6.1. TSUNAMI-1D: Control Module for One-Dimensional Cross-Section Sensitivity and Uncertainty

B. T. Rearden, M. A. Jessee, L. M. Petrie, and M. L. Williams

6.1.1. ABSTRACT

TSUNAMI-1D (Tools for Sensitivity and Uncertainty Analysis Methodology Implementation in One Dimension) is a SCALE control module that facilitates the application of sensitivity and uncertainty analysis theory to nuclear systems analyses. TSUNAMI-1D was originally developed to provide sensitivity and uncertainty analysis of k_eff for criticality safety applications, and subsequent updates provide for analysis of system responses other than k_eff, using generalized perturbation theory. TSUNAMI-1D provides for automated processing of material input, processing of cross-section data, calculation of neutron transport solutions, calculation of sensitivity coefficients, and the calculation of uncertainties in system responses due to cross-section-covariance data. The XSDRNPM module is used for the transport solver. XSDRNPM uses the method of discrete ordinates to calculate k_eff for applications that are appropriate for 1D modeling. The SAMS module is used to determine the sensitivities of the calculated value of k_eff and other system responses to the nuclear data used in the calculation as a function of nuclide, reaction type, and energy. The uncertainties in the calculated value of k_eff and other system responses, resulting from uncertainties in the basic nuclear data used in the calculation, are estimated using energy-dependent cross-section-covariance matrices. The implicit effects of the resonance self-shielding calculations are predicted using BONAMIST.

6.1.2. ACKNOWLEDGMENTS

The authors acknowledge Bryan Broadhead of 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 Calvin Hopper, Cecil Parks, and Don Mueller of Oak Ridge National Laboratory is also appreciated. Additionally, the authors wish to acknowledge Debbie Weaver and Sheila Walker for their assistance in preparing this document.

6.1.3. Introduction

TSUNAMI-1D (Tools for Sensitivity and Uncertainty Analysis Methodology Implementation in One Dimension) is a SCALE control module that facilitates the application of sensitivity and uncertainty theory to nuclear system analyses. The data computed with TSUNAMI-1D are the sensitivity of a system response, such as k_eff, to each constituent cross-section data component used in the calculation. The sensitivity data are coupled with cross-section uncertainty data, in the form of multigroup covariance matrices, to produce an uncertainty in the system response due to uncertainties in the underlying nuclear data. The group-wise sensitivity data computed with TSUNAMI-1D are stored in a sensitivity data file (.sdf file) that is suitable for use in assessing system similarity for code validation purposes using TSUNAMI-IP, (see TSUNAMI-IP chapter), and for advanced bias assessment using TSURFER, see the TSURFER chapter.

This manual is intended to provide the user with a detailed reference on code input options and provide some examples of the application of TSUNAMI-1D to generate sensitivity and uncertainty data. A detailed description of code input is provided in Sect. 6.1.4, three sample problems are given in Sect. 6.1.5 the techniques used in each computational sequence are described in Sect. 6.1.3.1, and additional information is provided in the appendices. A new user may wish to begin by reviewing the sample problems, and then refer to the input details in Sect. 6.1.4 to customize an input for his specific needs.

TSUNAMI-1D provides automated, problem-dependent cross sections using the same methods and input as the Criticality Safety Analysis Sequences (CSAS). The BONAMIST code computes the sensitivity of resonance self-shielded cross to the input data, the so-called “implicit sensitivities.”

After the cross sections are processed, the TSUNAMI-1D sequence performs two XSDRNPM criticality calculations, one forward and one adjoint. Finally, the sequence calls the SAMS module to calculate the sensitivity coefficients that indicate the sensitivity of the calculated values to changes in the cross sections and the uncertainty in the calculated value due to uncertainties in the basic nuclear data. SAMS prints energy-integrated sensitivity coefficients and their statistical uncertainties to the SCALE output file and generates a separate data file containing the energy-dependent sensitivity coefficients.

In addition to the sensitivity and uncertainty analysis sequence, the TSUNAMI-1DC sequence can be used to verify the accuracy of the TSUNAMI-1D calculations with direct perturbation criticality calculations. The verification of computed sensitivity coefficients is imported for systems where the cell-weighted material is not the only material used in the model. By default, TSUNAMI-1DC performs the same functions as the TSUNAMI-1D sequence with PARM=CENTRM, except that it does not perform the adjoint XSDRNPM calculation and does not call the SAMS module.

6.1.3.1. TSUNAMI-1D Techniques

TSUNAMI-1D is a SCALE control module. As such, its primary function is to control a sequence of calculations that are performed by other codes. The input for each of the TSUNAMI-1D sequences is very similar to that used for CSAS1, with the addition of the system model description and optional sensitivity calculation data. TSUNAMI-1D uses the same material and cell data input as all other SCALE sequences. The control sequences available in TSUNAMI-1D are summarized in Table 6.1.1, where the functional modules executed by each control sequence are also shown. A general flow diagram of TSUNAMI-1D is shown in Fig. 6.1.1.

Table 6.1.1 TSUNAMI-1D control sequences.

Control module	Functional modules executed by the control module
TSUNAMI-1D	XSProc	XSDRNPM forward	XSDRNPM adjoint*	BONAMIST	SAMS*
TSUNAMI-1DC	XSProc	XSDRNPM forward
* The XSDRNPM adjoint calculation and SAMS calculation on are repeated for each system response defined by the user.

TSUNAMI-1D and many other SCALE sequences apply a standardized procedure to provide appropriate cross sections for the calculation. This procedure is carried out by routines of the XSProc that generate number densities and related information, prepare geometry data for resonance self-shielding and flux-weighting cell calculations, and create data input files for the cross-section processing codes.

By default, the TSUNAMI-1D sequence performs cross-section processing with XSProc, exercising all available options there, performs the forward and adjoint XSDRNPM calculations, calls BONAMIST to produce implicit sensitivity coefficients, then calls SAMS to produce sensitivity and uncertainty output and sdf files. Optional sequence level parameters can be used to change methods applied in resonance self-shielding and exclude the implicit sensitivity calculation, which detailed later in this document. If additional system responses are requested in the input, TSUNAMI-1D executes additional generalized adjoint XSDRNPM and SAMS calculations for each system response.

The input requirements for the model description are very similar to those used for multiregion cell descriptions in the cell data section of input. The definition of system responses other than k_eff requires both the DEFINITIONS and SYSTEMRESPONSE block of input data. These blocks of data are described in Sect. 6.1.4.4. TSUNAMI-1D also reads and prepares inputs for the SAMS calculation. The additional input blocks for the SAMS module are optional. The input format of the SAMS blocks of data are described in the SAMS chapter.

Fig. 6.1.1 General flow diagram of TSUNAMI-1D.

6.1.4. TSUNAMI-1D Input Description

The input to TSUNAMI-1D consists of a SCALE Analytical Sequence Specification Record, SCALE XSProc data, model problem data, optional sensitivity and uncertainty calculation data, and optional system response characterization data. The data for each of these segments are entered using the SCALE free-form format, allowing alphanumeric data, floating-point data, and integer data to be entered in an unstructured manner. 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. Data can usually start or end in any column with a few exceptions. As an example, the word END beginning in column 1 and followed by two blank spaces will end the problem, any data following will be ignored. Each data entry must be followed by one or more blanks to terminate the data entry. For numeric data, either a comma or a blank can be used to terminate each data entry. Integers may be entered for floating values. For example, 10 will be interpreted as 10.0. Imbedded blanks are not allowed within a data entry unless an E precedes a single blank as in an unsigned exponent in a floating-point number. For example, 1.0E 4 would be correctly interpreted as 1.0 \(\times\) 10⁴. A comment is initiated with a single quote, '', and continues until the end of the input line.

6.1.4.1. Analytical sequence specification record

The analytical sequence specification begins in column 1 of the first line of the input file and must contain one of the following:

=TSUNAMI-1D

This sequence is used for sensitivity and uncertainty calculations.

=TSUNAMI-1DC

This sequence allows more flexibility than CSAS1 and is used for criticality calculations where the criticality problem description contains more detail than that specified in a single unit cell description.

Optional keyword input may be entered, starting after column 10 of the analytical sequence specification record. These keywords are

PARM=CHECK

This option allows the input data to be read and checked without executing any functional modules.

PARM=CHK

Alias for PARM=CHECK.

PARM=SIZE=n

The amount of memory requested in four-byte words may be set with this entry. The default value for n is 20000000. This value only affects calculations in BONAMIST, where this value of the SIZE parameter is used for allocation of storage for the derivatives. Please see the documentation on BONAMIST in the Sensitivity Utility Modules chapter for more details. All other codes use dynamic memory allocation and this value has no effect.

PARM=BONAMIST

This is the default configuration for MG TSUNAMI-1D calculations. XSProc with BONAMI and CENTRM is used for cross-section processing, and implicit sensitivities are produced with BONAMIST.

PARM=CENTRM

XSProc with BONAMI and CENTRM is used for cross-section processing, but BONAMIST is not run. TSUNAMI-1D sequence with PARM=CENTRM does not produce the implicit portions of the sensitivity coefficients, and should be used with caution.

PARM=BONAMI

XSProc with BONAMI is used for cross-section processing, but BONAMIST is not run. TSUNAMI-1D sequence with PARM=BONAMI does not produce the implicit portions of the sensitivity coefficients, and should be used with caution.

PARM=2REGION

XSProc with BONAMI and CENTRM are run where Dancoff factors are to compute the escape probabilities for an accelerated, yet more approximate, CENTRM calculation. Implicit sensitivities are computed with BONAMIST.

Multiple parameters can be used simultaneously by enclosing them in parentheses and separating them with commas such as PARM=(SIZE=2000000, CHECK).

Multiple parameters can be used simultaneously by enclosing them in parentheses and separating them with commas such as PARM=(SIZE=2000000, CHECK).

6.1.4.2. XSProc

XSProc reads the standard composition specification data and the unit cell geometry specifications. It then produces the mixing table and unit cell information necessary for processing the cross sections. The XSProc chapter provides a detailed description of the input data for the Material Information Processor.

6.1.4.3. Model problem data

The model problem data are used by the TSUNAMI-1D sequences to prepare input for the XSDRNPM transport calculation. This input section consists of two data blocks, one block contains a geometry description and one contains optional parameters.

6.1.4.3.1. Geometry data

The TSUNAMI-1D geometry data block begins with the keywords READ GEOM and ends with the keywords END GEOM. This data block is always required. The following data is contained within this data block:

1. A line containing the geometry and boundary conditions for the XSDRNPM criticality case. The first entry on this line describes the geometry and must be SLAB, CYLINDRICAL, or SPHERICAL. The second entry is optional and describes the right-boundary condition. The default value for the right-boundary condition is VACUUM. The third entry on this line is optional and describes the left-boundary condition. The default value for the left boundary condition is REFLECTED. The last entry on this line is END. Valid entries for the boundary conditions are the following:

VACUUM – No return at boundary

REFLECTED – Specular (mirror-like) return at boundary

PERIODIC – Infinite array of cells in slab geometry

WHITE – Isotropic return at boundary

2. A line containing the following two entries for each zone of the XSDRNPM case:

   1. mixture number in the zone and

   2. zone outer dimension (in cm).

Mixture numbers and zone dimensions are entered in pairs until the entire geometry is defined. The mixture numbers must be defined in the material input processor input. Mixture 0 is used for voids, and a mixture number defined with CELLMIX= in the MIP section of the input may be used here. It should be noted that, due to a restriction in XSDRNPM, the mixture number identified with CELLMIX= may not appear in the output file, even though it is input in this section. TSUNAMI-1D automatically renumbers the cell mixed mixture to the next available mixture number for use in XSDRNPM. A message is printed in the output identifying this change. TSUNAMI-1D uses the same techniques as CSAS1X to automatically prepare a spatial mesh appropriate for the input materials and dimensions.

6.1.4.3.2. Parameter data

An optional data block may be entered to change parameters of the XSDRNPM forward and adjoint calculations. This data block begins with the keywords READ PARA or READ PARM and must end with either END PARA or END PARM, corresponding to the read keyword. In this data block, the user may enter optional lines that contain entries for selected XSDRNPM input parameters. A list of the parameters and their default values are found in Table 6.1.2.

Table 6.1.2 Optional parameter input for the criticality problem data.

Name	Default	Meaning
ISN=	16	Order of angular quadrature
IIM=	20	Inner-iteration maximum
ICM=	100	Outer-iteration maximum
ID1=	-1	Flux-edit option:
		-1 no flux print
		0 scalar flux print
		1 scalar and angular flux print
SCT=	5	Order of Legendre expansion for cross sections
PRT=	-2	Cross-section print option:
		-2 no cross-section print
		-1 print 1-D cross sections
		0/N print 2-D cross sections through order N
PBT=	0	Balance table print option:
		-1 no balance table print
		0 fine group balance table print
EPS=	1.E-6	Outer-iteration convergence criteria
PTC=	1.E-6	Inner-iteration convergence criteria
DY=	0	First-transverse dimension (cm) for buckling correction (i.e., height of cylinder or slab)
DZ=	0	Second-transverse dimension (cm) for buckling correction (i.e., depth of slab)
SZF=	1.5	Size factor of spatial computational mesh intervals. Increasing this number will cause the forward and adjoint XSDRNPM calculations to be conducted with larger mesh intervals and fewer mesh points. 0.0<SZF<1.5 gives a finer mesh, SZF>1.5 gives a coarser mesh.

6.1.4.4. Sensitivity and uncertainty calculation data

The data blocks for controlling the sensitivity and uncertainty calculation are optional. The optional data blocks include the SAMS block, the HTML block, the COVARIANCE block, the DEFINITIONS block, and the SYSTEMRESPONSES block. These data blocks begin with the keywords READ BLOCKNAME and end with the keywords END BLOCKNAME, where BLOCKNAME is one of SAMS, HTML, COVARIANCE, DEFINITONS, or SYSTEMRESPONSES. These data blocks can be input in any order with the following two exceptions. First, all five data blocks must appear in the input file after the composition and cell data blocks of data. Second, if a SAMS block is specified, the HTML and COVARIANCE data blocks must come after the SAMS block, if they are to be specified. In addition, both the DEFINITONS and SYSTEMRESPONSES data blocks must be present for additional analysis of system responses other than k_eff. If only one or both of the data blocks are omitted, then analysis is only performed for k_eff. The following sub-sections describe these blocks of data in detail.

6.1.4.4.1. Response definition data

The DEFINITIONS and SYSTEMRESPONSES blocks are used to define system responses for additional sensitivities and uncertainty analysis in SAMS. For criticality calculations, the sensitivities of system responses other than k_eff are calculated in TSUNAMI-1D using generalized perturbation theory (GPT). The details of the GPT methodology are provided in General Perturbation Theory section of the SAMS chapter. Using GPT, a system response, denoted R, is defined as a ratio such as:

(6.1.1)\[R=\frac{\sum_{g} \int d \bar{r} H_{N, g}(\bar{r}) \phi_{g}(\bar{r})}{\sum_{g} \int d \bar{r} H_{D, g}(\bar{r}) \phi_{g}(\bar{r})}\]

In this equation, \(\phi_{g}(\bar{r})\) is the space-dependent multi-group scalar flux and \(H_{N, g}(\bar{r})\),\(H_{D, g}(\bar{r})\) are referred to as the space-dependent, multi-group response functions. In TSUNAMI-1D, the response functions are specified in the DEFINITIONS data block and the system responses are defined in the SYSTEMRESPONSES data block. Responses (other k_eff) treated in TSUNAMI-1D MUST be ratios.

The DEFINITIONS data block is used by TSUNAMI-1D similarly to that of the MAVRIC and MONACO modules in SCALE. The format of the DEFINITIONS block is as follows:

read definitions
  response I1
    (specifications for response I1)
  end response
  response I2
    (specifications for response I2)
  end response
  ...
end definitions

The DEFINITIONS block of data begins with READ DEFINITIONS and terminates with END DEFINITIONS. Likewise, each response function definition begins with RESPONSE - followed by a unique, positive integer identifier - and terminates with END RESPONSE. The keyword entries summarized in Table 6.1.3 are allowed for each response specification. Keywords ending with = must be followed by the value to be assigned to the corresponding variable. All keywords are optional and can be entered in any order. However certain keywords are required depending one of the seven basic types of response functions described in the following subsections. The required keywords are summarized for each of the seven basic response function types in Table 6.1.3 at the end of this section.

Table 6.1.3 Response function keywords in DEFINITIONS block.

Keyword	Type	Default value	Description
title=	String	” “	Response function title. The title must begin and end with quotes and have a maximum of 256 characters.
macro	Logical	F	Macroscopic cross-section flag. If specified, macroscopic cross-sections are used to define the response function.
micro	Logical	T	Microscopic cross-section flag. If specified, microscopic cross-sections are used to define the response function.
nuclide= or zaid=	Integer or string	Undefined	Nuclide identifier for which cross-sections are used to define the response function. The nuclide can be specified in integer format (92235) or in character string format (u-235).
reaction= or mt=	Integer or string	Undefined	Reaction identifier for which cross-sections are used to define the response function. The reaction can be specified as an MT number (18) or as a character string (fission). Supported reaction types are listed below.
material= or mixture=	Integer	Undefined	Mixture identifier for which cross-sections are used to define the response function.
multimix ... end or multimat ... end	Integer array	Undefined	Array of mixture identifiers for which cross-sections cross-sections are used to define the response function.
unity	Logical	F	Flux response function flag. If specified, cross-sections are not used to define the response function.
multiplier	Real	1.0	Response function multiplier.
ehigh=	Real	1025	Upper energy (eV) boundary of the response function.
elow=	Real	0.0	Lower energy (eV) boundary of the response function.
ehightransfer=	Real	1025	Upper energy (eV) boundary used for cross-sections with secondary particle distributions (elastic, inelastic, scatter, and n,2n).
elowtransfer=	Real	0.0	Lower energy (eV) boundary used for cross-sections with secondary particle distributions (elastic, inelastic, scatter, and n,2n).

6.1.4.4.1.1. Single-mixture flux response function

A single-mixture flux response is simply the integration of the neutron flux wherever a specified mixture is defined in the problem geometry. Therefore, the response function \(H_{g}(\bar{r})\) for a single mixture-j is defined as:

(6.1.2)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})\]

where

(6.1.3)\[\begin{split}\begin{aligned} &\delta_{g}=\left\{\begin{array}{cc} 1.0 & E_{\text {High}}>E_{g}^{\text {Lower}}, E_{\text {Low}}<E_{g}^{U \text {pper}} \\ 0 & \text {otherwise} \end{array}\right.\\ &\text { and }\\ &\delta_{j}(\bar{r})=\left\{\begin{array}{cc} 1.0 & \text { mixture jis used at } \bar{r} \\ 0 & \text { otherwise } \end{array}\right. \end{aligned}\end{split}\]

In this expression, the constant c is the scalar multiplier defined by the multiplier= keyword. For example, the “fast” and “thermal” flux responses for mixture 5 would be:

In this example, the energy cutoff between the fast group and the thermal group is defined as 0.625 eV. Response 3 reflects the total energy-integrated flux-scaled by a factor of 2.0-because the default values of eHigh and eLow are used.

For single-mixture flux responses, keywords unity and mixture are required; multiplier, eHigh, and eLow are optional; title, nuclide, reaction, micro, macro, eHighTransfer, and eLowTransfer are optional but are not used; and multimix is not allowed. These keyword dependencies are outlined for each response type in Table 6.1.4.

6.1.4.4.1.2. Multiple-mixture flux response

A multiple-mixture flux response is the integration of the neutron flux wherever a set of mixtures are defined in the problem geometry. Therefore, the response function \(H_{g}(\bar{r})\) for a set of mixtures “S” is defined as:

(6.1.4)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \sum_{j \in S} \delta_{j}(\bar{r})\]

For example, the following definition is for the energy-integrated flux response, and spatially-integrated wherever mixtures 5, 7, and 9 are used:

read definitions
  response 1 unity multimix 5 7 9 end end response
end definitions

or alternatively,

read definitions
  response 1 multimat 5 7 9 end unity end response
end definitions

For multiple-mixture flux responses, keywords unity and multimix are required; multiplier, eHigh, and eLow are optional; title, nuclide, reaction, micro, macro, eHighTransfer, and eLowTransfer are optional but are not used; and mixture is not allowed.

6.1.4.4.1.3. Single-mixture, single-nuclide, microscopic cross-section response

A single-mixture, single-nuclide, microscopic cross-section response is the integration of the neutron flux multiplied by a microscopic cross-section. The microscopic cross-section used in the integral is defined by a specific mixture, nuclide, and reaction type. Therefore, the response function \(H_{g}(\bar{r})\) is defined as:

(6.1.5)\[H_{g}(\bar{r})=c^{*} \delta_{g}^{*} \delta_{j}(\bar{r})^{*} \sigma_{x, g}^{j, n}\]

In this expression, \(\sigma_{x, g}^{j, n}\) is the microscopic cross-section for mixture-j, nuclide-n, reaction type-x, and energy group-g. For transfer reaction types-scatter, elastic, inelastic, and n,2n-the expression above is slightly modified so the user can define the energy range of the secondary particles, i.e.,

(6.1.6)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})^{*} \sum_{g^{\prime}} \delta_{g^{\prime}}^{T r a n s f e r} \sigma_{x, g \rightarrow g^{\prime}}^{j, n}\]

where

(6.1.7)\[\begin{split}\delta_{g}^{\text {Transfer}}=\left\{\begin{array}{cc} 1.0 & E_{\text {HighTranser}}>E_{g}^{\text {Lower}}, E_{\text {LowTransfer}}<E_{g}^{U \text {pper}} \\ 0 & \text {otherwise} \end{array}\right.\end{split}\]

Likewise, the expression for \(H_{g}(\bar{r})\) is also modified for fission distribution responses (chi), which are usually integrated by the energy-integrated neutron production rate rather than the neutron flux:

(6.1.8)\[H_{g}(\bar{r})=c^{*} \delta_{j}(\bar{r})^{*} \bar{v}_{f, g}^{j, n} * \sigma_{f, g}^{j, n} * \sum_{g^{\prime}} \delta_{g^{\prime}}^{*} \chi_{g^{\prime}}^{j, n}\]

For examples of this response type, the following DEFINITION block has response definitions for

• total nu-fission rate of U-235 in mixture 1,

• “fast” n,gamma capture rate of U-238 in mixture 1 (energy cutoff is 0.625 eV),

• downscatter rate of H-1 in mixture 2, and

• number fission neutrons born from Pu-239 fissions in mixture 1 in the intermediate energy range (E>0.625 eV and E<25 keV)

read definitions
  response 1
    reaction=nu-fission mixture=1 nuclide=92235
  end response
  response 2
    reaction=n,gamma mixture=1 nuclide=u-238 eLow=0.625
  end response
  response 3
    mt=0   mixture=2 zaid=1001  eLow=0.625 eHighTransfer=0.635
  end response
  response 4
    mt=chi mixture=1 zaid=pu-239 eHigh=25.0e3 eLow=0.625
  end response
end definitions

For single-mixture, single-nuclide microscopic cross-section responses, keywords mixture, nuclide, and reaction are required; multiplier, eHigh, eLow, eHighTransfer, eLowTransfer, and micro, are optional; title is optional but not used; and multimix, macro, and unity are not allowed. A list of supported cross-section reaction types is provided at the end of this section in Table 6.1.5.

6.1.4.4.1.4. Single-mixture, single-nuclide, macroscopic cross-section response

A single-mixture, single-nuclide, macroscopic cross-section response is the integration of the neutron flux multiplied by a macroscopic cross-section. The macroscopic cross-section used in the integral is defined by a specific mixture, nuclide, and reaction type. The response function \(H_{g}(\bar{r})\) is defined as:

(6.1.9)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j, n}\]

In this expression, \(\Sigma_{x, g}^{j, n}\) is the macroscopic cross-section (\(N^{j,n} * \sigma^{j,n}_{x,g}\)) for mixture-j, nuclide-n, reaction type-x, and energy group-g. The modifications to this expression for transfer reactions and chi are similar to that of single-mixture, single-nuclide, microscopic cross-section responses. Using the same example as above, the single-mixture, single-nuclide, macroscopic cross-section responses are given as:

read definitions
  response 1
    reaction=nu-fission mixture=1 nuclide=92235 macro
  end response
  response 2
    reaction=n,gamma mixture=1 nuclide=u-238 eLow=0.625 macro
  end response
  response 3
    mt=0   mixture=2 zaid=1001  eLow=0.625 eHighTransfer=0.635 macro
  end response
  response 4
    mt=chi mixture=1 zaid=pu-239 eHigh=25.0e3 eLow=0.625 macro
  end response
end definitions

For single-mixture, single-nuclide macroscopic cross-section responses, keywords mixture, nuclide, macro, and reaction are required; multiplier, eHigh, eLow, eHighTransfer, and eLowTransfer, are optional; title is optional but not used; and multimix, micro, and unity are not allowed.

6.1.4.4.1.5. Single-mixture, multiple-nuclide, macroscopic cross-section response

A single-mixture, multiple-nuclide, macroscopic cross-section response is the integration of the neutron flux multiplied by a macroscopic cross-section. The macroscopic cross-section used in the integral is defined by a specific mixture, and reaction type. The response function \(H_{g}(\bar{r})\) is defined as:

(6.1.10)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j}\]

In this expression, \(\Sigma_{x, g}^{j}\) is the mixture macroscopic cross-section defined as \(\sum_{n}N^{j,n} * \sigma^{j,n}_{x,g}\) for mixture-j, reaction type-x, and energy group-g. The modifications to this expression for transfer reactions is similar to that defined in previous subsections. For mixture chi responses, \(H_{g}(\bar{r})\) is rewritten as

(6.1.11)\[H_{g}(\bar{r})=c^{*} \delta_{j}(\bar{r})^{*} \sum_{n} \bar{v}_{f, g}^{j, n} * \Sigma_{f, g}^{j, n} * \sum_{g^{\prime}} \delta_{g^{\prime}} * \chi_{g^{\prime}}^{j, n}\]

For examples of this response type, the following DEFINITIONS block has response definitions for

• total nu-fission rate in mixture 1,

• “fast” n,gamma capture rate in mixture 1 (energy cutoff is 0.625 eV),

• downscatter rate in mixture 2, and

• number fission neutrons born in mixture 1 in the intermediate energy range (E>0.625 eV and E<25 keV)

read definitions
  response 1
    reaction=nu-fission mixture=1 macro
  end response
  response 2
    reaction=n,gamma mixture=1 eLow=0.625 macro
  end response
  response 3
    mt=0   mixture=2 eLow=0.625 eHighTransfer=0.635 macro
  end response
  response 4
    mt=chi mixture=1 eHigh=25.0e3 eLow=0.625 macro
  end response
end definitions

For single-mixture, multiple-nuclide macroscopic cross-section responses, keywords mixture, macro, and reaction are required; multiplier, eHigh, eLow, eHighTransfer, and eLowTransfer, are optional; title is optional but not used; and multimix, micro, nuclide, and unity are not allowed.

6.1.4.4.1.6. Multiple-mixture, single-nuclide, macroscopic cross-section response

A multiple-mixture, single-nuclide, macroscopic cross-section response is the integration of the neutron flux multiplied by a macroscopic cross-section over a set of mixtures defined in the problem geometry. The macroscopic cross-section used in the integral is defined by a specific mixture, nuclide, and reaction type. The response function \(H_{g}(\bar{r})\) is defined as:

(6.1.12)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \sum_{j \in S} \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j, n}\]

In this expression, \(\Sigma_{x, g}^{j, n}\) is the macroscopic cross-section (\(N^{j,n} * \sigma^{j,n}_{x,g}\)) for mixture-j, nuclide-n, reaction type-x, and energy group-g. The summation of mixtures in this expression is for a set of user-defined mixtures, denoted S. The modifications to this expression for transfer reactions and chi are applied similarly to previously defined response types above.

For examples of this response type, the following DEFINITIONS block has response definitions for

• total nu-fission rate of U-235 in the fuel mixtures (mixtures 1,3,5)

• “fast” n,gamma capture rate of U-238 in the fuel mixtures

• downscatter rate of H-1 in the moderator mixtures (mixtures 2,4)

• number fission neutrons born in the intermediate energy range (E>0.625 eV and E<25 keV) in Pu-239 in the fuel mixtures

read definitions
  response 1
    reaction=nu-fission multimix 1 3 5 end macro zaid=92235
  end response
  response 2
    reaction=n,gamma multimix 1 3 5 eLow=0.625 macro zaid=u-238
  end response
  response 3
    mt=0   multimix 2 4 end eLow=0.625 eHighTransfer=0.635 macro zaid=h-1
  end response
  response 4
    mt=chi multimix 1 3 5 end eHigh=25.0e3 eLow=0.625 macro zaid=pu-239
  end response
end definitions

For multiple-mixture, single-nuclide macroscopic cross-section responses, keywords multimix, nuclide, macro, and reaction are required; multiplier, eHigh, eLow, eHighTransfer, and eLowTransfer, are optional; title is optional but not used; and mixture, micro, and unity are not allowed.

6.1.4.4.1.7. Multiple-mixture, multiple-nuclide, macroscopic cross-section response

A multiple-mixture, multiple-nuclide, macroscopic cross-section response is the integration of the neutron flux multiplied by a macroscopic cross-section over a set of mixtures defined in the problem geometry. The macroscopic cross-section used in the integral is defined by a specific mixture, and reaction type. The response function \(H_{g}(\bar{r})\) is defined as:

(6.1.13)\[H_{g}(\bar{r})=c^{*} \delta_{g} * \sum_{j \in S} \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j}\]

In this expression, \(\Sigma_{x, g}^{j}\) is the mixture macroscopic cross-section for mixture-j and reaction type-x, and energy group-g. The summation of mixtures in this expression is for a set of user-defined mixtures, denoted S. The modifications to this expression for transfer reactions and chi are applied similarly to the previously defined response types above.

For examples of this response type, the following DEFINITIONS block has response definitions for

• total nu-fission rate in the fuel mixtures (mixtures 1,3,5)

• “fast” n,gamma capture rate in the fuel mixtures

• downscatter rate in the moderator mixtures (mixtures 2,4)

• number fission neutrons born in the intermediate energy range (E>0.625 eV and E<25 keV) in the fuel mixtures

read definitions
  response 1
    reaction=nu-fission multimix 1 3 5 end macro
  end response
  response 2
    reaction=n,gamma multimix 1 3 5 eLow=0.625 macro
  end response
  response 3
    mt=0   multimix 2 4 end eLow=0.625 eHighTransfer=0.635 macro
  end response
  response 4
    mt=chi multimix 1 3 5 end eHigh=25.0e3 eLow=0.625 macro
  end response
end definitions

For multiple-mixture, multiple-nuclide macroscopic cross-section responses, keywords multimix, macro, and reaction are required; multiplier, eHigh, eLow`, ``eHighTransfer, and eLowTransfer, are optional; title is optional but not used; and mixture, micro, nuclide, and unity are not allowed.

Table 6.1.4 Keyword dependencies for the DEFINITIONS block.

Response type	Required keywords	Unallowed keywords	Optional keywords	Optional, but not used keywords
Single-mixture flux	unity, mixture	multimix	multiplier, eHigh, eLow	title, nuclide, reaction, micro, macro, eHighTransfer, eLowTransfer
Multiple-mixture flux	unity, multimix	mixture	multiplier, eHigh, eLow	title, nuclide, reaction, micro, macro, eHighTransfer, eLowTransfer
Single-mixture, single-nuclide, microscopic cross-section	mixture, nuclide, reaction	unity, macro, multimix	multiplier, eHigh, eLow, micro, eHighTransfera , eLowTransfera	title
Single-mixture, single-nuclide, macroscopic cross-section	mixture, nuclide, reaction,macro	unity, micro, multimix	multiplier, eHigh, eLow, eHighTransfera , eLowTransfera	title
Single-mixture, multiple-nuclide, macroscopiccross-section	mixture, reaction,macro	unity, micro, multimix, nuclide	multiplier, eHigh, eLow, eHighTransfera , eLowTransfera	title
Multiple-mixture, single-nuclide, macroscopic cross-section	multimix, nuclide reaction,macro	unity, micro, mixture	multiplier, eHigh, eLow, eHighTransfera , eLowTransfera	title
Multiple-mixture, multiple-nuclide, macroscopic cross-section	multimix, reaction,macro	unity, micro, mixture, nuclide	multiplier, eHigh, eLow, eHighTransfera , eLowTransfera	title
aKeywords eHighTransfer and eLowTransfer are only used for the following reaction types: scatter (mt=0), elastic (mt=2), inelastic (mt=4), and n,2n (mt=16) For all other reaction types, these keywords are optional, but not used

Table 6.1.5 Supported Reaction Types in DEFINITIONS block.

MT	Reaction	String Identifier
1	total	Total
2	elastic scattering	Elastic
4	inelastic scattering	Inelastic
16a	effective n,2n	n,2n
0	sum of scattering (2+4+16)	Scatter
18	fission	Fission
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
101	Neutron disappearance (102+103+104+105+106+ 107)	capture
452	\(\bar{\nu}\)	nubar
1452	\(\bar{\nu}\) times fission	nu-fission
1018	\(\chi\)	chi
aThe effective n,2n is defined by the summation of transfer matrices of the following reaction types: (n,2n), (n,2n+\(\alpha\)), (n,2n+2\(\alpha\)), (n,3n), (n,3n+\(\alpha\)), and (n,4n). The individual transfer matrices are scaled by the number of exit channel neutrons, i.e., 2, 3, or 4.

6.1.4.4.2. System response definition data

The SYSTEMRESPONSES block is used to define the set of system responses for which TSUNAMI-1D will perform sensitivity and uncertainty analysis additional to k_eff. For SCALE 6.1, only system response ratios are supported in TSUNAMI-1D. The system response ratios are defined from the response function definitions created in the DEFINITIONS block. The format of the SYSTEMRESPONSES block is as follows:

read systemresponses
  ratio I1
    (specifications for response ratio I1)
  end ratio
  ratio I2
    (specifications for response ratio I2)
  end ratio
  ...
end systemresponses

The SYSTEMRESPONSES block of data begins with READ SYSTEMRESPONSES and terminates with END SYSTEMRESPONSES. Likewise, each system response ratio definition begins with RATIO - followed by a unique, positive integer identifier - and terminates with END RATIO. For each response ratio definition, the keywords title=, numer, and denom are allowed in any order. The title= specification is optional. However, if specified, the title must be begin and end with quotes and have a maximum of 20 characters. If omitted, the title of the ratio is “rsp ratio NNNNNNNNNN” where NNNNNNNNNN is a zero-padded 10-digit integer that is equal to the ratio identifier. The title is used as labels in both the TSUNAMI-1D text and html output. The title is also used by SAMS to generate the filename for the sensitivity data file for the ratio system response discussed further below.

The numer array is a list of integers that correlate to response function identifiers defined in the DEFINITIONS block. These response functions are added together to form the composite response function used in the numerator of the ratio. Likewise, the denom array is a list of integers that correlate to response function identifiers defined in the DEFINITIONS block. These response functions are added together to form the composite response function used in the denominator of the response ratio. Multiple response function ratios can be defined in a single input file.

For a simple example of the SYSTEMRESPONSES block, suppose the ratio system response of interest is the resonance escape probability for a given system. Using 2-group theory, this is equivalent to the following expression:

(6.1.14)\[p=\frac{\left\langle\Sigma_{s, 1 \rightarrow 2}\right\rangle}{\left\langle\Sigma_{r, 1}\right\rangle}=\frac{\int d \bar{r} \sum_{g \in 1} \phi_{g}(\bar{r}) \sum_{g^{\prime} \in 2} \Sigma_{s, g \rightarrow g^{\prime}}(\bar{r})}{\int d \bar{r} \sum_{g \in 1} \phi_{g}(\bar{r}) \Sigma_{r, g}(\bar{r})}\]

where \(\Sigma_{r, g}(\bar{r})\) is the removal cross-section defined as the total cross-section minus the within group cross-section — \(\Sigma_{t, g}(\bar{r})-\Sigma_{s, g \rightarrow g}(\bar{r})\). The TSUNAMI-1D model uses three mixtures whose ids are 6, 7, and 10. The thermal energy cutoff is 0.625 eV.

This ratio can be defined in multiple ways. First, the ratio can be defined with three response function definitions:

read definitions
  response 1 title="DownScatter"
     reaction=scatter
     multimix 6 7 10 end
     macro
     eLow=0.625 eHighTransfer=0.625
  end response
  response 2 title="Fast Total"
     reaction=total
     multimix 6 7 10 end
     macro
     eLow=0.625
  end response
  response 3 title="Fast Within Group (times -1)"
     reaction=scatter
     multimix 6 7 10 end
     macro
     eLow=0.625 eLowTransfer=0.625
     factor=-1.0
  end response
end definitions
read systemresponses
  ratio 100
     title="Res Escape"
     numer 1 end
     denom 2 3 end
  end ratio
end systemresponses

In the above input, the numerator of the response ratio is defined by a single response function (id=1), which represents the rate at which neutrons slow down from fast energies to slow energies. The denominator of the response ratio is defined by two response functions (id=2 and id=3). The addition of these two response functions represents the “total minus within group scattering” calculation to formulate the fast neutron removal rate. In this input, the title of the response ratio is set to “Res Escape”. Because only one response ratio is defined, TRITON will invoke SAMS twice, first for the k_eff sensitivity and uncertainty analysis and second for the analysis of the resonance escape probability. SAMS will generate two .sdf files, the first will be jobname.sdf for k_eff sensitivities and the second will be jobname.Res_Escape.sdf. jobname is the name of the input file. An underscore is used to replace blanks and special characters in the response ratio title in the sdf filename.

Similarly, the resonance escape probability can be defined in a variety of different ways. For example, the numerator response function can be expressed as the sum of individual mixture downscattering rates:

read definitions
  response 2 title="Fast Total"
     reaction=total
     multimix 6 7 10 end
     macro
     eLow=0.625
  end response
  response 3 title="Fast Within Group (times -1)"
     reaction=scatter
     multimix 6 7 10 end
     macro
     eLow=0.625 eLowTransfer=0.625
     factor=-1.0
  end response
  response  6 mt=0 mixture= 6 macro eLow=0.625 eHighTransfer=0.625
  end response
  response  7 mt=0 mixture= 7 macro eLow=0.625 eHighTransfer=0.625
  end response
  response 10 mt=0 mixture=10 macro eLow=0.625 eHighTransfer=0.625
  end response
end definitions
read systemresponses
  ratio 100
    numer 6 7 10 end
    denom 2 3        end
  end ratio
end systemresponses

In this input, the numerator of the response ratio is defined by adding the individual mixture downscattering rates together. Because a title was not given for the response ratio, SAMS will generate the filename of the response ratio sdf file as jobname.rsp_ratio_0000000100.sdf.

6.1.4.4.3. SAMS data

The SAMS block is used for controlling certain aspects of the sensitivity and uncertainty calculation. This data block begins with the keywords READ SAMS and ends with the keywords END SAMS. Any of the optional SAMS input data may be entered in free form format between the READ SAMS and END SAMS keywords. This optional SAMS input data is shown in:

Table 6.1.6, with the default values specific to TSUNAMI-1D. Parameters used to specify default covariance data to supplement or correct values on the files specified by coverx= are shown in Table 6.1.7. A more detailed explanation of the SAMS parameters may be found in the SAMS chapter.

Table 6.1.6 SAMS input keywords

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
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.
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
pn=	3	Legendre order for moment calculations
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
prtvols	F	Flag to cause the volumes of the regions to be printed by SAMS
unconstrainedchi	F	Flag to generate pre-SCALE 6 unconstrained chi (fission spectrum) sensitivities

Table 6.1.7 SAMS input keywords for default covariance data.

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 in cross-section data to use for all groups for nuclide-reaction pairs for which cross-section-covariance data are too large or not available on input covariance data library.
udcov_corr=	1.0	User-defined default correlation value to use for nuclide-reaction pairs for which cross-section-covariance data are not available on the input covariance library.
udcov_corr_type=	zone	User-defined default correlation to use for nuclide-reaction pairs for which cross-section-covariance data are not available on the input covariance library. Allowed values are long, zone, and short. See the table Input Data for Covariance Block of TSAR Input in the TSAR chapter for details on long, zone, and short.
udcov_therm=	0.0	User-defined default value of standard deviation in cross-section data to use for thermal data for nuclide-reaction pairs for which cross-section-covariance data are too large or not available on input covariance data library. If input, the udcov_therm overrides the udcov value in the thermal range (i.e. neutron energies below 0.625 eV).
udcov_inter=	0.0	User-defined default value of standard deviation in cross-section data to use for intermediate data for nuclide-reaction pairs for which cross-section-covariance data are too large or not available on input covariance data library. If input, the udcov_inter overrides the udcov value in the intermediate range (i.e. neutron energies above 0.625 eV and below 25 keV).
udcov_fast=	0.0	User-defined default value of standard deviation in cross-section data to use for fast data for nuclide-reaction pairs for which cross-section-covariance data are too large or not available on input covariance data library. If input, the udcov_fast overrides the udcov value in the fast range (i.e. neutron energies above 25 keV).

6.1.4.4.4. HTML and user-input covariance data

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 User Input Covariance Data of the TSUNAMI Utility Modules chapter.

As the SAMS module generates HTML output, the optional HTML data block will provides 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 TSUNAMI Utility Modules manual.

6.1.4.5. Input termination

The input specification for all TSUNAMI-1D sequences must terminate with a line containing END in column 1. This END terminates the control sequence.

6.1.5. Example Problems

Nine TSUNAMI-1D sample problems are included in the SCALE package to verify successful installation and to provide examples for users. They are provided in the smplprbs directory of the software distribution. Three example problems are presented in this section and comparisons among the different methods for cross-section processing are discussed. The first problem presented is a variant of the TSUNAMI-1D1 k_eff sensitivity sample problem with some addition input parameters in the READ SAMS data block and using INFHOMMEDIUM unit cell type. The second example problem presented in this section generates k_eff sensitivities using the MULTIREGION unit cell type. The third example problem is similar to the TSUNAMI-1D5 sample problem that demonstrates the GPT capabilities. The five sample problems in the software package are designed to run quickly and test most code features. The three examples presented here are designed to produce accurate results, but may require more computational resources.

For all problems the validity of the sensitivity coefficients should be confirmed through the use of direct perturbation sensitivity calculations. For each sensitivity coefficient examined by direct perturbation, the k_eff of the system is computed first with the nominal values of the input quantities, then with a selected input value increased by a certain percentage, and then with the value decreased by the same percentage. The direct perturbation sensitivity coefficient of k_eff to some input value \(\alpha\) is computed as

(6.1.15)\[S_{k, \alpha}=\frac{\alpha}{k} \times \frac{d k}{d \alpha}=\frac{\alpha}{k} \times \frac{k_{\alpha^{+}}-k_{\alpha^{-}}}{\alpha^{+}-\alpha^{-}} ,\]

where\(\alpha^{+}\) and \(\alpha^{-}\) represent the increased and decreased values, respectively, of the input quantity \(\alpha\) and \(k_{\alpha^{+}}\) and \(k_{\alpha^{-}}\) represent the corresponding values of k_eff.

The use of direct perturbation calculations to confirm the validity of sensitivity coefficients is strongly encouraged. Inconsistent modeling between the resonance-self shielding input and the criticality problem description can lead to erroneous sensitivity results. These erroneous results can be revealed through the use of direct perturbation confirmation of the energy-integrated sensitivity results for the total cross section. The total cross-section sensitivities are equivalent to number density sensitivities on an energy-integrated basis.

The results shown here were generated with a previous version of SCALE, so current data libraries and code implementations may product different results. However, the techniques demonstrated are applicable to the current version of TSUNAMI-1D.

6.1.5.1. INFHOMMEDIUM sample problem

The selected sample problem with INFHOMMEDIUM cross-section processing is based on an unreflected rectangular parallelepiped consisting of a homogeneous mixture of UF₄ and paraffin with an enrichment of 2% in ²³⁵U. The H/²³⁵U atomic ratio is 294:1. The dimensions of the experiment were 56.22 cm \(\times\) 56.22 cm \(\times\) 122.47 cm. [TS-1D-CLP86]. For the purposes of this exercise, this experiment was modeled as a sphere with a critical radius of 38.50 cm. This model is consistent with SCALE sample problem TSUNAMI-1D1, which utilizes the 238-group ENDF/B-VII cross-section library, and the default cross-section processing with BONAMIST and CENTRM/PMC/WORKER.

An annotated TSUNAMI-1D1 input for this experiment is shown in Sect. 6.1.3.1. The composition data is input as number densities for each nuclide. Because the material is treated as INFHOMMEDIUM, no explicit unit cell model is necessary, and the READ CELL block is omitted. The criticality description contains optional parameter data to change the default S₁₆ angular quadrature set to S₈. The change in angular quadrature is made only to demonstrate the input capabilities of TSUNAMI-1D and has little effect on this calculation. The criticality problem geometry uses a spherical coordinate system with the default boundary conditions (reflected left, vacuum right). The system consists of a single material zone containing mixture 1 with a radius of 38.50 cm. The optional sensitivity calculation data block was entered to request the extended edit of sensitivity by material zone (prtgeom), the extended edits of the explicit, implicit and complete sensitivity coefficients (prtimp), and corrections in the cross-section covariance data (use_dcov, cov_fix).

Prior to producing the output of the functional modules, TSUNAMI-1D produces output from the XSProc routines as it is processing the user input and creating internal inputs for the resonance processing codes. TSUNAMI-1D also prints information regarding the criticality description.

Fig. 6.1.2 TSUNAMI-1D INFHOMMEDIUM sample problem input.

For this problem, direct perturbation results were obtained for the number densities of each nuclide. In these calculations, the number density of each nuclide was perturbed by \(\pm2\%\) and the calculation was repeated using the TSUNAMI-1DC sequence. The sensitivity of k_eff to the number density is equivalent to the sensitivity of k_eff to the total cross section, integrated over energy. The direct perturbation sensitivity coefficients were computed by using the k_eff values from the unperturbed and perturbed cases in Eq. (6.1.15). To demonstrate the importance of the sensitivity to the resonance processing implicit sensitivity computed by BONAMIST, the same model shown in Fig. 6.1.2 was run with TSUNAMI-1D with PARM=CENTRM. The results from the INFHOMMEDIUM sample problem are given in Table 6.1.8. The TSUNAMI-1D results using the default codes for resonance processing show good agreement with the direct perturbation results for all nuclides. Due to omission of the implicit terms, the TSUNAMI-1D results with PARM=CENTRM do not show good agreement with the direct perturbation for this thermal system. The maximum difference between the direct perturbation results and the TSUNAMI-1D results occurs for ²³⁸U with a magnitude of 1.5%. The maximum difference between the direct perturbation results and the TSUNAMI-1D with PARM=CENTRM results occurs for ²³⁸U with a magnitude of 19%. Thus, the use of the default PARM=BONAMIST is recommended.

Table 6.1.8 Energy- and region-integrated sensitivity coefficients from TSUNAMI-1D INFHOMMEDIUM sample problem.

Isotope	Reaction	Direct perturbation	TSUNAMI-1D	TSUNAMI-1D PARM=CENTRM**
1H	total	2.20E-01	2.18E-01	2.52E-01
1H	scatter		3.19E-01	3.53E-01
1H	elastic		3.19E-01	3.53E-01
1H	capture		-1.01E-01	-1.01E-01
1H	n, \(gamma\)		-1.01E-01	-1.01E-01
12C	total	2.41E-02	2.38E-02	2.76E-02
12C	scatter		2.45E-02	2.83E-02
12C	elastic		2.43E-02	2.80E-02
12C	n,n’		2.20E-04	2.20E-04
12C	capture		-6.83E-04	-6.83E-04
12C	n, \(gamma\)		-4.98E-04	-4.98E-04
12C	n,p		-3.53E-08	-3.53E-08
12C	n,d		-7.33E-08	-7.33E-08
12C	n, \(alpha\)		-1.85E-04	-1.85E-04
19F	total	4.10E-02	4.06E-02	4.47E-02
19F	scatter		4.62E-02	5.03E-02
19F	elastic		2.94E-02	3.34E-02
19F	n,n’		1.58E-02	1.58E-02
19F	n,2n		2.89E-06	2.89E-06
19F	capture		-5.59E-03	-5.59E-03
19F	n, \(gamma\)		-2.39E-03	-2.39E-03
19F	n,p		-2.37E-04	-2.37E-04
19F	n,d		-1.27E-05	-1.27E-05
19F	n,t		-2.72E-06	-2.72E-06
19F	n, \(alpha\)		-2.96E-03	-2.96E-03
235U	total	2.52E-01	2.52E-01	2.53E-01
235U	scatter		4.32E-04	5.03E-04
235U	elastic		2.02E-04	2.73E-04
235U	n,n’		2.13E-04	2.13E-04
235U	n,2n		1.70E-05	1.70E-05
235U	fission		3.64E-01	3.65E-01
235U	capture		-1.13E-01	-1.12E-01
235U	n, \(gamma\)		-1.13E-01	-1.12E-01
235U	nubar		9.50E-01	9.50E-01
235U	\(\chi\)		8.52E-08	8.52E-08
238U	total	-2.08E-01	-2.05E-01	-2.47E-01
238U	scatter		4.81E-02	2.46E-02
238U	elastic		3.46E-02	1.10E-02
238U	n,n’		1.25E-02	1.25E-02
238U	n,2n		1.02E-03	1.02E-03
238U	fission		3.35E-02	3.35E-02
238U	capture		-2.86E-01	-3.05E-01
238U	n, \(gamma\)		-2.86E-01	-3.05E-01
238U	nubar		5.02E-02	5.02E-02
238U	\(\chi\)		4.54E-09	4.54E-09

The uncertainty information from SAMS for the INFHOMMEDIUM sample problem is shown in Example 6.1.1. Based on the 44GROUPCOV covariance data library, documented in the COVLIB chapter, the uncertainty in k_eff due to these covariance data is 0.6064% \(\Delta k/k\). A more detailed description of the uncertainty information is given in Chapter 6.3. Some plots of the energy-dependent sensitivity data were generated with Fulcrum. The energy-dependent data is available in the sensitivity data file, which is returned to the same directory as the input file and given the same name as the user’s input file with the extension .sdf. Energy-dependent sensitivity profiles for ²³⁵U fission and ¹H elastic scattering are shown in Fig. 6.1.3. The ²³⁸U capture sensitivity profiles from TSUNAMI-1D and TSUNAMI-1D with PARM=CENTRM are shown in Fig. 6.1.4. The effect of the implicit component of the sensitivity coefficients can be seen in the resonance region in the difference between the TSUNAMI-1D and TSUNAMI-1D PARM=CENTRM profiles.

Example 6.1.1 INFHOMMEDIUM sample problem.

 -----------------------------
    Uncertainty Information
 -----------------------------

  the relative standard deviation of k-eff (% delta-k/k) due
  to cross-section covariance data is:

    0.6064 % 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-238 n,gamma                       u-238 n,gamma                 3.8595E-01
         u-235 nubar                         u-235 nubar                   2.8506E-01
         u-238 n,n'                          u-238 n,n'                    2.1331E-01
         u-235 n,gamma                       u-235 n,gamma                 1.5963E-01
          f-19 elastic                        f-19 elastic                 1.3392E-01
         u-238 elastic                       u-238 n,n'                   -1.2469E-01
         u-235 fission                       u-235 n,gamma                 1.2396E-01
         u-235 fission                       u-235 fission                 1.2185E-01
           h-1 elastic                         h-1 elastic                 1.1625E-01
          f-19 elastic                        f-19 n,n'                   -1.1598E-01
          f-19 n,n'                           f-19 n,n'                    1.1072E-01
         u-235 chi                           u-235 chi                     8.4524E-02
         u-238 elastic                       u-238 elastic                 6.8573E-02
         u-238 nubar                         u-238 nubar                   5.8699E-02
           h-1 n,gamma                         h-1 n,gamma                 5.0686E-02
         u-238 elastic                       u-238 n,gamma                 4.9596E-02
          f-19 n,alpha                        f-19 n,alpha                 1.9853E-02
         u-238 fission                       u-238 fission                 1.7402E-02
             c elastic                           c elastic                 1.5259E-02
         u-238 n,2n                          u-238 n,2n                    1.3655E-02
          f-19 n,gamma                        f-19 n,gamma                 9.7725E-03
             c n,n'                              c elastic                -8.8958E-03
             c n,n'                              c n,n'                    8.4710E-03
          f-19 elastic                        f-19 n,alpha                 6.6444E-03
         u-238 chi                           u-238 chi                     5.6329E-03
         u-235 elastic                       u-235 n,gamma                 4.4651E-03
         u-235 elastic                       u-235 fission                -3.2889E-03
         u-238 fission                       u-238 n,gamma                 2.7666E-03
          f-19 n,p                            f-19 n,p                     2.0768E-03
         u-238 elastic                       u-238 n,2n                   -1.8932E-03
         u-238 elastic                       u-238 fission                -1.8189E-03
             c n,alpha                           c n,alpha                 1.6172E-03
             c n,gamma                           c n,gamma                 1.4880E-03
         u-235 n,n'                          u-235 n,n'                    1.3414E-03
         u-235 elastic                       u-235 n,n'                   -8.6275E-04
          f-19 elastic                        f-19 n,p                     5.8397E-04
          f-19 elastic                        f-19 n,gamma                 4.5179E-04
         u-235 elastic                       u-235 elastic                 4.3646E-04
          f-19 n,d                            f-19 n,d                     2.8169E-04
         u-235 n,2n                          u-235 n,2n                    1.5476E-04
             c n,n'                              c n,alpha                -1.4865E-04
          f-19 elastic                        f-19 n,2n                   -7.0280E-05
          f-19 elastic                        f-19 n,d                     6.6324E-05
          f-19 n,t                            f-19 n,t                     6.5613E-05
         u-235 elastic                       u-235 n,2n                   -2.7763E-05
          f-19 n,2n                           f-19 n,2n                    2.2764E-05
          f-19 n,n'                           f-19 n,2n                   -1.9276E-05
          f-19 elastic                        f-19 n,t                     1.4593E-05
             c n,n'                              c n,gamma                 6.9724E-06
             c n,d                               c n,d                     8.5422E-07
             c n,p                               c n,p                     4.5780E-07
             c n,n'                              c n,d                    -3.2157E-07
             c n,n'                              c n,p                    -1.5591E-07

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

Fig. 6.1.3 Energy-dependent sensitivity profiles from TSUNAMI-1D for INFHOMMEDIUM sample problem.

Fig. 6.1.4 Comparison of 238U capture sensitivities from TSUNAMI-1D and TSUNAMI-1D with PARM=CENTRM for INFHOMMEDIUM sample problem.

6.1.5.2. Multiregion sample problem

The sample problem selected to demonstrate the use of TSUNAMI-1D with MULTIREGION cross-section processing is the FLATTOP-25 metal system from the Cross-Section Evaluation Working Group benchmark specifications. [TS-1D-AKPZ74]. 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. The system is used for sample problems TSUNAMI-1D4 – TSUNAMI-1D7. For this example, input for TSUNAMI-1D4 was modified to use the SCALE 238-group ENDF/B-VII library with multiregion cell data as shown in Fig. 6.1.5. The multiregion cell data processes the cross sections in the same geometry as the criticality model. Therefore, the dimensions of the system are input twice in this model: once in the unit cell specification portion of the input and once in the criticality portion of the input. The unit cell specification geometry is used to generate input for BONAMIST and CENTRM/PMC/WORKER, and the criticality model is used to generate input for the forward and adjoint XSDRNPM calculations. The optional sensitivity calculation data block was entered to request the extended edit of sensitivity by material zone (prtgeom), the extended edits of the explicit, implicit and complete sensitivity coefficients (prtimp), and to allow larger implicit sensitivity values to be computed without producing warning messages (largeimp=1000).

This model was executed with TSUNAMI-1D and also with TSUNAMI-1D with PARM=CENTRM. Direct perturbation sensitivity results were obtained for the number densities of all nuclides, which correspond to the sensitivity of k_eff to the total cross section, integrated over energy. The energy-integrated sensitivity results are shown in Table 6.1.9. The TSUNAMI-1D results agree well with the direct perturbation results for this system. The maximum difference occurs for ²³⁸U in the reflector region with a magnitude of 0.9%. Because this is a fast system, the effect of the resonance processing on the sensitivity coefficients is minimal. Thus, the TSUNAMI-1D PARM=CENTRM results are almost identical to the default TSUNAMI-1D results with BONAMIST.

Fig. 6.1.5 TSUNAMI-1D MULTIREGION sample problem input.

Table 6.1.9 Energy- and region-integrated sensitivity coefficients from TSUNAMI-1D MULTIREGION sample problem.

The uncertainty information from SAMS HTML output for the multiregion sample problem is shown in Fig. 6.1.6. Based on the 44GROUPCOV covariance data file, the uncertainty in k_eff due to these covariance data is 1.2743% \(\Delta k/k\). The contributions to this uncertainty are listed by nuclide. These data are explained in more detail in the SAMS chapter.

Sensitivity profiles from TSUNAMI-1D for ²³⁵U fission in zone 1 (core) and zone 2 (reflector) were generated with Fulcrum and are shown in Fig. 6.1.7. Additionally, sensitivity profiles for ²³⁸U capture in zone 1 and zone 2 are shown in Fig. 6.1.8. Note that the capture sensitivities are negative, such that the lower curve has the greater magnitude. In ²³⁵U and ²³⁸U sensitivity profiles, the effect of the differing enrichments in the core and the reflector of this system are demonstrated with the much greater sensitivity to ²³⁵U fission in the core and to ²³⁸U capture in the reflector.

Fig. 6.1.6 Uncertainty information in HTML output from MULTIREGION sample problem.

Fig. 6.1.7 Sensitivity profiles from TSUNAMI-1D for ²³⁵U fission in zone 1 and zone 2 of MULTIREGION sample problem.

Fig. 6.1.8 Sensitivity profiles from TSUNAMI-1D for ²³⁸U capture in zone 1 and zone 2 of MULTIREGION sample problem.

6.1.5.3. GPT sample problem

The sample problem selected to demonstrate the use of TSUNAMI-1D with Generalized Perturbation theory is from the OECD LWR Uncertainty Analysis in Modeling benchmark specification [TS-1D-IAKS07]. The system consists of a 4.85% enriched uranium PWR fuel pin modeled at 551 K. This system is used for sample problem TSUNAMI-1D9. For this example, the DEFINITIONS and SYSTEMRESPONSES blocks are used to define six additional response ratios for sensitivity and uncertainty analysis. The requested responses in the benchmark were for the energy-integrated fission and absorption microscopic cross-sections for ²³⁴U, ²³⁵U, and ²³⁸U. The input for this sample problem is provided in Example 6.1.2. For this sample, seven separate sensitivity data files are generated, one for each of the six defined responses in addition to k_eff. Selected sensitivity profiles are shown in Fig. 6.1.9 for the ²³⁸U (n,\(\gamma\)) cross-section. This figure shows the negative sensitivity of k_eff due to ²³⁸U resonance absorption in the blue profile. The red profile shows the positive sensitivity of the energy-integrated ²³⁸U absorption cross-section due to the multigroup ²³⁸U (n,\(\gamma\)) cross-section. The large positive magnitude of this sensitivity is predominantly due to the presence of the ²³⁸U (n,\(\gamma\)) cross-section directly in the definition of the response ratio. In contrast, the black sensitivity profile shows the negative sensitivity of the energy-integrated ²³⁵U fission cross-section due to the multigroup ²³⁸U (n,\(\gamma\)) cross-section. In this response, positive perturbations to the ²³⁸U (n,\(\gamma\))  multigroup cross-sections induce changes in the flux spectra that lead to a decrease in the energy-integrated ²³⁵U fission cross-section. These indirect sensitivity effects are determined by the solution of the generalized adjoint calculations.

Example 6.1.2 Input for TSUNAMI-1D9 sample problem.

=tsunami-1d
PWR Unit Cell
v7-238
read comp
'fuel
uo2   10 den=10.283  1 551.0 92235  4.85 92234 0.045 92238 95.105 end
zirc4 20             1 551.0 end
h2o   30 den=0.766   1 551.0 end
he    40 den=0.00125 1 551.0 end
end comp
read celldata
latticecell squarepitch pitch=1.4427 30 fueld=0.9391 10 cladd=1.0928 20 gapd=0.9582 40  end
end celldata
read geom
cylindrical white reflected end
10 .46955 40 .4791 20 .5464 30 .813956
end geom
read definitions
  response 1 nuclide=92234 mt=102 mixture=10 micro end response
  response 2 nuclide=92234 mt= 18 mixture=10 micro end response
  response 3 nuclide=92235 mt=102 mixture=10 micro end response
  response 4 nuclide=92235 mt= 18 mixture=10 micro end response
  response 5 nuclide=92238 mt=102 mixture=10 micro end response
  response 6 nuclide=92238 mt= 18 mixture=10 micro end response
  response 7 unity multimix 10 20 30 40 end end response
end definitions
read systemresponses
  ratio 1 numer 1 2 end denom 7 end title='U234-abs' end ratio
  ratio 2 numer   2 end denom 7 end title='U234-fis' end ratio
  ratio 3 numer 3 4 end denom 7 end title='U235-abs' end ratio
  ratio 4 numer   4 end denom 7 end title='U235-fis' end ratio
  ratio 5 numer 5 6 end denom 7 end title='U238-abs' end ratio
  ratio 6 numer   6 end denom 7 end title='U238-fis' end ratio
end systemresponses
end

Fig. 6.1.9 Response Sensitivities to ²³⁸U n,gamma cross section for the TSUNAMI-1D9 sample problem.

References

TS-1D-AKPZ74

H. Alter, R. Kidman, R. B. abd Labauv, R. Protsik, and B. A. Zolotar. Cross Section Evaluation Working Group Benchmark Specification. Technical Report BNL 19302, National Neutron Cross Section Center (ENDF-202), Brookhaven National Laboratory , Upton, NY (USA), 11 1974.

TS-1D-CLP86

Wilfred C., N. F. Landers, and L. M. Petrie. Validation of KENO V.a Comparison with Critical Experiments. Technical Report ORNL/CSD/TM-238, Oak Ridge National Laboratory, Oak Ridge, TN (USA), 12 1986.

TS-1D-IAKS07

Kalcko Ivanov, M. Avramova, Ivan Aleksander Kodeli, and E. Sartori. Benchmark for uncertainty analysis in modeling (UAM) for design, operation and safety analysis of LWRs. Citeseer, 2007.

6.1.6. Appendices

• 6.1.6.1. XSDRNPM Data File Formats
• 6.1.6.2. XSDRNPM forward output file
• 6.1.6.3. XSDRNPM adjoint output file