Sensitivity Calculations

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Serpent relies on the collision history based first order GPT equivalent implementation[1] to calculate sensitivities of various responses to various perturbations. As a simple example, the sensitivity of the effective multiplication factor to the different nuclear cross sections can be calculated.

Input

Adding a response

Basic responses

Basic responses can be set up with the syntax

sens resp NAME FLAG

Where the different response names and flags are as follows

NAME explanation of NAME FLAG explanation of FLAG
keff sensitivity of effective multiplication factor to perturbation 0/1 OFF/ON
beff sensitivity of effective delayed neutron fraction to perturbation 0/1 OFF/ON
leff sensitivity of effective prompt generation time to perturbation 0/1 OFF/ON

Omitting (or forgetting) the flag from the definition, will turn the response on by default.

Reaction rate ratios

Sensitivities of reaction rate ratios to perturbations can be set up to be calculated with

sens resp detratio NAME DET1 DET2

Where the paramters are:

NAME : the name of the response. Will be used in output.
DET1 : Name of the numerator detector, i.e. the one whose reaction rate is being divided.
DET2 : Name of the denominator detector, i.e. the one whose reaction rate is the dividing value.

The two detectors have to be set up separately in normal Serpent fashion.

Notes:

  • Currently, each detector can have only one bin.
  • Currently, only the indirect effect of the perturbation is calculated and the direct effect must be calculated separately and added to the result.

Adding a perturbation

sens pert NAME FLAG

Where the different perturbation names and flags are as follows

NAME explanation of NAME FLAG explanation of FLAG
xs perturbation of basic cross sections 0/all/allmt OFF/sum reaction modes/partial (MT) reaction modes
chi perturbation of fission spectrum 0/1 OFF/ON
nubar perturbation of fission nubar 0/1 OFF/ON
eleg perturbation of Legendre moments of
elastic scattering angular distribution
0/1/2/3/4/5/6/7 Number of Legendre moments to perturb.

Omitting (or forgetting) the flag from the definition will turn the perturbation on by default. The default number of Legendre moments to include is 3.

Custom perturbations (xGPT)

Instead of adding or subtracting 1 in each accepted or rejected event, there is the possibility to, instead, add or subtract a user defined energy dependent value. This allows applying continuous energy perturbations such as the ones used in [2].

The syntax for adding custom perturbations is

sens pert custom NAME [efunc FILE] [zailist ZAI1 ZAI2 ...] [matlist MAT1 MAT2 ...] [mtlist MT1 MT2 ...]  [realist REA1 REA2 ...]

where NAME is simply a user-given name used for identification of the custom perturbation. By default, the value added or subtracted for the perturbation is 1, but an energy dependent function can be given with the efunc command, where FILE is the name of the file containing the energy dependent function in the following format:

INTERP_MODE
Npts
E1 VAL1
E2 VAL2
...
ENpts VALNpts

Accepted interpolation modes are lin-lin, lin-log, log-lin and log-log.

If the perturbation is only applied to certain nuclides (ZAIs) or materials, they can be given using the zailist and matlist options respectively. By default the perturbation is applied at each collision with the target nuclide(s). If the perturbation only affects certain reaction MT numbers, they can be given using the mtlist option. Certain sum-reactions can be flagged for the perturbation using the realist option, where the different sum reaction names are as follows:

ela  : Elastic scattering.
inl  : Inelastic scattering.
sab  : Thermal S(α,β) scattering.
capt  : Neutron capture (reaction neutron yield is zero).
fiss  : Fission.
nxn  : Neutron multiplying (n, xn) reactions.

Choosing nuclides to perturb

The perturbed nuclides can be specified by giving a list of the ZAI-numbers to include

sens pert zailist ZAI1 ZAI2 ... 

for example

sens pert zailist 922350 922380

for perturbations affecting 235U and 238U. By giving the control words total and/or sum in the list of ZAI-numbers, the total sensitivity over all ZAIs present in the calculation and the sum sensitivity of the listed ZAIs will be calculated, respectively.

All nuclides present in the calculation can be included with

sens pert zailist all

this will automatically include the calculation of the total sensitivity.

Choosing materials to perturb

The materials, where the perturbations are applied can similarly be specified by giving a list of the material names

sens pert matlist MAT1 MAT2 ...

for example

sens pert matlist fuel coolant

for perturbations affecting interactions in materials fuel and coolant. By giving the control words total and/or sum in the list of material names, the total sensitivity over all materials present in the calculation and the sum sensitivity of the listed materials will be calculated, respectively.

All materials present in the calculation can be included with

sens pert matlist all

this will automatically include the calculation of the total sensitivity.

Additional options

The options for sensitivity calculations can be set using the

sens opt ...

syntax.

Energy grid for scoring

The energy grid used for scoring the sensitivities can be set using

sens opt egrid ENAME

where ENAME is a name of an energy grid definition.

Number of latent generations

The number of latent generations used for k-effective sensitivity calculations can be set with

sens opt latgen NGEN

where NGEN is the number of generations to use. The number of summation generations for reaction rate ratios and bilinear-ratios will be NGEN+1

Direct scoring instead of event based scoring

There are two ways to go about calculating the net number of accepted collisions in the collision history of a particle.

  1. The default way used by Serpent is to create so-called event-objects during each accepted or rejected event of interest (e.g. collision, sampled fission neutron energy or sampled scattering angle). Then, when the net number of accepted collisions is needed, this list of events is traversed and the net number is calculated. The memory requirement of this method is dependent on the number of interesting events in Ngen generations of particle history, where Ngen is the total number of generations stored for the collision history. In order to save memory, the events are shared by each particle that has the event in their collision history. Traversing through the event tree in order to calculate the net number of a certain event in the collision history can be costly.
  2. There is the option to update the net number of a certain event on-the-fly, every time that event is sampled. Event objects are no longer stored. Instead the net number of accepted collisions is either incremented (accepted collision) or decremented (rejected collision) directly. This way each generation in the collision history is associated with a score matrix representing the net number of scores for each of the different events of interest. Again, to save memory each generational score matrix is shared by all neutrons that have that neutron history in their collision history.

It is usually beneficial to test both methods in order to find the faster (or more memory efficient) one for the specific application.

In order to use the direct scoring (option 2), the input is

sens opt direct F

where F indicates the number of score matrices to be allocated per source particle per generation. If the score matrices would not be shared across multiple source particles F would need to be set to 1.0. However, in reality fewer score matrices are needed and the F can be set to be smaller than unity (e.g. 0.2-0.5).

The score matrices require some additional processing after each cycle, which means that it is a good idea to try and make the F parameter as small as possible in order to avoid having to process an excess number of score matrices. If Serpent notices that the allocated score matrices are running out, it will automatically try and allocate more score matrices:

 ***** Warning: Adjusting event score matrix bank size for thread 5...

Even so, the score matrices can sometimes run out by surprise leading to an error:

Fatal error in function EBlockFromBank:

Event score matrix bank is empty. Increase the bank size with "sens opt direct <sz>".

Simulation aborted.

Event bank size

In a similar manner to the score matrices described in the previous section, Serpent also preallocates memory for the event objects used to store information about the accepted and rejected collisions. The event objects are also processed after each cycle, which means that setting a too large value for the event bank size will waste processing time.

The event bank size can be set using

set nbuf NBUF EBANK

where NBUF is the neutron buffer size (default is 5) and EBANK is the event bank size (default is 100). The event bank size is given by source particle, which means that if the events would not be shared between multiple particles, a reasonable value for the event bank size would be Nevent x Ngen, where Nevent is the mean number of events of interest in each generation and Ngen is the number of generations stored for the collision history. Due to the sharing of the events across multiple particles, much smaller values can typically be used (e.g. size of 20 for light water system with 16 stored generations with cross section and nubar perturbations for coolant and fuel nuclides).

If Serpent notices that the allocated score matrices are running out, it will automatically try and allocate more score matrices:

 ***** Warning: Adjusting event bank size for thread 2...

If the event banks run out completely the following error is given:

Fatal error in function EventFromBank:

Event bank is empty

Simulation aborted.

Output

The output of the sensitivity calculations is written into a MATLAB/OCTAVE readable file [input]_sens.m

General parameters

Parameter Size Description
SENS_N_MAT 1 Number of perturbed materials.
SENS_N_ZAI 1 Number of perturbed nuclides (ZAIs).
SENS_N_PERT 1 Number of different perturbation types (cross sections, fission spectra etc.).
SENS_N_ENE 1 Number of energy bins used for tallying the sensitivities.
SENS_MAT_LIST SENS_N_MAT strings Array of names for the perturbed materials.
SENS_ZAI_LIST SENS_N_ZAI Array of ZAI numbers for the perturbed nuclides.
SENS_PERT_LIST SENS_N_PERT strings Array of names for the perturbation types.
SENS_E SENS_N_ENE + 1 Grid points for the energy grid used for tallying the sensitivities.
SENS_LETHARGY_WIDTHS SENS_N_ENE Lethargy widths of the energy bins. Lethargy at the maximum energy bin boundary is 0.

Calculated sensitivities

The calculated sensitivities are printed out both in an energy dependent format (integrated to the energy grid defined by sens opt egrid) and an energy integrated format (integrated over all energies).

The standard sensitivity responses include:

Parameter Size Description
ADJ_PERT_KEFF_SENS SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times SENS_N_ENE \times 2 The energy dependent sensitivity of the multiplication factor and the associated statistical error.
ADJ_PERT_KEFF_SENS_E_INT SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times 2 The energy integrated sensitivity of the multiplication factor and the associated statistical error.
ADJ_PERT_BEFF_SENS SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times SENS_N_ENE \times 2 The energy dependent sensitivity of the delayed neutron fraction and the associated statistical error.
ADJ_PERT_BEFF_SENS_E_INT SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times 2 The energy integrated sensitivity of the delayed neutron fraction and the associated statistical error.
ADJ_PERT_LEFF_SENS SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times SENS_N_ENE \times 2 The energy dependent sensitivity of the prompt neutron generation time and the associated statistical error.
ADJ_PERT_LEFF_SENS_E_INT SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times 2 The energy integrated sensitivity of the prompt neutron generation time and the associated statistical error.
ADJ_PERT_VOID_SENS SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times SENS_N_ENE \times 2 The energy dependent sensitivity of the void reactivity coefficient and the associated statistical error.
ADJ_PERT_VOID_SENS_E_INT SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times 2 The energy integrated sensitivity of the void reactivity coefficient and the associated statistical error.

The reaction rate sensitivities defined using sens resp detratio are output as:

Parameter Size Description
ADJ_PERT_NAME_SENS SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times SENS_N_ENE \times 2 The energy dependent sensitivity of the reaction rate ratio NAME and the associated statistical error.
ADJ_PERT_NAME_SENS_E_INT SENS_N_MAT \times SENS_N_ZAI \times SENS_N_PERT \times 2 The energy integrated sensitivity of the reaction rate ratio NAME and the associated statistical error.

Helpful indices

The indices for different materials, ZAIs and perturbations are also included as separate variables of the following format

Parameter Size Description
iSENS_MAT_NAME 1 Index corresponding to material NAME.
iSENS_ZAI_NUMBER 1 Index corresponding to the nuclide with ZAI NUMBER.
iSENS_PERT_NAME 1 Index corresponding to the perturbation NAME.

These indices can be used to refer to the different materials, nuclides and perturbations in data processing without the need to manually figure out the indices. For example,

ADJ_PERT_KEFF_SENS(iSENS_MAT_fuel, iSENS_ZAI_922380, iSENS_PERT_CAPT_XS, :, 1)

will refer to the energy dependent sensitivity of the multiplication factor to the perturbation of the capture cross section of uranium 238 in the material fuel.

It should be noted that the indices for certain materials, ZAIs and perturbations only exist if they have been included in the sensitivity calculation using sens pert.

Notes

  • The energy dependent sensitivities are integrated over the energy bins and not divided by the bin energy- or lethargy-width.
  • The sensitivity output is mostly written as 1D arrays and reshaped and permuted using the appropriate MATLAB/OCTAVE commands (included in the .m file).
  • Perturbations of the scattering cosine are not yet calculated, although perturbations of Legendre moments of the elastic scattering distribution are.

Examples

References

  1. ^ Aufiero, M. et al. "A collision history-based approach to sensitivity/perturbation calculations in the continuous energy Monte Carlo code SERPENT", Ann. Nucl. Energy, 152 (2015) 245-258.
  2. ^ Aufiero, M., Martin, M. and Fratoni, M. "XGPT: Extending Monte Carlo Generalized Perturbation Theory capabilities to continuous-energy sensitivity functions." Ann. Nucl. Energy, 96 (2016) 295-306.