Sensitivity calculations

Serpent relies on the collision history based first order GPT equivalent implementation to calculate sensitivities of various responses to various perturbations. In version 2.1.31, these capabilities were extended to be used for uncertainty propagation of Serpent estimates . From version 2.2.1 and on, it applies not only to criticality calculations but also to external source calculations.

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
lambda	sensitivity of delayed neutron precursor decay constants to perturbation	0 0/1 1	OFF (total) OFF (group wise) / ON (total) ON (group wise)

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 [FRAC1 FRAC2]

Where the parameters 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.
FRAC1	: Fraction of contributions to DET1 to include in the sensitivity evaluation. Higher fraction yields better statistics but increases runtime. Default is 1 (all detector score contribute to the sensitivity).
FRAC2	: Fraction of contributions to DET2 to include in the sensitivity evaluation. Higher fraction yields better statistics but increases runtime. Default is 1 (all detector score contribute to the sensitivity).

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

Notes:

• Currently, the detectors must have the same number of bins and at most one response bin.
• Currently, the indirect effect of the perturbation is always calculated and the direct effect is calculated for the following detector responses:
  • dr -1 (macroscopic total XS)
  • dr -2 (macroscopic total capture XS)
  • dr -3 (macroscopic total elastic scattering XS)
  • dr -6 (macroscopic total fission XS)
  • dr -7 (macroscopic total fission neutron production XS)
  • Positive MT-numbers, however the sum reaction modes (e.g. MT=4) are not summed up from the partial reaction modes.

Void reactivity coefficient sensitivity

The sensitivity of the void reactivity coefficient to perturbations can be set up to be calculated with

sens resp void MATERIAL

Where the parameters are:

MATERIAL

: the name of the material corresponding to the void coefficient.

Adding a perturbation

Basic perturbations

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 See below for only including some specific reaction modes.
chi	perturbation of fission spectrum	0/1	OFF/ON
nubar	perturbation of fission nubar	0/1	OFF/ON
elamu	perturbation of elastic scattering cosine	0/1	OFF/ON
inlmu	perturbation of inelastic scattering cosine	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.
temperature	perturbation of temperature	0/1	OFF/ON

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.

Choosing specific reaction modes to perturb

Instead of including all sum reactions with sens pert xs all a subset of them can be included using

sens pert xs realist REA1 REA2 ... 

where the allowed names for the sum reactions REA_n are

• ela: Sum elastic scattering (excluding thermal scattering).
• sab: Sum thermal scattering.
• inl: Sum inelastic scattering (MTs 51-91).
• capt: Sum neutron capture (all reactions with no outgoing neutrons).
• fiss: Sum fission (MTs 19-21 and 38).
• nxn: Sum scattering production (non fission reactions with more than one outgoing neutron).

For example

sens pert xs realist capt fiss

will only include the capture and fission reaction modes for perturbation.

Similarly, it is possible to choose only certain partial (MT) reaction modes to perturb with

sens pert xs mtlist MT1 MT2 ... 

For example

sens pert xs mtlist 19 102

will only include the first-chance fission (MT 19) and radiative capture (MT 102) for perturbation.

Notes:

• Sum reaction modes and partial reaction modes cannot currently be perturbed at the same time (in the same input). For inputs containing both perturbations (e.g. a sens pert xs realist card and a sens pert xs mtlist card) only the latter perturbation is used.

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.

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 .

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.

Notes:

• The custom perturbations will only produce the total effect of the perturbation over all materials, nuclides and reaction modes that have not been excluded from the custom perturbation.
• The output for the custom perturbations can be obtained from the energy integrated (_E_INT) arrays printed to [input]_sens.m. The only nonzero value for the custom perturbations will be that with the lowest material and ZAI index.

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.

Spatial data mesh for scoring

The spatial data mesh used for scoring the sensitivities can be set using

sens opt dmesh DNAME

where DNAME is a name of a spatial data mesh definition.

Notes:

• The number of bins defining the energy grid and the spatial data mesh should match

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

Notes:

• The number of inactive generations should be set accordingly since collision history from NGEN+1 generations (cycles) ago will be used for scoring.
  • A good rule of thumb is to use <NNORMAL> + <NGEN> + 1 inactive generations, where <NNORMAL> is the number you would normally use for converging the fission source.

Effect of number of latent generations

Setting

sens opt history yes/1

will calculate and output the sensitivities with every number of latent generations up to the one indicated with sens opt latgen. This option can be used to estimate the required number of latent generations.

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 N_gen generations of particle history, where N_gen 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. Looping over this score matrix can also be costly.

Generally, direct scoring is the better option if the length of the collision history is large and the number of perturbations (number of materials X number of nuclides X number of reactions X number of energy groups) is small.

Conversely in systems with a short collision history length (fast systems) and a large number of total perturbations, the default event based scoring should fare better.

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 N_event × N_gen, where N_event is the mean number of events of interest in each generation and N_gen 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[time].m (dynamic/time-dependent calculations), [input]_sens[bu].m (burnup calculations) or [input]_sens[bu]b[idx].m (branch calculations), where "time" is the time-interval, "bu" is the burnup step, and "idx is the branch index.

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 × SENS_N_ZAI × SENS_N_PERT × SENS_N_ENE × 2	The energy dependent sensitivity of the multiplication factor and the associated statistical error.
ADJ_PERT_KEFF_SENS_E_INT	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × 2	The energy integrated sensitivity of the multiplication factor and the associated statistical error.
ADJ_PERT_BEFF_SENS	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × SENS_N_ENE × 2	The energy dependent sensitivity of the delayed neutron fraction and the associated statistical error.
ADJ_PERT_BEFF_SENS_E_INT	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × 2	The energy integrated sensitivity of the delayed neutron fraction and the associated statistical error.
ADJ_PERT_LEFF_SENS	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × SENS_N_ENE × 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 × SENS_N_ZAI × SENS_N_PERT × 2	The energy integrated sensitivity of the prompt neutron generation time and the associated statistical error.
ADJ_PERT_VOID_SENS	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × SENS_N_ENE × 2	The energy dependent sensitivity of the void reactivity coefficient and the associated statistical error.
ADJ_PERT_VOID_SENS_E_INT	SENS_N_MAT × SENS_N_ZAI × SENS_N_PERT × 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 × SENS_N_ZAI × SENS_N_PERT × SENS_N_ENE × 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 × SENS_N_ZAI × SENS_N_PERT × 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).

Examples

• Flattop k-eff sensitivity example
• xGPT example