NeutronAnalysis

Neutron analysis
Calculate the fusion and neutron power
Constants and parameters
Neutron attenuation by Enriched B4C
LiH
Direct comparison - LiH and B4C for fast neutrons
Mean number of collisions to absorb incoming energy of 2.5 MeV neutron
Attentuation Comparision of B4C to LiH
Polyethlyene

Neutron analysis

Want to first scatter the neutrons until they are thermalized, then capture them. Compute mean free path and attenuation for both fast and thermalized neutrons for LiH and B4C.

https://www.nndc.bnl.gov/sigma/

%--------------------------------------------------------------------------
% See Also: PowerDHe3
%--------------------------------------------------------------------------

%--------------------------------------------------------------------------
% Copyright 2017 Princeton Satellite Systems, Inc. All rights reserved.
%--------------------------------------------------------------------------

% 2015 IAC paper ("Reducing Neutron Emission..."):  5 MW reactor, 3-0.6 kW/m2
% neutrons

% Examples:
% fD = 1, fT = 1, ion temp = 70 keV, fN = 2%
% fD = 1, fT = 0, ion temp = 70 keV, fN = 0.38%
% fD = 1, fT = 0, ion temp = 70/140 keV, fN = 0.15%
% fD = 0.16, fT = 0, ion temp = 70/140 keV, fN = 0.02%

global PSS_NO_INTERACTIVE_DEMOS

Calculate the fusion and neutron power

d = PowerDHe3;
ratio  = 2;     % ratio 3He:D
ne     = 4.3e20;
d.tHe3 = 100;   % potentially uo to 140 keV
d.tD   = 100;   % potentially 70 keV
d.tE   = 30;   % electron temp
d.fD   = 1;    % as low as 0.25
d.fT   = 0;

% calculate ion densities from electron density assuming fuel ratio
den = 1 + 2*ratio;
densityBC = 1/den*ne;
d.nD = densityBC;
d.nHe3 = densityBC*ratio;

% these are powers in MW per cubic meter
[pF, pN, pB, nNd, nNt] = PowerDHe3( d )

% Calculate the volume and area
% Wall radius (average shielding radius)
radiusWall = 0.5;
% this is an elongation of 5 (K = l/2r)
rPlasma  = 0.3;
kappa    = 5;
lPlasma  = 2.5;  % 2*rPlasma*kappa; %
% This is the volume of the separatrix
volume   = (2/3)*pi*rPlasma^2*lPlasma;
% The wall technically extends past the separatrix plus has end caps, this is
% just a simple cylinder
wallArea = 2*pi*radiusWall*lPlasma;

pFusion = pF*volume;
pNeutron = pN*volume;
neutronWallLoad = pN/wallArea;

fprintf('\nTotal Fusion Power: %3.2f MW\n',pFusion);
fprintf('\nFusion Power per length: %3.2f MW/m\n',pFusion/lPlasma);
fprintf('%.2f kW      Neutron power\n',pNeutron*1e3);
fprintf('%.2f%%        Neutron power as %% of fusion power\n',pN/pF*100);
fprintf('%.2f kW/m2   Neutron wall load\n',neutronWallLoad*1e3);
fprintf('%.2g n/s  Neutron rate\n',nNd);
fprintf('%.2g n/m2/s Neutron flux\n',nNd/wallArea);
fprintf('1 year fluence: %g /m2\n',nNd/wallArea*365*86400)

pF =
        7.581
pN =
     0.034926
pB =
       1.1203
nNd =
   8.8987e+16
nNt =
     0

Total Fusion Power: 3.57 MW

Fusion Power per length: 1.43 MW/m
16.46 kW      Neutron power
0.46%        Neutron power as % of fusion power
4.45 kW/m2   Neutron wall load
8.9e+16 n/s  Neutron rate
1.1e+16 n/m2/s Neutron flux
1 year fluence: 3.57306e+23 /m2

Constants and parameters

Avogadro's number

NA = 6.022e23;  % 1/mol

thickness30 = linspace(0,30,101); % cm
thickness20 = linspace(0,20,101); % cm
thickness10 = linspace(0,10,101);
thickness1 = linspace(0,1,101);

Neutron attenuation by Enriched B4C

thermal capture B10: 3840 b number density: NA * rho / M, molecules/m3 Nboron = Nb4c * 4 (4 atoms per molecule) http://holbert.faculty.asu.edu/eee460/NumberDensity.pdf B10: M = 10.0129370 B: M = 10.81 (about 80% B11) Note: density of B4C is 2.52 g/cm3 and (10)B4C is 2.465 g/cm3

disp('Enriched B4C')

% Microscopic cross-sections - sigma; in barns (10e-24 cm-2)
sigmaC_H   = 0;         % H - no capture
sigmaC_B   = 760;       % Natural boron (10/11)
sigmaC_B10 = 3840;      % B10 only
sigmaS_B   = 2;         % fast - scatter
sigmaS_C   = 1.5;       % fast - scatter

% macro sections are additive - 4B and 1C
densityBC = 2.465;     % g/cm2
Mb4c      = 4*10.01 + 12.01;  % atomic weight of boron-carbide, g/mol
Nb        = NA*densityBC/Mb4c*4;   % atoms/cm3 of boron in boron-carbide - 4 atoms/molecule
Nc        = NA*densityBC/Mb4c;     % atoms/cm3 of carbon - 1 atom

% fast scattering: 2 barns for B11, 1.5 barns for B10, 1.5 for C
% mean free path: cm
SigmaFast_BC = sigmaS_B*1e-24*Nb + sigmaS_C*1e-24*Nc
lambdaS_BC = 1/SigmaFast_BC
% 2/1.5 barns for B4C gives a lambda of 3.7 cm

attenFast_BC = exp(-SigmaFast_BC*thickness20);
attenFast90_BC = -log(0.1)/SigmaFast_BC
Plot2D(thickness20,attenFast_BC,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Fast Neutron Attenuation by B4C')
% disp('select location on plot for text...')
% gtext('\sigma = 2 barnes')

% thermal capture: 3840 barns for B10, essentially 0 for the carbon
SigmaC_BC = sigmaC_B*1e-24*Nb + 0*1e-24*Nc;
lambdaC_BC = 1/SigmaC_BC
SigmaC_B10C = sigmaC_B10*1e-24*Nb + 0*1e-24*Nc;
lambdaC_B10C = 1/SigmaC_B10C

attenCap_BC = exp(-SigmaC_BC*thickness1);
attenCap_B10C = exp(-SigmaC_B10C*thickness1);
Plot2D(thickness1,[attenCap_BC;attenCap_B10C],'Thickness (cm)','Intensity (\Phi/\Phi_0)','Thermal Neutron Attenuation by B4C')
% disp('select location on plot for text...')
% gtext('\sigma = 760 barnes')

Enriched B4C
SigmaFast_BC =
      0.27093
lambdaS_BC =
        3.691
attenFast90_BC =
       8.4988
lambdaC_BC =
     0.011534
lambdaC_B10C =
    0.0022828

LiH

thermal scattering H: 82 b thermal scattering Li: 1.37 b thermal capture Li: 70.5 b thermal capture Li6: 940 b fast total Li: 1.5 b

disp('LiH')

% these cross sections are for THERMAL neutrons
sigmaS_H = 82;
sigmaC_H = 0;
sigmaC_Li = 70.5;  % natural Li (6/7)
sigmaC_Li6 = 1000; % Li6

% for FAST neutrons
sigmaFast_H = 2.8;
sigmaFast_Li = 1.9;

densityLH = 0.78; % g/cm3
Mlh = 6.94 + 1.008;         % atomic weight of LH
Nlh = NA*densityLH/Mlh      % molecules/cm3, same for both Li and H

% Macro cross-sections
% fast - scattering: both elements contribute
SigmaFast_LH = (sigmaFast_H + sigmaFast_Li)*1e-24*Nlh;
attenFast90_LH = -log(0.1)/SigmaFast_LH

% capture - Li only
SigmaS_LH = sigmaS_H*1e-24*Nlh;
SigmaC_LH = sigmaC_Li*1e-24*Nlh;
SigmaC_L6H = sigmaC_Li6*1e-24*Nlh;

% mean free paths
lambdaFast_LH = 1/SigmaFast_LH
lambdaS_LH = 1/SigmaS_LH
lambdaC_LH = 1/SigmaC_LH
lambdaC_L6H = 1/SigmaC_L6H

% plots
attenFast_LH = exp(-SigmaFast_LH*thickness20);
Plot2D(thickness20,attenFast_LH,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Fast Neutron Attenuation by LiH')
if ~PSS_NO_INTERACTIVE_DEMOS
  disp('select location on plot for text...')
  gtext('\sigma = 2 barns')
end

scatterH = exp(-sigmaS_H*1e-24*Nlh*thickness10);
lambdaH = 1/(sigmaS_H*1e-24*Nlh)
Plot2D(thickness10,scatterH,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Thermal Neutron Scattering by H in LiH')
captureLi = exp(-sigmaC_Li*1e-24*Nlh*thickness10);
Plot2D(thickness10,captureLi,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Thermal Neutron Capture by Li in LiH')
captureLi6 = exp(-SigmaC_L6H*thickness10);
Plot2D(thickness10,captureLi6,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Thermal Neutron Capture by Li6 in LiH')
SigmaLH_t = (sigmaS_H + sigmaC_Li)*1e-24*Nlh;
Plot2D(thickness10,exp(-SigmaLH_t*thickness10),'Thickness (cm)','Intensity (\Phi/\Phi_0)','Total Transmission (Thermal) in LiH')

Plot2D(thickness10,exp(-SigmaC_L6H*thickness10),'Thickness (cm)','Intensity (\Phi/\Phi_0)','Total Transmission (Fast) in LiH')
hold on
yy = axis;
plot(yy(1:2),0.1*[1 1],'r')

LiH
Nlh =
   5.9099e+22
attenFast90_LH =
       8.2897
lambdaFast_LH =
       3.6002
lambdaS_LH =
      0.20635
lambdaC_LH =
      0.24001
lambdaC_L6H =
     0.016921
lambdaH =
      0.20635

Direct comparison - LiH and B4C for fast neutrons

Plot2D(thickness20,[attenFast_BC;attenFast_LH],'Thickness (cm)','Intensity (\Phi/\Phi_0)',...
  'Fast Neutron Attenuation (~2.5 MeV)')
legend('B4C','LiH')

Mean number of collisions to absorb incoming energy of 2.5 MeV neutron

thermal neutron: 0.025 eV

fprintf('Number of Collisions To Thermalize 2.5 MeV Neutron:\n')
% Boron
A = 5;
zeta = 1 + (A-1)^2/2/A*log((A-1)/(A+1));
n_B = 1/zeta*(log(2.5e6)-log(0.025))
% Lithium
A = 3;
zeta = 1 + (A-1)^2/2/A*log((A-1)/(A+1));
n_Li = 1/zeta*(log(2.5e6)-log(0.025))
% Hydrogen, zeta = 1
n_H = (log(2.5e6)-log(0.025))

Number of Collisions To Thermalize 2.5 MeV Neutron:
n_B =
       52.442
n_Li =
       34.245
n_H =
       18.421

Attentuation Comparision of B4C to LiH

aBC = densityBC*thickness30;
aLH = densityLH*thickness30;

NewFig('Comparison')
plot(aBC,attenFast_BC);
hold on;
plot(aLH,attenFast_LH);
legend('B4C','LH')
grid on
xlabel('Areal Mass (g/cm2)')
ylabel('Attenuation')
title('Comparison of LiH and B4C for Same Cross-Section')
if ~PSS_NO_INTERACTIVE_DEMOS
  gtext('\sigma = 2 barns')
end

% compare thermalized capture
sigmaCaptureB = 3840;
sigmaCaptureLi = 1000;
SigmaFast_BC = sigmaCaptureB*1e-24*Nb + 0*1e-24*Nc;
SigmaLiH = sigmaCaptureLi*1e-24*Nb + 0*1e-24*Nc;
lambdaCap = 1/SigmaFast_BC         % Mean free path (cm)
thickness = linspace(0,10,101); % cm
attenCap = exp(-SigmaFast_BC*thickness);
Plot2D(thickness,attenCap,'Thickness (cm)','Intensity (\Phi/\Phi_0)','Neutron Attenuation by 2 barns B4C')
if ~PSS_NO_INTERACTIVE_DEMOS
  disp('select location on plot for text...')
  gtext('\sigma = 3840 barns')
end

lambdaCap =
    0.0022828

Polyethlyene

http://www.shieldwerx.com/assets/swx-213.pdf

Nch2 = 7.9e22;

% Todo!


%--------------------------------------

Contents