent the difference between the values and expectation values of the nuclear data parameters and \(\left\langle {} \right\rangle\) represents integration over the ranges of \(\alpha _{x,g}^{i}\) and \(\alpha _{y,{g}'}^{j}\) weighted with a probability density function. A rigorous definition of the cross-section-covariance data is given in [SAMSDun00]. SAMS simply reads the covariance data from a standard COVERX data file.
The vector of length M containing sensitivities of the calculated keff to the \(\mathbf{\alpha}\) parameters is represented by S\(_{\bf{k}}\), where each element is
For the purposes of SAMS uncertainty calculations, the \(\alpha _{x,g}^{i}\) parameters are simply the group-wise cross section data. If a particular material is present in more than one material region, the sensitivity coefficients for all regions are summed prior to creating the S\(_{\bf{k}}\) vector.
The variance for the system keff value is given as
where T indicates a transpose.
The covariance in keff due to the energy correlations of two particular processes can be assessed by examining a subset of the elements of \(\mathbf{\mathrm{C}_{\alpha \alpha}}\), where i, j, x and y are held constant. If G is the number of energy groups, the covariance data for a particular process is represented as the G \(\times\) G matrix \({{\mathbf{C}}_{\alpha _{x}^{i}\alpha _{\mathbf{y}}^{\mathbf{j}}}}\) and the group-wise sensitivity vectors, of length G, for the processes are represented as \({{\mathbf{S}}_{\mathbf{k},\alpha _{\mathbf{x}}^{\mathbf{i}}}}\) and \({{\mathbf{S}}_{\mathbf{k},\alpha _{\mathbf{y}}^{\mathbf{j}}}}\). The relative covariance in keff due to the particular process or processes is given as
In actuality, the COVERX data file represents the covariance data in the form of multiple \({{\mathbf{C}}_{\alpha _{x}^{i}\alpha _{\mathbf{y}}^{\mathbf{j}}}}\) matrices. Thus, although commonly used for its mathematical convenience, C\(_{\mathbf{\alpha \alpha}}\) does not exist as a continuous matrix. In the COVERX format, if \({{\mathbf{C}}_{\alpha _{x}^{i}\alpha _{\mathbf{y}}^{\mathbf{j}}}}\) is present on the data file with \(i \neq j\) and/or \(x \neq y\), then the transpose matrix \({{\mathbf{C}}_{\alpha _{\mathbf{y}}^{\mathbf{j}}\alpha _{x}^{i}}}\) is not present. Thus, using each matrix on the COVERX file only once, an upper (or lower) triangular C\(_{\mathbf{\alpha \alpha}}\) matrix could be constructed, but not a full matrix.
In SAMS, the value of \(\sigma _{k}^{2}\) is calculated by first determining the values of the variances or covariance as in Eq. (6.6.35) for all processes in the system under consideration, excluding the total reaction. The total reaction is excluded because it is the sum of the other processes and its inclusion would increase the variance from its actual value. The value of is then computed as the sum of the variances (diagonal elements of C\(_{\mathbf{\alpha \alpha}}\) plus twice the sum of the covariances (off-diagonal elements of C\(_{\mathbf{\alpha \alpha}}\). The standard deviation of keff is simply the square root of \(\sigma _{k}^{2}\).
6.6.2.5. Problem characterization
SAMS computes a number of characteristic parameters of fissile systems for the convenience of the user for multigroup sensitivity calculations. These characterization parameters are not calculated in continuous-energy mode. These include the median fission group, average fission group, average energy causing fission, and energy of average lethargy causing fission. Corresponding parameters are also computed for capture and scattering.
The median fission group is computed as the first group M that satisfies the inequality
where
z = index for material regions or zones,
Z = total number of regions or zones in the system model,
G = total number of energy groups.
The average fission group (AFG) is computed as
The average energy causing fission (AECF) is computed as
where
\({{\bar{E}}_{g}}\) = the average energy of group g, computed as (Eg +Eg+1)/2.
Lethargy is defined as u = ln(E0/E*), where E0 is the maximum energy considered, which is assumed for the lethargy calculation to be 10 MeV. The average lethargy causing fission is computed as
where
\({{\bar{u}}_{g}}\) = the average lethargy of group g, computed as (ug+ug+1)/2.
The energy of the average lethargy causing fission (EALF) is
The above definitions are applicable to determining corresponding parameters for capture and scattering reactions where the capture and 1-D scattering cross-section data are substituted for the fission cross section data.
6.6.2.5.1. Generalized perturbation theory
General Responses for Critical Systems
The previous sections have presented perturbation theory expressions for the explicit and implicit sensitivity coefficients relating changes in cross section data to changes in the fundamental eigenvalue of the neutron transport equation in Eq. (6.6.1). Generalized perturbation theory (GPT) can be used to extend the deterministic S/U methods in TSUNAMI-1D to address more general responses that depend on the solution of the neutron transport equation. There is currently no TSUNAMI-3D GPT capability based on MG KENO Monte Carlo calculations, but a CE capability exists, as described in the TSUNAMI-3D section of the SCALE manual. Here the basic expressions used in GPT computations are described. More details and background material on GPT can be found in [SAMSWil86].
The neutron transport equation is homogeneous because the fundamental eigenvalue effectively forces the multiplying medium to satisfy a pseudo-critical condition by scaling the fission production term to exactly balance the neutron losses. Since the solution of a homogeneous equation can only be found within an arbitrary normalization factor, the responses of interest in a critical system are generally ratios of linear functionals of the neutron flux, such as
where HN and HD are response functions defining the response of interest. The value for a ratio response is independent of the flux normalization since it cancels from the numerator and denominator; thus the response is uniquely defined. The following are some examples of ratio responses:
reaction rate ratios such as the conversion factor:
relative power density in fuel region Vz (e.g., fuel pin):
fast-to-thermal flux ratio:
collapsed/homogenized thermal absorption cross section:
Generalized Perturbation Theory for Response Ratios.
GPT is used to relate variations in input data to changes in the calculated ratio-response. Let \(\alpha\) be an arbitrary input parameter (number density, multigroup cross section, nubar, chi, etc.) that affects the response ratio because the response functions HN and/or HD depend on \(\alpha\), or because the parameter appears in the transport equation, so that the flux is a function of \(\alpha\); i.e.,
The response perturbation due to a sufficiently small variation in \(\alpha\) can be approximated by the first-order term in a functional Taylor series expansion,
In GPT it is common to express the response perturbation as the sum of a “direct” and an “indirect” component. Eq. (6.6.46) is rearranged slightly to obtain:
where
The direct component accounts for response changes due to perturbations in the response functions HN and/or HD that depend explicitly (or implicitly through self-shielding) on data parameter \(\alpha\). The second term, called the indirect component, accounts for the response change due to a flux perturbation in the numerator or denominator of Eq. (6.6.40), which is caused by varying \(\alpha\) in the transport equation.
If the response functions depend explicitly on \(\alpha\), then the functional derivatives appearing in the direct effect component can be evaluated to give
In order to evaluate the indirect effect it is necessary to determine the impact of the data perturbation on the neutron balance equation. The operators A and B, as well as the flux \(\phi\) and eigenvalue \(\lambda\) may all be functions of \(\alpha\). Taking the first-order expansion for each of these functions and neglecting higher order terms gives
An equation called the generalized adjoint equation is introduced as follows
where \(\,{{\Gamma }^{*}}\) is known as the generalized adjoint, or generalized importance, function. The adjoint source term in Eq. (6.6.52) is equal to the functional derivative of the response ratio with respect to the neutron flux, which corresponds to
Taking the inner product of \(\Gamma^{*}\) with Eq. (6.6.51), and the inner product of \(\left(\frac{d \phi}{d \alpha} \delta \alpha\right)\) with eq:eq6-3-53, and then applying the property of adjoint operators and rearranging, gives:
The left side of Eq. (6.6.54) is identical to the indirect component defined in Eq. (6.6.49). As shown in the next section, the generalized adjoint function is calculated such that the term \(\left\langle \Gamma^{*} B\phi \right\rangle\) is equal to zero; thus
Combining Eq. (6.6.50) and Eq. (6.6.55) gives the response perturbation due to both direct and indirect components:
Calculation of the Generalized Adjoint Function
The forward flux \(\phi\) and adjoint function \(\phi^*\)are fundamental eigenfunctions of the homogeneous equations in Eq. (6.6.1) and Eq. (6.6.3), respectively. In contrast, the generalized adjoint function \(\Gamma ^*\) obeys an inhomogeneous equation. Furthermore, the operator \(\left( {{A}^{\dagger }}-\lambda {{B}^{\dagger }} \right)\) in Eq. (6.6.52) is singular, since by definition \(\lambda\) is an eigenvalue. Taking the inner production of \(\phi\) with Eq. (6.6.52) and applying the property of adjoint operators gives
The above relation is necessary for the singular inhomogeneous equation to have a solution; i.e., the adjoint source must be orthogonal to the forward flux. It is easily shown that the terms in Eq. (6.6.53) satisfy this condition. However a solution to Eq. (6.6.52) is not unique—the general solution for \(\Gamma^*\) is the sum of a particular and a homogeneous solution. If \(\Gamma_{\text{p}}^{*}\) is any particular solution, then the function \(\left(\Gamma _{\text{p}}^{*}\ +\ \,a\,{{\phi }^{\,\dagger }}\, \right)\) is also a solution, where “a” is any constant. This can be proved by direct substitution into Eq. (6.6.52). For GPT calculations it is convenient to define an auxiliary condition that “normalizes” the generalized fission source \({B}^{\,\dagger}\,{\Gamma }^{*}\) to contain no fundamental mode; i.e.,
The above relation is satisfied by defining the constant to be \(\,\text{a}\ \,=\ \ \frac{\left\langle \phi \ {{B}^{\dagger }}\,\Gamma _{\text{p}}^{*}\, \right\rangle }{\left\langle \phi \ {{B}^{\dagger }}\,{{\phi }^{\,\dagger }}\, \right\rangle }\), so that:
The normalization in Eq. (6.6.58) is done for two reasons. First, the 2nd term on the right side of Eq. (6.6.54) vanishes, so that the indirect effect is stationary with respect to eigenvalue perturbations introduced by \(\delta \alpha\). The second reason is due to numerical difficulties in converging the generalized fission source. During outer iterations the fundamental mode component multiplies like a resonance frequency compared to other harmonics. This can lead to loss of numerical significance in the general solution. Theoretically, if the initial outer iteration has no fundamental mode in \({{B}^{\,\dagger}}\,{{\Gamma }^{*}}\), then subsequent outer iterations also will not. However in practice numerical approximations and incomplete convergence of inner iterations tend to re-introduce the fundamental mode. Therefore the operation in Eq. (6.6.59) is applied after each outer iteration to remove fundamental mode “contamination.”
Sensitivity Coefficients for General Responses
Analogous to the eigenvalue sensitivities, the response sensitivity coefficient at phase space coordinate \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\xi}\) is defined as the relative change in R due to a relative change in \(\alpha(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\xi})\):
The total response perturbation is obtained by summing over the response perturbations due to arbitrary \(\alpha\)-variations throughout phase space
(6.6.61)\[\frac{\delta R}{R} \cong\left\langle S_{R, \alpha}(\vec{\xi}) \frac{\delta \alpha}{\alpha}(\vec{\xi})\right\rangle\]
Comparing Eq. (6.6.46) and Eq. (6.6.56) shows that the sensitivity coefficient is equal to
As in the case for eigenvalue sensitivity theory, the response perturbations introduced by \(\delta \alpha\) may include explicit as well as implicit effects caused by changes in self-shielded cross sections \(\tilde{\alpha }\) that are impacted by the value of parameter \(\alpha\). This can be treated by extending the partial derivative operator with respect to \(\alpha\) to be a total derivative, so that
When the above substitution is made into Eq. (6.6.62), the complete sensitivity coefficient is obtained
It can be seen that for GPT there are four distinct contributions to the complete sensitivity coefficient:
The SAMS code evaluates sensitivity coefficients for GPT as well as eigenvalue perturbation theory. Direct sensitivity coefficients for GPT are easily determined since they depend only on the response functions and do not require an adjoint calculation. On the other hand, the expressions for indirect explicit sensitivities are very similar to those presented in Sect. 6.6.2.1 for eigenvalue responses. The detailed expressions in Eq. (6.6.10) through Eq. (6.6.23) for eigenvalue sensitivity coefficients with respect to each type of nuclear data are also valid for general responses, by replacing the fundamental mode adjoint function \(\phi^{\dagger}\) with the generalized adjoint \(\Gamma^*\), and setting the value of the denominator D equal to the response R. The treatment of implicit effects described in Section Sect. 6.6.2.2 for eigenvalue responses is also the same for general responses.
6.6.3. SAMS Input Description
The input to SAMS consists of a SCALE Analytical Sequence Specification Record, a single block of data that contains keyword input that controls the code execution, and a SCALE input termination END record. Very little input is required to obtain sensitivity and uncertainty results, given the availability of appropriate data files, as the default values are suitable for a basic analysis. The input data for SAMS are entered using the SCALE free-form reading routines. The input is not case sensitive, so either upper- or lowercase letters may be used. A maximum of 252 columns per line may be used for input.
6.6.3.1. Analytical sequence specification record
The analytical sequence specification begins in column 1 of the first line of the input file and must contain the following line.
=SAMS5 (for XSDRN or KENO V.a calculations)
=SAMS6 (for KENO-VI calculations)
This signals the SCALE driver to execute the appropriate SAMS functional module. Input data following the Analytical Sequence Specification Record and preceding the END record are passed to SAMS as input.
6.6.3.2. SAMS input data
SAMS input data consists of a single SCALE input data block, the INITIAL block. The input must begin with the keywords READ INITIAL and end with the keywords END INITIAL. The keyword entries summarized in Table 6.6.2 and Table 6.6.3 are allowed between the READ INITIAL and END INITIAL keywords. Keywords ending with ‘=’ must be followed by the value to be assigned to the corresponding variable. Default values listed in Table 6.6.2 and Table 6.6.3 may be set or altered if SAMS is executed as part of a sequence, such as TSUNAMI-3D-K5 or TSUNAMI-3D-K6; in these cases, the sequence manual for that sequence should also be consulted for default variable values.
Keyword |
Default value |
Description |
binsen |
F |
Produces SENPRO formatted binary sensitivity data file on unit 40 |
coverx= |
56groupcov7.1 |
Name of covariance data file to use for uncertainty analysis |
kenova |
Kenova |
Indicates whether KENO V.a or KENO-VI data files are present on logical unit number kunit and kunita |
kenovi |
kenova |
Indicates whether KENO V.a or KENO-VI data files are present on logical unit number kunit and kunita |
kunit= |
35 |
Logical unit number of the binary data file from the forward calculation |
kunita= |
36 |
Logical unit number of the binary data file from the adjoint case |
largeimp= |
100.0 |
Value for the absolute value of implicit sensitivities, which if exceeded, will be reset to 0.0 and print a warning message. |
makeimp |
F |
Flag to cause implicit sensitivity coefficients to be generated. |
newt |
Kenova |
Indicates that NEWT data files are present on logical unit number kunit and kunita |
nocovar |
T |
Flag to cause uncertainty edit to be turned off (sets print_covar to F) |
nohtml |
F |
Flag to cause HTML output to not be produced. |
nomix |
F |
Flag to cause the sensitivities by mixture to be turned off |
pmeshvol |
F |
Flag to print mesh volumes |
pn= |
3 |
Legendre order for moment calculations |
pltp |
F |
Flag to cause a Javapeño 3D file containing \(\phi \phi^{*}\) matrices to be written inside the .htmd directory of the output. |
prtgeom |
F |
Flag to cause the sensitivities to be output for each geometry region |
prtimp |
F |
Prints explicit sensitivities coefficients, implicit sensitivity coefficients and complete sensitivity coefficients |
prtp |
F |
Flag to cause \(\phi \phi^{*}\) matrices to be written to the output file. |
prtvols |
F |
Flag to cause the volumes of the regions to be printed by SAMS |
useang |
F |
Flag to cause the angular flux calculated in KENO to be used to calculate flux moments for sensitivity calculations. If angular fluxes were not computed by KENO, useang is set to false internally. |
usemom |
T |
Flag to cause the flux moments calculated with KENO to be used to in sensitivity coefficient generation. If flux moments were not computed by KENO, usemom is set to false internally. |
xsdrn |
kenova |
Indicates that XSDRN data files are present on kunit and kunita logical unit numbers |
xunit= |
4 |
Logical unit number of the AMPX working formatted cross-section library |
xunitm= |
42 |
Logical unit number for master sensitivity library |
shortx= |
11 |
Logical unit number of the AMPX short master formatted cross-section library. |
unconstrainedchi |
F |
Flag to generate pre-SCALE 6 unconstrained chi (fission spectrum) sensitivities |
sensitivity_format= |
txt |
Specifies desired format for the resulting sensitivity data file (SDF). May be ‘txt’ for text-based format or ‘hdf5’ for HDF5-based format. |
Due to differences in the continuous-energy and multigroup sensitivity coefficient calculation methods, SAMS input files that use continuous-energy TSUNAMI do not use most of the input parameters that are described in Table 6.6.2. In fact, continuous-energy TSUNAMI calculations will only use the nomix, unconstrainedchi, and sensitivity_format input parameters; all other parameters that are input will be ignored.
Keyword |
Default value |
Description |
use_dcov |
F |
Use default covariance data |
use_icov |
F |
Use user-input covariance data |
cov_fix |
F |
Correct covariance data if the uncertainty is large >1000% or zero |
large_cov |
10.0 |
Relative Standard deviation to apply cov_fix |
return_work_cov |
F |
Create a new covariance data file with only the cross section covariance data used in the analysis |
udcov= |
0.05 |
User-defined default value of standard deviation for all energy groups |
udcov_corr= |
1.0 |
User-defined default correlation value |
udcov_corr_type= |
zone |
User-defined default correlation type; allowed values are long, zone, and short |
udcov_therm= |
0.0 |
User-defined default value of standard deviation for thermal energy groups |
udcov_inter= |
0.0 |
User-defined default value of standard deviation for intermediate energy groups |
udcov_fast= |
0.0 |
User-defined default value of standard deviation for fast energy groups |
Additionally, user-defined covariance data can be specified for individual nuclides and reactions using the COVARIANCE data block. This data begins with the keywords READ COVARIANCE and ends with the keywords END COVARIANCE. Any of the optional COVARIANCE input data may be entered in free form format between the READ COVARIANCE and END COVARIANCE keywords. The specifications for the COVARIANCE data block are described in the “User Input Covariance Data” section of the TSUNAMI Utility Modules chapter of the SCALE manual.
As the SAMS module generates HTML output, the optional HTML data block will provide user control over some formats of the output. This data begins with the keywords READ HTML and ends with the keywords END HTML. Any of the optional HTML input data may be entered in free form format between the READ HTML and END HTML keywords. The specifications for the HTML data block are described in the HTML Data section of the TSUNAMI Utility Modules chapter of the SCALE manual.
6.6.3.3. Data files required by SAMS
SAMS requires a number of data files to be available in the directory in which in the program executes to calculate MG sensitivities. These files are summarized in Table 6.6.4, but are not needed for CE sensitivity calculations.
Filename |
When required |
Description |
ftkunitf001 |
Always |
Data from forward transport solution |
ftkunitaf001 |
Always |
Data from adjoint transport solution |
ftxunitf001 |
Always |
AMPX working formatted cross-section data library containing resonance self-shielded cross-section data used in transport calculations |
ftxunitmf001 |
Always |
Master sensitivity cross-section data library; this library contains resonance self-shielded data for more reactions than are available on the working formatted library |
ftxunitsxf001 |
Implicit Sensitivities |
Short master cross-section data library containing infinitely dilute cross-section data |
senlib.sen |
Implicit Sensitivities |
Sensitivity data file produced by SENLIB; contains sensitivity of values input to resonance self-shielding codes to input material number densities |
bonamist.sen |
Implicit Sensitivities |
Sensitivity data file produced by BONAMIST; contains the sensitivity of shielded group-wise cross sections to data input to BONAMIST |
coverx |
Uncertainty Analysis |
COVERX formatted cross-section-covariance data file |
6.6.3.4. Example input
An example input file is given in Example 6.6.1. In this example, XSDRNPM was used as the transport solver (xsdrn), the data file from the forward transport solution is read from file ft33f001 (kunit=33), and the data file from the adjoint transport solution is read from file ft32f001 (kunita=32). SAMS will generate implicit sensitivity coefficients (makeimp) and print them in detail (prtimp). SAMS will compute and print sensitivity coefficients by region (prtgeom) and print the fluxes (prtflx).
=sams5
read initial
xsdrn
kunit=33 kunita=32
makeimp prtimp prtgeom
end initial
end
6.6.4. Sample Problems And Output Description
6.6.4.1. Example problem using XSDRNPM
This sample problem is the Flattop-25 metal system from the Cross-Section Evaluation Working Group benchmark specifications [SAMSNationalNDCenter74]. The system consists of a 6.116-cm sphere of 93%-enriched uranium with a natural uranium reflector. The outer radius of the reflector is 24.13 cm.
Prior to running SAMS, the data files listed in Table 6.6.4 were generated for this system. It is recommended that users perform sensitivity analyses on 1-D systems using the TSUNAMI-1D control module, which executes the appropriate codes to generate the required data files in an automated manner from a single input. SAMS is executed automatically as a part of the TSUNAMI-1D control sequence. However, given appropriate data files SAMS can be run as stand-alone program.
Input and text output
The SAMS input is shown in Example 6.6.2 and the output is shown in Example 6.6.3. Each section of the output is described below. Note that output shown in this section is intended only to represent the format of output that the user will encounter. The actual computed results may vary.
Parameter Table and Summary of Transport Calculation
In the output listing, the SAMS Parameter Table gives the values of the SAMS input data. Next, the title from the XSDRNPM input file is given and a brief summary of the transport solution is given.
Energy-, Region-, and Mixture Integrated Sensitivity Coefficients
Next the energy-, region-, and mixture integrated sensitivity coefficients are given for each reaction of each nuclide. The nuclide symbol and reaction name are given. The data given in the Sensitivity column represents the expected change in keff of this system due to a uniform fractional change in the cross section of the given nuclide and reaction in all energy groups for all regions and mixtures that contain this nuclide. For example, a uniform fractional change of 0.01 (or 1%) in the fission cross section of 235U would produce a fractional change in keff of \(5.7367 \cdot 10^{-1} \times 0.01\) or \(5.7367 \cdot 10^{-3}\) (0.57367% \(\Delta k/k\)). Because the keyword PRTIMP was entered in the SAMS input, the explicit, implicit and complete sensitivity coefficients are listed in separate columns. If PRTIMP were not entered, only the complete sensitivity would be given in the Sensitivity column.
Energy- and Region-Integrated Sensitivity Coefficients
Next the energy- and region-integrated sensitivity coefficients are given for each reaction of each nuclide. The mixture number, nuclide symbol and reaction name are given. The data given in the Sensitivity column represents the expected change in keff of this system due to a uniform fractional change in the cross section of the given nuclide and reaction in all energy groups for all regions that contain the particular mixture. For example, a uniform fractional change of 0.01 (or 1%) in the fission cross section of 235U in mixture 1 would produce a fractional change in keff of \(5.6594 \cdot 10^{-1} \times 0.01\) or \(5.6594 \cdot 10^{-3}\) (0.56594% \(\Delta k/k\)). Because the keyword PRTIMP was entered in the SAMS input, the explicit, implicit and complete sensitivity coefficients are listed in separate columns. If PRTIMP were not entered, only the complete sensitivity would be given in the Sensitivity column.
Total Sensitivity Coefficients by Nuclide
The next section of the output summarizes the total sensitivity by nuclide and mixture. The sensitivity values given here are the same as the values given in the previous section for the sensitivity of keff to the total cross section. Also included in this edit are the atom densities for each nuclide in each mixture.
Total Sensitivity Coefficients by Mixture
The next output edit gives the total sensitivity of keff to the mixture. Here, the sensitivity of keff to the total cross section is summed over each nuclide in a given mixture.
Problem Characterization (MG Calculations Only)
The next output edit is for the problem characterization data. This section includes the median group of neutrons causing fission, capture and scattering; average group for fission, capture and scattering; the average energy causing fission, capture and scattering; and the energy of the average lethargy causing fission, capture and scattering.
Sensitivity Coefficients by Material Zone
Because the keyword PRTGEOM was entered in the input, the next output edit gives the energy-integrated sensitivity coefficients for each material zone defined in the XSDRNPM criticality model. First, the zone number, material number and volume of the zone are given. Next, the sensitivity coefficients are given in the same format as for the region-integrated data. The edit is repeated for each material zone in the XSDRNPM criticality model.
Uncertainty Information
The next output edit contains the uncertainty information. First, the percent relative standard deviation in keff (\(\Delta\)keff/keff\(\times\) 100%) due to cross-section-covariance data is given. Next, a listing of the uncertainty terms as produced by Eq. (6.6.58) are given. The nuclide-reaction to nuclide-reaction covariance matrix responsible for the uncertainty contribution is given followed by the contribution to the uncertainty in terms of % \(\Delta \mathrm{k} / \mathrm{k}\). The total uncertainty can be computed from individual values by adding the square of the values with positive signs and subtracting the square of the values with negative signs, then taking the square root. The negative values are the result of anti-correlations in the cross-section-covariance data.
Execution Complete
The final edit produced simply states that the execution is complete and gives the elapsed time.
=sams5
read initial
xsdrn kunit= 31 kunita= 32 xunit= 4 xunitm= 42 makeimp prtgeom prtimp largeimp= 1.0000E+03
end initial
end
SAMS: Sensitivity Analysis Module for SCALE
SAMS Parameter Table
====================
transport solution from: XSDRNPM
forward transport solution on unit: 31
adjoint transport solution on unit: 32
working cross-section library on unit: 4
master sensitivity cross-section library on unit: 42
short master cross-section library on unit: 11
covariance data library read from file: 56groupcov7.1
order of flux moments requested for sensitivity calculations: 5
make implicit sensitivities: true
print implicit sensitivities: true
print sensitivities by mixture: true
print sensitivities for each region/zone: true
print uncertainty data: true
print mesh volumes: false
output binary SENPRO data file: false
implicit sensitivity threshold: 1000.0000
HTML output: true
Chi sensitivities are constrained: true
Fixup large/zero covariance data: false
Use Default Covariance Data: false
Use User Input Covariance Data: false
HTML Format Options
===================
Background color: PapayaWhip
Major Headings color: Maroon
Sub-headings color: Navy
Plain text color: Black
Hyperlink color: Navy
Hypelink style: none
Visited link color: Navy
Default Covariance color: Blue
User Input Covariance color: Red
Corrected with default covariance data color: RoyalBlue
Corrected with user input covariance data color: Green
tsunami-1d flattop
==================
Summary of Transport Solution
-----------------------------
number of neutron groups: 238
number of gamma groups: 0
number of regions: 2
maximum mixture number: 2
number of mixing table entries: 5
quadrature order of angular fluxes: 32
maximum order of flux moments: 5
Forward Calculation: k-eff = 1.00533677
Adjoint Calculation: k-eff = 1.00528886
-------------------------------------------------------------------------------------
Energy, Region and Mixture Integrated Sensitivity Coefficients for this Problem
-------------------------------------------------------------------------------------
Nuclide Reaction Explicit Implicit Sensitivity
------------- --------------- ----------- ----------- -----------
u-234 total 4.5108E-03 1.4614E-07 4.5110E-03
u-234 scatter 6.6431E-04 1.2145E-07 6.6443E-04
u-234 elastic 4.0086E-04 1.1594E-07 4.0097E-04
u-234 n,n' 2.6201E-04 5.5685E-09 2.6202E-04
u-234 n,2n 1.4354E-06 -5.7151E-11 1.4354E-06
u-234 fission 4.3022E-03 1.6639E-08 4.3022E-03
u-234 capture -4.5566E-04 8.0468E-09 -4.5565E-04
u-234 n,gamma -4.5566E-04 8.0468E-09 -4.5565E-04
u-234 nubar 6.5581E-03 0.0000E+00 6.5581E-03
u-234 chi 1.7178E-10 0.0000E+00 1.7178E-10
u-235 total 5.9204E-01 -4.7099E-06 5.9204E-01
u-235 scatter 6.6931E-02 -3.6751E-06 6.6927E-02
u-235 elastic 3.2362E-02 -3.6651E-06 3.2358E-02
u-235 n,n' 3.3483E-02 -8.6100E-09 3.3483E-02
u-235 n,2n 1.0842E-03 -1.4191E-09 1.0842E-03
u-235 fission 5.7367E-01 -7.7390E-07 5.7367E-01
u-235 capture -4.8559E-02 -2.6087E-07 -4.8559E-02
u-235 n,gamma -4.8559E-02 -2.6087E-07 -4.8559E-02
u-235 nubar 9.1500E-01 0.0000E+00 9.1500E-01
u-235 chi -2.3075E-08 0.0000E+00 -2.3075E-08
u-238 total 2.1733E-01 6.2352E-05 2.1740E-01
u-238 scatter 2.1045E-01 6.0318E-05 2.1051E-01
u-238 elastic 1.4370E-01 5.9909E-05 1.4376E-01
u-238 n,n' 6.5971E-02 3.8252E-07 6.5971E-02
u-238 n,2n 7.6541E-04 2.6330E-08 7.6544E-04
u-238 fission 5.5624E-02 4.9512E-08 5.5624E-02
u-238 capture -4.8736E-02 1.9847E-06 -4.8734E-02
u-238 n,gamma -4.8736E-02 1.9847E-06 -4.8734E-02
u-238 nubar 7.8442E-02 0.0000E+00 7.8442E-02
u-238 chi 7.5020E-10 0.0000E+00 7.5020E-10
----------------------------------------------------------------------------
Energy and Region Integrated Sensitivity Coefficients for this Problem
----------------------------------------------------------------------------
Mixture Nuclide Reaction Explicit Implicit Sensitivity
------- ------------- --------------- ----------- ----------- -----------
1 u-234 total 4.5108E-03 1.4614E-07 4.5110E-03
1 u-234 scatter 6.6431E-04 1.2145E-07 6.6443E-04
1 u-234 elastic 4.0086E-04 1.1594E-07 4.0097E-04
1 u-234 n,n' 2.6201E-04 5.5685E-09 2.6202E-04
1 u-234 n,2n 1.4354E-06 -5.7151E-11 1.4354E-06
1 u-234 fission 4.3022E-03 1.6639E-08 4.3022E-03
1 u-234 capture -4.5566E-04 8.0468E-09 -4.5565E-04
1 u-234 n,gamma -4.5566E-04 8.0468E-09 -4.5565E-04
1 u-234 nubar 6.5581E-03 0.0000E+00 6.5581E-03
1 u-234 chi 1.7178E-10 0.0000E+00 1.7178E-10
1 u-235 total 5.8379E-01 2.3611E-06 5.8379E-01
1 u-235 scatter 6.5770E-02 1.8838E-06 6.5772E-02
1 u-235 elastic 3.1559E-02 1.8623E-06 3.1561E-02
1 u-235 n,n' 3.3131E-02 2.1512E-08 3.3131E-02
1 u-235 n,2n 1.0797E-03 -1.7622E-11 1.0797E-03
1 u-235 fission 5.6593E-01 3.5945E-07 5.6594E-01
1 u-235 capture -4.7915E-02 1.1785E-07 -4.7914E-02
1 u-235 n,gamma -4.7915E-02 1.1785E-07 -4.7914E-02
1 u-235 nubar 9.0395E-01 0.0000E+00 9.0395E-01
1 u-235 chi -2.3539E-08 0.0000E+00 -2.3539E-08
1 u-238 total 8.0059E-03 3.2531E-09 8.0059E-03
1 u-238 scatter 4.6599E-03 3.1531E-09 4.6599E-03
1 u-238 elastic 2.3732E-03 3.0891E-09 2.3732E-03
1 u-238 n,n' 2.2134E-03 5.9442E-11 2.2134E-03
1 u-238 n,2n 7.2528E-05 4.5161E-12 7.2528E-05
1 u-238 fission 5.0328E-03 9.2930E-12 5.0328E-03
1 u-238 capture -1.6868E-03 9.0770E-11 -1.6868E-03
1 u-238 n,gamma -1.6868E-03 9.0770E-11 -1.6868E-03
1 u-238 nubar 7.6590E-03 0.0000E+00 7.6590E-03
1 u-238 chi -4.8497E-10 0.0000E+00 -4.8497E-10
2 u-235 total 8.2517E-03 -7.0711E-06 8.2446E-03
2 u-235 scatter 1.1604E-03 -5.5590E-06 1.1548E-03
2 u-235 elastic 8.0337E-04 -5.5275E-06 7.9785E-04
2 u-235 n,n' 3.5255E-04 -3.0122E-08 3.5252E-04
2 u-235 n,2n 4.4336E-06 -1.4015E-09 4.4322E-06
2 u-235 fission 7.7357E-03 -1.1333E-06 7.7346E-03
2 u-235 capture -6.4439E-04 -3.7872E-07 -6.4477E-04
2 u-235 n,gamma -6.4439E-04 -3.7872E-07 -6.4477E-04
2 u-235 nubar 1.1046E-02 0.0000E+00 1.1046E-02
2 u-235 chi 4.6475E-10 0.0000E+00 4.6475E-10
2 u-238 total 2.0933E-01 6.2348E-05 2.0939E-01
2 u-238 scatter 2.0579E-01 6.0314E-05 2.0585E-01
2 u-238 elastic 1.4133E-01 5.9906E-05 1.4139E-01
2 u-238 n,n' 6.3758E-02 3.8246E-07 6.3758E-02
2 u-238 n,2n 6.9288E-04 2.6325E-08 6.9291E-04
2 u-238 fission 5.0591E-02 4.9503E-08 5.0591E-02
2 u-238 capture -4.7050E-02 1.9846E-06 -4.7048E-02
2 u-238 n,gamma -4.7050E-02 1.9846E-06 -4.7048E-02
2 u-238 nubar 7.0783E-02 0.0000E+00 7.0783E-02
2 u-238 chi 1.2352E-09 0.0000E+00 1.2352E-09
-----------------------------------------------
Total Sensitivity Coefficients by Nuclide
-----------------------------------------------
Mixture Nuclide Atom Density Sensitivity
------- ------------- ------------ -----------
1 u-234 4.9000E-04 4.5110E-03
1 u-235 4.4490E-02 5.8379E-01
2 u-235 3.4000E-04 8.2446E-03
1 u-238 2.7000E-03 8.0059E-03
2 u-238 4.7740E-02 2.0939E-01
-----------------------------------------------
Total Sensitivity Coefficients by Mixture
-----------------------------------------------
Mixture Sensitivity
------- -----------
1 5.9631E-01
2 2.1763E-01
------------------------------
Problem Characterization
------------------------------
median fission group is 25 9.000E+05 to 8.750E+05(eV)
average fission group 2.682E+01
average energy(eV) causing fission 1.516E+06
energy(eV) of average lethargy causing fission 7.402E+05
median capture group is 42 2.000E+05 to 1.500E+05(eV)
average capture group 4.216E+01
average energy(eV) causing capture 3.014E+05
energy(eV) of average lethargy causing capture 1.498E+05
median scatter group is 41 2.700E+05 to 2.000E+05(eV)
average scatter group 3.915E+01
average energy(eV) causing scatter 4.743E+05
energy(eV) of average lethargy causing scatter 2.231E+05
-----------------------------------------------
Sensitivity Coefficients by Material Zone
-----------------------------------------------
Zone 1 Material 1 Volume of zone = 9.58277E+02 cm^3
Mixture Nuclide Reaction Explicit Implicit Sensitivity
------- ------------- --------------- ----------- ----------- -----------
1 u-234 total 4.5108E-03 1.4614E-07 4.5110E-03
1 u-234 scatter 6.6431E-04 1.2145E-07 6.6443E-04
1 u-234 elastic 4.0086E-04 1.1594E-07 4.0097E-04
1 u-234 n,n' 2.6201E-04 5.5685E-09 2.6202E-04
1 u-234 n,2n 1.4354E-06 -5.7151E-11 1.4354E-06
1 u-234 fission 4.3022E-03 1.6639E-08 4.3022E-03
1 u-234 capture -4.5566E-04 8.0468E-09 -4.5565E-04
1 u-234 n,gamma -4.5566E-04 8.0468E-09 -4.5565E-04
1 u-234 nubar 6.5581E-03 0.0000E+00 6.5581E-03
1 u-234 chi 1.7178E-10 0.0000E+00 1.7178E-10
1 u-235 total 5.8379E-01 2.3611E-06 5.8379E-01
1 u-235 scatter 6.5770E-02 1.8838E-06 6.5772E-02
1 u-235 elastic 3.1559E-02 1.8623E-06 3.1561E-02
1 u-235 n,n' 3.3131E-02 2.1512E-08 3.3131E-02
1 u-235 n,2n 1.0797E-03 -1.7622E-11 1.0797E-03
1 u-235 fission 5.6593E-01 3.5945E-07 5.6594E-01
1 u-235 capture -4.7915E-02 1.1785E-07 -4.7914E-02
1 u-235 n,gamma -4.7915E-02 1.1785E-07 -4.7914E-02
1 u-235 nubar 9.0395E-01 0.0000E+00 9.0395E-01
1 u-235 chi -2.3539E-08 0.0000E+00 -2.3539E-08
1 u-238 total 8.0059E-03 3.2531E-09 8.0059E-03
1 u-238 scatter 4.6599E-03 3.1531E-09 4.6599E-03
1 u-238 elastic 2.3732E-03 3.0891E-09 2.3732E-03
1 u-238 n,n' 2.2134E-03 5.9442E-11 2.2134E-03
1 u-238 n,2n 7.2528E-05 4.5161E-12 7.2528E-05
1 u-238 fission 5.0328E-03 9.2930E-12 5.0328E-03
1 u-238 capture -1.6868E-03 9.0770E-11 -1.6868E-03
1 u-238 n,gamma -1.6868E-03 9.0770E-11 -1.6868E-03
1 u-238 nubar 7.6590E-03 0.0000E+00 7.6590E-03
1 u-238 chi -4.8497E-10 0.0000E+00 -4.8497E-10
Zone 2 Material 2 Volume of zone = 5.78936E+04 cm^3
Mixture Nuclide Reaction Explicit Implicit Sensitivity
------- ------------- --------------- ----------- ----------- -----------
2 u-235 total 8.2517E-03 -7.0711E-06 8.2446E-03
2 u-235 scatter 1.1604E-03 -5.5590E-06 1.1548E-03
2 u-235 elastic 8.0337E-04 -5.5275E-06 7.9785E-04
2 u-235 n,n' 3.5255E-04 -3.0122E-08 3.5252E-04
2 u-235 n,2n 4.4336E-06 -1.4015E-09 4.4322E-06
2 u-235 fission 7.7357E-03 -1.1333E-06 7.7346E-03
2 u-235 capture -6.4439E-04 -3.7872E-07 -6.4477E-04
2 u-235 n,gamma -6.4439E-04 -3.7872E-07 -6.4477E-04
2 u-235 nubar 1.1046E-02 0.0000E+00 1.1046E-02
2 u-235 chi 4.6475E-10 0.0000E+00 4.6475E-10
2 u-238 total 2.0933E-01 6.2348E-05 2.0939E-01
2 u-238 scatter 2.0579E-01 6.0314E-05 2.0585E-01
2 u-238 elastic 1.4133E-01 5.9906E-05 1.4139E-01
2 u-238 n,n' 6.3758E-02 3.8246E-07 6.3758E-02
2 u-238 n,2n 6.9288E-04 2.6325E-08 6.9291E-04
2 u-238 fission 5.0591E-02 4.9503E-08 5.0591E-02
2 u-238 capture -4.7050E-02 1.9846E-06 -4.7048E-02
2 u-238 n,gamma -4.7050E-02 1.9846E-06 -4.7048E-02
2 u-238 nubar 7.0783E-02 0.0000E+00 7.0783E-02
2 u-238 chi 1.2352E-09 0.0000E+00 1.2352E-09
Generating working covariance matrix ...
Working covariance matrix created for future processing.
-----------------------------
Uncertainty Information
-----------------------------
the relative standard deviation of k-eff (% delta-k/k) due to cross-section covariance data is:
1.2743 % delta-k/k
contributions to uncertainty in k-eff (% delta-k/k) by individual energy covariance matrices:
covariance matrix
nuclide-reaction with nuclide-reaction % delta-k/k due to this matrix
------------------------------ ------------------------------- -----------------------------------
u-235 n,gamma u-235 n,gamma 1.1253E+00
u-238 elastic u-238 n,n' -7.6667E-01
u-238 n,n' u-238 n,n' 7.2823E-01
u-238 elastic u-238 elastic 4.5961E-01
u-235 n,n' u-235 n,n' 2.7423E-01
u-235 fission u-235 fission 2.3563E-01
u-235 elastic u-235 n,n' -2.1997E-01
u-235 elastic u-235 n,gamma 1.8368E-01
u-235 chi u-235 chi 1.8056E-01
u-235 nubar u-235 nubar 1.3559E-01
u-234 fission u-234 fission 1.1526E-01
u-235 elastic u-235 elastic 1.0107E-01
u-238 nubar u-238 nubar 9.1147E-02
u-238 n,gamma u-238 n,gamma 6.1302E-02
u-235 elastic u-235 fission -4.0658E-02
u-238 chi u-238 chi 3.7177E-02
u-238 fission u-238 fission 2.9009E-02
u-238 elastic u-238 n,gamma 1.9436E-02
u-238 n,2n u-238 n,2n 1.2044E-02
u-235 n,2n u-235 n,2n 9.4214E-03
u-238 elastic u-238 fission -7.8074E-03
u-238 elastic u-238 n,2n -5.2898E-03
u-234 n,gamma u-234 n,gamma 4.4184E-03
u-234 n,n' u-234 n,n' 3.9180E-03
u-235 elastic u-235 n,2n -2.4536E-03
u-234 elastic u-234 elastic 1.8845E-03
u-235 fission u-235 n,gamma 1.6297E-04
u-238 fission u-238 n,gamma 9.5794E-05
u-234 n,2n u-234 n,2n 4.1081E-05
Note: relative standard deviation in k-eff can be computed from
individual values by adding the square of the values with positive signs and
subtracting the square of the values with negative signs, then taking the square root
-----------------------------------
SAMS execution complete
Elapsed time: 0.02717 minutes
-----------------------------------
Sensitivity data file
The energy-dependent sensitivity data computed by SAMS is written to a sensitivity data file (SDF), which SCALE returns to the same directory as the output file with the extension “.sdf”. The SDF is generated directly in KENO in CE TSUNAMI-3D calculations. When XSDRNPM is used for the transport solution, the data file is presented in the TSUNAMI-A format described in Appendix A. This data file is suitable for visualization with the Fulcrum SCALE user interface or for system comparison with the TSUNAMI-IP code, which is described in section 6.5. A sample section of the TSUNAMI-A formatted sensitivity data file is show in Example 6.6.4.
tsunami-1d flattop
238 number of neutron groups
130 number of sensitivity profiles 30 are region integrated
1.005337 k-eff from the forward case
energy boundaries:
2.000000E+07 1.733300E+07 1.568300E+07 1.455000E+07 1.384000E+07
1.284000E+07 1.000000E+07 8.187300E+06 6.434000E+06 4.800000E+06
4.304000E+06 3.000000E+06 2.479000E+06 2.354000E+06 1.850000E+06
1.500000E+06 1.400000E+06 1.356000E+06 1.317000E+06 1.250000E+06
1.200000E+06 1.100000E+06 1.010000E+06 9.200000E+05 9.000000E+05
8.750000E+05 8.611000E+05 8.200000E+05 7.500000E+05 6.790000E+05
6.700000E+05 6.000000E+05 5.730000E+05 5.500000E+05 4.995200E+05
4.700000E+05 4.400000E+05 4.200000E+05 4.000000E+05 3.300000E+05
2.700000E+05 2.000000E+05 1.500000E+05 1.283000E+05 1.000000E+05
8.500000E+04 8.200000E+04 7.500000E+04 7.300000E+04 6.000000E+04
5.200000E+04 5.000000E+04 4.500000E+04 3.000000E+04 2.500000E+04
1.700000E+04 1.300000E+04 9.500000E+03 8.030000E+03 6.000000E+03
3.900000E+03 3.740000E+03 3.000000E+03 2.580000E+03 2.290000E+03
2.200000E+03 1.800000E+03 1.550000E+03 1.500000E+03 1.150000E+03
9.500000E+02 6.830000E+02 6.700000E+02 5.500000E+02 3.050000E+02
2.850000E+02 2.400000E+02 2.100000E+02 2.075000E+02 1.925000E+02
1.860000E+02 1.220000E+02 1.190000E+02 1.150000E+02 1.080000E+02
1.000000E+02 9.000000E+01 8.200000E+01 8.000000E+01 7.600000E+01
7.200000E+01 6.750000E+01 6.500000E+01 6.100000E+01 5.900000E+01
5.340000E+01 5.200000E+01 5.060000E+01 4.920000E+01 4.830000E+01
4.700000E+01 4.520000E+01 4.400000E+01 4.240000E+01 4.100000E+01
3.960000E+01 3.910000E+01 3.800000E+01 3.700000E+01 3.550000E+01
3.460000E+01 3.375000E+01 3.325000E+01 3.175000E+01 3.125000E+01
3.000000E+01 2.750000E+01 2.500000E+01 2.250000E+01 2.100000E+01
2.000000E+01 1.900000E+01 1.850000E+01 1.700000E+01 1.600000E+01
1.509990E+01 1.440000E+01 1.375000E+01 1.290000E+01 1.190000E+01
1.150000E+01 1.000000E+01 9.099990E+00 8.099990E+00 7.150000E+00
7.000000E+00 6.750000E+00 6.500000E+00 6.250000E+00 6.000000E+00
5.400000E+00 5.000000E+00 4.750000E+00 4.000000E+00 3.730000E+00
3.500000E+00 3.150000E+00 3.049990E+00 3.000000E+00 2.969990E+00
2.870000E+00 2.770000E+00 2.669990E+00 2.570000E+00 2.469990E+00
2.379990E+00 2.299990E+00 2.209990E+00 2.120000E+00 2.000000E+00
1.940000E+00 1.860000E+00 1.770000E+00 1.679990E+00 1.589990E+00
1.500000E+00 1.450000E+00 1.400000E+00 1.349990E+00 1.299990E+00
1.250000E+00 1.224990E+00 1.200000E+00 1.174990E+00 1.150000E+00
1.139990E+00 1.129990E+00 1.120000E+00 1.110000E+00 1.099990E+00
1.089990E+00 1.080000E+00 1.070000E+00 1.059990E+00 1.049990E+00
1.040000E+00 1.030000E+00 1.020000E+00 1.009990E+00 1.000000E+00
9.750000E-01 9.500000E-01 9.250000E-01 9.000000E-01 8.500000E-01
8.000000E-01 7.500000E-01 7.000000E-01 6.500000E-01 6.250000E-01
6.000000E-01 5.500000E-01 5.000000E-01 4.500000E-01 4.000000E-01
3.750000E-01 3.500000E-01 3.250000E-01 3.000000E-01 2.750000E-01
2.500000E-01 2.250000E-01 2.000000E-01 1.750000E-01 1.500000E-01
1.250000E-01 1.000000E-01 9.000000E-02 8.000000E-02 7.000000E-02
6.000000E-02 5.000000E-02 4.000000E-02 3.000000E-02 2.530000E-02
1.000000E-02 7.500000E-03 5.000000E-03 4.000000E-03 3.000000E-03
2.500000E-03 2.000000E-03 1.500000E-03 1.200000E-03 1.000000E-03
7.500000E-04 5.000000E-04 1.000000E-04 1.000000E-05
u-234 total 92234 1 0 0.000000E+00
4.510986E-03 4.757363E-03 -1.231885E-04
5.814569E-10 9.742489E-08 2.063851E-07 2.319431E-07 6.176152E-07
9.231530E-06 2.764173E-05 8.188761E-05 1.996819E-04 1.188430E-04
5.678897E-04 3.723359E-04 1.044565E-04 5.127857E-04 4.478409E-04
1.387182E-04 6.498478E-05 5.994327E-05 1.059632E-04 8.117370E-05
1.721682E-04 1.626100E-04 1.709000E-04 4.117717E-05 5.369000E-05
3.138744E-05 9.733480E-05 1.855348E-04 1.821296E-04 2.169121E-05
1.616294E-04 5.783673E-05 4.866144E-05 1.037796E-04 4.962128E-05
4.391558E-05 2.489447E-05 2.192546E-05 5.810353E-05 3.308191E-05
1.757015E-05 -9.932299E-07 -5.762635E-06 -1.347860E-05 -1.218557E-05
-2.017047E-06 -6.224814E-06 -1.817214E-06 -1.336107E-05 -8.857124E-06
-2.534309E-06 -6.268505E-06 -1.942463E-05 -6.952287E-06 -9.270964E-06
-5.016695E-06 -3.539706E-06 -1.224633E-06 -1.550815E-06 -1.328389E-06
-1.087067E-07 -4.089939E-07 -1.942041E-07 -1.483071E-07 -3.014540E-08
-1.437365E-07 -9.007177E-08 -1.296454E-08 -1.098961E-07 -4.759211E-08
-3.958266E-08 -2.063948E-09 -1.005130E-08 -2.381676E-08 -1.047572E-09
-1.118473E-09 1.241990E-10 -5.366082E-11 -5.346411E-11 -2.923313E-09
-3.338868E-09 -1.493598E-11 -2.279208E-11 -7.199807E-10 -8.987974E-11
-2.189838E-10 -2.175781E-11 -9.396965E-12 -1.682207E-10 1.138659E-11
-4.801163E-11 4.277212E-14 6.418566E-11 -3.590361E-12 1.551486E-11
-7.022784E-12 4.754449E-13 1.262907E-12 -1.061301E-10 -3.115855E-12
-3.998677E-11 5.218320E-14 2.886770E-12 9.972963E-12 3.602206E-13
-3.544641E-14 -4.565340E-13 -3.906762E-13 -4.094176E-12 1.509958E-12
9.896480E-13 -8.890050E-15 -9.980198E-12 -1.834534E-11 -6.660895E-11
-8.361293E-12 -2.314084E-12 7.910957E-12 -3.747421E-12 -4.674509E-13
-2.783726E-13 5.640714E-13 -1.263619E-12 1.876653E-12 -3.458782E-13
3.964266E-13 -8.096029E-14 -3.014990E-13 3.357317E-13 7.684112E-13
4.710800E-14 2.347336E-13 -8.513845E-13 -2.995314E-13 -5.691168E-15
-1.575343E-14 1.975820E-15 3.589577E-14 -3.283024E-14 -4.554053E-12
-1.031342E-11 -3.311645E-13 -1.711711E-12 -2.871135E-13 -2.395040E-14
-4.988314E-14 -8.021950E-15 -6.806139E-15 -6.283709E-15 -1.928744E-14
-2.273676E-14 -2.624242E-14 -2.940419E-14 -2.932818E-14 -2.487096E-14
-2.055185E-14 -2.090153E-14 -1.568286E-14 4.044556E-15 -1.100004E-15
-1.006007E-14 -1.512058E-14 -1.528490E-14 -1.476455E-14 -1.377876E-14
-6.998115E-15 -6.414986E-15 -5.551053E-15 -4.272736E-15 -3.051582E-15
-1.120191E-15 -8.480302E-16 -6.478704E-16 -5.176351E-16 -6.748717E-17
-6.293938E-17 -6.058076E-17 -5.969218E-17 -6.119141E-17 -6.332008E-17
-6.634259E-17 -7.015929E-17 -7.458915E-17 -7.908430E-17 -8.370476E-17
-8.862749E-17 -9.090348E-17 -9.294229E-17 -1.020032E-16 -8.841811E-16
-9.690290E-16 -1.031417E-15 -1.062247E-15 -2.241879E-15 -2.303293E-15
-2.306673E-15 -2.165671E-15 -2.024708E-15 -6.316471E-16 -5.119917E-16
-1.643515E-15 -1.461638E-15 -1.228501E-15 -9.115838E-16 -9.803894E-17
-5.921025E-17 -3.726506E-17 -2.782963E-17 -2.087060E-17 -3.530473E-18
-8.656642E-19 -2.791854E-19 -8.745909E-20 -2.513338E-20 -6.505315E-21
-1.451612E-21 -7.438999E-23 -2.611310E-23 -8.538001E-24 -2.522094E-24
-6.532481E-25 -1.437914E-25 -2.575603E-26 -2.525173E-27 -8.675912E-28
-1.414196E-29 -2.457059E-30 -1.730775E-31 -5.537974E-32 -3.241248E-38
0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00
u-234 scatter 92234 0 0 0.000000E+00
6.644297E-04 6.789372E-04 -7.253796E-06
1.624068E-11 1.498993E-09 2.210061E-09 3.019234E-09 1.485854E-08
4.147500E-07 4.994740E-07 -2.057688E-06 8.260277E-06 6.384048E-06
3.416607E-05 2.314843E-05 6.702604E-06 3.465375E-05 3.799558E-05
1.587156E-05 8.343846E-06 8.102940E-06 1.545759E-05 1.265593E-05
2.843251E-05 2.820164E-05 2.933217E-05 6.597646E-06 8.259760E-06
4.585853E-06 1.321025E-05 2.173617E-05 2.408941E-05 3.326117E-06
2.761796E-05 1.187137E-05 1.046844E-05 2.436908E-05 1.536275E-05
1.655217E-05 1.118884E-05 1.113633E-05 3.968209E-05 3.674090E-05
3.947471E-05 2.495301E-05 8.168438E-06 8.267026E-06 2.511112E-06
8.989297E-07 9.842643E-07 2.826410E-07 5.933744E-07 1.081451E-07
-1.158760E-07 -3.228441E-07 -1.829070E-06 -8.369636E-07 -6.930253E-07
-6.464461E-07 -3.126912E-07 -7.491714E-08 -1.256803E-07 -8.790425E-08
-6.682722E-09 -3.177474E-08 -1.358181E-08 -2.448430E-08 -7.405575E-10
-2.381531E-08 -1.677138E-08 1.087897E-09 -1.518392E-08 -8.140176E-09
-5.731765E-09 -2.551186E-10 -8.359771E-10 -1.978769E-09 -4.494942E-11
1.914997E-10 2.502574E-10 3.854115E-11 -1.587687E-11 -3.741346E-10
-1.176212E-10 -1.434239E-11 -2.200907E-11 -5.662974E-12 1.412261E-11
1.968906E-10 -1.946603E-11 -8.655245E-12 -2.396444E-12 1.215475E-11
-4.719829E-11 1.913950E-13 6.404752E-11 -3.486064E-12 1.559158E-11
-6.699422E-12 5.604126E-13 2.181300E-12 -2.223242E-12 4.962365E-14
-3.415680E-13 3.069277E-13 2.900106E-12 9.854989E-12 4.147955E-13
-2.949746E-14 -4.034699E-13 -3.260798E-13 -3.978481E-12 1.474672E-12
9.657468E-13 7.884317E-15 -7.658717E-12 4.440999E-13 -6.924095E-14
-5.525669E-12 -2.119092E-12 3.596589E-12 -3.585189E-12 -4.470271E-13
-2.687707E-13 5.603641E-13 -1.206110E-12 1.865397E-12 -3.252544E-13
4.068594E-13 -6.950363E-14 -2.712372E-13 3.388504E-13 7.461941E-13
1.470861E-13 2.514678E-13 -7.298236E-13 -6.930753E-14 -1.983845E-15
-8.764347E-15 4.590198E-15 2.890316E-14 9.989641E-15 -2.343592E-13
-2.819649E-14 9.121820E-16 1.567449E-16 3.853913E-16 -9.749661E-16
7.412544E-16 6.365694E-16 -2.435351E-16 -2.771577E-16 -1.120208E-15
-1.646584E-15 3.543834E-16 6.820250E-17 -5.691048E-16 -7.483003E-16
-9.871186E-16 -1.800469E-15 -2.429645E-15 5.703603E-15 1.945347E-15
-1.574540E-17 -5.370011E-16 -5.154802E-16 -5.423486E-16 -5.361593E-16
-2.856485E-16 -2.903410E-16 -2.902574E-16 -1.815125E-16 -4.625630E-17
-2.416642E-17 -2.314575E-17 -2.404208E-17 -1.934835E-17 -4.763757E-18
-4.636201E-18 -4.626402E-18 -4.616198E-18 -4.937270E-18 -5.223761E-18
-5.573777E-18 -5.975374E-18 -6.465877E-18 -6.933935E-18 -7.378606E-18
-7.813760E-18 -7.974083E-18 -7.988734E-18 -8.346500E-18 -2.236693E-17
-2.469848E-17 -2.698569E-17 -2.763953E-17 -5.769021E-17 -5.864681E-17
-5.662232E-17 -5.234773E-17 -4.781621E-17 -2.247383E-17 -2.022061E-17
-3.282721E-17 -2.644715E-17 -1.936636E-17 -8.471592E-18 -3.742548E-18
-1.898564E-18 -8.735266E-19 -4.796991E-19 -3.169119E-19 -1.568985E-19
-4.404778E-20 -1.393036E-20 -4.185272E-21 -1.127225E-21 -2.686736E-22
-5.494400E-23 -2.394512E-24 -7.783942E-25 -2.344133E-25 -6.314507E-26
-1.460224E-26 -2.787798E-27 -4.105965E-28 -3.404075E-29 -2.367579E-30
-6.950740E-32 -2.676098E-33 -8.730089E-35 -3.622198E-35 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
0.000000E+00 0.000000E+00 0.000000E+00
Sample plots of the energy-dependent sensitivity profiles from the SDF are shown below. Fig. 6.6.1 and Fig. 6.6.2 depict the region-dependent sensitivity profiles for 235U fission and 238U capture, respectively, in zones 1 and 2 of the Flattop-25 sample problem, which are the core and the reflector regions, respectively.
Fig. 6.6.1 Region-dependent fission sensitivity profiles for Flattop-25 sample problem.
Fig. 6.6.2 Region-dependent capture sensitivity profiles for Flattop-25 sample problem.
HTML formatted output
SAMS will, by default, present the sensitivity results in HTML formatted output for interactive viewing in a web browser. SCALE will return an HTML file to the users’ directory with the same name as the output file, but with the extension “.html”. Opening this file in a web browser will allow the user to begin viewing the output. Other files necessary for the HTML output are stored in the “.htmd” directory associated with the output file and some utilities for data visualization are stored in the applet_resources directory, which is created in the same directory as the output file.
When the .html file is opened, an index page like the one in Fig. 6.6.3 will be shown. In this example, only one SCALE module with HTML output, SAMS, was executed from the input file. If more than one set of HTML output was available, additional modules would be listed on this page. To begin viewing the HTML output, click on the word SAMS in the middle of the page. Once SAMS is selected, the program verification information page is displayed, as shown in Fig. 6.6.4. The output data is sorted into four sections, three of which are shown on the left side of the window in this example. General Information consists of the Program Verification Information and the Elapsed Time for program execution. Input Data lists data input to the calculation including user input and data read from the required forward and adjoint transport calculations. Results contains the computed results, and an additional Messages section, not shown, would contain any error or warning messages generated during the SAMS calculation.
Clicking on the Input Data menu item reveals the menu of pages containing input data. Clicking on the Input Parameters item reveals the page shown in Fig. 6.6.5. Clicking on the Summary of Transport Calculations item reveals the page shown in Fig. 6.6.6.
Clicking on the Results menu item reveals the results available for viewing. Clicking on Energy, Region and Mixture Integrated Sensitivity Coefficients for this Problem under the results menu reveals the energy-, region- and mixture integrated sensitivity coefficients for this problem, as shown in Fig. 6.6.7. As with most of the tables in the HTML output, the table of sensitivity coefficients on this page can be sorted in ascending and descending order by clicking on the heading of the column for which the sorting is desired. When clicking twice on the sensitivity column, the sensitivity results are sorted in descending order as shown in Fig. 6.6.8.
Skipping several output edits for brevity and selecting the Sensitivity Data Plot item will open a page with an applet version of Javapeño with the sensitivity data file preloaded as shown in Fig. 6.6.9. To create a plot of sensitivity data, double-click on the desired information on the right side of the window. Multiple items can be simultaneously selected in the list using standard platform specific modifier keys such as shift or control. Right-clicking on an item in the list will present a menu with various options for plotting. For more information on the use of Javapeño, please select Help from the Javapeño help menu.
Fig. 6.6.3 Index page of SAMS HTML output.
Fig. 6.6.4 Program verification page of SAMS HTML output.
Fig. 6.6.5 Input parameters from SAMS HTML output.
Fig. 6.6.6 Summary of transport calculations from SAMS HTML output.
Fig. 6.6.7 Energy- region- and mixture-integrated sensitivity coefficients from SAMS HTML output.
Fig. 6.6.8 Energy- region- and mixture-integrated sensitivity coefficients from SAMS HTML output sorted by sensitivity.
Fig. 6.6.9 Sensitivity data plot from SAMS HTML output.
6.6.4.2. Example problem using KENO V.a
This sample problem is critical experiment number 10 from evaluation LEU-COMP-THERM-009 of the ICSBEP Handbook [SAMSCom99]. The critical assembly consists of three 8 \(\times\) 15 fuel-rod arrays separated by 6.46 mm thick copper plates. The U(4.31)O2 fuel rods are 92.075-cm long. A KENO3D rendering of the KENO V.a geometry model with the water moderator removed is shown in Fig. 6.6.10.
Fig. 6.6.10 KENO3D rendering of KENO V.a model of LEU-COMP-THERM-009 case 10.
Prior to executing SAMS, the data files listed in Table 6.6.4 were generated for this system. It is recommended that users perform sensitivity analyses on 3-D systems using the TSUNAMI-3D control module, which executes the appropriate codes to generate the required data files in an automated manner from a single input. SAMS is executed as a part of the TSUNAMI-3D control sequence. However, given appropriate data files SAMS can be run as a stand-alone program.
Input data and text output
The SAMS input is shown in Example 6.6.5 and the output is shown in Example 6.6.6. Each section of the output is described below. Note that output shown in this section is intended only to represent the format of output that the user will encounter. The actual computed results may vary.
Parameter Table and Summary of Transport Calculation
In the output listing, the SAMS Parameter Table gives the values of the SAMS input data. Next, the title from the KENO V.a input file is given and a brief summary of the transport solution is given.
Energy-, Region-, and Mixture Integrated Sensitivity Coefficients
Next the energy-, region-, and mixture integrated sensitivity coefficients are given for each reaction of each nuclide. The nuclide symbol and reaction name are given. The data given in the Sensitivity column represents the expected change in keff of this system due to a uniform fractional change in the cross section of the given nuclide and reaction in all energy groups for all regions and mixtures that contain this nuclide. The standard deviations represent stochastic uncertainties from the Monte Carlo calculation. For example, a uniform fractional change of 0.01 (or 1%) in the fission cross section of 235U would produce a fractional change in keff of \(3.2433 \cdot 10^{-1} \times 0.01\) or \(3.2433 \cdot 10^{-3}\) (0.32433% \(\Delta k/k\)). Because the keyword PRTIMP was not entered in the SAMS input, only the complete sensitivity would be given in the Sensitivity column. If PRTIMP was entered, the explicit, implicit and complete sensitivity coefficients are listed in separate columns. Region-integrated sensitivity profiles for 1H elastic scattering, 238U capture and 235U fission are shown in Fig. 6.6.11.
Energy- and Region-Integrated Sensitivity Coefficients
Next the energy- and region-integrated sensitivity coefficients are given for each reaction of each nuclide. The mixture number, nuclide symbol and reaction name are given. The data given in the Sensitivity column represents the expected change in keff of this system due to a uniform fractional change in the cross section of the given nuclide and reaction in all energy groups for all regions that contain the particular mixture. The standard deviations represent stochastic uncertainties from the Monte Carlo calculation. Because the keyword PRTIMP was not entered in the SAMS input, only the complete sensitivity would be given in the Sensitivity column. If PRTIMP was entered the explicit, implicit and complete sensitivity coefficients are listed in separate columns.
Total Sensitivity Coefficients by Nuclide
The next section of the output summarizes the total sensitivity by nuclide. The sensitivity values given here are the same as the values given in the previous section for the sensitivity of keff to the total cross section. Also included in this edit are the atom densities for each nuclide in each mixture.
Total Sensitivity Coefficients by Mixture
The next output edit gives the total sensitivity of keff to the mixture. Here, the sensitivity of keff to the total cross section is summed over each nuclide in a given mixture.
Problem Characterization (MG Calculations Only)
The next output edit is for the problem characterization data. This section includes the median neutron energy group causing fission, capture and scattering, average group for fission, capture and scattering, the average energy causing fission, capture and scattering, and the energy of the average lethargy causing fission, capture and scattering.
Sensitivity Coefficients by Region
If the keyword, PRTGEOM was entered in the SAMS input, the next output edit would give the energy-integrated sensitivity coefficients for each region of each unit defined in the KENO V.a geometry input. Note that for brevity, this output edit was not requested.
Uncertainty Information
The next output edit contains the uncertainty information. First, the percent relative standard deviation in \(k_{e f f}\left(\Delta k_{e f f} / k_{e f f} \times 100 \%\right)\) due to cross-section-covariance data is given. Next a listing of the uncertainty terms as produced by Eq. (6.6.38) are given. The nuclide-reaction to nuclide-reaction covariance matrix responsible for the uncertainty contribution is given followed by the contribution to the uncertainty in terms of % \(\Delta k/k\). The total uncertainty can be computed from individual values by adding the square of the values with positive signs and subtracting the square of the values with negative signs, then taking the square root. The negative values are the result of anti-correlations in the cross-section-covariance data.
Execution Complete
The final edit produced simply states that the execution is complete and gives the elapsed time.
=sams5
read initial xunit= 4 xunitm= 42
makeimp prtgeom end initial
end
----------------------------------------------------------
SAMS: Sensitivity Analysis Module for SCALE
----------------------------------------------------------
SAMS Parameter Table
transport solution from: KENO V.a
use angular fluxes to compute flux moments: false
forward transport solution on unit: 35
adjoint transport solution on unit: 36
working cross-section library on unit: 4
master sensitivity cross-section library on unit: 42
short master cross-section library on unit: 11
covariance data library read from file: 56groupcov7.1
order of flux moments requested for sensitivity calculations: 3
make implicit sensitivities: true
print implicit sensitivities: false
print sensitivities by mixture: true
print sensitivities for each region/zone: false
print uncertainty data: true
print mesh volumes: false
output binary SENPRO data file: false
implicit sensitivity threshold: 100.0000
HTML output: true
Chi sensitivities are constrained: true
Fixup large/zero covariance data: false
Use Default Covariance Data: false
Use User Input Covariance Data: false
HTML Format Options
Background color: PapayaWhip
Major Headings color: Maroon
Sub-headings color: Navy
Plain text color: Black
Hyperlink color: Navy
Hypelink style: none
Visitied link color: Navy
Default Covariance color: Blue
User Input Covariance color: Red
Corrected with default covariance data color: RoyalBlue
Corrected with user input covariance data color: Green
f410p three 15x8 clusters, 2.54 cm pitch, 8.15 cm separation
Summary of Transport Solution
number of neutron groups: 238
number of gamma groups: 0
number of regions: 17
maximum mixture number: 7
number of mixing table entries: 70
quadrature order of angular fluxes: 0
maximum order of flux moments: 3
mesh fluxes generated: true
differential sensitivity: false
Forward Calculation
histories per generation: 10000
number of generations: 1885
k-eff = 0.99783 +/- 0.00019 which occurs for 101 generations skipped
Adjoint Calculation
histories per generation: 30000
number of generations: 10200
k-eff = 0.9988 +/- 0.0013 which occurs for 304 generations skipped
-------------------------------------------------------------------------------------
Energy, Region and Mixture Integrated Sensitivity Coefficients for this Problem
-------------------------------------------------------------------------------------
Nuclide Reaction Sensitivity Std. Dev. % Std. Dev.
------------- --------------- ----------- ---------- -----------
h-1 total 1.6868E-01 +\- 7.6109E-04 ( 0.45%)
h-1 scatter 3.6624E-01 +\- 7.5366E-04 ( 0.21%)
h-1 elastic 3.6625E-01 +\- 7.5366E-04 ( 0.21%)
h-1 capture -1.9756E-01 +\- 1.3139E-05 ( 0.01%)
h-1 n,gamma -1.9756E-01 +\- 1.3139E-05 ( 0.01%)
c total 3.0487E-04 +\- 4.0117E-06 ( 1.32%)
c scatter 3.0745E-04 +\- 3.9729E-06 ( 1.29%)
c elastic 3.0569E-04 +\- 3.9725E-06 ( 1.30%)
c n,n' 1.7567E-06 +\- 2.2595E-08 ( 1.29%)
c capture -2.5788E-06 +\- 5.6099E-09 ( 0.22%)
c n,gamma -2.4558E-06 +\- 3.2370E-09 ( 0.13%)
c n,p -2.8483E-11 +\- 0.0000E+00 ( 0.00%)
c n,d -7.1686E-11 +\- 0.0000E+00 ( 0.00%)
c n,alpha -1.2290E-07 +\- 4.5772E-09 ( 3.72%)
o-16 total 7.4111E-02 +\- 5.3610E-05 ( 0.07%)
o-16 scatter 7.5947E-02 +\- 5.3599E-05 ( 0.07%)
o-16 elastic 7.5552E-02 +\- 5.3596E-05 ( 0.07%)
o-16 n,n' 3.8299E-04 +\- 2.5943E-07 ( 0.07%)
o-16 n,2n 3.7303E-11 +\- 1.5211E-17 ( 0.00%)
o-16 capture -1.8367E-03 +\- 2.2526E-07 ( 0.01%)
o-16 n,gamma -6.6536E-05 +\- 3.9010E-09 ( 0.01%)
o-16 n,p -8.9997E-06 +\- 8.7659E-09 ( 0.10%)
o-16 n,d -9.2619E-07 +\- 1.7089E-09 ( 0.18%)
o-16 n,t -3.3277E-12 +\- 2.7563E-18 ( 0.00%)
o-16 n,alpha -1.7602E-03 +\- 2.2246E-07 ( 0.01%)
na-23 total 2.3175E-08 +\- 1.4447E-09 ( 6.23%)
na-23 scatter 5.7809E-08 +\- 1.4371E-09 ( 2.49%)
na-23 elastic 5.2606E-08 +\- 1.4371E-09 ( 2.73%)
na-23 n,n' 5.2023E-09 +\- 7.8251E-15 ( 0.00%)
na-23 n,2n 4.5318E-13 +\- 0.0000E+00 ( 0.00%)
na-23 capture -3.4634E-08 +\- 2.0188E-11 ( 0.06%)
na-23 n,gamma -3.4409E-08 +\- 2.0188E-11 ( 0.06%)
na-23 n,p -1.5160E-10 +\- 3.2974E-18 ( 0.00%)
na-23 n,alpha -7.2671E-11 +\- 5.0907E-18 ( 0.00%)
mg-24 total 4.5199E-05 +\- 4.5339E-07 ( 1.00%)
mg-24 scatter 5.5375E-05 +\- 4.5356E-07 ( 0.82%)
mg-24 elastic 4.3580E-05 +\- 4.5024E-07 ( 1.03%)
mg-24 n,n' 1.1783E-05 +\- 3.3434E-08 ( 0.28%)
mg-24 n,2n 5.8604E-13 +\- 1.0107E-22 ( 0.00%)
mg-24 capture -1.0176E-05 +\- 3.6669E-09 ( 0.04%)
mg-24 n,gamma -9.1064E-06 +\- 3.1240E-09 ( 0.03%)
mg-24 n,p -2.5472E-07 +\- 5.7541E-10 ( 0.23%)
mg-24 n,alpha -8.1482E-07 +\- 1.3622E-09 ( 0.17%)
mg-25 total 1.2162E-06 +\- 3.2733E-08 ( 2.69%)
mg-25 scatter 5.8494E-06 +\- 3.1369E-08 ( 0.54%)
mg-25 elastic 4.2267E-06 +\- 3.0490E-08 ( 0.72%)
mg-25 n,n' 1.5857E-06 +\- 3.7904E-09 ( 0.24%)
mg-25 n,2n 3.6950E-08 +\- 9.6404E-11 ( 0.26%)
mg-25 capture -4.6331E-06 +\- 1.5204E-09 ( 0.03%)
mg-25 n,gamma -4.2713E-06 +\- 1.4848E-09 ( 0.03%)
mg-25 n,p -3.2385E-08 +\- 3.3458E-11 ( 0.10%)
mg-25 n,alpha -3.2950E-07 +\- 2.8796E-10 ( 0.09%)
mg-26 total 5.3028E-06 +\- 3.2698E-08 ( 0.62%)
mg-26 scatter 6.2567E-06 +\- 3.2364E-08 ( 0.52%)
mg-26 elastic 4.8131E-06 +\- 3.1809E-08 ( 0.66%)
mg-26 n,n' 1.4401E-06 +\- 3.4023E-09 ( 0.24%)
mg-26 n,2n 3.6129E-09 +\- 2.0677E-18 ( 0.00%)
mg-26 capture -9.5390E-07 +\- 3.2481E-10 ( 0.03%)
mg-26 n,gamma -9.4936E-07 +\- 3.2481E-10 ( 0.03%)
mg-26 n,p -5.2412E-10 +\- 1.6179E-19 ( 0.00%)
mg-26 n,alpha -4.0217E-09 +\- 5.7744E-19 ( 0.00%)
al-27 total -6.7582E-05 +\- 2.3201E-05 ( 34.33%)
al-27 scatter 4.6080E-03 +\- 2.2663E-05 ( 0.49%)
al-27 elastic 3.2847E-03 +\- 2.1856E-05 ( 0.67%)
al-27 n,n' 1.3189E-03 +\- 3.0167E-06 ( 0.23%)
al-27 n,2n 5.9944E-08 +\- 1.4100E-09 ( 2.35%)
al-27 capture -4.6756E-03 +\- 1.6107E-06 ( 0.03%)
al-27 n,gamma -4.5949E-03 +\- 1.6079E-06 ( 0.03%)
al-27 n,p -6.7911E-05 +\- 6.9477E-08 ( 0.10%)
al-27 n,d -1.9733E-07 +\- 1.4218E-09 ( 0.72%)
al-27 n,t -6.6132E-09 +\- 5.0053E-16 ( 0.00%)
al-27 n,alpha -1.2531E-05 +\- 3.2986E-08 ( 0.26%)
si-28 total 4.6433E-06 +\- 1.2249E-07 ( 2.64%)
si-28 scatter 2.3947E-05 +\- 1.2025E-07 ( 0.50%)
si-28 elastic 1.9104E-05 +\- 1.1760E-07 ( 0.62%)
si-28 n,n' 4.8407E-06 +\- 1.3004E-08 ( 0.27%)
si-28 capture -1.9303E-05 +\- 6.5597E-09 ( 0.03%)
si-28 n,gamma -1.8340E-05 +\- 6.3655E-09 ( 0.03%)
si-28 n,p -6.6369E-07 +\- 1.0533E-09 ( 0.16%)
si-28 n,d -8.9840E-10 +\- 1.1828E-18 ( 0.00%)
si-28 n,alpha -2.9909E-07 +\- 5.3824E-10 ( 0.18%)
si-29 total 5.5254E-07 +\- 5.6554E-09 ( 1.02%)
si-29 scatter 1.2730E-06 +\- 4.0360E-09 ( 0.32%)
si-29 elastic 8.2855E-07 +\- 3.6454E-09 ( 0.44%)
si-29 n,n' 4.4317E-07 +\- 9.9759E-10 ( 0.23%)
si-29 n,2n 1.2964E-09 +\- 8.4820E-19 ( 0.00%)
si-29 capture -7.2047E-07 +\- 2.3029E-10 ( 0.03%)
si-29 n,gamma -6.7259E-07 +\- 2.2493E-10 ( 0.03%)
si-29 n,p -1.7997E-08 +\- 1.7962E-11 ( 0.10%)
si-29 n,alpha -2.9884E-08 +\- 2.2339E-11 ( 0.07%)
si-30 total 1.4343E-07 +\- 4.4925E-09 ( 3.13%)
si-30 scatter 7.5256E-07 +\- 3.2216E-09 ( 0.43%)
si-30 elastic 5.6016E-07 +\- 3.1391E-09 ( 0.56%)
si-30 n,n' 1.9152E-07 +\- 4.2667E-10 ( 0.22%)
si-30 n,2n 8.8318E-10 +\- 5.2234E-19 ( 0.00%)
si-30 capture -6.0913E-07 +\- 8.1712E-10 ( 0.13%)
si-30 n,gamma -6.0862E-07 +\- 8.1712E-10 ( 0.13%)
si-30 n,p -9.3872E-11 +\- 1.2377E-19 ( 0.00%)
si-30 n,alpha -4.1605E-10 +\- 3.2227E-19 ( 0.00%)
s-32 total -9.4450E-07 +\- 1.5023E-08 ( 1.59%)
s-32 scatter 1.1163E-06 +\- 1.1617E-08 ( 1.04%)
s-32 elastic 1.0496E-06 +\- 1.1393E-08 ( 1.09%)
s-32 n,n' 6.6511E-08 +\- 8.6196E-10 ( 1.30%)
s-32 n,2n 1.0953E-14 +\- 0.0000E+00 ( 0.00%)
s-32 capture -2.0608E-06 +\- 4.4706E-09 ( 0.22%)
s-32 n,gamma -1.9066E-06 +\- 4.2718E-09 ( 0.22%)
s-32 n,p -4.8964E-08 +\- 5.3423E-10 ( 1.09%)
s-32 n,alpha -1.0530E-07 +\- 5.3326E-10 ( 0.51%)
s-33 total -1.2276E-08 +\- 2.3332E-14 ( 0.00%)
s-33 scatter 8.0190E-09 +\- 2.1686E-14 ( 0.00%)
s-33 elastic 7.2275E-09 +\- 2.1662E-14 ( 0.00%)
s-33 n,n' 7.8504E-10 +\- 1.7509E-16 ( 0.00%)
s-33 n,2n 5.0222E-12 +\- 2.0474E-20 ( 0.00%)
s-33 capture -2.0295E-08 +\- 1.6612E-15 ( 0.00%)
s-33 n,gamma -9.9685E-09 +\- 3.3428E-17 ( 0.00%)
s-33 n,p -4.7345E-10 +\- 1.0045E-16 ( 0.00%)
s-33 n,alpha -9.8528E-09 +\- 1.6143E-15 ( 0.00%)
s-34 total 1.2906E-08 +\- 3.9241E-10 ( 3.04%)
s-34 scatter 4.8732E-08 +\- 1.6902E-10 ( 0.35%)
s-34 elastic 4.4628E-08 +\- 1.1281E-10 ( 0.25%)
s-34 n,n' 4.0885E-09 +\- 7.8533E-16 ( 0.00%)
s-34 n,2n 6.7127E-12 +\- 1.8198E-20 ( 0.00%)
s-34 capture -3.5826E-08 +\- 6.1670E-11 ( 0.17%)
s-34 n,gamma -3.5723E-08 +\- 6.1670E-11 ( 0.17%)
s-34 n,p -1.9406E-11 +\- 1.4282E-18 ( 0.00%)
s-34 n,alpha -8.3072E-11 +\- 9.4595E-18 ( 0.00%)
s-36 total 1.3056E-10 +\- 5.1210E-16 ( 0.00%)
s-36 scatter 2.4907E-10 +\- 5.1183E-16 ( 0.00%)
s-36 elastic 2.3590E-10 +\- 5.1180E-16 ( 0.00%)
s-36 n,n' 1.2962E-11 +\- 1.1331E-18 ( 0.00%)
s-36 n,2n 2.0465E-13 +\- 8.5709E-22 ( 0.00%)
s-36 capture -1.1851E-10 +\- 3.2691E-19 ( 0.00%)
s-36 n,gamma -1.1850E-10 +\- 3.2691E-19 ( 0.00%)
s-36 n,p -7.0565E-17 +\- 1.4329E-24 ( 0.00%)
s-36 n,alpha -5.9656E-15 +\- 2.0535E-22 ( 0.00%)
ca-40 total -1.7949E-06 +\- 1.1894E-07 ( 6.63%)
ca-40 scatter 6.7330E-06 +\- 9.7909E-08 ( 1.45%)
ca-40 elastic 6.5226E-06 +\- 9.7589E-08 ( 1.50%)
ca-40 n,n' 2.0785E-07 +\- 3.3003E-09 ( 1.59%)
ca-40 n,2n 4.5607E-14 +\- 0.0000E+00 ( 0.00%)
ca-40 capture -8.5279E-06 +\- 1.9716E-08 ( 0.23%)
ca-40 n,gamma -7.9727E-06 +\- 1.8069E-08 ( 0.23%)
ca-40 n,p -3.7411E-07 +\- 5.5225E-09 ( 1.48%)
ca-40 n,d -5.9766E-11 +\- 6.5478E-19 ( 0.00%)
ca-40 n,t -4.2787E-15 +\- 0.0000E+00 ( 0.00%)
ca-40 n,he-3 -5.7096E-14 +\- 0.0000E+00 ( 0.00%)
ca-40 n,alpha -1.8104E-07 +\- 2.0743E-09 ( 1.15%)
ca-42 total -3.3798E-08 +\- 4.2370E-10 ( 1.25%)
ca-42 scatter 5.5595E-08 +\- 1.8840E-10 ( 0.34%)
ca-42 elastic 4.4749E-08 +\- 7.7226E-14 ( 0.00%)
ca-42 n,n' 1.0831E-08 +\- 3.5480E-15 ( 0.00%)
ca-42 n,2n 4.2807E-12 +\- 0.0000E+00 ( 0.00%)
ca-42 capture -8.9393E-08 +\- 1.9154E-10 ( 0.21%)
ca-42 n,gamma -8.9285E-08 +\- 1.9154E-10 ( 0.21%)
ca-42 n,p -6.3001E-11 +\- 6.1936E-18 ( 0.00%)
ca-42 n,d -3.9571E-14 +\- 0.0000E+00 ( 0.00%)
ca-42 n,t -1.1401E-16 +\- 0.0000E+00 ( 0.00%)
ca-42 n,he-3 -2.8628E-18 +\- 0.0000E+00 ( 0.00%)
ca-42 n,alpha -4.4886E-11 +\- 4.9613E-18 ( 0.00%)
ca-43 total -2.9992E-07 +\- 8.5309E-10 ( 0.28%)
ca-43 scatter 1.7308E-08 +\- 9.3864E-15 ( 0.00%)
ca-43 elastic 1.4035E-08 +\- 8.9886E-15 ( 0.00%)
ca-43 n,n' 3.2593E-09 +\- 1.0415E-15 ( 0.00%)
ca-43 n,2n 1.3992E-11 +\- 1.0619E-22 ( 0.00%)
ca-43 capture -3.1723E-07 +\- 7.1329E-10 ( 0.22%)
ca-43 n,gamma -3.1720E-07 +\- 7.1328E-10 ( 0.22%)
ca-43 n,p -1.2736E-11 +\- 1.4953E-18 ( 0.00%)
ca-43 n,d -3.9211E-15 +\- 0.0000E+00 ( 0.00%)
ca-43 n,t -2.0903E-16 +\- 0.0000E+00 ( 0.00%)
ca-43 n,he-3 -8.8381E-21 +\- 0.0000E+00 ( 0.00%)
ca-43 n,alpha -1.8078E-11 +\- 1.6658E-18 ( 0.00%)
ca-44 total -1.9121E-07 +\- 2.9350E-09 ( 1.53%)
ca-44 scatter 1.8201E-07 +\- 9.8994E-10 ( 0.54%)
ca-44 elastic 1.4003E-07 +\- 6.5307E-10 ( 0.47%)
ca-44 n,n' 4.1934E-08 +\- 3.9319E-10 ( 0.94%)
ca-44 n,2n 3.7701E-11 +\- 0.0000E+00 ( 0.00%)
ca-44 capture -3.7322E-07 +\- 8.4317E-10 ( 0.23%)
ca-44 n,gamma -3.7321E-07 +\- 8.4317E-10 ( 0.23%)
ca-44 n,p -4.3462E-12 +\- 1.1952E-19 ( 0.00%)
ca-44 n,d -1.9015E-14 +\- 0.0000E+00 ( 0.00%)
ca-44 n,t -3.6521E-17 +\- 0.0000E+00 ( 0.00%)
ca-44 n,he-3 -5.7133E-22 +\- 0.0000E+00 ( 0.00%)
ca-44 n,alpha -2.1833E-12 +\- 5.2524E-20 ( 0.00%)
ca-46 total -9.1789E-11 +\- 2.9314E-16 ( 0.00%)
ca-46 scatter 5.0404E-10 +\- 2.9308E-16 ( 0.00%)
ca-46 elastic 4.3313E-10 +\- 2.8501E-16 ( 0.00%)
ca-46 n,n' 7.0696E-11 +\- 2.2304E-17 ( 0.00%)
ca-46 n,2n 2.0718E-13 +\- 0.0000E+00 ( 0.00%)
ca-46 capture -5.9583E-10 +\- 5.9929E-20 ( 0.00%)
ca-46 n,gamma -5.9583E-10 +\- 5.9929E-20 ( 0.00%)
ca-46 n,p -3.2267E-16 +\- 3.6714E-26 ( 0.00%)
ca-46 n,d -6.8722E-18 +\- 0.0000E+00 ( 0.00%)
ca-46 n,t -3.3633E-21 +\- 0.0000E+00 ( 0.00%)
ca-46 n,alpha -2.2633E-17 +\- 0.0000E+00 ( 0.00%)
ca-48 total -2.4743E-08 +\- 1.8724E-10 ( 0.76%)
ca-48 scatter 1.6341E-08 +\- 8.2057E-15 ( 0.00%)
ca-48 elastic 1.5512E-08 +\- 8.2022E-15 ( 0.00%)
ca-48 n,n' 8.1356E-10 +\- 6.9284E-17 ( 0.00%)
ca-48 n,2n 1.5225E-11 +\- 0.0000E+00 ( 0.00%)
ca-48 capture -4.1084E-08 +\- 7.9141E-11 ( 0.19%)
ca-48 n,gamma -4.1084E-08 +\- 7.9141E-11 ( 0.19%)
ca-48 n,p -1.0552E-16 +\- 0.0000E+00 ( 0.00%)
ca-48 n,d -2.1494E-17 +\- 0.0000E+00 ( 0.00%)
ca-48 n,t -5.0004E-22 +\- 0.0000E+00 ( 0.00%)
ca-48 n,alpha -1.9945E-17 +\- 0.0000E+00 ( 0.00%)
ti-46 total -1.8856E-07 +\- 8.6800E-10 ( 0.46%)
ti-46 scatter 2.1933E-07 +\- 4.7014E-10 ( 0.21%)
ti-46 elastic 1.2896E-07 +\- 3.2795E-10 ( 0.25%)
ti-46 n,n' 9.0312E-08 +\- 1.9591E-10 ( 0.22%)
ti-46 n,2n 4.8498E-12 +\- 2.8827E-21 ( 0.00%)
ti-46 capture -4.0789E-07 +\- 1.3319E-10 ( 0.03%)
ti-46 n,gamma -3.9961E-07 +\- 1.3319E-10 ( 0.03%)
ti-46 n,p -7.7485E-09 +\- 3.8514E-18 ( 0.00%)
ti-46 n,d -3.1806E-13 +\- 2.3627E-22 ( 0.00%)
ti-46 n,t -9.9763E-17 +\- 3.2655E-26 ( 0.00%)
ti-46 n,he-3 -4.2439E-16 +\- 1.9547E-25 ( 0.00%)
ti-46 n,alpha -5.2794E-10 +\- 2.5965E-19 ( 0.00%)
ti-47 total -8.7007E-07 +\- 2.4463E-09 ( 0.28%)
ti-47 scatter 2.4268E-07 +\- 2.2853E-09 ( 0.94%)
ti-47 elastic 1.3531E-07 +\- 2.2669E-09 ( 1.68%)
ti-47 n,n' 1.0717E-07 +\- 1.8872E-10 ( 0.18%)
ti-47 n,2n 1.9211E-10 +\- 4.0015E-20 ( 0.00%)
ti-47 capture -1.1128E-06 +\- 3.6552E-10 ( 0.03%)
ti-47 n,gamma -1.1003E-06 +\- 3.6552E-10 ( 0.03%)
ti-47 n,p -1.0646E-08 +\- 1.4197E-16 ( 0.00%)
ti-47 n,d -2.0197E-13 +\- 1.5135E-22 ( 0.00%)
ti-47 n,t -4.3601E-15 +\- 3.0922E-24 ( 0.00%)
ti-47 n,he-3 -2.3300E-17 +\- 7.4079E-27 ( 0.00%)
ti-47 n,alpha -1.8482E-09 +\- 1.3222E-17 ( 0.00%)
ti-48 total -4.5957E-05 +\- 3.3263E-08 ( 0.07%)
ti-48 scatter 3.0030E-06 +\- 2.3419E-08 ( 0.78%)
ti-48 elastic 2.1566E-06 +\- 2.3249E-08 ( 1.08%)
ti-48 n,n' 8.4526E-07 +\- 1.8984E-09 ( 0.22%)
ti-48 n,2n 1.0593E-09 +\- 5.3938E-19 ( 0.00%)
ti-48 capture -4.8960E-05 +\- 1.7297E-08 ( 0.04%)
ti-48 n,gamma -4.8958E-05 +\- 1.7297E-08 ( 0.04%)
ti-48 n,p -1.6058E-09 +\- 9.4652E-19 ( 0.00%)
ti-48 n,d -1.6876E-12 +\- 1.6673E-21 ( 0.00%)
ti-48 n,t -8.0952E-16 +\- 2.7332E-25 ( 0.00%)
ti-48 n,he-3 -1.9723E-17 +\- 6.7861E-27 ( 0.00%)
ti-48 n,alpha -1.5654E-10 +\- 4.0991E-20 ( 0.00%)
ti-49 total -6.8733E-07 +\- 6.2660E-10 ( 0.09%)
ti-49 scatter 1.6784E-07 +\- 2.2439E-09 ( 1.34%)
ti-49 elastic 1.0265E-07 +\- 2.2349E-09 ( 2.18%)
ti-49 n,n' 6.4463E-08 +\- 1.2032E-10 ( 0.19%)
ti-49 n,2n 7.2472E-10 +\- 7.6807E-20 ( 0.00%)
ti-49 capture -8.5517E-07 +\- 2.9502E-10 ( 0.03%)
ti-49 n,gamma -8.5476E-07 +\- 2.9502E-10 ( 0.03%)
ti-49 n,p -3.3313E-10 +\- 3.2208E-19 ( 0.00%)
ti-49 n,d -7.4647E-14 +\- 6.7611E-23 ( 0.00%)
ti-49 n,t -2.1162E-15 +\- 9.3763E-25 ( 0.00%)
ti-49 n,he-3 -9.3750E-20 +\- 0.0000E+00 ( 0.00%)
ti-49 n,alpha -7.1068E-11 +\- 4.3628E-20 ( 0.00%)
ti-50 total 5.1758E-08 +\- 4.1741E-10 ( 0.81%)
ti-50 scatter 1.3378E-07 +\- 1.9577E-10 ( 0.15%)
ti-50 elastic 8.8832E-08 +\- 1.1755E-10 ( 0.13%)
ti-50 n,n' 4.4731E-08 +\- 8.8290E-11 ( 0.20%)
ti-50 n,2n 2.1770E-10 +\- 7.0963E-20 ( 0.00%)
ti-50 capture -8.2023E-08 +\- 2.5593E-11 ( 0.03%)
ti-50 n,gamma -8.2018E-08 +\- 2.5593E-11 ( 0.03%)
ti-50 n,p -2.9356E-12 +\- 9.4805E-22 ( 0.00%)
ti-50 n,d -1.7101E-14 +\- 1.1908E-23 ( 0.00%)
ti-50 n,t -1.0953E-17 +\- 2.8155E-27 ( 0.00%)
ti-50 n,he-3 -4.0903E-23 +\- 0.0000E+00 ( 0.00%)
ti-50 n,alpha -1.7272E-12 +\- 6.5310E-22 ( 0.00%)
cr-50 total -1.3040E-05 +\- 9.1891E-09 ( 0.07%)
cr-50 scatter 1.3376E-06 +\- 7.2895E-09 ( 0.54%)
cr-50 elastic 1.2307E-06 +\- 7.2762E-09 ( 0.59%)
cr-50 n,n' 1.0681E-07 +\- 2.3841E-10 ( 0.22%)
cr-50 n,2n 3.1922E-12 +\- 2.3793E-21 ( 0.00%)
cr-50 capture -1.4377E-05 +\- 5.0788E-09 ( 0.04%)
cr-50 n,gamma -1.4333E-05 +\- 5.0787E-09 ( 0.04%)
cr-50 n,p -4.1672E-08 +\- 3.2114E-11 ( 0.08%)
cr-50 n,d -7.5849E-12 +\- 5.4355E-19 ( 0.00%)
cr-50 n,alpha -2.7989E-09 +\- 1.7847E-15 ( 0.00%)
cr-52 total -8.7501E-06 +\- 2.6630E-08 ( 0.30%)
cr-52 scatter 4.4850E-06 +\- 2.1763E-08 ( 0.49%)
cr-52 elastic 2.3489E-06 +\- 2.0602E-08 ( 0.88%)
cr-52 n,n' 2.1346E-06 +\- 4.5774E-09 ( 0.21%)
cr-52 n,2n 1.3180E-09 +\- 9.1937E-19 ( 0.00%)
cr-52 capture -1.3235E-05 +\- 4.5982E-09 ( 0.03%)
cr-52 n,gamma -1.3217E-05 +\- 4.5982E-09 ( 0.03%)
cr-52 n,p -1.5832E-08 +\- 1.8313E-11 ( 0.12%)
cr-52 n,alpha -1.9304E-09 +\- 3.8775E-17 ( 0.00%)
cr-53 total -3.4245E-05 +\- 2.1670E-08 ( 0.06%)
cr-53 scatter 1.3865E-06 +\- 1.4475E-08 ( 1.04%)
cr-53 elastic 1.0474E-06 +\- 1.4432E-08 ( 1.38%)
cr-53 n,n' 3.3527E-07 +\- 7.1251E-10 ( 0.21%)
cr-53 n,2n 3.8152E-09 +\- 3.6614E-17 ( 0.00%)
cr-53 capture -3.5632E-05 +\- 1.2596E-08 ( 0.04%)
cr-53 n,gamma -3.5630E-05 +\- 1.2596E-08 ( 0.04%)
cr-53 n,p -6.9509E-10 +\- 5.2410E-17 ( 0.00%)
cr-53 n,alpha -1.5386E-09 +\- 1.4981E-16 ( 0.00%)
cr-54 total -1.6695E-08 +\- 4.1104E-10 ( 2.46%)
cr-54 scatter 1.6318E-07 +\- 2.3409E-10 ( 0.14%)
cr-54 elastic 9.3303E-08 +\- 7.9919E-11 ( 0.09%)
cr-54 n,n' 6.9373E-08 +\- 1.3815E-10 ( 0.20%)
cr-54 n,2n 5.0486E-10 +\- 8.3567E-20 ( 0.00%)
cr-54 capture -1.7988E-07 +\- 5.9710E-11 ( 0.03%)
cr-54 n,gamma -1.7985E-07 +\- 5.9710E-11 ( 0.03%)
cr-54 n,p -4.2582E-12 +\- 1.1391E-21 ( 0.00%)
cr-54 n,alpha -2.1413E-11 +\- 1.0788E-20 ( 0.00%)
mn-55 total -9.6360E-05 +\- 1.4079E-07 ( 0.15%)
mn-55 scatter 6.8704E-06 +\- 1.3262E-07 ( 1.93%)
mn-55 elastic 5.7292E-06 +\- 1.3254E-07 ( 2.31%)
mn-55 n,n' 1.1366E-06 +\- 2.5436E-09 ( 0.22%)
mn-55 n,2n 4.4184E-09 +\- 1.0180E-18 ( 0.00%)
mn-55 capture -1.0323E-04 +\- 3.5222E-08 ( 0.03%)
mn-55 n,gamma -1.0323E-04 +\- 3.5222E-08 ( 0.03%)
mn-55 n,p -3.7300E-09 +\- 9.1810E-18 ( 0.00%)
mn-55 n,d -3.5791E-11 +\- 6.7992E-21 ( 0.00%)
mn-55 n,t -1.3299E-12 +\- 1.2646E-21 ( 0.00%)
mn-55 n,he-3 -4.6031E-14 +\- 4.8445E-23 ( 0.00%)
mn-55 n,alpha -8.3737E-10 +\- 6.0947E-19 ( 0.00%)
fe-54 total -3.9443E-06 +\- 8.5217E-09 ( 0.22%)
fe-54 scatter 7.5526E-07 +\- 8.1028E-09 ( 1.07%)
fe-54 elastic 5.5856E-07 +\- 8.0723E-09 ( 1.45%)
fe-54 n,n' 1.9618E-07 +\- 4.3649E-10 ( 0.22%)
fe-54 n,2n 3.8024E-12 +\- 1.3165E-21 ( 0.00%)
fe-54 capture -4.6996E-06 +\- 1.5839E-09 ( 0.03%)
fe-54 n,gamma -4.5471E-06 +\- 1.5807E-09 ( 0.03%)
fe-54 n,p -1.5084E-07 +\- 9.9241E-11 ( 0.07%)
fe-54 n,d -2.5872E-11 +\- 1.0217E-20 ( 0.00%)
fe-54 n,alpha -1.5885E-09 +\- 6.8285E-19 ( 0.00%)
fe-56 total -6.9531E-05 +\- 1.3507E-07 ( 0.19%)
fe-56 scatter 1.1558E-05 +\- 1.1066E-07 ( 0.96%)
fe-56 elastic 7.5772E-06 +\- 1.0976E-07 ( 1.45%)
fe-56 n,n' 3.9750E-06 +\- 9.5385E-09 ( 0.24%)
fe-56 n,2n 5.5701E-09 +\- 3.2182E-18 ( 0.00%)
fe-56 capture -8.1090E-05 +\- 2.8413E-08 ( 0.04%)
fe-56 n,gamma -8.1049E-05 +\- 2.8413E-08 ( 0.04%)
fe-56 n,p -2.8313E-08 +\- 4.3821E-11 ( 0.15%)
fe-56 n,d -6.3279E-11 +\- 5.4153E-20 ( 0.00%)
fe-56 n,t -1.5517E-13 +\- 1.2163E-22 ( 0.00%)
fe-56 n,he-3 -1.7491E-14 +\- 1.2250E-23 ( 0.00%)
fe-56 n,alpha -1.1979E-08 +\- 1.2530E-11 ( 0.10%)
fe-57 total -1.4596E-06 +\- 2.4578E-09 ( 0.17%)
fe-57 scatter 2.9573E-07 +\- 2.2278E-09 ( 0.75%)
fe-57 elastic 1.6174E-07 +\- 2.1139E-09 ( 1.31%)
fe-57 n,n' 1.3200E-07 +\- 2.4140E-10 ( 0.18%)
fe-57 n,2n 1.9892E-09 +\- 3.7761E-19 ( 0.00%)
fe-57 capture -1.7553E-06 +\- 6.0368E-10 ( 0.03%)
fe-57 n,gamma -1.7542E-06 +\- 6.0368E-10 ( 0.03%)
fe-57 n,p -4.5820E-10 +\- 1.9581E-19 ( 0.00%)
fe-57 n,alpha -6.5747E-10 +\- 3.5652E-19 ( 0.00%)
fe-58 total -8.6319E-08 +\- 9.2863E-11 ( 0.11%)
fe-58 scatter 3.2999E-08 +\- 2.1157E-13 ( 0.00%)
fe-58 elastic 1.9560E-08 +\- 2.1157E-13 ( 0.00%)
fe-58 n,n' 1.3357E-08 +\- 1.7645E-16 ( 0.00%)
fe-58 n,2n 8.1863E-11 +\- 2.4070E-20 ( 0.00%)
fe-58 capture -1.1932E-07 +\- 3.5967E-11 ( 0.03%)
fe-58 n,gamma -1.1931E-07 +\- 3.5967E-11 ( 0.03%)
fe-58 n,p -8.9843E-13 +\- 3.3131E-22 ( 0.00%)
fe-58 n,alpha -5.5353E-12 +\- 1.7256E-21 ( 0.00%)
cu-63 total -3.2291E-03 +\- 6.7644E-06 ( 0.21%)
cu-63 scatter 5.2401E-04 +\- 4.5049E-06 ( 0.86%)
cu-63 elastic 4.4000E-04 +\- 4.1846E-06 ( 0.95%)
cu-63 n,n' 8.2502E-05 +\- 1.1401E-06 ( 1.38%)
cu-63 n,2n 2.1792E-07 +\- 7.0126E-09 ( 3.22%)
cu-63 capture -3.7531E-03 +\- 3.1714E-06 ( 0.08%)
cu-63 n,gamma -3.7267E-03 +\- 3.1712E-06 ( 0.09%)
cu-63 n,p -2.5132E-05 +\- 3.5516E-08 ( 0.14%)
cu-63 n,d -5.3278E-07 +\- 1.5798E-09 ( 0.30%)
cu-63 n,he-3 -3.1134E-12 +\- 3.9260E-19 ( 0.00%)
cu-63 n,alpha -6.8776E-07 +\- 2.7471E-09 ( 0.40%)
cu-65 total -6.2532E-04 +\- 4.7383E-06 ( 0.76%)
cu-65 scatter 1.8352E-04 +\- 4.2466E-06 ( 2.31%)
cu-65 elastic 1.4901E-04 +\- 4.1861E-06 ( 2.81%)
cu-65 n,n' 3.4114E-05 +\- 4.9055E-07 ( 1.44%)
cu-65 n,2n 3.8621E-07 +\- 7.8368E-09 ( 2.03%)
cu-65 capture -8.0884E-04 +\- 6.8255E-07 ( 0.08%)
cu-65 n,gamma -8.0848E-04 +\- 6.8255E-07 ( 0.08%)
cu-65 n,p -3.1527E-07 +\- 7.8886E-10 ( 0.25%)
cu-65 n,d -4.1498E-08 +\- 1.8559E-10 ( 0.45%)
cu-65 n,t -1.0755E-11 +\- 2.7949E-18 ( 0.00%)
cu-65 n,he-3 -3.9455E-13 +\- 1.3313E-20 ( 0.00%)
cu-65 n,alpha -9.0375E-09 +\- 1.3569E-15 ( 0.00%)
u-234 total -9.0979E-04 +\- 3.1204E-07 ( 0.03%)
u-234 scatter 6.2441E-06 +\- 1.2041E-08 ( 0.19%)
u-234 elastic 4.0083E-06 +\- 1.1854E-08 ( 0.30%)
u-234 n,n' 2.2050E-06 +\- 1.4630E-09 ( 0.07%)
u-234 n,2n 3.0794E-08 +\- 1.7472E-11 ( 0.06%)
u-234 fission 2.2909E-05 +\- 1.7241E-09 ( 0.01%)
u-234 capture -9.3894E-04 +\- 3.0291E-07 ( 0.03%)
u-234 n,gamma -9.3894E-04 +\- 3.0291E-07 ( 0.03%)
u-234 nubar 3.6021E-05 +\- 2.6348E-09 ( 0.01%)
u-234 chi 8.7265E-13 +\- 1.5382E-09 (-999.99%)
u-235 total 2.0616E-01 +\- 7.0501E-05 ( 0.03%)
u-235 scatter 8.8279E-04 +\- 1.0661E-06 ( 0.12%)
u-235 elastic 3.1799E-04 +\- 9.6547E-07 ( 0.30%)
u-235 n,n' 5.3198E-04 +\- 3.3888E-07 ( 0.06%)
u-235 n,2n 3.2795E-05 +\- 1.0524E-08 ( 0.03%)
u-235 fission 3.2433E-01 +\- 6.2460E-05 ( 0.02%)
u-235 capture -1.1906E-01 +\- 1.1810E-05 ( 0.01%)
u-235 n,gamma -1.1906E-01 +\- 1.1810E-05 ( 0.01%)
u-235 nubar 9.6382E-01 +\- 8.0129E-05 ( 0.01%)
u-235 chi -6.6884E-08 +\- 4.6474E-05 (-999.99%)
u-236 total -2.1805E-04 +\- 1.2725E-07 ( 0.06%)
u-236 scatter 5.6471E-06 +\- 9.6565E-09 ( 0.17%)
u-236 elastic 2.3929E-06 +\- 9.2543E-09 ( 0.39%)
u-236 n,n' 3.1626E-06 +\- 2.0863E-09 ( 0.07%)
u-236 n,2n 9.1002E-08 +\- 4.1257E-11 ( 0.05%)
u-236 fission 1.2829E-05 +\- 1.7760E-09 ( 0.01%)
u-236 capture -2.3653E-04 +\- 1.2003E-07 ( 0.05%)
u-236 n,gamma -2.3653E-04 +\- 1.2003E-07 ( 0.05%)
u-236 nubar 2.1111E-05 +\- 1.6677E-09 ( 0.01%)
u-236 chi -4.2095E-13 +\- 1.0077E-09 (-999.99%)
u-238 total -5.9609E-02 +\- 2.6857E-05 ( 0.05%)
u-238 scatter 2.9406E-02 +\- 2.0373E-05 ( 0.07%)
u-238 elastic 1.3217E-02 +\- 1.6047E-05 ( 0.12%)
u-238 n,n' 1.5251E-02 +\- 9.7098E-06 ( 0.06%)
u-238 n,2n 9.2840E-04 +\- 3.7033E-07 ( 0.04%)
u-238 fission 2.4727E-02 +\- 2.4196E-06 ( 0.01%)
u-238 capture -1.1374E-01 +\- 9.9579E-06 ( 0.01%)
u-238 n,gamma -1.1374E-01 +\- 9.9579E-06 ( 0.01%)
u-238 nubar 3.6122E-02 +\- 3.8010E-06 ( 0.01%)
u-238 chi -2.9925E-09 +\- 1.7124E-06 (-999.99%)
----------------------------------------------------------------------------
Energy and Region Integrated Sensitivity Coefficients for this Problem
----------------------------------------------------------------------------
Mixture Nuclide Reaction Sensitivity Std. Dev. % Std. Dev.
------- ------------- --------------- ----------- ---------- -----------
1 o-16 total 1.1444E-02 +/- 3.3431E-05 ( 0.29%)
1 o-16 scatter 1.2013E-02 +/- 3.3421E-05 ( 0.28%)
1 o-16 elastic 1.1922E-02 +/- 3.3419E-05 ( 0.28%)
1 o-16 n,n' 8.7378E-05 +/- 1.9449E-07 ( 0.22%)
1 o-16 n,2n 1.0127E-11 +/- 1.6420E-20 ( 0.00%)
1 o-16 capture -5.6947E-04 +/- 1.5760E-07 ( 0.03%)
1 o-16 n,gamma -1.0052E-05 +/- 1.0479E-09 ( 0.01%)
1 o-16 n,p -2.8703E-06 +/- 6.5195E-09 ( 0.23%)
1 o-16 n,d -2.9590E-07 +/- 1.2700E-09 ( 0.43%)
1 o-16 n,t -1.0698E-12 +/- 2.9755E-21 ( 0.00%)
1 o-16 n,alpha -5.5625E-04 +/- 1.5540E-07 ( 0.03%)
1 u-234 total -9.0979E-04 +/- 3.1204E-07 ( 0.03%)
1 u-234 scatter 6.2441E-06 +/- 1.2041E-08 ( 0.19%)
1 u-234 elastic 4.0083E-06 +/- 1.1854E-08 ( 0.30%)
1 u-234 n,n' 2.2050E-06 +/- 1.4630E-09 ( 0.07%)
1 u-234 n,2n 3.0794E-08 +/- 1.7472E-11 ( 0.06%)
1 u-234 fission 2.2909E-05 +/- 1.7241E-09 ( 0.01%)
1 u-234 capture -9.3894E-04 +/- 3.0291E-07 ( 0.03%)
1 u-234 n,gamma -9.3894E-04 +/- 3.0291E-07 ( 0.03%)
1 u-234 nubar 3.6021E-05 +/- 2.6348E-09 ( 0.01%)
1 u-234 chi 8.7265E-13 +/- 1.5382E-09 (-999.99%)
1 u-235 total 2.0616E-01 +/- 7.0501E-05 ( 0.03%)
1 u-235 scatter 8.8279E-04 +/- 1.0661E-06 ( 0.12%)
1 u-235 elastic 3.1799E-04 +/- 9.6547E-07 ( 0.30%)
1 u-235 n,n' 5.3198E-04 +/- 3.3888E-07 ( 0.06%)
1 u-235 n,2n 3.2795E-05 +/- 1.0524E-08 ( 0.03%)
1 u-235 fission 3.2433E-01 +/- 6.2460E-05 ( 0.02%)
1 u-235 capture -1.1906E-01 +/- 1.1810E-05 ( 0.01%)
1 u-235 n,gamma -1.1906E-01 +/- 1.1810E-05 ( 0.01%)
1 u-235 nubar 9.6382E-01 +/- 8.0129E-05 ( 0.01%)
1 u-235 chi -6.6884E-08 +/- 4.6474E-05 (-999.99%)
1 u-236 total -2.1805E-04 +/- 1.2725E-07 ( 0.06%)
1 u-236 scatter 5.6471E-06 +/- 9.6565E-09 ( 0.17%)
1 u-236 elastic 2.3929E-06 +/- 9.2543E-09 ( 0.39%)
1 u-236 n,n' 3.1626E-06 +/- 2.0863E-09 ( 0.07%)
1 u-236 n,2n 9.1002E-08 +/- 4.1257E-11 ( 0.05%)
1 u-236 fission 1.2829E-05 +/- 1.7760E-09 ( 0.01%)
1 u-236 capture -2.3653E-04 +/- 1.2003E-07 ( 0.05%)
1 u-236 n,gamma -2.3653E-04 +/- 1.2003E-07 ( 0.05%)
1 u-236 nubar 2.1111E-05 +/- 1.6677E-09 ( 0.01%)
1 u-236 chi -4.2095E-13 +/- 1.0077E-09 (-999.99%)
1 u-238 total -5.9609E-02 +/- 2.6857E-05 ( 0.05%)
1 u-238 scatter 2.9406E-02 +/- 2.0373E-05 ( 0.07%)
1 u-238 elastic 1.3217E-02 +/- 1.6047E-05 ( 0.12%)
1 u-238 n,n' 1.5251E-02 +/- 9.7098E-06 ( 0.06%)
1 u-238 n,2n 9.2840E-04 +/- 3.7033E-07 ( 0.04%)
1 u-238 fission 2.4727E-02 +/- 2.4196E-06 ( 0.01%)
1 u-238 capture -1.1374E-01 +/- 9.9579E-06 ( 0.01%)
1 u-238 n,gamma -1.1374E-01 +/- 9.9579E-06 ( 0.01%)
1 u-238 nubar 3.6122E-02 +/- 3.8010E-06 ( 0.01%)
1 u-238 chi -2.9925E-09 +/- 1.7124E-06 (-999.99%)
2 h-1 total 2.0159E-01 +/- 4.6480E-04 ( 0.23%)
2 h-1 scatter 3.4592E-01 +/- 4.6062E-04 ( 0.13%)
2 h-1 elastic 3.4592E-01 +/- 4.6062E-04 ( 0.13%)
2 h-1 capture -1.4433E-01 +/- 1.0386E-05 ( 0.01%)
2 h-1 n,gamma -1.4433E-01 +/- 1.0386E-05 ( 0.01%)
2 o-16 total 3.6005E-02 +/- 2.9143E-05 ( 0.08%)
2 o-16 scatter 3.7111E-02 +/- 2.9137E-05 ( 0.08%)
2 o-16 elastic 3.6871E-02 +/- 2.9136E-05 ( 0.08%)
2 o-16 n,n' 2.3245E-04 +/- 1.5061E-07 ( 0.06%)
2 o-16 n,2n 2.3322E-11 +/- 1.2879E-17 ( 0.00%)
2 o-16 capture -1.1055E-03 +/- 1.4843E-07 ( 0.01%)
2 o-16 n,gamma -4.1305E-05 +/- 2.9719E-09 ( 0.01%)
2 o-16 n,p -5.3552E-06 +/- 5.0619E-09 ( 0.09%)
2 o-16 n,d -5.5135E-07 +/- 9.9279E-10 ( 0.18%)
2 o-16 n,t -2.0353E-12 +/- 2.3337E-18 ( 0.00%)
2 o-16 n,alpha -1.0583E-03 +/- 1.4700E-07 ( 0.01%)
3 mg-24 total 4.5166E-05 +/- 4.5339E-07 ( 1.00%)
3 mg-24 scatter 5.5339E-05 +/- 4.5356E-07 ( 0.82%)
3 mg-24 elastic 4.3547E-05 +/- 4.5024E-07 ( 1.03%)
3 mg-24 n,n' 1.1781E-05 +/- 3.3434E-08 ( 0.28%)
3 mg-24 n,2n 5.8600E-13 +/- 1.0107E-22 ( 0.00%)
3 mg-24 capture -1.0173E-05 +/- 3.6669E-09 ( 0.04%)
3 mg-24 n,gamma -9.1039E-06 +/- 3.1240E-09 ( 0.03%)
3 mg-24 n,p -2.5459E-07 +/- 5.7541E-10 ( 0.23%)
3 mg-24 n,alpha -8.1441E-07 +/- 1.3622E-09 ( 0.17%)
3 mg-25 total 1.2140E-06 +/- 3.2733E-08 ( 2.70%)
3 mg-25 scatter 5.8458E-06 +/- 3.1369E-08 ( 0.54%)
3 mg-25 elastic 4.2233E-06 +/- 3.0490E-08 ( 0.72%)
3 mg-25 n,n' 1.5854E-06 +/- 3.7904E-09 ( 0.24%)
3 mg-25 n,2n 3.6936E-08 +/- 9.6404E-11 ( 0.26%)
3 mg-25 capture -4.6318E-06 +/- 1.5204E-09 ( 0.03%)
3 mg-25 n,gamma -4.2701E-06 +/- 1.4848E-09 ( 0.03%)
3 mg-25 n,p -3.2369E-08 +/- 3.3458E-11 ( 0.10%)
3 mg-25 n,alpha -3.2935E-07 +/- 2.8796E-10 ( 0.09%)
3 mg-26 total 5.2992E-06 +/- 3.2698E-08 ( 0.62%)
3 mg-26 scatter 6.2529E-06 +/- 3.2364E-08 ( 0.52%)
3 mg-26 elastic 4.8094E-06 +/- 3.1809E-08 ( 0.66%)
3 mg-26 n,n' 1.4399E-06 +/- 3.4023E-09 ( 0.24%)
3 mg-26 n,2n 3.6117E-09 +/- 2.0668E-18 ( 0.00%)
3 mg-26 capture -9.5364E-07 +/- 3.2481E-10 ( 0.03%)
3 mg-26 n,gamma -9.4910E-07 +/- 3.2481E-10 ( 0.03%)
3 mg-26 n,p -5.2381E-10 +/- 1.5687E-19 ( 0.00%)
3 mg-26 n,alpha -4.0194E-09 +/- 5.3317E-19 ( 0.00%)
3 al-27 total -6.7582E-05 +/- 2.3201E-05 ( 34.33%)
3 al-27 scatter 4.6080E-03 +/- 2.2663E-05 ( 0.49%)
3 al-27 elastic 3.2847E-03 +/- 2.1856E-05 ( 0.67%)
3 al-27 n,n' 1.3189E-03 +/- 3.0167E-06 ( 0.23%)
3 al-27 n,2n 5.9944E-08 +/- 1.4100E-09 ( 2.35%)
3 al-27 capture -4.6756E-03 +/- 1.6107E-06 ( 0.03%)
3 al-27 n,gamma -4.5949E-03 +/- 1.6079E-06 ( 0.03%)
3 al-27 n,p -6.7911E-05 +/- 6.9477E-08 ( 0.10%)
3 al-27 n,d -1.9733E-07 +/- 1.4218E-09 ( 0.72%)
3 al-27 n,t -6.6132E-09 +/- 5.0053E-16 ( 0.00%)
3 al-27 n,alpha -1.2531E-05 +/- 3.2986E-08 ( 0.26%)
3 si-28 total 4.3107E-06 +/- 1.2244E-07 ( 2.84%)
3 si-28 scatter 2.3404E-05 +/- 1.2022E-07 ( 0.51%)
3 si-28 elastic 1.8601E-05 +/- 1.1757E-07 ( 0.63%)
3 si-28 n,n' 4.8005E-06 +/- 1.3001E-08 ( 0.27%)
3 si-28 capture -1.9094E-05 +/- 6.5542E-09 ( 0.03%)
3 si-28 n,gamma -1.8139E-05 +/- 6.3599E-09 ( 0.04%)
3 si-28 n,p -6.5750E-07 +/- 1.0533E-09 ( 0.16%)
3 si-28 n,d -8.8870E-10 +/- 3.5446E-19 ( 0.00%)
3 si-28 n,alpha -2.9630E-07 +/- 5.3824E-10 ( 0.18%)
3 si-29 total 5.3687E-07 +/- 5.6554E-09 ( 1.05%)
3 si-29 scatter 1.2496E-06 +/- 4.0360E-09 ( 0.32%)
3 si-29 elastic 8.0857E-07 +/- 3.6454E-09 ( 0.45%)
3 si-29 n,n' 4.3972E-07 +/- 9.9759E-10 ( 0.23%)
3 si-29 n,2n 1.2847E-09 +/- 2.7509E-19 ( 0.00%)
3 si-29 capture -7.1270E-07 +/- 2.3029E-10 ( 0.03%)
3 si-29 n,gamma -6.6526E-07 +/- 2.2493E-10 ( 0.03%)
3 si-29 n,p -1.7830E-08 +/- 1.7962E-11 ( 0.10%)
3 si-29 n,alpha -2.9617E-08 +/- 2.2339E-11 ( 0.08%)
3 si-30 total 1.3408E-07 +/- 4.4925E-09 ( 3.35%)
3 si-30 scatter 7.3741E-07 +/- 3.2216E-09 ( 0.44%)
3 si-30 elastic 5.4630E-07 +/- 3.1391E-09 ( 0.57%)
3 si-30 n,n' 1.9023E-07 +/- 4.2667E-10 ( 0.22%)
3 si-30 n,2n 8.7604E-10 +/- 2.6538E-19 ( 0.00%)
3 si-30 capture -6.0334E-07 +/- 8.1712E-10 ( 0.14%)
3 si-30 n,gamma -6.0283E-07 +/- 8.1712E-10 ( 0.14%)
3 si-30 n,p -9.2891E-11 +/- 2.3442E-20 ( 0.00%)
3 si-30 n,alpha -4.1193E-10 +/- 7.5633E-20 ( 0.00%)
3 ti-46 total -1.8856E-07 +/- 8.6800E-10 ( 0.46%)
3 ti-46 scatter 2.1933E-07 +/- 4.7014E-10 ( 0.21%)
3 ti-46 elastic 1.2896E-07 +/- 3.2795E-10 ( 0.25%)
3 ti-46 n,n' 9.0312E-08 +/- 1.9591E-10 ( 0.22%)
3 ti-46 n,2n 4.8498E-12 +/- 2.8827E-21 ( 0.00%)
3 ti-46 capture -4.0789E-07 +/- 1.3319E-10 ( 0.03%)
3 ti-46 n,gamma -3.9961E-07 +/- 1.3319E-10 ( 0.03%)
3 ti-46 n,p -7.7485E-09 +/- 3.8514E-18 ( 0.00%)
3 ti-46 n,d -3.1806E-13 +/- 2.3627E-22 ( 0.00%)
3 ti-46 n,t -9.9763E-17 +/- 3.2655E-26 ( 0.00%)
3 ti-46 n,he-3 -4.2439E-16 +/- 1.9547E-25 ( 0.00%)
3 ti-46 n,alpha -5.2794E-10 +/- 2.5965E-19 ( 0.00%)
3 ti-47 total -8.7007E-07 +/- 2.4463E-09 ( 0.28%)
3 ti-47 scatter 2.4268E-07 +/- 2.2853E-09 ( 0.94%)
3 ti-47 elastic 1.3531E-07 +/- 2.2669E-09 ( 1.68%)
3 ti-47 n,n' 1.0717E-07 +/- 1.8872E-10 ( 0.18%)
3 ti-47 n,2n 1.9211E-10 +/- 4.0015E-20 ( 0.00%)
3 ti-47 capture -1.1128E-06 +/- 3.6552E-10 ( 0.03%)
3 ti-47 n,gamma -1.1003E-06 +/- 3.6552E-10 ( 0.03%)
3 ti-47 n,p -1.0646E-08 +/- 1.4197E-16 ( 0.00%)
3 ti-47 n,d -2.0197E-13 +/- 1.5135E-22 ( 0.00%)
3 ti-47 n,t -4.3601E-15 +/- 3.0922E-24 ( 0.00%)
3 ti-47 n,he-3 -2.3300E-17 +/- 7.4079E-27 ( 0.00%)
3 ti-47 n,alpha -1.8482E-09 +/- 1.3222E-17 ( 0.00%)
3 ti-48 total -4.5957E-05 +/- 3.3263E-08 ( 0.07%)
3 ti-48 scatter 3.0030E-06 +/- 2.3419E-08 ( 0.78%)
3 ti-48 elastic 2.1566E-06 +/- 2.3249E-08 ( 1.08%)
3 ti-48 n,n' 8.4526E-07 +/- 1.8984E-09 ( 0.22%)
3 ti-48 n,2n 1.0593E-09 +/- 5.3938E-19 ( 0.00%)
3 ti-48 capture -4.8960E-05 +/- 1.7297E-08 ( 0.04%)
3 ti-48 n,gamma -4.8958E-05 +/- 1.7297E-08 ( 0.04%)
3 ti-48 n,p -1.6058E-09 +/- 9.4652E-19 ( 0.00%)
3 ti-48 n,d -1.6876E-12 +/- 1.6673E-21 ( 0.00%)
3 ti-48 n,t -8.0952E-16 +/- 2.7332E-25 ( 0.00%)
3 ti-48 n,he-3 -1.9723E-17 +/- 6.7861E-27 ( 0.00%)
3 ti-48 n,alpha -1.5654E-10 +/- 4.0991E-20 ( 0.00%)
3 ti-49 total -6.8733E-07 +/- 6.2660E-10 ( 0.09%)
3 ti-49 scatter 1.6784E-07 +/- 2.2439E-09 ( 1.34%)
3 ti-49 elastic 1.0265E-07 +/- 2.2349E-09 ( 2.18%)
3 ti-49 n,n' 6.4463E-08 +/- 1.2032E-10 ( 0.19%)
3 ti-49 n,2n 7.2472E-10 +/- 7.6807E-20 ( 0.00%)
3 ti-49 capture -8.5517E-07 +/- 2.9502E-10 ( 0.03%)
3 ti-49 n,gamma -8.5476E-07 +/- 2.9502E-10 ( 0.03%)
3 ti-49 n,p -3.3313E-10 +/- 3.2208E-19 ( 0.00%)
3 ti-49 n,d -7.4647E-14 +/- 6.7611E-23 ( 0.00%)
3 ti-49 n,t -2.1162E-15 +/- 9.3763E-25 ( 0.00%)
3 ti-49 n,he-3 -9.3750E-20 +/- 0.0000E+00 ( 0.00%)
3 ti-49 n,alpha -7.1068E-11 +/- 4.3628E-20 ( 0.00%)
3 ti-50 total 5.1758E-08 +/- 4.1741E-10 ( 0.81%)
3 ti-50 scatter 1.3378E-07 +/- 1.9577E-10 ( 0.15%)
3 ti-50 elastic 8.8832E-08 +/- 1.1755E-10 ( 0.13%)
3 ti-50 n,n' 4.4731E-08 +/- 8.8290E-11 ( 0.20%)
3 ti-50 n,2n 2.1770E-10 +/- 7.0963E-20 ( 0.00%)
3 ti-50 capture -8.2023E-08 +/- 2.5593E-11 ( 0.03%)
3 ti-50 n,gamma -8.2018E-08 +/- 2.5593E-11 ( 0.03%)
3 ti-50 n,p -2.9356E-12 +/- 9.4805E-22 ( 0.00%)
3 ti-50 n,d -1.7101E-14 +/- 1.1908E-23 ( 0.00%)
3 ti-50 n,t -1.0953E-17 +/- 2.8155E-27 ( 0.00%)
3 ti-50 n,he-3 -4.0903E-23 +/- 0.0000E+00 ( 0.00%)
3 ti-50 n,alpha -1.7272E-12 +/- 6.5310E-22 ( 0.00%)
3 cr-50 total -1.3040E-05 +/- 9.1891E-09 ( 0.07%)
3 cr-50 scatter 1.3376E-06 +/- 7.2895E-09 ( 0.54%)
3 cr-50 elastic 1.2307E-06 +/- 7.2762E-09 ( 0.59%)
3 cr-50 n,n' 1.0681E-07 +/- 2.3841E-10 ( 0.22%)
3 cr-50 n,2n 3.1922E-12 +/- 2.3793E-21 ( 0.00%)
3 cr-50 capture -1.4377E-05 +/- 5.0788E-09 ( 0.04%)
3 cr-50 n,gamma -1.4333E-05 +/- 5.0787E-09 ( 0.04%)
3 cr-50 n,p -4.1672E-08 +/- 3.2114E-11 ( 0.08%)
3 cr-50 n,d -7.5849E-12 +/- 5.4355E-19 ( 0.00%)
3 cr-50 n,alpha -2.7989E-09 +/- 1.7847E-15 ( 0.00%)
3 cr-52 total -8.7501E-06 +/- 2.6630E-08 ( 0.30%)
3 cr-52 scatter 4.4850E-06 +/- 2.1763E-08 ( 0.49%)
3 cr-52 elastic 2.3489E-06 +/- 2.0602E-08 ( 0.88%)
3 cr-52 n,n' 2.1346E-06 +/- 4.5774E-09 ( 0.21%)
3 cr-52 n,2n 1.3180E-09 +/- 9.1937E-19 ( 0.00%)
3 cr-52 capture -1.3235E-05 +/- 4.5982E-09 ( 0.03%)
3 cr-52 n,gamma -1.3217E-05 +/- 4.5982E-09 ( 0.03%)
3 cr-52 n,p -1.5832E-08 +/- 1.8313E-11 ( 0.12%)
3 cr-52 n,alpha -1.9304E-09 +/- 3.8775E-17 ( 0.00%)
3 cr-53 total -3.4245E-05 +/- 2.1670E-08 ( 0.06%)
3 cr-53 scatter 1.3865E-06 +/- 1.4475E-08 ( 1.04%)
3 cr-53 elastic 1.0474E-06 +/- 1.4432E-08 ( 1.38%)
3 cr-53 n,n' 3.3527E-07 +/- 7.1251E-10 ( 0.21%)
3 cr-53 n,2n 3.8152E-09 +/- 3.6614E-17 ( 0.00%)
3 cr-53 capture -3.5632E-05 +/- 1.2596E-08 ( 0.04%)
3 cr-53 n,gamma -3.5630E-05 +/- 1.2596E-08 ( 0.04%)
3 cr-53 n,p -6.9509E-10 +/- 5.2410E-17 ( 0.00%)
3 cr-53 n,alpha -1.5386E-09 +/- 1.4981E-16 ( 0.00%)
3 cr-54 total -1.6695E-08 +/- 4.1104E-10 ( 2.46%)
3 cr-54 scatter 1.6318E-07 +/- 2.3409E-10 ( 0.14%)
3 cr-54 elastic 9.3303E-08 +/- 7.9919E-11 ( 0.09%)
3 cr-54 n,n' 6.9373E-08 +/- 1.3815E-10 ( 0.20%)
3 cr-54 n,2n 5.0486E-10 +/- 8.3567E-20 ( 0.00%)
3 cr-54 capture -1.7988E-07 +/- 5.9710E-11 ( 0.03%)
3 cr-54 n,gamma -1.7985E-07 +/- 5.9710E-11 ( 0.03%)
3 cr-54 n,p -4.2582E-12 +/- 1.1391E-21 ( 0.00%)
3 cr-54 n,alpha -2.1413E-11 +/- 1.0788E-20 ( 0.00%)
3 mn-55 total -9.6360E-05 +/- 1.4079E-07 ( 0.15%)
3 mn-55 scatter 6.8704E-06 +/- 1.3262E-07 ( 1.93%)
3 mn-55 elastic 5.7292E-06 +/- 1.3254E-07 ( 2.31%)
3 mn-55 n,n' 1.1366E-06 +/- 2.5436E-09 ( 0.22%)
3 mn-55 n,2n 4.4184E-09 +/- 1.0180E-18 ( 0.00%)
3 mn-55 capture -1.0323E-04 +/- 3.5222E-08 ( 0.03%)
3 mn-55 n,gamma -1.0323E-04 +/- 3.5222E-08 ( 0.03%)
3 mn-55 n,p -3.7300E-09 +/- 9.1810E-18 ( 0.00%)
3 mn-55 n,d -3.5791E-11 +/- 6.7992E-21 ( 0.00%)
3 mn-55 n,t -1.3299E-12 +/- 1.2646E-21 ( 0.00%)
3 mn-55 n,he-3 -4.6031E-14 +/- 4.8445E-23 ( 0.00%)
3 mn-55 n,alpha -8.3737E-10 +/- 6.0947E-19 ( 0.00%)
3 fe-54 total -3.9407E-06 +/- 8.5216E-09 ( 0.22%)
3 fe-54 scatter 7.5145E-07 +/- 8.1027E-09 ( 1.08%)
3 fe-54 elastic 5.5492E-07 +/- 8.0723E-09 ( 1.45%)
3 fe-54 n,n' 1.9601E-07 +/- 4.3649E-10 ( 0.22%)
3 fe-54 n,2n 3.7980E-12 +/- 1.3165E-21 ( 0.00%)
3 fe-54 capture -4.6921E-06 +/- 1.5839E-09 ( 0.03%)
3 fe-54 n,gamma -4.5400E-06 +/- 1.5807E-09 ( 0.03%)
3 fe-54 n,p -1.5046E-07 +/- 9.9241E-11 ( 0.07%)
3 fe-54 n,d -2.5789E-11 +/- 3.7341E-21 ( 0.00%)
3 fe-54 n,alpha -1.5839E-09 +/- 6.7397E-19 ( 0.00%)
3 fe-56 total -6.9426E-05 +/- 1.3507E-07 ( 0.19%)
3 fe-56 scatter 1.1537E-05 +/- 1.1066E-07 ( 0.96%)
3 fe-56 elastic 7.5606E-06 +/- 1.0976E-07 ( 1.45%)
3 fe-56 n,n' 3.9702E-06 +/- 9.5385E-09 ( 0.24%)
3 fe-56 n,2n 5.5596E-09 +/- 3.1334E-18 ( 0.00%)
3 fe-56 capture -8.0963E-05 +/- 2.8413E-08 ( 0.04%)
3 fe-56 n,gamma -8.0923E-05 +/- 2.8413E-08 ( 0.04%)
3 fe-56 n,p -2.8231E-08 +/- 4.3821E-11 ( 0.16%)
3 fe-56 n,d -6.3061E-11 +/- 4.9191E-20 ( 0.00%)
3 fe-56 n,t -1.5472E-13 +/- 1.2160E-22 ( 0.00%)
3 fe-56 n,he-3 -1.7440E-14 +/- 1.2169E-23 ( 0.00%)
3 fe-56 n,alpha -1.1945E-08 +/- 1.2530E-11 ( 0.10%)
3 fe-57 total -1.4580E-06 +/- 2.4578E-09 ( 0.17%)
3 fe-57 scatter 2.9453E-07 +/- 2.2278E-09 ( 0.76%)
3 fe-57 elastic 1.6082E-07 +/- 2.1139E-09 ( 1.31%)
3 fe-57 n,n' 1.3172E-07 +/- 2.4140E-10 ( 0.18%)
3 fe-57 n,2n 1.9852E-09 +/- 1.4642E-19 ( 0.00%)
3 fe-57 capture -1.7526E-06 +/- 6.0368E-10 ( 0.03%)
3 fe-57 n,gamma -1.7515E-06 +/- 6.0368E-10 ( 0.03%)
3 fe-57 n,p -4.5690E-10 +/- 1.9448E-19 ( 0.00%)
3 fe-57 n,alpha -6.5570E-10 +/- 3.5601E-19 ( 0.00%)
3 fe-58 total -8.6249E-08 +/- 9.2863E-11 ( 0.11%)
3 fe-58 scatter 3.2882E-08 +/- 2.1096E-13 ( 0.00%)
3 fe-58 elastic 1.9457E-08 +/- 2.1096E-13 ( 0.00%)
3 fe-58 n,n' 1.3343E-08 +/- 1.7621E-16 ( 0.00%)
3 fe-58 n,2n 8.1702E-11 +/- 1.5798E-20 ( 0.00%)
3 fe-58 capture -1.1913E-07 +/- 3.5967E-11 ( 0.03%)
3 fe-58 n,gamma -1.1912E-07 +/- 3.5967E-11 ( 0.03%)
3 fe-58 n,p -8.9556E-13 +/- 2.0142E-22 ( 0.00%)
3 fe-58 n,alpha -5.5186E-12 +/- 1.4040E-21 ( 0.00%)
3 cu-63 total -9.5973E-05 +/- 7.0063E-08 ( 0.07%)
3 cu-63 scatter 8.9795E-06 +/- 3.9903E-08 ( 0.44%)
3 cu-63 elastic 5.1356E-06 +/- 3.8382E-08 ( 0.75%)
3 cu-63 n,n' 3.8154E-06 +/- 7.8316E-09 ( 0.21%)
3 cu-63 n,2n 4.8437E-09 +/- 1.9022E-18 ( 0.00%)
3 cu-63 capture -1.0495E-04 +/- 3.5784E-08 ( 0.03%)
3 cu-63 n,gamma -1.0450E-04 +/- 3.5783E-08 ( 0.03%)
3 cu-63 n,p -4.3880E-07 +/- 2.3679E-10 ( 0.05%)
3 cu-63 n,d -7.9573E-09 +/- 9.1280E-17 ( 0.00%)
3 cu-63 n,he-3 -3.8432E-14 +/- 1.6090E-23 ( 0.00%)
3 cu-63 n,alpha -1.0048E-08 +/- 7.9401E-17 ( 0.00%)
3 cu-65 total -1.7689E-05 +/- 4.4209E-08 ( 0.25%)
3 cu-65 scatter 4.5920E-06 +/- 3.3439E-08 ( 0.73%)
3 cu-65 elastic 2.9304E-06 +/- 3.3117E-08 ( 1.13%)
3 cu-65 n,n' 1.6531E-06 +/- 3.3430E-09 ( 0.20%)
3 cu-65 n,2n 8.2988E-09 +/- 2.5459E-11 ( 0.31%)
3 cu-65 capture -2.2281E-05 +/- 7.6516E-09 ( 0.03%)
3 cu-65 n,gamma -2.2276E-05 +/- 7.6516E-09 ( 0.03%)
3 cu-65 n,p -4.8468E-09 +/- 2.3482E-16 ( 0.00%)
3 cu-65 n,d -5.9497E-10 +/- 1.8412E-17 ( 0.00%)
3 cu-65 n,t -1.3215E-13 +/- 7.8888E-23 ( 0.00%)
3 cu-65 n,he-3 -4.7968E-15 +/- 6.6710E-24 ( 0.00%)
3 cu-65 n,alpha -1.2252E-10 +/- 1.7122E-18 ( 0.00%)
4 h-1 total -3.3487E-05 +/- 3.3383E-05 ( 99.69%)
4 h-1 scatter 1.1785E-04 +/- 3.3082E-05 ( 28.07%)
4 h-1 elastic 1.1785E-04 +/- 3.3082E-05 ( 28.07%)
4 h-1 capture -1.5134E-04 +/- 3.4431E-07 ( 0.23%)
4 h-1 n,gamma -1.5134E-04 +/- 3.4431E-07 ( 0.23%)
4 c total 1.4324E-04 +/- 3.4393E-06 ( 2.40%)
4 c scatter 1.4444E-04 +/- 3.4379E-06 ( 2.38%)
4 c elastic 1.4356E-04 +/- 3.4374E-06 ( 2.39%)
4 c n,n' 8.7941E-07 +/- 2.1229E-08 ( 2.41%)
4 c capture -1.2004E-06 +/- 5.0249E-09 ( 0.42%)
4 c n,gamma -1.1534E-06 +/- 2.6118E-09 ( 0.23%)
4 c n,p -1.2812E-11 +/- 0.0000E+00 ( 0.00%)
4 c n,d -3.3016E-11 +/- 0.0000E+00 ( 0.00%)
4 c n,alpha -4.6915E-08 +/- 4.2884E-09 ( 9.14%)
4 o-16 total 3.9480E-05 +/- 8.0523E-07 ( 2.04%)
4 o-16 scatter 3.9628E-05 +/- 8.0503E-07 ( 2.03%)
4 o-16 elastic 3.9526E-05 +/- 8.0498E-07 ( 2.04%)
4 o-16 n,n' 1.0040E-07 +/- 3.9571E-09 ( 3.94%)
4 o-16 n,2n 5.9840E-15 +/- 0.0000E+00 ( 0.00%)
4 o-16 capture -1.4730E-07 +/- 2.8085E-09 ( 1.91%)
4 o-16 n,gamma -1.8551E-08 +/- 3.2427E-11 ( 0.17%)
4 o-16 n,p -4.6597E-10 +/- 2.6874E-13 ( 0.06%)
4 o-16 n,d -4.2225E-11 +/- 1.3904E-14 ( 0.03%)
4 o-16 n,t -3.5750E-16 +/- 0.0000E+00 ( 0.00%)
4 o-16 n,alpha -1.2824E-07 +/- 2.8083E-09 ( 2.19%)
4 si-28 total 1.5659E-07 +/- 3.1168E-09 ( 1.99%)
4 si-28 scatter 2.7569E-07 +/- 2.1958E-09 ( 0.80%)
4 si-28 elastic 2.4930E-07 +/- 2.0614E-09 ( 0.83%)
4 si-28 n,n' 2.6386E-08 +/- 2.5243E-10 ( 0.96%)
4 si-28 capture -1.1911E-07 +/- 2.5877E-10 ( 0.22%)
4 si-28 n,gamma -1.1814E-07 +/- 2.5877E-10 ( 0.22%)
4 si-28 n,p -6.7032E-10 +/- 6.8669E-17 ( 0.00%)
4 si-28 n,d -7.0401E-13 +/- 0.0000E+00 ( 0.00%)
4 si-28 n,alpha -2.9677E-10 +/- 2.4154E-17 ( 0.00%)
4 si-29 total 8.3353E-09 +/- 7.5001E-15 ( 0.00%)
4 si-29 scatter 1.2660E-08 +/- 7.4989E-15 ( 0.00%)
4 si-29 elastic 1.0360E-08 +/- 7.1408E-15 ( 0.00%)
4 si-29 n,n' 2.2970E-09 +/- 7.6907E-16 ( 0.00%)
4 si-29 n,2n 3.6953E-12 +/- 0.0000E+00 ( 0.00%)
4 si-29 capture -4.3250E-09 +/- 5.9139E-18 ( 0.00%)
4 si-29 n,gamma -4.2731E-09 +/- 1.0615E-18 ( 0.00%)
4 si-29 n,p -1.8703E-11 +/- 2.1885E-18 ( 0.00%)
4 si-29 n,alpha -3.3206E-11 +/- 3.9362E-18 ( 0.00%)
4 si-30 total 5.0684E-09 +/- 6.4263E-15 ( 0.00%)
4 si-30 scatter 8.0334E-09 +/- 6.4252E-15 ( 0.00%)
4 si-30 elastic 7.1395E-09 +/- 6.3753E-15 ( 0.00%)
4 si-30 n,n' 8.9160E-10 +/- 2.1621E-16 ( 0.00%)
4 si-30 n,2n 2.2872E-12 +/- 0.0000E+00 ( 0.00%)
4 si-30 capture -2.9651E-09 +/- 1.4537E-18 ( 0.00%)
4 si-30 n,gamma -2.9646E-09 +/- 1.4536E-18 ( 0.00%)
4 si-30 n,p -7.7936E-14 +/- 3.6748E-26 ( 0.00%)
4 si-30 n,alpha -3.6427E-13 +/- 5.9486E-21 ( 0.00%)
4 s-32 total -9.3027E-07 +/- 1.5023E-08 ( 1.61%)
4 s-32 scatter 1.0965E-06 +/- 1.1617E-08 ( 1.06%)
4 s-32 elastic 1.0303E-06 +/- 1.1393E-08 ( 1.11%)
4 s-32 n,n' 6.5967E-08 +/- 8.6196E-10 ( 1.31%)
4 s-32 n,2n 1.0859E-14 +/- 0.0000E+00 ( 0.00%)
4 s-32 capture -2.0268E-06 +/- 4.4706E-09 ( 0.22%)
4 s-32 n,gamma -1.8835E-06 +/- 4.2718E-09 ( 0.23%)
4 s-32 n,p -4.4415E-08 +/- 5.3423E-10 ( 1.20%)
4 s-32 n,alpha -9.8944E-08 +/- 5.3326E-10 ( 0.54%)
4 s-33 total -1.1925E-08 +/- 1.0473E-14 ( 0.00%)
4 s-33 scatter 7.9053E-09 +/- 9.6281E-15 ( 0.00%)
4 s-33 elastic 7.1213E-09 +/- 9.5751E-15 ( 0.00%)
4 s-33 n,n' 7.7770E-10 +/- 1.7507E-16 ( 0.00%)
4 s-33 n,2n 4.8458E-12 +/- 0.0000E+00 ( 0.00%)
4 s-33 capture -1.9830E-08 +/- 8.4932E-16 ( 0.00%)
4 s-33 n,gamma -9.8480E-09 +/- 1.5761E-18 ( 0.00%)
4 s-33 n,p -4.3988E-10 +/- 9.9510E-17 ( 0.00%)
4 s-33 n,alpha -9.5421E-09 +/- 8.0138E-16 ( 0.00%)
4 s-34 total 1.2596E-08 +/- 3.9241E-10 ( 3.12%)
4 s-34 scatter 4.7978E-08 +/- 1.6902E-10 ( 0.35%)
4 s-34 elastic 4.3911E-08 +/- 1.1281E-10 ( 0.26%)
4 s-34 n,n' 4.0515E-09 +/- 7.8532E-16 ( 0.00%)
4 s-34 n,2n 6.4633E-12 +/- 0.0000E+00 ( 0.00%)
4 s-34 capture -3.5381E-08 +/- 6.1670E-11 ( 0.17%)
4 s-34 n,gamma -3.5292E-08 +/- 6.1670E-11 ( 0.17%)
4 s-34 n,p -1.6841E-11 +/- 1.4263E-18 ( 0.00%)
4 s-34 n,alpha -7.2939E-11 +/- 9.4585E-18 ( 0.00%)
4 s-36 total 1.2812E-10 +/- 1.6526E-16 ( 0.00%)
4 s-36 scatter 2.4510E-10 +/- 1.6524E-16 ( 0.00%)
4 s-36 elastic 2.3204E-10 +/- 1.6517E-16 ( 0.00%)
4 s-36 n,n' 1.2864E-11 +/- 1.1331E-18 ( 0.00%)
4 s-36 n,2n 1.9756E-13 +/- 0.0000E+00 ( 0.00%)
4 s-36 capture -1.1697E-10 +/- 2.0921E-20 ( 0.00%)
4 s-36 n,gamma -1.1697E-10 +/- 2.0919E-20 ( 0.00%)
4 s-36 n,p -5.8200E-17 +/- 0.0000E+00 ( 0.00%)
4 s-36 n,alpha -5.1249E-15 +/- 2.0171E-22 ( 0.00%)
4 ca-40 total -1.7949E-06 +/- 1.1894E-07 ( 6.63%)
4 ca-40 scatter 6.7330E-06 +/- 9.7909E-08 ( 1.45%)
4 ca-40 elastic 6.5226E-06 +/- 9.7589E-08 ( 1.50%)
4 ca-40 n,n' 2.0785E-07 +/- 3.3003E-09 ( 1.59%)
4 ca-40 n,2n 4.5607E-14 +/- 0.0000E+00 ( 0.00%)
4 ca-40 capture -8.5279E-06 +/- 1.9716E-08 ( 0.23%)
4 ca-40 n,gamma -7.9727E-06 +/- 1.8069E-08 ( 0.23%)
4 ca-40 n,p -3.7411E-07 +/- 5.5225E-09 ( 1.48%)
4 ca-40 n,d -5.9766E-11 +/- 6.5478E-19 ( 0.00%)
4 ca-40 n,t -4.2787E-15 +/- 0.0000E+00 ( 0.00%)
4 ca-40 n,he-3 -5.7096E-14 +/- 0.0000E+00 ( 0.00%)
4 ca-40 n,alpha -1.8104E-07 +/- 2.0743E-09 ( 1.15%)
4 ca-42 total -3.3798E-08 +/- 4.2370E-10 ( 1.25%)
4 ca-42 scatter 5.5595E-08 +/- 1.8840E-10 ( 0.34%)
4 ca-42 elastic 4.4749E-08 +/- 7.7226E-14 ( 0.00%)
4 ca-42 n,n' 1.0831E-08 +/- 3.5480E-15 ( 0.00%)
4 ca-42 n,2n 4.2807E-12 +/- 0.0000E+00 ( 0.00%)
4 ca-42 capture -8.9393E-08 +/- 1.9154E-10 ( 0.21%)
4 ca-42 n,gamma -8.9285E-08 +/- 1.9154E-10 ( 0.21%)
4 ca-42 n,p -6.3001E-11 +/- 6.1936E-18 ( 0.00%)
4 ca-42 n,d -3.9571E-14 +/- 0.0000E+00 ( 0.00%)
4 ca-42 n,t -1.1401E-16 +/- 0.0000E+00 ( 0.00%)
4 ca-42 n,he-3 -2.8628E-18 +/- 0.0000E+00 ( 0.00%)
4 ca-42 n,alpha -4.4886E-11 +/- 4.9613E-18 ( 0.00%)
4 ca-43 total -2.9992E-07 +/- 8.5309E-10 ( 0.28%)
4 ca-43 scatter 1.7308E-08 +/- 9.3864E-15 ( 0.00%)
4 ca-43 elastic 1.4035E-08 +/- 8.9886E-15 ( 0.00%)
4 ca-43 n,n' 3.2593E-09 +/- 1.0415E-15 ( 0.00%)
4 ca-43 n,2n 1.3992E-11 +/- 1.0619E-22 ( 0.00%)
4 ca-43 capture -3.1723E-07 +/- 7.1329E-10 ( 0.22%)
4 ca-43 n,gamma -3.1720E-07 +/- 7.1328E-10 ( 0.22%)
4 ca-43 n,p -1.2736E-11 +/- 1.4953E-18 ( 0.00%)
4 ca-43 n,d -3.9211E-15 +/- 0.0000E+00 ( 0.00%)
4 ca-43 n,t -2.0903E-16 +/- 0.0000E+00 ( 0.00%)
4 ca-43 n,he-3 -8.8381E-21 +/- 0.0000E+00 ( 0.00%)
4 ca-43 n,alpha -1.8078E-11 +/- 1.6658E-18 ( 0.00%)
4 ca-44 total -1.9121E-07 +/- 2.9350E-09 ( 1.53%)
4 ca-44 scatter 1.8201E-07 +/- 9.8994E-10 ( 0.54%)
4 ca-44 elastic 1.4003E-07 +/- 6.5307E-10 ( 0.47%)
4 ca-44 n,n' 4.1934E-08 +/- 3.9319E-10 ( 0.94%)
4 ca-44 n,2n 3.7701E-11 +/- 0.0000E+00 ( 0.00%)
4 ca-44 capture -3.7322E-07 +/- 8.4317E-10 ( 0.23%)
4 ca-44 n,gamma -3.7321E-07 +/- 8.4317E-10 ( 0.23%)
4 ca-44 n,p -4.3462E-12 +/- 1.1952E-19 ( 0.00%)
4 ca-44 n,d -1.9015E-14 +/- 0.0000E+00 ( 0.00%)
4 ca-44 n,t -3.6521E-17 +/- 0.0000E+00 ( 0.00%)
4 ca-44 n,he-3 -5.7133E-22 +/- 0.0000E+00 ( 0.00%)
4 ca-44 n,alpha -2.1833E-12 +/- 5.2524E-20 ( 0.00%)
4 ca-46 total -9.1789E-11 +/- 2.9314E-16 ( 0.00%)
4 ca-46 scatter 5.0404E-10 +/- 2.9308E-16 ( 0.00%)
4 ca-46 elastic 4.3313E-10 +/- 2.8501E-16 ( 0.00%)
4 ca-46 n,n' 7.0696E-11 +/- 2.2304E-17 ( 0.00%)
4 ca-46 n,2n 2.0718E-13 +/- 0.0000E+00 ( 0.00%)
4 ca-46 capture -5.9583E-10 +/- 5.9929E-20 ( 0.00%)
4 ca-46 n,gamma -5.9583E-10 +/- 5.9929E-20 ( 0.00%)
4 ca-46 n,p -3.2267E-16 +/- 3.6714E-26 ( 0.00%)
4 ca-46 n,d -6.8722E-18 +/- 0.0000E+00 ( 0.00%)
4 ca-46 n,t -3.3633E-21 +/- 0.0000E+00 ( 0.00%)
4 ca-46 n,alpha -2.2633E-17 +/- 0.0000E+00 ( 0.00%)
4 ca-48 total -2.4743E-08 +/- 1.8724E-10 ( 0.76%)
4 ca-48 scatter 1.6341E-08 +/- 8.2057E-15 ( 0.00%)
4 ca-48 elastic 1.5512E-08 +/- 8.2022E-15 ( 0.00%)
4 ca-48 n,n' 8.1356E-10 +/- 6.9284E-17 ( 0.00%)
4 ca-48 n,2n 1.5225E-11 +/- 0.0000E+00 ( 0.00%)
4 ca-48 capture -4.1084E-08 +/- 7.9141E-11 ( 0.19%)
4 ca-48 n,gamma -4.1084E-08 +/- 7.9141E-11 ( 0.19%)
4 ca-48 n,p -1.0552E-16 +/- 0.0000E+00 ( 0.00%)
4 ca-48 n,d -2.1494E-17 +/- 0.0000E+00 ( 0.00%)
4 ca-48 n,t -5.0004E-22 +/- 0.0000E+00 ( 0.00%)
4 ca-48 n,alpha -1.9945E-17 +/- 0.0000E+00 ( 0.00%)
5 h-1 total -4.6343E-05 +/- 2.5632E-05 ( 55.31%)
5 h-1 scatter 1.4599E-04 +/- 2.5374E-05 ( 17.38%)
5 h-1 elastic 1.4599E-04 +/- 2.5374E-05 ( 17.38%)
5 h-1 capture -1.9233E-04 +/- 2.9929E-07 ( 0.16%)
5 h-1 n,gamma -1.9233E-04 +/- 2.9929E-07 ( 0.16%)
5 c total 1.5186E-04 +/- 2.0605E-06 ( 1.36%)
5 c scatter 1.5313E-04 +/- 1.9864E-06 ( 1.30%)
5 c elastic 1.5229E-04 +/- 1.9863E-06 ( 1.30%)
5 c n,n' 8.3833E-07 +/- 7.6391E-09 ( 0.91%)
5 c capture -1.2702E-06 +/- 2.4828E-09 ( 0.20%)
5 c n,gamma -1.2318E-06 +/- 1.9115E-09 ( 0.16%)
5 c n,p -9.4682E-12 +/- 0.0000E+00 ( 0.00%)
5 c n,d -2.8081E-11 +/- 0.0000E+00 ( 0.00%)
5 c n,alpha -3.8362E-08 +/- 1.5828E-09 ( 4.13%)
5 o-16 total 5.7694E-05 +/- 6.3654E-07 ( 1.10%)
5 o-16 scatter 5.7871E-05 +/- 6.3645E-07 ( 1.10%)
5 o-16 elastic 5.7729E-05 +/- 6.3644E-07 ( 1.10%)
5 o-16 n,n' 1.3911E-07 +/- 2.0477E-09 ( 1.47%)
5 o-16 n,2n 1.1747E-14 +/- 0.0000E+00 ( 0.00%)
5 o-16 capture -1.7668E-07 +/- 1.5046E-09 ( 0.85%)
5 o-16 n,gamma -2.7736E-08 +/- 3.4230E-11 ( 0.12%)
5 o-16 n,p -6.3458E-10 +/- 1.2415E-13 ( 0.02%)
5 o-16 n,d -5.2519E-11 +/- 6.4231E-15 ( 0.01%)
5 o-16 n,t -4.4058E-16 +/- 0.0000E+00 ( 0.00%)
5 o-16 n,alpha -1.4826E-07 +/- 1.4006E-09 ( 0.94%)
6 c total 9.7668E-06 +/- 1.3853E-07 ( 1.42%)
6 c scatter 9.8750E-06 +/- 1.3850E-07 ( 1.40%)
6 c elastic 9.8360E-06 +/- 1.3846E-07 ( 1.41%)
6 c n,n' 3.8987E-08 +/- 1.2258E-09 ( 3.14%)
6 c capture -1.0818E-07 +/- 2.4021E-10 ( 0.22%)
6 c n,gamma -7.0538E-08 +/- 5.6578E-11 ( 0.08%)
6 c n,p -6.2023E-12 +/- 0.0000E+00 ( 0.00%)
6 c n,d -1.0588E-11 +/- 0.0000E+00 ( 0.00%)
6 c n,alpha -3.7621E-08 +/- 2.3320E-10 ( 0.62%)
6 o-16 total 6.9207E-07 +/- 7.5312E-09 ( 1.09%)
6 o-16 scatter 7.0981E-07 +/- 7.5282E-09 ( 1.06%)
6 o-16 elastic 7.0868E-07 +/- 7.5282E-09 ( 1.06%)
6 o-16 n,n' 1.0108E-09 +/- 2.8433E-16 ( 0.00%)
6 o-16 n,2n 6.5082E-17 +/- 0.0000E+00 ( 0.00%)
6 o-16 capture -1.7739E-08 +/- 2.1801E-11 ( 0.12%)
6 o-16 n,gamma -2.6146E-10 +/- 2.2204E-17 ( 0.00%)
6 o-16 n,p -1.0955E-10 +/- 1.7204E-17 ( 0.00%)
6 o-16 n,d -1.2082E-11 +/- 8.9008E-19 ( 0.00%)
6 o-16 n,t -2.2926E-17 +/- 0.0000E+00 ( 0.00%)
6 o-16 n,alpha -1.7356E-08 +/- 2.1801E-11 ( 0.13%)
6 na-23 total 2.3175E-08 +/- 1.4447E-09 ( 6.23%)
6 na-23 scatter 5.7809E-08 +/- 1.4371E-09 ( 2.49%)
6 na-23 elastic 5.2606E-08 +/- 1.4371E-09 ( 2.73%)
6 na-23 n,n' 5.2023E-09 +/- 7.8251E-15 ( 0.00%)
6 na-23 n,2n 4.5318E-13 +/- 0.0000E+00 ( 0.00%)
6 na-23 capture -3.4634E-08 +/- 2.0188E-11 ( 0.06%)
6 na-23 n,gamma -3.4409E-08 +/- 2.0188E-11 ( 0.06%)
6 na-23 n,p -1.5160E-10 +/- 3.2974E-18 ( 0.00%)
6 na-23 n,alpha -7.2671E-11 +/- 5.0907E-18 ( 0.00%)
6 mg-24 total 3.2809E-08 +/- 5.8412E-12 ( 0.02%)
6 mg-24 scatter 3.5854E-08 +/- 5.8398E-12 ( 0.02%)
6 mg-24 elastic 3.3595E-08 +/- 5.8398E-12 ( 0.02%)
6 mg-24 n,n' 2.2531E-09 +/- 3.1539E-16 ( 0.00%)
6 mg-24 n,2n 4.2480E-17 +/- 0.0000E+00 ( 0.00%)
6 mg-24 capture -3.0449E-09 +/- 2.0795E-15 ( 0.00%)
6 mg-24 n,gamma -2.5024E-09 +/- 2.0794E-15 ( 0.00%)
6 mg-24 n,p -1.3151E-10 +/- 4.3984E-18 ( 0.00%)
6 mg-24 n,alpha -4.1094E-10 +/- 8.3640E-18 ( 0.00%)
6 mg-25 total 2.2861E-09 +/- 1.0962E-12 ( 0.05%)
6 mg-25 scatter 3.6225E-09 +/- 1.0956E-12 ( 0.03%)
6 mg-25 elastic 3.3170E-09 +/- 1.0956E-12 ( 0.03%)
6 mg-25 n,n' 2.9207E-10 +/- 1.2629E-16 ( 0.00%)
6 mg-25 n,2n 1.3416E-11 +/- 1.1115E-18 ( 0.00%)
6 mg-25 capture -1.3364E-09 +/- 7.2936E-16 ( 0.00%)
6 mg-25 n,gamma -1.1676E-09 +/- 7.2936E-16 ( 0.00%)
6 mg-25 n,p -1.5856E-11 +/- 1.8444E-19 ( 0.00%)
6 mg-25 n,alpha -1.5294E-10 +/- 1.4511E-18 ( 0.00%)
6 mg-26 total 3.6092E-09 +/- 4.4230E-13 ( 0.01%)
6 mg-26 scatter 3.8717E-09 +/- 4.4228E-13 ( 0.01%)
6 mg-26 elastic 3.6643E-09 +/- 4.4228E-13 ( 0.01%)
6 mg-26 n,n' 2.0617E-10 +/- 2.7499E-17 ( 0.00%)
6 mg-26 n,2n 1.2053E-12 +/- 6.0525E-20 ( 0.00%)
6 mg-26 capture -2.6241E-10 +/- 2.5730E-17 ( 0.00%)
6 mg-26 n,gamma -2.5990E-10 +/- 2.5729E-17 ( 0.00%)
6 mg-26 n,p -3.0666E-13 +/- 3.9578E-20 ( 0.00%)
6 mg-26 n,alpha -2.2079E-12 +/- 2.2172E-19 ( 0.00%)
6 si-28 total 1.7607E-07 +/- 1.6638E-09 ( 0.94%)
6 si-28 scatter 2.6676E-07 +/- 1.5896E-09 ( 0.60%)
6 si-28 elastic 2.5293E-07 +/- 1.5241E-09 ( 0.60%)
6 si-28 n,n' 1.3815E-08 +/- 1.7317E-15 ( 0.00%)
6 si-28 capture -9.0687E-08 +/- 6.7580E-11 ( 0.07%)
6 si-28 n,gamma -8.2663E-08 +/- 6.7580E-11 ( 0.08%)
6 si-28 n,p -5.5220E-09 +/- 1.0643E-16 ( 0.00%)
6 si-28 n,d -8.9972E-12 +/- 1.1284E-18 ( 0.00%)
6 si-28 n,alpha -2.4934E-09 +/- 5.7790E-17 ( 0.00%)
6 si-29 total 7.3370E-09 +/- 1.5194E-12 ( 0.02%)
6 si-29 scatter 1.0779E-08 +/- 1.5176E-12 ( 0.01%)
6 si-29 elastic 9.6163E-09 +/- 1.5176E-12 ( 0.02%)
6 si-29 n,n' 1.1542E-09 +/- 1.7759E-16 ( 0.00%)
6 si-29 n,2n 8.0511E-12 +/- 8.0235E-19 ( 0.00%)
6 si-29 capture -3.4416E-09 +/- 7.1336E-15 ( 0.00%)
6 si-29 n,gamma -3.0602E-09 +/- 7.1336E-15 ( 0.00%)
6 si-29 n,p -1.4767E-10 +/- 2.5051E-18 ( 0.00%)
6 si-29 n,alpha -2.3373E-10 +/- 3.3600E-18 ( 0.00%)
6 si-30 total 4.2864E-09 +/- 9.7142E-13 ( 0.02%)
6 si-30 scatter 7.1185E-09 +/- 9.6378E-13 ( 0.01%)
6 si-30 elastic 6.7125E-09 +/- 9.6378E-13 ( 0.01%)
6 si-30 n,n' 4.0116E-10 +/- 5.5897E-17 ( 0.00%)
6 si-30 n,2n 4.8546E-12 +/- 4.4990E-19 ( 0.00%)
6 si-30 capture -2.8321E-09 +/- 1.8920E-14 ( 0.00%)
6 si-30 n,gamma -2.8275E-09 +/- 1.8920E-14 ( 0.00%)
6 si-30 n,p -9.0348E-13 +/- 1.2153E-19 ( 0.00%)
6 si-30 n,alpha -3.7528E-12 +/- 3.1321E-19 ( 0.00%)
6 s-32 total -1.4228E-08 +/- 1.7286E-12 ( 0.01%)
6 s-32 scatter 1.9787E-08 +/- 1.7227E-12 ( 0.01%)
6 s-32 elastic 1.9229E-08 +/- 1.7227E-12 ( 0.01%)
6 s-32 n,n' 5.4483E-10 +/- 8.3488E-17 ( 0.00%)
6 s-32 n,2n 9.4026E-17 +/- 0.0000E+00 ( 0.00%)
6 s-32 capture -3.4015E-08 +/- 9.3425E-15 ( 0.00%)
6 s-32 n,gamma -2.3112E-08 +/- 3.3672E-15 ( 0.00%)
6 s-32 n,p -4.5487E-09 +/- 5.5174E-17 ( 0.00%)
6 s-32 n,alpha -6.3539E-09 +/- 7.3703E-15 ( 0.00%)
6 s-33 total -3.5098E-10 +/- 2.0849E-14 ( 0.01%)
6 s-33 scatter 1.1379E-10 +/- 1.9431E-14 ( 0.02%)
6 s-33 elastic 1.0619E-10 +/- 1.9431E-14 ( 0.02%)
6 s-33 n,n' 7.3395E-12 +/- 2.0279E-18 ( 0.00%)
6 s-33 n,2n 1.7646E-13 +/- 2.0474E-20 ( 0.00%)
6 s-33 capture -4.6477E-10 +/- 1.4277E-15 ( 0.00%)
6 s-33 n,gamma -1.2051E-10 +/- 3.3390E-17 ( 0.00%)
6 s-33 n,p -3.3567E-11 +/- 1.3744E-17 ( 0.00%)
6 s-33 n,alpha -3.1070E-10 +/- 1.4014E-15 ( 0.00%)
6 s-34 total 3.0984E-10 +/- 8.4943E-14 ( 0.03%)
6 s-34 scatter 7.5435E-10 +/- 8.4929E-14 ( 0.01%)
6 s-34 elastic 7.1661E-10 +/- 8.4929E-14 ( 0.01%)
6 s-34 n,n' 3.6985E-11 +/- 4.3271E-18 ( 0.00%)
6 s-34 n,2n 2.4941E-13 +/- 1.8198E-20 ( 0.00%)
6 s-34 capture -4.4451E-10 +/- 3.1938E-17 ( 0.00%)
6 s-34 n,gamma -4.3181E-10 +/- 3.1937E-17 ( 0.00%)
6 s-34 n,p -2.5649E-12 +/- 7.4283E-20 ( 0.00%)
6 s-34 n,alpha -1.0132E-11 +/- 1.3880E-19 ( 0.00%)
6 s-36 total 2.4367E-12 +/- 4.8470E-16 ( 0.02%)
6 s-36 scatter 3.9693E-12 +/- 4.8442E-16 ( 0.01%)
6 s-36 elastic 3.8641E-12 +/- 4.8442E-16 ( 0.01%)
6 s-36 n,n' 9.8043E-14 +/- 1.0060E-20 ( 0.00%)
6 s-36 n,2n 7.0907E-15 +/- 8.5709E-22 ( 0.00%)
6 s-36 capture -1.5325E-12 +/- 3.2624E-19 ( 0.00%)
6 s-36 n,gamma -1.5317E-12 +/- 3.2624E-19 ( 0.00%)
6 s-36 n,p -1.2365E-17 +/- 1.4329E-24 ( 0.00%)
6 s-36 n,alpha -8.4074E-16 +/- 3.8508E-23 ( 0.00%)
6 fe-54 total -3.6694E-09 +/- 6.3742E-12 ( 0.17%)
6 fe-54 scatter 3.8072E-09 +/- 6.3646E-12 ( 0.17%)
6 fe-54 elastic 3.6383E-09 +/- 6.3646E-12 ( 0.17%)
6 fe-54 n,n' 1.6766E-10 +/- 2.5736E-17 ( 0.00%)
6 fe-54 n,2n 4.3302E-15 +/- 0.0000E+00 ( 0.00%)
6 fe-54 capture -7.4766E-09 +/- 9.6690E-15 ( 0.00%)
6 fe-54 n,gamma -7.0941E-09 +/- 9.6690E-15 ( 0.00%)
6 fe-54 n,p -3.7781E-10 +/- 4.3570E-18 ( 0.00%)
6 fe-54 n,d -8.2739E-14 +/- 9.5098E-21 ( 0.00%)
6 fe-54 n,alpha -4.5819E-12 +/- 1.0972E-19 ( 0.00%)
6 fe-56 total -1.0500E-07 +/- 4.7404E-10 ( 0.45%)
6 fe-56 scatter 2.1456E-08 +/- 7.5931E-12 ( 0.04%)
6 fe-56 elastic 1.6631E-08 +/- 7.5931E-12 ( 0.05%)
6 fe-56 n,n' 4.8137E-09 +/- 2.9475E-15 ( 0.00%)
6 fe-56 n,2n 1.0449E-11 +/- 7.3407E-19 ( 0.00%)
6 fe-56 capture -1.2645E-07 +/- 1.0772E-10 ( 0.09%)
6 fe-56 n,gamma -1.2633E-07 +/- 1.0772E-10 ( 0.09%)
6 fe-56 n,p -8.2373E-11 +/- 2.2336E-18 ( 0.00%)
6 fe-56 n,d -2.1784E-13 +/- 2.2646E-20 ( 0.00%)
6 fe-56 n,t -4.5091E-16 +/- 2.5804E-24 ( 0.00%)
6 fe-56 n,he-3 -5.1613E-17 +/- 1.4066E-24 ( 0.00%)
6 fe-56 n,alpha -3.4811E-11 +/- 9.9336E-19 ( 0.00%)
6 fe-57 total -1.5428E-09 +/- 1.5175E-12 ( 0.10%)
6 fe-57 scatter 1.1965E-09 +/- 1.5103E-12 ( 0.13%)
6 fe-57 elastic 9.2071E-10 +/- 1.5085E-12 ( 0.16%)
6 fe-57 n,n' 2.7180E-10 +/- 1.9306E-14 ( 0.01%)
6 fe-57 n,2n 3.9813E-12 +/- 3.4807E-19 ( 0.00%)
6 fe-57 capture -2.7393E-09 +/- 7.2413E-15 ( 0.00%)
6 fe-57 n,gamma -2.7362E-09 +/- 7.2413E-15 ( 0.00%)
6 fe-57 n,p -1.2950E-12 +/- 2.2824E-20 ( 0.00%)
6 fe-57 n,alpha -1.7737E-12 +/- 1.9032E-20 ( 0.00%)
6 fe-58 total -7.0000E-11 +/- 1.6196E-14 ( 0.02%)
6 fe-58 scatter 1.1679E-10 +/- 1.6041E-14 ( 0.01%)
6 fe-58 elastic 1.0304E-10 +/- 1.6040E-14 ( 0.02%)
6 fe-58 n,n' 1.3592E-11 +/- 9.0752E-18 ( 0.00%)
6 fe-58 n,2n 1.6128E-13 +/- 1.8159E-20 ( 0.00%)
6 fe-58 capture -1.8679E-10 +/- 7.3090E-16 ( 0.00%)
6 fe-58 n,gamma -1.8677E-10 +/- 7.3090E-16 ( 0.00%)
6 fe-58 n,p -2.8627E-15 +/- 2.6305E-22 ( 0.00%)
6 fe-58 n,alpha -1.6727E-14 +/- 1.0033E-21 ( 0.00%)
6 cu-63 total -3.1331E-03 +/- 6.7640E-06 ( 0.22%)
6 cu-63 scatter 5.1503E-04 +/- 4.5047E-06 ( 0.87%)
6 cu-63 elastic 4.3487E-04 +/- 4.1844E-06 ( 0.96%)
6 cu-63 n,n' 7.8687E-05 +/- 1.1401E-06 ( 1.45%)
6 cu-63 n,2n 2.1308E-07 +/- 7.0126E-09 ( 3.29%)
6 cu-63 capture -3.6481E-03 +/- 3.1712E-06 ( 0.09%)
6 cu-63 n,gamma -3.6222E-03 +/- 3.1710E-06 ( 0.09%)
6 cu-63 n,p -2.4693E-05 +/- 3.5515E-08 ( 0.14%)
6 cu-63 n,d -5.2482E-07 +/- 1.5798E-09 ( 0.30%)
6 cu-63 n,he-3 -3.0750E-12 +/- 3.9260E-19 ( 0.00%)
6 cu-63 n,alpha -6.7771E-07 +/- 2.7471E-09 ( 0.41%)
6 cu-65 total -6.0763E-04 +/- 4.7381E-06 ( 0.78%)
6 cu-65 scatter 1.7893E-04 +/- 4.2465E-06 ( 2.37%)
6 cu-65 elastic 1.4608E-04 +/- 4.1860E-06 ( 2.87%)
6 cu-65 n,n' 3.2461E-05 +/- 4.9054E-07 ( 1.51%)
6 cu-65 n,2n 3.7791E-07 +/- 7.8368E-09 ( 2.07%)
6 cu-65 capture -7.8656E-04 +/- 6.8251E-07 ( 0.09%)
6 cu-65 n,gamma -7.8620E-04 +/- 6.8251E-07 ( 0.09%)
6 cu-65 n,p -3.1043E-07 +/- 7.8886E-10 ( 0.25%)
6 cu-65 n,d -4.0903E-08 +/- 1.8559E-10 ( 0.45%)
6 cu-65 n,t -1.0623E-11 +/- 2.7949E-18 ( 0.00%)
6 cu-65 n,he-3 -3.8976E-13 +/- 1.3313E-20 ( 0.00%)
6 cu-65 n,alpha -8.9150E-09 +/- 1.3569E-15 ( 0.00%)
7 h-1 total -3.2827E-02 +/- 6.0121E-04 ( 1.83%)
7 h-1 scatter 2.0058E-02 +/- 5.9506E-04 ( 2.97%)
7 h-1 elastic 2.0058E-02 +/- 5.9506E-04 ( 2.97%)
7 h-1 capture -5.2884E-02 +/- 8.0351E-06 ( 0.02%)
7 h-1 n,gamma -5.2884E-02 +/- 8.0351E-06 ( 0.02%)
7 o-16 total 2.6563E-02 +/- 3.0100E-05 ( 0.11%)
7 o-16 scatter 2.6725E-02 +/- 3.0097E-05 ( 0.11%)
7 o-16 elastic 2.6660E-02 +/- 3.0097E-05 ( 0.11%)
7 o-16 n,n' 6.2924E-05 +/- 8.2297E-08 ( 0.13%)
7 o-16 n,2n 3.8366E-12 +/- 8.0943E-18 ( 0.00%)
7 o-16 capture -1.6136E-04 +/- 6.2177E-08 ( 0.04%)
7 o-16 n,gamma -1.5133E-05 +/- 2.2991E-09 ( 0.02%)
7 o-16 n,p -7.7305E-07 +/- 2.9519E-09 ( 0.38%)
7 o-16 n,d -7.8828E-08 +/- 5.6744E-10 ( 0.72%)
7 o-16 n,t -2.2174E-13 +/- 1.4667E-18 ( 0.00%)
7 o-16 n,alpha -1.4538E-04 +/- 6.1000E-08 ( 0.04%)
-----------------------------------------------
Total Sensitivity Coefficients by Nuclide
-----------------------------------------------
Mixture Nuclide Atom Density Sensitivity Std. Dev. % Std. Dev.
------- ------------- ------------ ----------- ---------- -----------
1 u-234 5.1835E-06 -9.0979E-04 +/- 3.1204E-07 ( 0.03%)
1 u-235 1.0102E-03 2.0616E-01 +/- 7.0501E-05 ( 0.03%)
1 u-236 5.1395E-06 -2.1805E-04 +/- 1.2725E-07 ( 0.06%)
1 u-238 2.2157E-02 -5.9609E-02 +/- 2.6857E-05 ( 0.05%)
1 o-16 4.6753E-02 1.1444E-02 +/- 3.3431E-05 ( 0.29%)
2 o-16 3.3338E-02 3.6005E-02 +/- 2.9143E-05 ( 0.08%)
4 o-16 1.2461E-02 3.9480E-05 +/- 8.0523E-07 ( 2.04%)
5 o-16 1.4273E-02 5.7694E-05 +/- 6.3654E-07 ( 1.10%)
6 o-16 1.0064E-04 6.9207E-07 +/- 7.5312E-09 ( 1.09%)
7 o-16 3.3338E-02 2.6563E-02 +/- 3.0100E-05 ( 0.11%)
2 h-1 6.6675E-02 2.0159E-01 +/- 4.6480E-04 ( 0.23%)
4 h-1 5.8178E-02 -3.3487E-05 +/- 3.3383E-05 ( 99.69%)
5 h-1 5.6642E-02 -4.6343E-05 +/- 2.5632E-05 ( 55.31%)
7 h-1 6.6675E-02 -3.2827E-02 +/- 6.0121E-04 ( 1.83%)
3 al-27 5.8433E-02 -6.7582E-05 +/- 2.3201E-05 ( 34.33%)
3 cr-50 2.7074E-06 -1.3040E-05 +/- 9.1891E-09 ( 0.07%)
3 cr-52 5.2210E-05 -8.7501E-06 +/- 2.6630E-08 ( 0.30%)
3 cr-53 5.9194E-06 -3.4245E-05 +/- 2.1670E-08 ( 0.06%)
3 cr-54 1.4736E-06 -1.6695E-08 +/- 4.1104E-10 ( 2.46%)
3 cu-63 6.5503E-05 -9.5973E-05 +/- 7.0063E-08 ( 0.07%)
6 cu-63 5.8191E-02 -3.1331E-03 +/- 6.7640E-06 ( 0.22%)
3 cu-65 2.9195E-05 -1.7689E-05 +/- 4.4209E-08 ( 0.25%)
6 cu-65 2.5937E-02 -6.0763E-04 +/- 4.7381E-06 ( 0.78%)
3 mg-24 5.2648E-04 4.5166E-05 +/- 4.5339E-07 ( 1.00%)
6 mg-24 3.4888E-06 3.2809E-08 +/- 5.8412E-12 ( 0.02%)
3 mg-25 6.6651E-05 1.2140E-06 +/- 3.2733E-08 ( 2.70%)
6 mg-25 4.4168E-07 2.2861E-09 +/- 1.0962E-12 ( 0.05%)
3 mg-26 7.3383E-05 5.2992E-06 +/- 3.2698E-08 ( 0.62%)
6 mg-26 4.8629E-07 3.6092E-09 +/- 4.4230E-13 ( 0.01%)
3 mn-55 2.2115E-05 -9.6360E-05 +/- 1.4079E-07 ( 0.15%)
3 ti-46 2.0300E-06 -1.8856E-07 +/- 8.6800E-10 ( 0.46%)
3 ti-47 1.8524E-06 -8.7007E-07 +/- 2.4463E-09 ( 0.28%)
3 ti-48 1.8727E-05 -4.5957E-05 +/- 3.3263E-08 ( 0.07%)
3 ti-49 1.3956E-06 -6.8733E-07 +/- 6.2660E-10 ( 0.09%)
3 ti-50 1.3702E-06 5.1758E-08 +/- 4.1741E-10 ( 0.81%)
3 si-28 3.1918E-04 4.3107E-06 +/- 1.2244E-07 ( 2.84%)
4 si-28 8.8873E-05 1.5659E-07 +/- 3.1168E-09 ( 1.99%)
6 si-28 3.5253E-05 1.7607E-07 +/- 1.6638E-09 ( 0.94%)
3 si-29 1.6161E-05 5.3687E-07 +/- 5.6554E-09 ( 1.05%)
4 si-29 4.5000E-06 8.3353E-09 +/- 7.5001E-15 ( 0.00%)
6 si-29 1.7850E-06 7.3370E-09 +/- 1.5194E-12 ( 0.02%)
3 si-30 1.0728E-05 1.3408E-07 +/- 4.4925E-09 ( 3.35%)
4 si-30 2.9872E-06 5.0684E-09 +/- 6.4263E-15 ( 0.00%)
6 si-30 1.1849E-06 4.2864E-09 +/- 9.7142E-13 ( 0.02%)
3 fe-54 5.9897E-06 -3.9407E-06 +/- 8.5216E-09 ( 0.22%)
6 fe-54 2.2682E-07 -3.6694E-09 +/- 6.3742E-12 ( 0.17%)
3 fe-56 9.3114E-05 -6.9426E-05 +/- 1.3507E-07 ( 0.19%)
6 fe-56 3.5261E-06 -1.0500E-07 +/- 4.7404E-10 ( 0.45%)
3 fe-57 2.1319E-06 -1.4580E-06 +/- 2.4578E-09 ( 0.17%)
6 fe-57 8.0732E-08 -1.5428E-09 +/- 1.5175E-12 ( 0.10%)
3 fe-58 2.8426E-07 -8.6249E-08 +/- 9.2863E-11 ( 0.11%)
6 fe-58 1.0764E-08 -7.0000E-11 +/- 1.6196E-14 ( 0.02%)
4 c 4.3562E-02 1.4324E-04 +/- 3.4393E-06 ( 2.40%)
5 c 3.5648E-02 1.5186E-04 +/- 2.0605E-06 ( 1.36%)
6 c 1.5194E-03 9.7668E-06 +/- 1.3853E-07 ( 1.42%)
4 ca-40 2.4875E-03 -1.7949E-06 +/- 1.1894E-07 ( 6.63%)
4 ca-42 1.6602E-05 -3.3798E-08 +/- 4.2370E-10 ( 1.25%)
4 ca-43 3.4641E-06 -2.9992E-07 +/- 8.5309E-10 ( 0.28%)
4 ca-44 5.3527E-05 -1.9121E-07 +/- 2.9350E-09 ( 1.53%)
4 ca-46 1.0264E-07 -9.1789E-11 +/- 2.9314E-16 ( 0.00%)
4 ca-48 4.7984E-06 -2.4743E-08 +/- 1.8724E-10 ( 0.76%)
4 s-32 4.5439E-04 -9.3027E-07 +/- 1.5023E-08 ( 1.61%)
6 s-32 3.1807E-06 -1.4228E-08 +/- 1.7286E-12 ( 0.01%)
4 s-33 3.5865E-06 -1.1925E-08 +/- 1.0473E-14 ( 0.00%)
6 s-33 2.5106E-08 -3.5098E-10 +/- 2.0849E-14 ( 0.01%)
4 s-34 2.0132E-05 1.2596E-08 +/- 3.9241E-10 ( 3.12%)
6 s-34 1.4093E-07 3.0984E-10 +/- 8.4943E-14 ( 0.03%)
4 s-36 9.5640E-08 1.2812E-10 +/- 1.6526E-16 ( 0.00%)
6 s-36 6.6948E-10 2.4367E-12 +/- 4.8470E-16 ( 0.02%)
6 na-23 4.6695E-06 2.3175E-08 +/- 1.4447E-09 ( 6.23%)
-----------------------------------------------
Total Sensitivity Coefficients by Mixture
-----------------------------------------------
Mixture Sensitivity Std. Dev. % Std. Dev.
------- ----------- ---------- -----------
1 1.5686E-01 +/- 8.2519E-05 ( 0.05%)
2 2.3760E-01 +/- 4.6571E-04 ( 0.20%)
3 -3.9956E-04 +/- 2.3207E-05 ( 5.81%)
4 1.4613E-04 +/- 3.3569E-05 ( 22.97%)
5 1.6321E-04 +/- 2.5723E-05 ( 15.76%)
6 -3.7302E-03 +/- 8.2596E-06 ( 0.22%)
7 -6.2632E-03 +/- 6.0196E-04 ( 9.61%)
------------------------------
Problem Characterization
------------------------------
median fission group is 222 5.000E-02 to 4.000E-02(eV)
average fission group 2.077E+02 +/- 3.212E-01
average energy(eV) causing fission 1.115E+05 +/- 2.059E+02
energy(eV) of average lethargy causing fission 1.128E-01 +/- 1.749E-04
median capture group is 223 4.000E-02 to 3.000E-02(eV)
average capture group 2.072E+02 +/- 1.110E-01
average energy(eV) causing capture 3.771E+04 +/- 4.509E+01
energy(eV) of average lethargy causing capture 8.751E-02 +/- 4.773E-05
Figure 6.3.16. SAMS output for LEU-COMP-THERM-009 case 10 sample problem. (continued)
median scatter group is 221 6.000E-02 to 5.000E-02(eV)
average scatter group 1.903E+02 +/- 5.331E-02
average energy(eV) causing scatter 6.721E+04 +/- 2.875E+01
energy(eV) of average lethargy causing scatter 3.839E-01 +/- 1.069E-04
Generating working covariance matrix ...
Working covariance matrix created for future processing.
-----------------------------
Uncertainty Information
-----------------------------
the relative standard deviation of k-eff (% delta-k/k) due to cross-section covariance data is:
0.6109 +/- 0.0000 % delta-k/k
contributions to uncertainty in k-eff (% delta-k/k) by individual energy covariance matrices:
covariance matrix
nuclide-reaction with nuclide-reaction % delta-k/k due to this matrix
------------------------------ ------------------------------- -----------------------------------
u-235 nubar u-235 nubar 2.8973E-01 +/- 3.9856E-06
u-238 n,n' u-238 n,n' 2.6630E-01 +/- 2.9468E-05
u-235 chi u-235 chi 2.4752E-01 +/- 1.6035E-05
h-1 elastic h-1 elastic 2.4045E-01 +/- 8.1628E-06
u-235 n,gamma u-235 n,gamma 1.7330E-01 +/- 1.8063E-06
u-238 n,gamma u-238 n,gamma 1.5959E-01 +/- 1.1160E-06
u-235 fission u-235 n,gamma 1.2186E-01 +/- 9.5110E-07
o-16 elastic o-16 elastic 1.1639E-01 +/- 4.4365E-06
u-235 fission u-235 fission 1.0865E-01 +/- 1.2931E-06
u-238 elastic u-238 n,n' -9.9362E-02 +/- 5.3145E-06
h-1 n,gamma h-1 n,gamma 9.8744E-02 +/- 3.7005E-07
u-238 nubar u-238 nubar 4.2236E-02 +/- 1.3385E-07
u-238 elastic u-238 elastic 2.2575E-02 +/- 4.8689E-07
al-27 n,gamma al-27 n,gamma 1.9265E-02 +/- 7.1802E-08
al-27 n,n' al-27 n,n' 1.7017E-02 +/- 5.0822E-07
cu-63 n,gamma cu-63 n,gamma 1.4853E-02 +/- 1.0721E-07
u-238 n,2n u-238 n,2n 1.3281E-02 +/- 1.1785E-07
u-238 fission u-238 fission 1.2837E-02 +/- 1.1196E-08
u-238 chi u-238 chi 1.1469E-02 +/- 3.3598E-08
al-27 elastic al-27 n,n' -1.0157E-02 +/- 2.1624E-07
u-238 elastic u-238 n,gamma 8.5476E-03 +/- 7.8644E-07
u-235 elastic u-235 n,gamma 6.0964E-03 +/- 7.9059E-08
u-235 elastic u-235 fission -4.4946E-03 +/- 7.3563E-08
o-16 n,alpha o-16 n,alpha 4.0401E-03 +/- 2.6708E-09
u-234 n,gamma u-234 n,gamma 3.8813E-03 +/- 2.0326E-08
o-16 n,n' o-16 n,n' 3.8417E-03 +/- 1.1648E-08
al-27 elastic al-27 elastic 3.6255E-03 +/- 4.5685E-08
cu-65 n,gamma cu-65 n,gamma 3.4344E-03 +/- 5.5487E-09
u-235 n,n' u-235 n,n' 3.3168E-03 +/- 4.5674E-09
cu-63 elastic cu-63 elastic 3.2014E-03 +/- 4.7334E-08
o-16 n,n' o-16 elastic -2.1989E-03 +/- 3.3250E-09
cu-65 elastic cu-65 elastic 1.6081E-03 +/- 1.1779E-08
u-235 elastic u-235 n,n' -1.5190E-03 +/- 1.0084E-09
u-238 elastic u-238 fission -1.2258E-03 +/- 8.6756E-10
u-238 fission u-238 n,gamma 1.1071E-03 +/- 1.6581E-10
u-238 elastic u-238 n,2n -1.0603E-03 +/- 8.3265E-09
o-16 n,gamma o-16 n,gamma 6.6497E-04 +/- 1.4657E-11
u-235 elastic u-235 elastic 6.4491E-04 +/- 5.5253E-10
u-236 n,gamma u-236 n,gamma 6.4328E-04 +/- 4.8310E-10
al-27 elastic al-27 n,gamma 6.3680E-04 +/- 4.9874E-08
u-234 fission u-234 fission 5.5055E-04 +/- 1.4300E-11
fe-56 n,gamma fe-56 n,gamma 4.6510E-04 +/- 4.5069E-11
mn-55 n,gamma mn-55 n,gamma 4.2641E-04 +/- 3.3567E-11
cu-63 n,p cu-63 n,p 4.1549E-04 +/- 2.0900E-10
ti-48 n,gamma ti-48 n,gamma 4.1164E-04 +/- 5.1774E-11
al-27 n,p al-27 n,p 4.0075E-04 +/- 1.5438E-10
cu-63 elastic cu-63 n,gamma 3.5387E-04 +/- 6.2656E-10
cu-63 elastic cu-63 n,n' -3.5353E-04 +/- 8.2827E-10
u-236 fission u-236 fission 3.0090E-04 +/- 5.0084E-12
u-235 n,2n u-235 n,2n 2.9426E-04 +/- 7.2192E-11
cr-53 n,gamma cr-53 n,gamma 2.8295E-04 +/- 1.8411E-11
cu-63 n,n' cu-63 n,n' 2.7900E-04 +/- 9.5920E-10
mg-24 elastic mg-24 elastic 2.2982E-04 +/- 3.3498E-10
c elastic c elastic 1.8308E-04 +/- 1.3865E-10
cu-63 n,n' cu-63 n,p 1.8186E-04 +/- 2.1087E-10
cu-65 elastic cu-65 n,gamma 1.7182E-04 +/- 1.7540E-10
cu-65 elastic cu-65 n,n' -1.4020E-04 +/- 1.3870E-10
cu-65 n,n' cu-65 n,n' 1.1168E-04 +/- 1.6558E-10
al-27 elastic al-27 n,p 1.0924E-04 +/- 5.0806E-11
cr-52 n,gamma cr-52 n,gamma 9.4888E-05 +/- 2.0408E-12
mg-24 n,n' mg-24 n,n' 8.9307E-05 +/- 1.6296E-11
si-28 n,gamma si-28 n,gamma 8.3099E-05 +/- 1.6055E-12
c n,n' c elastic -7.0959E-05 +/- 3.4172E-11
cr-50 n,gamma cr-50 n,gamma 6.1797E-05 +/- 7.7139E-13
al-27 n,alpha al-27 n,alpha 5.1627E-05 +/- 6.3487E-12
u-234 n,n' u-234 n,n' 4.9075E-05 +/- 1.3555E-12
c n,n' c n,n' 4.8570E-05 +/- 3.6800E-11
o-16 n,p o-16 n,p 4.7167E-05 +/- 3.5470E-12
mn-55 elastic mn-55 elastic 4.3763E-05 +/- 6.4296E-11
cu-63 n,n' cu-63 n,gamma 4.1035E-05 +/- 1.8996E-11
u-236 n,n' u-236 n,n' 3.8927E-05 +/- 1.4655E-12
ca-40 n,gamma ca-40 n,gamma 3.8300E-05 +/- 2.0407E-12
al-27 n,n' al-27 n,p 3.8185E-05 +/- 1.9522E-11
fe-54 n,gamma fe-54 n,gamma 3.2900E-05 +/- 2.4380E-13
mg-26 elastic mg-26 elastic 3.0966E-05 +/- 4.0780E-12
mg-24 n,gamma mg-24 n,gamma 2.9109E-05 +/- 1.6944E-13
o-16 elastic o-16 n,gamma 2.9062E-05 +/- 6.2232E-12
mg-25 elastic mg-25 elastic 2.5701E-05 +/- 3.0122E-12
u-235 elastic u-235 n,2n -2.5517E-05 +/- 6.3158E-12
ca-40 elastic ca-40 elastic 2.4956E-05 +/- 1.5456E-11
si-28 elastic si-28 elastic 2.3690E-05 +/- 1.8751E-12
al-27 n,n' al-27 n,alpha 1.8266E-05 +/- 3.6345E-12
si-28 n,n' si-28 n,n' 1.8068E-05 +/- 7.2723E-13
fe-57 n,gamma fe-57 n,gamma 1.7622E-05 +/- 8.0437E-14
cu-63 n,d cu-63 n,d 1.7414E-05 +/- 9.7220E-13
u-234 elastic u-234 elastic 1.6232E-05 +/- 1.9361E-12
o-16 n,d o-16 n,d 1.3977E-05 +/- 5.9002E-13
fe-56 elastic fe-56 elastic 1.3490E-05 +/- 1.2632E-12
al-27 elastic al-27 n,alpha 1.1830E-05 +/- 1.6930E-12
mn-55 elastic mn-55 n,gamma 1.1605E-05 +/- 1.0979E-12
ti-46 n,gamma ti-46 n,gamma 1.1500E-05 +/- 2.6908E-14
u-236 elastic u-236 elastic 1.1425E-05 +/- 8.1487E-13
cr-52 elastic cr-52 elastic 1.1243E-05 +/- 7.7884E-13
si-28 elastic si-28 n,n' -1.0955E-05 +/- 3.0843E-13
fe-56 n,n' fe-56 n,n' 1.0721E-05 +/- 2.4123E-13
al-27 n,n' al-27 n,gamma 1.0698E-05 +/- 7.9823E-13
mg-25 n,gamma mg-25 n,gamma 1.0650E-05 +/- 1.6936E-14
si-28 n,p si-28 n,p 1.0124E-05 +/- 1.7907E-13
cu-65 n,n' cu-65 n,gamma 9.9993E-06 +/- 9.1006E-13
mg-25 n,n' mg-25 n,n' 9.7352E-06 +/- 1.3961E-13
ca-43 n,gamma ca-43 n,gamma 9.5474E-06 +/- 1.2570E-13
cr-50 elastic cr-50 elastic 9.4119E-06 +/- 7.7575E-13
cu-63 n,n' cu-63 n,d 9.0483E-06 +/- 9.7927E-13
mg-26 n,n' mg-26 n,n' 8.6371E-06 +/- 1.2850E-13
c n,gamma c n,gamma 7.3604E-06 +/- 4.2545E-14
cu-65 n,p cu-65 n,p 7.2791E-06 +/- 1.2170E-13
s-32 elastic s-32 elastic 6.8662E-06 +/- 8.4130E-13
si-28 n,n' si-28 n,p 6.6795E-06 +/- 1.9459E-13
fe-56 elastic fe-56 n,n' -6.2701E-06 +/- 7.5901E-14
cr-52 n,n' cr-52 n,n' 6.0843E-06 +/- 8.2734E-14
cr-53 elastic cr-53 elastic 5.4910E-06 +/- 9.0167E-13
s-32 n,gamma s-32 n,gamma 5.0179E-06 +/- 3.4498E-14
mn-55 n,n' mn-55 n,n' 5.0115E-06 +/- 4.1826E-14
si-28 n,alpha si-28 n,alpha 4.8192E-06 +/- 5.1984E-14
si-30 n,gamma si-30 n,gamma 4.7027E-06 +/- 1.0526E-13
ti-48 n,n' ti-48 n,n' 3.9882E-06 +/- 2.8137E-14
u-236 n,2n u-236 n,2n 3.8665E-06 +/- 6.9405E-15
cu-65 n,n' cu-65 n,p 3.6181E-06 +/- 1.2145E-13
cr-53 elastic cr-53 n,gamma 3.2232E-06 +/- 4.5743E-14
cr-52 elastic cr-52 n,n' -3.1662E-06 +/- 2.3274E-14
si-28 n,n' si-28 n,alpha 3.1124E-06 +/- 5.5246E-14
si-29 n,gamma si-29 n,gamma 2.9509E-06 +/- 0.0000E+00
ti-47 elastic ti-47 elastic 2.9274E-06 +/- 2.6958E-13
fe-56 elastic fe-56 n,gamma 2.7043E-06 +/- 3.8368E-14
ti-47 n,gamma ti-47 n,gamma 2.6204E-06 +/- 0.0000E+00
si-29 elastic si-29 elastic 2.5109E-06 +/- 2.3146E-14
fe-54 elastic fe-54 elastic 2.4053E-06 +/- 1.4319E-13
ti-49 elastic ti-49 elastic 2.2186E-06 +/- 1.6191E-13
si-29 n,n' si-29 n,n' 2.1944E-06 +/- 9.7431E-15
ca-44 n,gamma ca-44 n,gamma 2.0769E-06 +/- 5.0792E-15
cu-63 n,alpha cu-63 n,alpha 1.9167E-06 +/- 1.6316E-14
c n,n' c n,alpha 1.9077E-06 +/- 1.1726E-13
ti-49 n,gamma ti-49 n,gamma 1.8792E-06 +/- 0.0000E+00
ti-46 elastic ti-46 elastic 1.8174E-06 +/- 1.9222E-14
cr-54 n,gamma cr-54 n,gamma 1.8052E-06 +/- 0.0000E+00
cr-50 elastic cr-50 n,gamma -1.6885E-06 +/- 2.2389E-15
al-27 n,2n al-27 n,2n 1.6378E-06 +/- 1.0328E-13
mg-26 n,gamma mg-26 n,gamma 1.4435E-06 +/- 0.0000E+00
ca-40 n,n' ca-40 n,n' 1.3555E-06 +/- 3.3924E-14
si-29 elastic si-29 n,n' -1.3141E-06 +/- 2.9905E-15
cu-65 n,d cu-65 n,d 1.2882E-06 +/- 8.1066E-15
si-30 elastic si-30 elastic 1.2448E-06 +/- 3.8759E-15
cr-52 elastic cr-52 n,gamma -1.2364E-06 +/- 2.5560E-15
cu-63 n,2n cu-63 n,2n 1.2318E-06 +/- 7.9923E-14
c n,alpha c n,alpha 1.0668E-06 +/- 6.1427E-14
ti-50 elastic ti-50 elastic 1.0218E-06 +/- 0.0000E+00
cu-63 n,n' cu-63 n,alpha 1.0189E-06 +/- 1.5895E-14
cr-53 n,n' cr-53 n,n' 9.9320E-07 +/- 1.7846E-15
ca-44 elastic ca-44 elastic 9.9005E-07 +/- 9.0779E-15
ti-46 n,n' ti-46 n,n' 9.6859E-07 +/- 0.0000E+00
fe-57 elastic fe-57 elastic 9.5215E-07 +/- 9.6326E-15
ti-47 n,n' ti-47 n,n' 9.0613E-07 +/- 0.0000E+00
ca-42 n,gamma ca-42 n,gamma 8.9913E-07 +/- 0.0000E+00
si-30 n,n' si-30 n,n' 8.6223E-07 +/- 0.0000E+00
al-27 n,n' al-27 n,2n -8.2603E-07 +/- 1.6896E-14
cu-65 n,n' cu-65 n,d 6.9595E-07 +/- 8.0999E-15
fe-54 n,n' fe-54 n,n' 6.7558E-07 +/- 0.0000E+00
u-234 n,2n u-234 n,2n 6.5672E-07 +/- 0.0000E+00
cu-63 n,n' cu-63 n,2n -5.9352E-07 +/- 2.0173E-14
cr-53 elastic cr-53 n,n' -5.8452E-07 +/- 0.0000E+00
fe-58 n,gamma fe-58 n,gamma 5.8445E-07 +/- 0.0000E+00
si-30 elastic si-30 n,gamma 5.6922E-07 +/- 2.2346E-14
fe-56 n,n' fe-56 n,gamma 5.2558E-07 +/- 0.0000E+00
ca-48 n,gamma ca-48 n,gamma 5.1948E-07 +/- 0.0000E+00
cu-65 n,2n cu-65 n,2n 4.8462E-07 +/- 7.8007E-15
ca-44 n,n' ca-44 n,n' 4.5030E-07 +/- 0.0000E+00
si-30 elastic si-30 n,n' -4.4830E-07 +/- 0.0000E+00
ti-49 n,n' ti-49 n,n' 4.4467E-07 +/- 0.0000E+00
mg-25 n,2n mg-25 n,2n 4.2964E-07 +/- 0.0000E+00
cr-52 n,n' cr-52 n,gamma 4.1052E-07 +/- 0.0000E+00
fe-57 n,n' fe-57 n,n' 3.9365E-07 +/- 0.0000E+00
si-29 n,alpha si-29 n,alpha 3.8432E-07 +/- 0.0000E+00
cr-54 elastic cr-54 elastic 3.6988E-07 +/- 0.0000E+00
si-28 elastic si-28 n,gamma 3.6913E-07 +/- 0.0000E+00
cr-50 n,n' cr-50 n,n' 3.5176E-07 +/- 0.0000E+00
fe-54 elastic fe-54 n,n' -3.4272E-07 +/- 0.0000E+00
fe-54 n,p fe-54 n,p 3.2528E-07 +/- 0.0000E+00
cr-50 n,p cr-50 n,p 3.2089E-07 +/- 0.0000E+00
s-32 n,n' s-32 n,n' 3.1845E-07 +/- 0.0000E+00
ti-50 n,n' ti-50 n,n' 3.1488E-07 +/- 0.0000E+00
fe-57 elastic fe-57 n,n' -2.9570E-07 +/- 0.0000E+00
si-28 n,n' si-28 n,gamma 2.8067E-07 +/- 0.0000E+00
si-29 n,n' si-29 n,alpha 2.6995E-07 +/- 0.0000E+00
s-34 elastic s-34 elastic 2.5320E-07 +/- 0.0000E+00
s-34 n,gamma s-34 n,gamma 2.3668E-07 +/- 0.0000E+00
ca-42 elastic ca-42 elastic 2.3320E-07 +/- 0.0000E+00
fe-54 n,n' fe-54 n,p 2.3313E-07 +/- 0.0000E+00
cr-50 n,n' cr-50 n,p 2.3306E-07 +/- 0.0000E+00
cu-65 n,n' cu-65 n,2n -2.3141E-07 +/- 2.7300E-15
si-29 n,p si-29 n,p 2.2789E-07 +/- 0.0000E+00
ca-43 elastic ca-43 elastic 2.2169E-07 +/- 0.0000E+00
cr-50 elastic cr-50 n,n' -2.1587E-07 +/- 0.0000E+00
na-23 elastic na-23 elastic 2.0908E-07 +/- 2.2505E-15
cu-65 n,alpha cu-65 n,alpha 2.0897E-07 +/- 0.0000E+00
cr-54 n,n' cr-54 n,n' 2.0240E-07 +/- 0.0000E+00
fe-56 n,alpha fe-56 n,alpha 1.8941E-07 +/- 0.0000E+00
cr-52 n,p cr-52 n,p 1.8760E-07 +/- 0.0000E+00
fe-54 elastic fe-54 n,gamma 1.8132E-07 +/- 0.0000E+00
si-29 n,n' si-29 n,p 1.6557E-07 +/- 0.0000E+00
cr-52 n,n' cr-52 n,p 1.5092E-07 +/- 0.0000E+00
fe-56 n,n' fe-56 n,alpha 1.4803E-07 +/- 0.0000E+00
fe-56 n,2n fe-56 n,2n 1.4651E-07 +/- 0.0000E+00
ca-48 elastic ca-48 elastic 1.4271E-07 +/- 0.0000E+00
ti-50 n,gamma ti-50 n,gamma 1.3989E-07 +/- 0.0000E+00
s-33 n,gamma s-33 n,gamma 1.2657E-07 +/- 0.0000E+00
fe-58 elastic fe-58 elastic 1.2168E-07 +/- 0.0000E+00
cr-54 elastic cr-54 n,n' -1.2057E-07 +/- 0.0000E+00
cu-65 n,n' cu-65 n,alpha 1.1181E-07 +/- 0.0000E+00
ca-42 n,n' ca-42 n,n' 1.0111E-07 +/- 0.0000E+00
fe-57 elastic fe-57 n,gamma 9.3777E-08 +/- 0.0000E+00
cr-53 n,n' cr-53 n,gamma 8.4457E-08 +/- 0.0000E+00
fe-57 n,n' fe-57 n,gamma 7.6615E-08 +/- 0.0000E+00
fe-56 n,n' fe-56 n,2n -7.4856E-08 +/- 0.0000E+00
na-23 n,gamma na-23 n,gamma 6.9611E-08 +/- 0.0000E+00
mn-55 n,p mn-55 n,p 5.8834E-08 +/- 0.0000E+00
fe-56 n,p fe-56 n,p 5.6198E-08 +/- 0.0000E+00
na-23 elastic na-23 n,n' -4.8025E-08 +/- 0.0000E+00
fe-56 n,n' fe-56 n,p 4.5081E-08 +/- 0.0000E+00
cr-50 n,alpha cr-50 n,alpha 4.4847E-08 +/- 0.0000E+00
cr-52 n,alpha cr-52 n,alpha 4.3919E-08 +/- 0.0000E+00
fe-58 n,n' fe-58 n,n' 4.2403E-08 +/- 0.0000E+00
na-23 n,n' na-23 n,n' 3.9418E-08 +/- 0.0000E+00
s-33 elastic s-33 elastic 3.7973E-08 +/- 0.0000E+00
cr-50 n,n' cr-50 n,alpha 3.6143E-08 +/- 0.0000E+00
cr-53 n,alpha cr-53 n,alpha 3.5247E-08 +/- 0.0000E+00
cr-52 n,n' cr-52 n,alpha 3.4289E-08 +/- 0.0000E+00
cr-50 n,n' cr-50 n,gamma 3.3993E-08 +/- 0.0000E+00
mg-26 n,2n mg-26 n,2n 3.2987E-08 +/- 0.0000E+00
cr-53 n,2n cr-53 n,2n 3.0637E-08 +/- 0.0000E+00
ca-43 n,n' ca-43 n,n' 2.9945E-08 +/- 0.0000E+00
o-16 n,n' o-16 n,gamma 2.8202E-08 +/- 0.0000E+00
mn-55 n,2n mn-55 n,2n 2.8181E-08 +/- 0.0000E+00
si-29 elastic si-29 n,gamma 2.7325E-08 +/- 0.0000E+00
cr-53 n,n' cr-53 n,alpha 2.6753E-08 +/- 0.0000E+00
fe-54 n,n' fe-54 n,gamma 2.5858E-08 +/- 0.0000E+00
si-29 n,2n si-29 n,2n 2.4668E-08 +/- 0.0000E+00
fe-58 elastic fe-58 n,n' -2.3267E-08 +/- 0.0000E+00
si-28 n,d si-28 n,d 2.2815E-08 +/- 0.0000E+00
si-30 n,n' si-30 n,gamma 2.1466E-08 +/- 0.0000E+00
fe-58 elastic fe-58 n,gamma 1.9178E-08 +/- 0.0000E+00
c n,n' c n,gamma 1.7660E-08 +/- 0.0000E+00
na-23 elastic na-23 n,gamma -1.6839E-08 +/- 0.0000E+00
cr-53 n,n' cr-53 n,2n -1.6046E-08 +/- 0.0000E+00
fe-57 n,2n fe-57 n,2n 1.5101E-08 +/- 0.0000E+00
s-34 n,n' s-34 n,n' 1.4729E-08 +/- 0.0000E+00
si-30 n,2n si-30 n,2n 1.3545E-08 +/- 0.0000E+00
fe-54 n,alpha fe-54 n,alpha 1.3005E-08 +/- 0.0000E+00
fe-54 n,n' fe-54 n,alpha 1.0111E-08 +/- 0.0000E+00
si-29 n,n' si-29 n,2n -1.0081E-08 +/- 0.0000E+00
si-30 n,alpha si-30 n,alpha 8.4209E-09 +/- 0.0000E+00
cr-54 elastic cr-54 n,gamma 8.3271E-09 +/- 0.0000E+00
fe-57 n,n' fe-57 n,2n -7.9525E-09 +/- 0.0000E+00
cr-53 n,p cr-53 n,p 7.9053E-09 +/- 0.0000E+00
ca-46 elastic ca-46 elastic 7.6477E-09 +/- 0.0000E+00
fe-58 n,n' fe-58 n,gamma 7.2381E-09 +/- 0.0000E+00
si-30 n,n' si-30 n,2n -7.0930E-09 +/- 0.0000E+00
mn-55 n,alpha mn-55 n,alpha 7.0079E-09 +/- 0.0000E+00
cr-52 n,2n cr-52 n,2n 6.5167E-09 +/- 0.0000E+00
si-30 n,n' si-30 n,alpha 6.4283E-09 +/- 0.0000E+00
ti-48 n,2n ti-48 n,2n 6.2875E-09 +/- 0.0000E+00
cr-53 n,n' cr-53 n,p 6.2628E-09 +/- 0.0000E+00
ti-49 n,2n ti-49 n,2n 5.8476E-09 +/- 0.0000E+00
ca-46 n,gamma ca-46 n,gamma 5.5651E-09 +/- 0.0000E+00
fe-57 n,alpha fe-57 n,alpha 5.3090E-09 +/- 0.0000E+00
cr-54 n,n' cr-54 n,gamma 4.9562E-09 +/- 0.0000E+00
ca-48 n,n' ca-48 n,n' 4.9102E-09 +/- 0.0000E+00
fe-56 n,d fe-56 n,d 4.7681E-09 +/- 0.0000E+00
fe-57 n,n' fe-57 n,alpha 3.9611E-09 +/- 0.0000E+00
s-33 n,n' s-33 n,n' 3.7151E-09 +/- 0.0000E+00
fe-56 n,n' fe-56 n,d 3.6804E-09 +/- 0.0000E+00
cr-54 n,2n cr-54 n,2n 3.6592E-09 +/- 0.0000E+00
si-29 n,n' si-29 n,gamma 3.4777E-09 +/- 0.0000E+00
fe-57 n,p fe-57 n,p 3.4511E-09 +/- 0.0000E+00
cr-52 n,n' cr-52 n,2n -3.4308E-09 +/- 0.0000E+00
ti-48 n,p ti-48 n,p 2.9116E-09 +/- 0.0000E+00
fe-57 n,n' fe-57 n,p 2.6805E-09 +/- 0.0000E+00
ti-47 n,2n ti-47 n,2n 2.6563E-09 +/- 0.0000E+00
s-36 n,gamma s-36 n,gamma 2.5689E-09 +/- 0.0000E+00
na-23 n,p na-23 n,p 2.1365E-09 +/- 0.0000E+00
cr-54 n,n' cr-54 n,2n -2.0358E-09 +/- 0.0000E+00
si-30 n,p si-30 n,p 1.9130E-09 +/- 0.0000E+00
ti-50 n,2n ti-50 n,2n 1.4840E-09 +/- 0.0000E+00
si-30 n,n' si-30 n,p 1.4354E-09 +/- 0.0000E+00
c n,n' c n,d 1.3648E-09 +/- 0.0000E+00
na-23 n,alpha na-23 n,alpha 1.3262E-09 +/- 0.0000E+00
s-36 elastic s-36 elastic 1.2923E-09 +/- 0.0000E+00
fe-58 n,2n fe-58 n,2n 1.2017E-09 +/- 0.0000E+00
o-16 n,n' o-16 n,2n -9.4097E-10 +/- 0.0000E+00
mn-55 n,d mn-55 n,d 8.7967E-10 +/- 0.0000E+00
c n,d c n,d 8.3498E-10 +/- 0.0000E+00
fe-54 n,d fe-54 n,d 7.9457E-10 +/- 0.0000E+00
na-23 elastic na-23 n,p 7.7621E-10 +/- 0.0000E+00
ca-46 n,n' ca-46 n,n' 7.5538E-10 +/- 0.0000E+00
fe-58 n,n' fe-58 n,2n -6.7419E-10 +/- 0.0000E+00
cu-65 n,t cu-65 n,t 6.5557E-10 +/- 0.0000E+00
ti-48 n,alpha ti-48 n,alpha 6.2508E-10 +/- 0.0000E+00
c n,n' c n,p 6.0127E-10 +/- 0.0000E+00
na-23 elastic na-23 n,alpha 4.5436E-10 +/- 0.0000E+00
fe-54 n,2n fe-54 n,2n 3.8024E-10 +/- 0.0000E+00
c n,p c n,p 3.6950E-10 +/- 0.0000E+00
cr-54 n,alpha cr-54 n,alpha 3.3662E-10 +/- 0.0000E+00
cu-65 n,n' cu-65 n,t 3.3605E-10 +/- 0.0000E+00
ca-44 n,2n ca-44 n,2n 3.3085E-10 +/- 0.0000E+00
fe-54 n,n' fe-54 n,2n -3.2000E-10 +/- 0.0000E+00
cr-54 n,n' cr-54 n,alpha 2.7557E-10 +/- 0.0000E+00
o-16 n,2n o-16 n,2n 2.4315E-10 +/- 0.0000E+00
ca-43 n,2n ca-43 n,2n 1.7648E-10 +/- 0.0000E+00
fe-56 n,n' fe-56 n,t 1.5852E-10 +/- 0.0000E+00
cr-50 n,d cr-50 n,d 1.5158E-10 +/- 0.0000E+00
ca-48 n,2n ca-48 n,2n 1.3661E-10 +/- 0.0000E+00
cr-50 n,n' cr-50 n,d 1.2412E-10 +/- 0.0000E+00
cu-63 n,he-3 cu-63 n,he-3 1.1688E-10 +/- 0.0000E+00
s-33 n,2n s-33 n,2n 1.1455E-10 +/- 0.0000E+00
ti-46 n,2n ti-46 n,2n 8.3916E-11 +/- 0.0000E+00
fe-58 n,alpha fe-58 n,alpha 8.3616E-11 +/- 0.0000E+00
cr-50 n,2n cr-50 n,2n 7.9460E-11 +/- 0.0000E+00
s-36 n,n' s-36 n,n' 7.3825E-11 +/- 0.0000E+00
ca-42 n,2n ca-42 n,2n 7.0527E-11 +/- 0.0000E+00
fe-58 n,n' fe-58 n,alpha 6.8070E-11 +/- 0.0000E+00
cu-63 n,n' cu-63 n,he-3 5.9819E-11 +/- 0.0000E+00
s-34 n,2n s-34 n,2n 5.5409E-11 +/- 0.0000E+00
cr-50 n,n' cr-50 n,2n -4.5867E-11 +/- 0.0000E+00
cr-54 n,p cr-54 n,p 3.4617E-11 +/- 0.0000E+00
fe-56 n,n' fe-56 n,he-3 2.9868E-11 +/- 0.0000E+00
mn-55 n,t mn-55 n,t 2.9356E-11 +/- 0.0000E+00
o-16 n,t o-16 n,t 2.9008E-11 +/- 0.0000E+00
cr-54 n,n' cr-54 n,p 2.7365E-11 +/- 0.0000E+00
fe-56 n,t fe-56 n,t 1.5517E-11 +/- 0.0000E+00
fe-58 n,p fe-58 n,p 1.3800E-11 +/- 0.0000E+00
fe-58 n,n' fe-58 n,p 1.1055E-11 +/- 0.0000E+00
cu-65 n,he-3 cu-65 n,he-3 6.2232E-12 +/- 0.0000E+00
mg-24 n,2n mg-24 n,2n 3.6825E-12 +/- 0.0000E+00
cu-65 n,n' cu-65 n,he-3 3.1211E-12 +/- 0.0000E+00
na-23 n,2n na-23 n,2n 2.9396E-12 +/- 0.0000E+00
s-36 n,2n s-36 n,2n 1.8272E-12 +/- 0.0000E+00
fe-56 n,he-3 fe-56 n,he-3 1.7491E-12 +/- 0.0000E+00
ca-46 n,2n ca-46 n,2n 1.7030E-12 +/- 0.0000E+00
na-23 elastic na-23 n,2n -1.0631E-12 +/- 0.0000E+00
mn-55 n,he-3 mn-55 n,he-3 8.9376E-13 +/- 0.0000E+00
ca-40 n,2n ca-40 n,2n 4.9330E-13 +/- 0.0000E+00
s-32 n,2n s-32 n,2n 1.1554E-13 +/- 0.0000E+00
Note: relative standard deviation in k-eff can be computed from
individual values by adding the square of the values with positive signs and
subtracting the square of the values with negative signs, then taking the square root
-----------------------------------
SAMS execution complete
Elapsed time: 8.10467 minutes
-----------------------------------
Fig. 6.6.11 Region-integrated sensitivity profiles from test case LEU-COMP-THERM-009 case 10.
6.6.5. Error And Warning Messages
SAMS will display a message when a predetermined condition is encountered. Error messages are displayed prior to code termination. Warning messages are displayed when a potentially abnormal condition is encountered, but code execution continues.
6.6.5.1. Error Messages
- SAMS Error 001: Selected calculation type ______ is not valid.
This message is displayed by VIP when the user enters a calculation type other than kenova or xsdrn.
- SAMS Error 002: There was a zero response value.
This message is displayed by SENSITIVITY_SUMMATION when there was no fissile material in the problem or the forward or adjoint flux in all regions containing fissile material was zero. This message can be encountered when the adjoint solution fails to proceed far enough to accumulate active generations for flux accumulation in TSUNAMI-3D models.
- SAMS Error 003: Error reading covariance data io=___ mat1=____
- mt1=____ mat2=____ mt2=____ nblok=____ ngroup=____
This error is displayed by SAMCAVA when a non-zero return code is obtained when reading from the covariance data library.
- SAMS Error 004: Adjoint flux file from KENO was not found on unit ____
This error is displayed by GET_FLUXES when the adjoint flux unit cannot be opened.
- SAMS Error 005: Forward flux file from KENO was not found on unit ____
This error is displayed by GET_FLUXES when the forward flux unit cannot be opened.
- SAMS Error 006: Adjoint fluxes were not calculated in the source case
This error is displayed by GET_FLUXES when an adjoint data file is found, but it does not contain flux data. The adjoint case should be rerun with flux accumulation turned on.
- SAMS Error 007: Only the scalar fluxes were calculated. Execution is stopping.
This message is displayed by GET_FLUXES when the KENO calculation was run with both PNM=0 and NQD=0. The KENO case does not contain sufficient information to produce sensitivity coefficients.
- SAMS Error 008: Error reading the forward restart file, record type ____
This message is displayed by GET_FLUXES when an unknown record type from the KENO forward solution is encountered.
- SAMS Error 009: Error reading the adjoint restart file, record type ____
This message is displayed by GET_FLUXES if an unknown record type from the KENO adjoint solution is encountered.
- SAMS Error 010: Invalid read key ____
This message is displayed by GET_INPUT when an invalid word is encountered in the input before a “read” card is encountered.
- SAMS Error 011: Invalid block key ____. The first block must be initial.
This message is displayed by GET_INPUT when the first word after the first “read” is not “initial”.
- SAMS Error 012: End of input data reached. Expecting more data.
This message is displayed by GET_INPUT when the end of the input stream is unexpectedly encountered.
- SAMS Error 013: Ending block keyword ___
- does not equal the beginning block keyword ___
This message is displayed by GET_INPUT when the word following “END” does not match the word after “READ” for the current input block.
- SAMS Error 014: Unknown read return flag ____
This message is displayed by GET_INPUT when a reading error occurs. This is likely due to an input error.
- Error allocating ______
This message is displayed by GET_FLUXES when insufficient computational resources exist to allocate memory for processing the problem. If this message is encountered, try reducing the size of the calculation by reducing the number of mesh intervals (e.g., increase the mesh size, MSH in KENO), reducing the order of flux moments or angular fluxes accumulated (PNM and NQD in KENO), or using a cross-section data library with fewer energy groups.
6.6.5.2. Warning Messages
- SAMS Warning 001: Use of mesh flux was requested (mfx=yes), but a
- mesh size of ____ was requested. Please enter a positive value for msh= in the
- KENO parameter data. SAMS will continue its calculation with the region fluxes,
- equivalent to mfx=no.
This message is displayed by GET_FLUXES when MFX=YES is entered in KENO parameter data, but the mesh size MSH is not entered or is non-positive. However, if the user specifies another mesh in a different section of a TSUNAMI-3D input (i.e. in the read GridGeometry block), then TSUNAMI will default to use that mesh.
- SAMS Warning 002: The k-eff values for the forward and adjoint
- calculations differ by more than one percent.
This message is displayed by PRINT_TRANS when the k-eff values from the forward and adjoint calculations differ by more than one percent. This typically indicates poor convergence of the adjoint Monte Carlo calculation. The calculation should be rerun with more histories to improve agreement between the forward and adjoint k-eff values.
- SAMS Warning 003: Implicit sensitivities were requested but no
- implicit datafiles were found. Proceeding with explicit sensitivity coefficients
- only.
This message is displayed by IMPLICIT_SENSITIVITIES when MAKEIMP is present in the SAMS input, but no datafiles from SENLIB or BONAMIST are found. The calculation will continue with explicit sensitivity coefficients only.
- SAMS Warning 004: Implicit sensitivity of ____ ____ to ____ group
- ____ from ____ had a value of ____ and has been reset to 0.0.
This message is displayed by IMPLICIT_SENSITIVITIES when an implicit sensitivity value greater than that set by LARGEIMP is encountered. The large implicit value is likely due to an inconsistency between the CELLDATA specification for the material identified in the message and the use of that material in the geometry of the model.
- SAMS Warning 005: Implicit sensitivity for mixture ____ ____ ____
- exceeds the explicit sensitivity by more than an order of magnitude. Implicit
- sensitivity: ____Explicit sensitivity: ____ Please check for appropriate
- resonance self-shielding models in celldata.
This message is displayed by IMPLICIT_SENSITIVITIES when the implicit sensitivity value exceeds the explicit sensitivity value by more than an order of magnitude. The large implicit value is possibly due to an inconsistency between the CELLDATA specification for the material identified in the message and the use of that material in the geometry of the model.
- SAMS Warning 006: An implicit sensitivity was generated by the
- resonance self-shielding calculation in ____ for mixture ____ ____ ___ with
- mixture ____ ____ which was not found in the model. This potentially important
- information will be skipped. Please check the celldata input.
This message is displayed by IMPLICIT_SENSITIVITIES when implicit data is generated by the resonance self-shielding codes, but a material needed for the calculation is not present in the geometry model. Please check for consistent use of materials in the CELLDATA input and in the geometry model.
- SAMS Warning 007: USEANG was requested in the input, but angular
- fluxes were not computed. USEMOM has been set to true, and flux moments will be
- used to compute sensitivity coefficients.
This message is displayed by GET_FLUXES when USEANG is true but angular fluxes were not computed in the transport solution. In this case the flux moments computed in the transport solution are used to compute sensitivity coefficients. If neither angular fluxes nor flux moments are found, SAMS Error 007 will stop code execution prior to reaching this warning.
- SAMS Warning 008: USEMOM was requested in the input, but flux
- moments were not computed. USEANG has been set to true, and angular fluxes will
- be used to compute sensitivity coefficients.
This message is displayed by GET_FLUXES when USEMOM is true but flux moments were not computed in the transport solution. In this case the angular fluxes computed in the transport solution are used to compute sensitivity coefficients. If neither angular fluxes nor flux moments are found, SAMS Error 007 will stop code execution prior to reaching this warning.
References
- SAMSBev69
P. Bevington. R.(1969) Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill, 1969.
- SAMSCom99
NEA Nuclear Science Committe. International Handbook of Evaluated Criticality Safety Benchmarks Experiments. OECD Nuclear Energy Agency, 1999.
- SAMSDun00
M. E. Dunn. PUFF-III: A Code for Processing ENDF Uncertainty Data into Multigroup Covariance Matrices. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 2000.
- SAMSGan67
A. Gandini. A generalized perturbation method for bi-linear functionals of the real and adjoint neutron fluxes. Journal of Nuclear Energy, 21(10):755–765, 1967. Publisher: Pergamon.
- SAMSHTK84
Akihiro Hara, Toshikazu Takeda, and Yasuyuki Kikuchi. SAGEP: two-dimensional sensitivity analysis code based on generalized perturbation theory. Technical Report, Japan Atomic Energy Research Inst., 1984.
- SAMSLWM+81
J. L. Lucius, C. R. Weisbin, J. H. Marable, J. D. Drischler, R. Q. Wright, and J. E. White. Users manual for the FORSS sensitivity and uncertainty analysis code system. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 1981.
- SAMSObl76
E. M. Oblow. Sensitivity theory from a differential viewpoint. Nuclear Science and Engineering, 59(2):187–189, 1976. Publisher: Taylor & Francis.
- SAMSSJ72
Weston M. Stacey Jr. Variational estimates and generalized perturbation theory for the ratios of linear and bilinear functionals. Journal of Mathematical Physics, 13(8):1119–1125, 1972. Publisher: American Institute of Physics.
- SAMSUsa64
L. N. Usachev. Perturbation theory for the breeding ratio and for other number ratios pertaining to various reactor processes. Journal of Nuclear Energy. Parts A/B. Reactor Science and Technology, 18(10):571–583, 1964. Publisher: Elsevier.
- SAMSWML+76(1,2,3,4)
C. R. Weisbin, J. H. Marable, J. L. Lucius, E. M. Oblow, F. R. Mynatt, R. W. Peelle, and F. G. Perey. Application of FORSS sensitivity and uncertainty methodology to fast reactor benchmark analysis. Technical Report, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 1976.
- SAMSWOM+78
C. R. Weisbin, E. M. Oblow, J. H. Marable, R. W. Peelle, and J. L. Lucius. Application of sensitivity and uncertainty methodology to fast reactor integral experiment analysis. Nuclear Science and Engineering, 66(3):307–333, 1978. Publisher: Taylor & Francis.
- SAMSWBP01(1,2)
M. L. Williams, B. L. Broadhead, and C. V. Parks. Eigenvalue sensitivity theory for resonance-shielded cross sections. Nuclear Science and Engineering, 138(2):177–191, 2001.
- SAMSWil86
Mark L. Williams. Perturbation theory for nuclear reactor analysis. CRC Press, Inc., 1986.
- SAMSNationalNDCenter74
National Nuclear Data Center. Cross Section Evaluation Working Group Benchmark Specifications. Brookhaven National Laboratory, 1974.