# Flattop k-eff sensitivity example

This example demonstrates some basic sensitivity calculation capabilities for the Popsy (Flattop) critical experiment (PU-MET-FAST-006 in the ICSBEP handbook).

## Contents

## Description

The experimental configuration consists of a plutonium alloy sphere (radius 4.5332 cm) reflected by a natural uranium blanket (outer radius 24.1420 cm). The multiplication factor of the system is close to unity (being a critical experiment). This input calculates the sensitivity of the multiplication factor to perturbations applied to the cross sections of ^{235}U, ^{238}U, ^{239}Pu and ^{240}Pu. A processing script for MATLAB/OCTAVE is provided for printing and plotting out the sensitivities.

## Input

**Input for Popsy (Flattop) sensitivity calculation example:**

% --- Core outer surface and blanket outer surface surf s1 sph 0.0 0.0 0.0 4.5332 surf s2 sph 0.0 0.0 0.0 24.1420 % --- Divide space to core, blanket and outside cell c1 0 core -s1 cell c3 0 blanket s1 -s2 cell c4 0 outside s2 % --- Core is made from plutonium alloy mat core sum 94239.03c 3.6697e-2 94240.03c 1.8700e-3 94241.03c 1.1639e-4 31000.03c 1.4755e-3 % --- Blanket is made from natural uranium mat blanket sum 92234.03c 2.6438e-6 92235.03c 3.4610e-4 92238.03c 4.7721e-2 % --- Cross section libraries %set acelib "<path-to-acelib>" % --- Neutron population set pop 10000 100000 50 1 100 100 % --- Use a central point source as the initial source src point sp 0.0 0.0 0.0 % --- Write output each 100 cycles set outp 100 % --- Modify event buffer size to 3 events per neutron set nbuf 3 3 % --- Use unresolved resonance sampling for the fast system set ures 1 2 % --- Define Vitamin-J energy grid (one of the defaults) ene myvit-j 4 nj23 % %%% % %%% --- Sensitivity options % %%% % --- Run sensitivity calculation using Vitamin-J energy grid sens opt egrid myvit-j % --- Use 15 latent generations for the sensitivity calculations sens opt latgen 15 % --- Use 10 generations for Iterated Fission Probability set ifp 10 % %%% % %%% --- Sensitivity responses % %%% % --- Calculate sensitivity of k-effective to perturbations sens resp keff % %%% % %%% --- Sensitivity perturbations % %%% % --- Separate perturbations for the different nuclides and total sens pert zailist 922350 922380 942390 942400 total % --- Do not calculate material-wise perturbations (only total) sens pert matlist total % --- Perturb cross sections separately for each sum reaction mode (fission, capture etc.) sens pert xs all % --- Score sensitivity results directly to a matrix for each particle % Assume 0.2 matrices needed per generation per particle sens opt direct 0.2

## Output

The contents of some long arrays have been replaced with "...".

**Output for Popsy (Flattop) sensitivity calculation example:**

% Number of different bins in sensitivity calculation: SENS_N_MAT = 1; SENS_N_ZAI = 5; SENS_N_PERT = 7; SENS_N_ENE = 175; % Materials included in sensitivity calculation: SENS_MAT_LIST = [ 'total ' ]; % Indices for different materials: iSENS_MAT_TOT = 1; % Nuclides included in sensitivity calculation: SENS_ZAI_LIST = [ 0 % total 922350 922380 942390 942400 ]; % Indices for different ZAIs: iSENS_ZAI_TOT = 1; iSENS_ZAI_922350 = 2; iSENS_ZAI_922380 = 3; iSENS_ZAI_942390 = 4; iSENS_ZAI_942400 = 5; % Reactions included in sensitivity calculation: SENS_PERT_LIST = [ 'total xs ' 'ela scatt xs ' 'sab scatt xs ' 'inl scatt xs ' 'capture xs ' 'fission xs ' 'nxn xs ' ]; % Indices for different perturbations: iSENS_PERT_TOT_XS = 1; iSENS_PERT_ELA_XS = 2; iSENS_PERT_SAB_XS = 3; iSENS_PERT_INL_XS = 4; iSENS_PERT_CAPT_XS = 5; iSENS_PERT_FISS_XS = 6; iSENS_PERT_NXN_XS = 7; % Sensitivity calculation energy group boundaries: SENS_E = [ ... ]; % Sensitivity calculation energy group lethargy widths: SENS_LETHARGY_WIDTHS = [ ... ]; % Sensitivities with 15 latent generations: % Effective multiplication factor: ADJ_PERT_KEFF_SENS = [ ... ]; ADJ_PERT_KEFF_SENS = reshape(ADJ_PERT_KEFF_SENS, [2, SENS_N_ENE, SENS_N_PERT, SENS_N_ZAI, SENS_N_MAT]); ADJ_PERT_KEFF_SENS = permute(ADJ_PERT_KEFF_SENS, [5, 4, 3, 2, 1]); ADJ_PERT_KEFF_SENS_E_INT = [ ... ]; ADJ_PERT_KEFF_SENS_E_INT = reshape(ADJ_PERT_KEFF_SENS_E_INT, [2, SENS_N_PERT, SENS_N_ZAI, SENS_N_MAT]); ADJ_PERT_KEFF_SENS_E_INT = permute(ADJ_PERT_KEFF_SENS_E_INT, [4, 3, 2, 1]);

## Data-processing and plotting

**Script for printing and plotting the results tested with MATLAB R2014a and OCTAVE 4.0.0:**

%% --- Flag to print out zero values printzeros = false; %% --- Input file name fname = 'input'; %% --- Run the sensitivity output file run([fname '_sens0.m']); %% --- Print some basic output data disp(sprintf('Results from Serpent sensitivity calculation using:\n')); %% --- Materials disp([num2str(SENS_N_MAT) ' perturbed materials:']); for i=1:1:SENS_N_MAT disp(SENS_MAT_LIST(i,:)); end %% --- Nuclides disp(sprintf(['\n' num2str(SENS_N_ZAI) ' perturbed nuclides (ZAIs):'])); for i=1:1:SENS_N_ZAI disp(sprintf('%d',SENS_ZAI_LIST(i,:))); end %% --- Perturbations disp(sprintf(['\n' num2str(SENS_N_PERT) ' different perturbations:'])); for i=1:1:SENS_N_PERT disp(SENS_PERT_LIST(i,:)); end %% --- Energy grid disp(sprintf(['\nEnergy grid with ' num2str(SENS_N_ENE) ' bins.'])); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% -- Output sensitivities one by one %%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(sprintf('\n********************\n\nSensitivity results:\n')) txt = sprintf('%-20s %-6s %-20s : %12s | 2 sigma confidence interval', 'Material', 'ZAI', ' Perturbation', 'sensitivity'); disp(txt) %% --- This script only prints/plots the k-eff sensitivities %% You can change the following three lines to plot something else res = ADJ_PERT_KEFF_SENS; resE = ADJ_PERT_KEFF_SENS_E_INT; resStr = 'K-eff'; for iMat=1:1:SENS_N_MAT matStr = SENS_MAT_LIST(iMat,:); for iZai=1:1:SENS_N_ZAI myZai = SENS_ZAI_LIST(iZai); %% --- Turn ZAI into a string for printing if (myZai == 0) zaiStr = sprintf('%-6s ', 'total'); else zaiStr = sprintf('%6d ', SENS_ZAI_LIST(iZai)); end %% --- Last loop is over perturbations for iPert=1:1:SENS_N_PERT %% --- Turn perturbation into a string for printing pertStr = sprintf('%-20s ', SENS_PERT_LIST(iPert,:)); val = resE(iMat,iZai,iPert,1); err = resE(iMat,iZai,iPert,2); %% --- Calculate 2 sigma confidence interval assuming %% normally distributed random variable minV = min([val*(1-2*err), val*(1+2*err)]); maxV = max([val*(1-2*err), val*(1+2*err)]); %% --- Create a formatted string for printing out the sensitivity txt = sprintf('%-20s %-6s %-20s: %+12.8f | [%+12.8f, %+12.8f]', matStr, zaiStr, pertStr, val, minV, maxV); %% --- Print formatted string if (printzeros || (val ~= 0) || (err ~= 0)) disp(txt); %% --- Move on to plotting a stairstep plot %% --- Convert energy grid to eV xvals = SENS_E*1e6; %% --- Get energy dependent sensitivity and relative error yvals = squeeze(res(iMat, iZai, iPert, :, 1))./SENS_LETHARGY_WIDTHS'; yerr = squeeze(res(iMat, iZai, iPert, :, 2)); %% --- We'll need to re-append the last value if we want to %% get the stairstep plot correct yvals = [yvals; yvals(end)]; yerr = [yerr; yerr(end)]; %% --- Create the figure for plotting fig1 = figure('visible', 'off'); %% --- Plot the sample mean h1 = stairs(xvals, yvals, 'k-'); set(h1, 'LineWidth', 2); %% --- Plot the sample mean +- 2*sqrt(sample variance) hold on; h2 = stairs(xvals, yvals.*(1+2*yerr), 'r-'); h3 = stairs(xvals, yvals.*(1-2*yerr), 'r-'); %% --- Fill the area of non-zero-sensitivity N = size(xvals,2)*2-2; xx = zeros(N,1); yy = zeros(N,1); xx(1:2:N) = xvals(1:1:end-1); xx(2:2:N) = xvals(2:1:end); yy(1:2:N) = yvals(1:1:end-1); yy(2:2:N) = yvals(1:1:end-1); h4 = fill(xx, yy, [1.0 0.8 0.8]); hold off; %% --- Make the picture a bit nicer set(gca,'xscale','log'); set(gca,'XTick', [1e4 1e5 1e6 1e7]); grid on; box on; xlim([1e4, SENS_E(end)*1e6]); ylim([-0.1, 0.5]); %% --- Label the axes and add a title xlabel('Energy (eV)'); ylabel('Sensitivity per unit lethargy'); txt = sprintf('Response: %s\nMaterial: %s\nZAI: %s\n perturbation %s', resStr, matStr, zaiStr, pertStr); title(txt); %% --- Save the plot to file filename = sprintf('%s_sens_%ld_%ld_%ld.png', resStr, iMat, iZai, iPert); print(filename,'-dpng'); %% --- Close figure close(fig1); end end %% --- Add empty line after each ZAI disp(sprintf(' ')); end end

### Scripts for visualizing with Python

The following two scripts rely on the serpentTools python package for the parsing of the output and plotting https://serpent-tools.readthedocs.io/en/latest/

Fonts are configured with matplotlib to look like LaTeX https://matplotlib.org/tutorials/introductory/customizing.html

Full example of the sensitivity reader can be found at https://serpent-tools.readthedocs.io/en/latest/examples/Sensitivity.html

**Simple python processing script:**

"""Script for visualizing k-eff sensitivities for Flattop problem""" from matplotlib import pyplot import serpentTools s = serpentTools.read("flattop_sens.m") # Only response is required, but then it plots against all # permutations of materials, perturbations, and materials # Pass material and/or isotope ZAI and/or specific perturbation to plot # sensitivities due to those properties only # s.plot handles log scaling, step plotting, and uncertainty s.plot( "keff", mat="total", zai=922380, pert="fission xs", ) # Format the plot to look like the MATLAB / OCTAVE example # No y axis limits pyplot.xlim(1E4, s.energies[-1]*1E6) # Turn on minor grids for the x-axis to show the log scale better pyplot.grid("x", which="both") pyplot.grid("y") # Add a title to the top of the figure pyplot.title("Response: K-eff Material: total ZAI: 922380 perturbation: fission xs") # Show the figure pyplot.show()

**Detailed python processing script:**

This script seeks to fully replicate the figures provided in the Wiki made using the MATLAB / OCTAVE programs

from matplotlib import pyplot import serpentTools s = serpentTools.read("flattop_sens.m") # Extract the response of interest from the sensitivities dictionary ks = s.sensitivities["keff"] # Obtain the expected value and uncertainty by indexing into the multi-dimensional # response. The matrix is structured [materials, isotopes, perturbations, energies, 2] # where the last dimension stores value and uncertainty # We can use the materials, zais, and perts dictionaries to obtain the correct index # in each dimension for the quantity of interest kslice = ks[ s.materials["total"], # index for sensitivity due to all materials s.zais[922380], # index for sensitivity due to U238 s.perts["fission xs"], # index for sensitivity due to fission xs ] # Normalize per unit lethargy # Python arrays are zero-indexed, so the expected value is in the 0 index value = kslice[:, 0] / s.lethargyWidths # Compute 2-sigma uncertainty unc = kslice[:, 1] * 2 * value # Draw errorbars # The energy vector has one additional entry, so we will instead drop the first item # by slicing from the second position forward pyplot.errorbar( s.energies[1:] * 1E6, # convert to MeV value, # expected value yerr=unc, # uncertainty drawstyle="steps-mid", # step-plot c="k", # draw using a black line ) # Shade the region under the curve red # Use a mild transparency pyplot.fill_between(s.energies[1:]*1E6, value, color="tab:red", alpha=0.3, step="mid") # Format the plot pyplot.xscale("log") pyplot.xlim(1E4, s.energies[-1]*1E6) # Major and minor grids for the log-scaled x axis pyplot.grid("x", which="both") pyplot.grid("y") pyplot.xlabel("Energy (eV)") pyplot.ylabel("Sensitivity per unit lethargy") pyplot.title("Response: K-eff Material: total ZAI: 922380 perturbation: fission xs") pyplot.show()

## Results

Runtime of the input on a computer cluster node with a Intel Xeon E5-2690 v2 processor (10 physical cores, 20 logical cores) using 20 OpenMP threads took approximately 4 hours 50 minutes and used (allocated) 6117 MB of RAM. The run produced an 133 KB `[input]_sens.m` file.

Executing the processing and plotting script produced the following output using OCTAVE 4.0.0

**Output from the processing script:**

Results from Serpent sensitivity calculation using: 1 perturbed materials: total 5 perturbed nuclides (ZAIs): 0 922350 922380 942390 942400 7 different perturbations: total xs ela scatt xs sab scatt xs inl scatt xs capture xs fission xs nxn xs Energy grid with 175 bins. ******************** Sensitivity results: Material ZAI Perturbation : sensitivity | 2 sigma confidence interval total total total xs : +0.90527000 | [ +0.90411125, +0.90642875] total total ela scatt xs : +0.16188300 | [ +0.16084695, +0.16291905] total total inl scatt xs : +0.07863120 | [ +0.07815941, +0.07910299] total total capture xs : -0.05447680 | [ -0.05459665, -0.05435695] total total fission xs : +0.71923200 | [ +0.71897308, +0.71949092] total total nxn xs : +0.00125871 | [ +0.00122598, +0.00129144] total 922350 total xs : +0.00818980 | [ +0.00809971, +0.00827989] total 922350 ela scatt xs : +0.00081898 | [ +0.00074855, +0.00088941] total 922350 inl scatt xs : +0.00038832 | [ +0.00035803, +0.00041861] total 922350 capture xs : -0.00054016 | [ -0.00054913, -0.00053120] total 922350 fission xs : +0.00752267 | [ +0.00747753, +0.00756781] total 922350 nxn xs : +0.00000661 | [ +0.00000463, +0.00000859] total 922380 total xs : +0.21901900 | [ +0.21805532, +0.21998268] total 922380 ela scatt xs : +0.13649800 | [ +0.13562441, +0.13737159] total 922380 inl scatt xs : +0.06499990 | [ +0.06459690, +0.06540290] total 922380 capture xs : -0.03974930 | [ -0.03986060, -0.03963800] total 922380 fission xs : +0.05727050 | [ +0.05714450, +0.05739650] total 922380 nxn xs : +0.00096484 | [ +0.00093783, +0.00099186] total 942390 total xs : +0.65488800 | [ +0.65424621, +0.65552979] total 942390 ela scatt xs : +0.02212830 | [ +0.02159722, +0.02265938] total 942390 inl scatt xs : +0.01241460 | [ +0.01216631, +0.01266289] total 942390 capture xs : -0.01294550 | [ -0.01299469, -0.01289631] total 942390 fission xs : +0.63329000 | [ +0.63301135, +0.63356865] total 942390 nxn xs : +0.00027191 | [ +0.00025614, +0.00028768] total 942400 total xs : +0.02006720 | [ +0.01991469, +0.02021971] total 942400 ela scatt xs : +0.00129135 | [ +0.00117513, +0.00140757] total 942400 inl scatt xs : +0.00063176 | [ +0.00056606, +0.00069746] total 942400 capture xs : -0.00102246 | [ -0.00103534, -0.00100958] total 942400 fission xs : +0.01916650 | [ +0.01909367, +0.01923933] total 942400 nxn xs : +0.00000945 | [ +0.00000567, +0.00001323]

These results were obtained using ENDF/B-VII.1 based cross section libraries.

A number of produced figures are also shown here for ease of comparison: