.. |keff| replace:: *k*\ :sub:`eff` .. _sec-tsunami.1d: 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* ABSTRACT -------- TSUNAMI-1D (**T**\ ools for **S**\ ensitivity and **Un**\ certainty **A**\ nalysis **M**\ ethodology **I**\ mplementation in **One** **D**\ imension) 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 |keff| for criticality safety applications, and subsequent updates provide for analysis of system responses other than |keff|, 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 |keff| for applications that are appropriate for 1D modeling. The SAMS module is used to determine the sensitivities of the calculated value of |keff| 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 |keff| 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. 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. Introduction ------------ TSUNAMI-1D (**T**\ ools for **S**\ ensitivity and **Un**\ certainty **A**\ nalysis **M**\ ethodology **I**\ mplementation in **One** **D**\ imension) 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 |keff|, 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 :numref:`6-1-3`, three sample problems are given in :numref:`6-1-4` the techniques used in each computational sequence are described in :numref:`6-1-2`, 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 :numref:`6-1-3` to customize an input for his specific needs. TSUNAMI-1D provides automated, problem-dependent cross sections using the same methods and input as the **C**\ riticality **S**\ afety **A**\ nalysis **S**\ equences (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. .. _6-1-2: 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 :numref:`tab6-1-1`, where the functional modules executed by each control sequence are also shown. A general flow diagram of TSUNAMI-1D is shown in :numref:`fig6-1-1`. .. tabularcolumns:: |\Y{0.25}|\Y{0.15}|\Y{0.15}|\Y{0.15}|\Y{0.15}|\Y{0.15}| .. table:: TSUNAMI-1D control sequences. :name: tab6-1-1 :class: longtable +--------------+----------------------------------------------------------------+ | **Control** | **Functional modules executed by the control module** | | | | | **module** | | +==============+================+===========+===========+===========+===========+ | TSUNAMI-1D | XSProc | XSDRNPM | XSDRNPM | BONAMIST | SAMS\ | | | | | | | :sup:`\*` | | | | forward | adjoint\ | | | | | | | :sup:`\*` | | | +--------------+----------------+-----------+-----------+-----------+-----------+ | TSUNAMI-1DC | XSProc | XSDRNPM | | | | | forward | | +--------------+----------------+-----------+-----------------------------------+ | :sup:`\*` 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 |keff| requires both the ``DEFINITIONS`` and ``SYSTEMRESPONSE`` block of input data. These blocks of data are described in :numref:`6-1-3-4`. 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. .. figure:: figs/TSUNAMI-1D/fig1.png :align: center :width: 600 :name: fig6-1-1 General flow diagram of TSUNAMI-1D. .. _6-1-3: 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 :math:`\times` 10\ :sup:`4`. A comment is initiated with a single quote, ``''``, and continues until the end of the input line. 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: .. describe:: =TSUNAMI-1D This sequence is used for sensitivity and uncertainty calculations. .. describe:: =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 .. describe:: PARM=CHECK This option allows the input data to be read and checked without executing any functional modules. .. describe:: PARM=CHK Alias for ``PARM=CHECK``. .. describe:: 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. .. describe:: 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. .. describe:: 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.** .. describe:: 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.** .. describe:: 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). 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. 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. 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: a. mixture number in the zone and b. 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. 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 :numref:`tab6-1-2`. .. tabularcolumns:: |\Y{0.15}|\Y{0.20}|\Y{0.65}| .. table:: Optional parameter input for the criticality problem data. :class: longtable :name: tab6-1-2 :widths: 15 20 65 +-----------------------+-----------------------+-----------------------+ | **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. 0.01.5 gives a | | | | coarser mesh. | +-----------------------+-----------------------+-----------------------+ .. _6-1-3-4: 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 |keff|. If only one or both of the data blocks are omitted, then analysis is only performed for |keff|. The following sub-sections describe these blocks of data in detail. 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 |keff| 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: .. math:: :label: eq6-1-1 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, :math:`\phi_{g}(\bar{r})` is the space-dependent multi-group scalar flux and :math:`H_{N, g}(\bar{r})`,\ :math:`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 |keff|) 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: .. highlight:: scale :: 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 :numref:`tab6-1-3` 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 :numref:`tab6-1-3` at the end of this section. .. tabularcolumns:: |\Y{0.15}|\Y{0.15}|\Y{0.15}|\Y{0.55}| .. table:: Response function keywords in ``DEFINITIONS`` block. :class: longtable :name: tab6-1-3 :widths: 15 15 15 55 +----------------------+-----------------+-----------------+-----------------+ | **Keyword** | **Type** | **Default** | **Description** | | | | **value** | | +======================+=================+=================+=================+ | ``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=`` | Integer or | Undefined | Nuclide | | | string | | identifier for | | or | | | which | | | | | cross-sections | | ``zaid=`` | | | are used to | | | | | define the | | | | | response | | | | | function. The | | | | | nuclide can be | | | | | specified in | | | | | integer format | | | | | (92235) or in | | | | | character | | | | | string format | | | | | (u-235). | +----------------------+-----------------+-----------------+-----------------+ | ``reaction=`` | Integer or | Undefined | Reaction | | | string | | identifier for | | or | | | which | | | | | cross-sections | | ``mt=`` | | | 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=`` | Integer | Undefined | Mixture | | | | | identifier for | | or | | | which | | | | | cross-sections | | ``mixture=`` | | | are used to | | | | | define the | | | | | response | | | | | function. | +----------------------+-----------------+-----------------+-----------------+ | ``multimix ... end`` | Integer array | Undefined | Array of | | | | | mixture | | or | | | identifiers for | | | | | which | | ``multimat ... end`` | | | 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 | 10\ :sup:`25` | Upper energy | | | | | (eV) boundary | | | | | of the response | | | | | function. | +----------------------+-----------------+-----------------+-----------------+ | ``elow=`` | Real | 0.0 | Lower energy | | | | | (eV) boundary | | | | | of the response | | | | | function. | +----------------------+-----------------+-----------------+-----------------+ | ``ehightransfer=`` | Real | 10\ :sup:`25` | 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). | +----------------------+-----------------+-----------------+-----------------+ 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 :math:`H_{g}(\bar{r})` for a single mixture-\ *j* is defined as: .. math:: :label: eq6-1-2 H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r}) where .. math:: \begin{aligned} &\delta_{g}=\left\{\begin{array}{cc} 1.0 & E_{\text {High}}>E_{g}^{\text {Lower}}, E_{\text {Low}}E_{g}^{\text {Lower}}, E_{\text {LowTransfer}}0.625 eV and E<25 keV) .. code-block:: scale 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 :numref:`tab6-1-5`. 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 :math:`H_{g}(\bar{r})` is defined as: .. math:: :label: eq6-1-7 H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j, n} In this expression, :math:`\Sigma_{x, g}^{j, n}` is the macroscopic cross-section (:math:`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: .. code-block:: scale 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. 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 :math:`H_{g}(\bar{r})` is defined as: .. math:: :label: eq6-1-8 H_{g}(\bar{r})=c^{*} \delta_{g} * \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j} In this expression, :math:`\Sigma_{x, g}^{j}` is the mixture macroscopic cross-section defined as :math:`\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, :math:`H_{g}(\bar{r})` is rewritten as .. math:: :label: eq6-1-9 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) .. code-block:: scale 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. 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 :math:`H_{g}(\bar{r})` is defined as: .. math:: :label: eq6-1-10 H_{g}(\bar{r})=c^{*} \delta_{g} * \sum_{j \in S} \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j, n} In this expression, :math:`\Sigma_{x, g}^{j, n}` is the macroscopic cross-section (:math:`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 .. code-block:: scale 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. 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 :math:`H_{g}(\bar{r})` is defined as: .. math:: :label: eq6-1-11 H_{g}(\bar{r})=c^{*} \delta_{g} * \sum_{j \in S} \delta_{j}(\bar{r})^{*} \Sigma_{x, g}^{j} In this expression, :math:`\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 .. code-block:: scale 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. .. tabularcolumns:: |\Y{0.2}|\Y{0.2}|\Y{0.2}|\Y{0.2}|\Y{0.2}| .. table:: Keyword dependencies for the ``DEFINITIONS`` block. :class: longtable :name: tab6-1-4 :widths: auto +-------------+-------------+-------------+-------------+-------------+ | **Response | **Required | **Unallowed | **Optional | **Optional, | | type** | keywords** | keywords** | keywords** | but not | | | | | | used | | | | | | keywords** | +=============+=============+=============+=============+=============+ | Single-mixt\| unity, | multimix | multiplier, | title, | | ure | mixture | | eHigh, eLow | nuclide, | | flux | | | | reaction, | | | | | | micro, | | | | | | macro, | | | | | | eHighTransf\| | | | | | er, | | | | | | | | | | | | eLowTransfe\| | | | | | r | +-------------+-------------+-------------+-------------+-------------+ | Multiple-mi\| unity, | mixture | multiplier, | title, | | xture | multimix | | eHigh, eLow | nuclide, | | flux | | | | reaction, | | | | | | micro, | | | | | | macro, | | | | | | eHighTransf\| | | | | | er, | | | | | | eLowTransfe\| | | | | | r | +-------------+-------------+-------------+-------------+-------------+ | Single-mixt\| mixture, | unity, | multiplier, | title | | ure, | nuclide, | macro, | eHigh, | | | single-nucl\| reaction | multimix | eLow, | | | ide, | | | micro, | | | microscopic | | | eHighTransf\| | | cross-secti\| | | er\ :sup:`a`| | | on | | | , | | | | | | eLowTransfe\| | | | | | r\ :sup:`a` | | +-------------+-------------+-------------+-------------+-------------+ | Single-mixt\| mixture, | unity, | multiplier, | title | | ure, | nuclide, | micro, | eHigh, | | | single-nucl\| reaction,ma\| multimix | eLow, | | | ide, | cro | | eHighTransf\| | | macroscopic | | | er\ :sup:`a`| | | cross-secti\| | | , | | | on | | | eLowTransfe\| | | | | | r\ :sup:`a` | | +-------------+-------------+-------------+-------------+-------------+ | Single-mixt\| mixture, | unity, | multiplier, | title | | ure, | reaction,ma\| micro, | eHigh, | | | multiple-nu\| cro | multimix, | eLow, | | | clide, | | nuclide | eHighTransf\| | | macroscopic\| | | er\ :sup:`a`| | | cross-secti\| | | , | | | on | | | eLowTransfe\| | | | | | r\ :sup:`a` | | +-------------+-------------+-------------+-------------+-------------+ | Multiple-mi\| multimix, | unity, | multiplier, | title | | xture, | nuclide | micro, | eHigh, | | | single-nucl\| reaction,ma\| mixture | eLow, | | | ide, | cro | | eHighTransf\| | | macroscopic | | | er\ :sup:`a`| | | cross-secti\| | | , | | | on | | | eLowTransfe\| | | | | | r\ :sup:`a` | | +-------------+-------------+-------------+-------------+-------------+ | Multiple-mi\| multimix, | unity, | multiplier, | title | | xture, | reaction,ma\| micro, | eHigh, | | | multiple-nu\| cro | mixture, | eLow, | | | clide, | | nuclide | eHighTransf\| | | macroscopic | | | er\ :sup:`a`| | | cross-secti\| | | , | | | on | | | eLowTransfe\| | | | | | r\ :sup:`a` | | +-------------+-------------+-------------+-------------+-------------+ | :sup:`a`\ Keywords ``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 | +-------------+-------------+-------------+-------------+-------------+ .. raw:: latex \clearpage .. tabularcolumns:: |\Y{0.3}|\Y{0.3}|\Y{0.4}| .. table:: Supported Reaction Types in ``DEFINITIONS`` block. :name: tab6-1-5 :class: longtable +-----------------------+-----------------------+-----------------------+ | **MT** | **Reaction** | **String Identifier** | +=======================+=======================+=======================+ | 1 | total | Total | +-----------------------+-----------------------+-----------------------+ | 2 | elastic scattering | Elastic | +-----------------------+-----------------------+-----------------------+ | 4 | inelastic scattering | Inelastic | +-----------------------+-----------------------+-----------------------+ | 16\ :sup:`a` | effective n,2n | n,2n | +-----------------------+-----------------------+-----------------------+ | 0 | sum of scattering | Scatter | | | (2+4+16) | | +-----------------------+-----------------------+-----------------------+ | 18 | fission | Fission | +-----------------------+-----------------------+-----------------------+ | 102 | n, |gam| | n,gamma | +-----------------------+-----------------------+-----------------------+ | 103 | n,p | n,p | +-----------------------+-----------------------+-----------------------+ | 104 | n,d | n,d | +-----------------------+-----------------------+-----------------------+ | 105 | n,t | n,t | +-----------------------+-----------------------+-----------------------+ | 106 | n,\ :sup:`3`\ he | n,he-3 | +-----------------------+-----------------------+-----------------------+ | 107 | n, |alph| | n,alpha | +-----------------------+-----------------------+-----------------------+ | 101 | Neutron disappearance | capture | | | (102+103+104+105+106+ | | | | 107) | | +-----------------------+-----------------------+-----------------------+ | 452 | :math:`\bar{\nu}` | nubar | +-----------------------+-----------------------+-----------------------+ | 1452 | :math:`\bar{\nu}` | nu-fission | | | times fission | | +-----------------------+-----------------------+-----------------------+ | 1018 | :math:`\chi` | chi | +-----------------------+-----------------------+-----------------------+ | :sup:`a`\ The effective n,2n is defined by the summation of transfer | | matrices of the following reaction types: (n,2n), | | (n,2n+\ :math:`\alpha`), (n,2n+2\ :math:`\alpha`), (n,3n), | | (n,3n+\ :math:`\alpha`), and (n,4n). The individual transfer matrices | | are scaled by the number of exit channel neutrons, i.e., 2, 3, or 4. | +-----------------------+-----------------------+-----------------------+ 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 |keff|. 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: .. code-block:: scale 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: .. math:: :label: eq6-1-12 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 :math:`\Sigma_{r, g}(\bar{r})` is the removal cross-section defined as the total cross-section minus the within group cross-section --- :math:`\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: .. code-block:: scale 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 |keff| 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 |keff| 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: .. code-block:: scale 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 :file:`jobname.rsp_ratio_0000000100.sdf`. 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: :numref:`tab6-1-6`, 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 :numref:`tab6-1-7`. A more detailed explanation of the SAMS parameters may be found in the :ref:`SAMS chapter `. .. tabularcolumns:: |\Y{0.15}|\Y{0.25}|\Y{0.6}| .. table:: SAMS input keywords :name: tab6-1-6 :class: longtable :widths: 15 25 60 +-----------------------+-----------------------+-----------------------+ | **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 | +-----------------------+-----------------------+-----------------------+ .. .. tabularcolumns:: |\Y{0.3}|\Y{0.15}|\Y{0.55}| .. table:: SAMS input keywords for default covariance data. :name: tab6-1-7 :class: longtable :widths: 30 15 55 +-----------------------+-----------------------+------------------------+ | **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-covaria\ | | | | nce | | | | 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-covaria\ | | | | nce | | | | 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-covaria\ | | | | nce | | | | 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-covaria\ | | | | nce | | | | 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-covaria\ | | | | nce | | | | 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-covaria\ | | | | nce | | | | 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). | +-----------------------+-----------------------+------------------------+ 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. 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. .. _6-1-4: 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 |keff| 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 |keff| 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 |keff| 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 |keff| to some input value :math:`\alpha` is computed as .. math:: :label: eq6-1-13 S_{k, \alpha}=\frac{\alpha}{k} \times \frac{d k}{d \alpha}=\frac{\alpha}{k} \times \frac{k_{\alpha^{+}}-k_{\alpha^{-}}}{\alpha^{+}-\alpha^{-}} , where\ :math:`\alpha^{+}` and :math:`\alpha^{-}` represent the increased and decreased values, respectively, of the input quantity :math:`\alpha` and :math:`k_{\alpha^{+}}` and :math:`k_{\alpha^{-}}` represent the corresponding values of |keff|. 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. 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 UF\ :sub:`4` and paraffin with an enrichment of 2% in :sup:`235`\ U. The H/\ :sup:`235`\ U atomic ratio is 294:1. The dimensions of the experiment were 56.22 cm :math:`\times` 56.22 cm :math:`\times` 122.47 cm. :cite:`TS-1D-jordan_validation_1986`. 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 :numref:`6-1-2`. 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 S\ :sub:`16` angular quadrature set to S\ :sub:`8`. 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. .. figure:: figs/TSUNAMI-1D/fig2.png :align: center :width: 500 :name: fig6-1-2 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 :math:`\pm2\%` and the calculation was repeated using the TSUNAMI-1DC sequence. The sensitivity of |keff| to the number density is equivalent to the sensitivity of |keff| to the total cross section, integrated over energy. The direct perturbation sensitivity coefficients were computed by using the |keff| values from the unperturbed and perturbed cases in :eq:`eq6-1-13`. To demonstrate the importance of the sensitivity to the resonance processing implicit sensitivity computed by BONAMIST, the same model shown in :numref:`fig6-1-2` was run with TSUNAMI-1D with ``PARM=CENTRM``. The results from the ``INFHOMMEDIUM`` sample problem are given in :numref:`tab6-1-8`. 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 :sup:`238`\ U with a magnitude of 1.5%. The maximum difference between the direct perturbation results and the TSUNAMI-1D with ``PARM=CENTRM`` results occurs for :sup:`238`\ U with a magnitude of 19%. Thus, the use of the default ``PARM=BONAMIST`` is recommended. .. |alph| replace:: :math:`alpha` .. |gam| replace:: :math:`gamma` .. tabularcolumns:: |\Y{0.10}|\Y{0.15}|\Y{0.15}|\Y{0.15}|\Y{0.25}| .. table:: Energy- and region-integrated sensitivity coefficients from TSUNAMI-1D INFHOMMEDIUM sample problem. :name: tab6-1-8 :class: longtable :widths: 10 15 15 15 25 +-------------+--------------+-------------+----------------+-----------------+ | **Isotope** | **Reaction** | **Direct** | **TSUNAMI-1D** | **TSUNAMI-1D** | | | | | | **PARM=**\ | | | | **perturbat\| | CENTRM** | | | | ion** | | | +=============+==============+=============+================+=================+ | :sup:`1`\ H | total | 2.20E-01 | 2.18E-01 | 2.52E-01 | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`1`\ H | scatter | | 3.19E-01 | 3.53E-01 | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`1`\ H | elastic | | 3.19E-01 | 3.53E-01 | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`1`\ H | capture | | -1.01E-01 | -1.01E-01 | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`1`\ H | n, |gam| | | -1.01E-01 | -1.01E-01 | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | total | 2.41E-02 | 2.38E-02 | 2.76E-02 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | scatter | | 2.45E-02 | 2.83E-02 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | elastic | | 2.43E-02 | 2.80E-02 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | n,n' | | 2.20E-04 | 2.20E-04 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | capture | | -6.83E-04 | -6.83E-04 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | n, |gam| | | -4.98E-04 | -4.98E-04 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | n,p | | -3.53E-08 | -3.53E-08 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | n,d | | -7.33E-08 | -7.33E-08 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`12`\ | n, |alph| | | -1.85E-04 | -1.85E-04 | | C | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | total | 4.10E-02 | 4.06E-02 | 4.47E-02 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | scatter | | 4.62E-02 | 5.03E-02 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | elastic | | 2.94E-02 | 3.34E-02 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n,n' | | 1.58E-02 | 1.58E-02 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n,2n | | 2.89E-06 | 2.89E-06 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | capture | | -5.59E-03 | -5.59E-03 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n, |gam| | | -2.39E-03 | -2.39E-03 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n,p | | -2.37E-04 | -2.37E-04 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n,d | | -1.27E-05 | -1.27E-05 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n,t | | -2.72E-06 | -2.72E-06 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`19`\ | n, |alph| | | -2.96E-03 | -2.96E-03 | | F | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | total | 2.52E-01 | 2.52E-01 | 2.53E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | scatter | | 4.32E-04 | 5.03E-04 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | elastic | | 2.02E-04 | 2.73E-04 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | n,n' | | 2.13E-04 | 2.13E-04 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | n,2n | | 1.70E-05 | 1.70E-05 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | fission | | 3.64E-01 | 3.65E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | capture | | -1.13E-01 | -1.12E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | n, |gam| | | -1.13E-01 | -1.12E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | nubar | | 9.50E-01 | 9.50E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`235`\ | :math:`\chi` | | 8.52E-08 | 8.52E-08 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | total | -2.08E-01 | -2.05E-01 | -2.47E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | scatter | | 4.81E-02 | 2.46E-02 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | elastic | | 3.46E-02 | 1.10E-02 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | n,n' | | 1.25E-02 | 1.25E-02 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | n,2n | | 1.02E-03 | 1.02E-03 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | fission | | 3.35E-02 | 3.35E-02 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | capture | | -2.86E-01 | -3.05E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | n, |gam| | | -2.86E-01 | -3.05E-01 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | nubar | | 5.02E-02 | 5.02E-02 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ | :sup:`238`\ | :math:`\chi` | | 4.54E-09 | 4.54E-09 | | U | | | | | +-------------+--------------+-------------+----------------+-----------------+ The uncertainty information from SAMS for the INFHOMMEDIUM sample problem is shown in :numref:`list6-1-1`. Based on the 44GROUPCOV covariance data library, documented in the COVLIB chapter, the uncertainty in |keff| due to these covariance data is 0.6064% :math:`\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 :sup:`235`\ U fission and :sup:`1`\ H elastic scattering are shown in :numref:`fig6-1-4`. The :sup:`238`\ U capture sensitivity profiles from TSUNAMI-1D and TSUNAMI-1D with PARM=CENTRM are shown in :numref:`fig6-1-5`. 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. .. code-block:: none :name: list6-1-1 :caption: 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 .. figure:: figs/TSUNAMI-1D/fig4.png :align: center :width: 600 :name: fig6-1-4 Energy-dependent sensitivity profiles from TSUNAMI-1D for INFHOMMEDIUM sample problem. .. figure:: figs/TSUNAMI-1D/fig5.png :align: center :width: 600 :name: fig6-1-5 Comparison of 238U capture sensitivities from TSUNAMI-1D and TSUNAMI-1D with PARM=CENTRM for INFHOMMEDIUM sample problem. 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. :cite:`TS-1D-BNL19302`. 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 :numref:`fig6-1-6`. 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 |keff| to the total cross section, integrated over energy. The energy-integrated sensitivity results are shown in :numref:`tab6-1-9`. The TSUNAMI-1D results agree well with the direct perturbation results for this system. The maximum difference occurs for :sup:`238`\ U 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. .. figure:: figs/TSUNAMI-1D/fig6.png :align: center :width: 500 :name: fig6-1-6 TSUNAMI-1D MULTIREGION sample problem input. .. raw:: latex \clearpage .. list-table:: Energy- and region-integrated sensitivity coefficients from TSUNAMI-1D MULTIREGION sample problem. :align: center :name: tab6-1-9 * - .. image:: figs/TSUNAMI-1D/tab9.svg :width: 700 The uncertainty information from SAMS HTML output for the multiregion sample problem is shown in :numref:`fig6-1-7`. Based on the 44GROUPCOV covariance data file, the uncertainty in |keff| due to these covariance data is 1.2743% :math:`\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 :sup:`235`\ U fission in zone 1 (core) and zone 2 (reflector) were generated with Fulcrum and are shown in :numref:`fig6-1-8`. Additionally, sensitivity profiles for :sup:`238`\ U capture in zone 1 and zone 2 are shown in :numref:`fig6-1-9`. Note that the capture sensitivities are negative, such that the lower curve has the greater magnitude. In :sup:`235`\ U and :sup:`238`\ U 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 :sup:`235`\ U fission in the core and to :sup:`238`\ U capture in the reflector. .. figure:: figs/TSUNAMI-1D/fig7.png :align: center :width: 500 :name: fig6-1-7 Uncertainty information in HTML output from MULTIREGION sample problem. .. figure:: figs/TSUNAMI-1D/fig8.png :align: center :width: 600 :name: fig6-1-8 Sensitivity profiles from TSUNAMI-1D for :sup:`235`\ U fission in zone 1 and zone 2 of MULTIREGION sample problem. .. figure:: figs/TSUNAMI-1D/fig9.png :align: center :width: 600 :name: fig6-1-9 Sensitivity profiles from TSUNAMI-1D for :sup:`238`\ U capture in zone 1 and zone 2 of MULTIREGION sample problem. 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 :cite:`TS-1D-ivanov_benchmark_2007`. 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 :sup:`234`\ U, :sup:`235`\ U, and :sup:`238`\ U. The input for this sample problem is provided in :numref:`list6-1-2`. For this sample, seven separate sensitivity data files are generated, one for each of the six defined responses in addition to |keff|. Selected sensitivity profiles are shown in :numref:`fig6-1-11` for the :sup:`238`\ U (n,\ :math:`\gamma`) cross-section. This figure shows the negative sensitivity of |keff| due to :sup:`238`\ U resonance absorption in the blue profile. The red profile shows the positive sensitivity of the energy-integrated :sup:`238`\ U absorption cross-section due to the multigroup :sup:`238`\ U (n,\ :math:`\gamma`) cross-section. The large positive magnitude of this sensitivity is predominantly due to the presence of the :sup:`238`\ U (n,\ :math:`\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 :sup:`235`\ U fission cross-section due to the multigroup :sup:`238`\ U (n,\ :math:`\gamma`) cross-section. In this response, positive perturbations to the :sup:`238`\ U (n,\ :math:`\gamma`)  multigroup cross-sections induce changes in the flux spectra that lead to a decrease in the energy-integrated :sup:`235`\ U fission cross-section. These indirect sensitivity effects are determined by the solution of the generalized adjoint calculations. .. code-block:: scale :name: list6-1-2 :caption: 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 .. figure:: figs/TSUNAMI-1D/fig11.png :align: center :width: 600 :name: fig6-1-11 Response Sensitivities to :sup:`238`\ U n,gamma cross section for the TSUNAMI-1D9 sample problem. .. only:: html .. rubric:: References .. bibliography:: zSCALE.bib :cited: :keyprefix: TS-1D- :labelprefix: TS-1D- Appendices ---------- .. toctree:: tsunami-1d-appA