PHGN590
Introduction to Nuclear Reactor Physics
Modeling Neutron Slowing in Reactors
J. A. McNeil
Physics Department
Colorado School of Mines
2/2009
Task: Role of moderation
In this task we explore the role of moderating (slowing) the neutrons. As can be seen from the data lists, for
-1
235
U the macro-
-1
scopic cross section for fission by a fast neutron is 0.068 cm , while that for a thermal (slow) neutron is 28.4 cm . A moderator
is a material that slows the neutrons down without absorbing them. Graphite ( 12 C) is an excellent moderator. It has a macroscopic scattering cross section of 0.381cm-1 and an absorption cross section of 0.00027 cm-1 . The addition of a moderating
material, like graphite, can alter the critical reactivity. Enrico Fermi used this to construct the first sustained chain reaction using
natural uranium as the fuel.
ü Analytic tasks
(a) To illustrate this calculate k¶, Eq.(8), for natural Uranium (235U (.72% ) and 238U
(99.28%)). Since Uranium is so much heavier than a neutron, elastic scattering does
not slow the neutron down and the fast cross sections must be used.)
H* Constants *L
Cons = 8kB Ø 1.38066 µ 10 ^ -23 , Troom Ø 293.15,
e -> 1.60219 µ 10 ^ -19, mn Ø 1.674929 µ 10 ^ -27<;
H* Thermal neutron values *L
U235Th = 8r Ø .01886, nd Ø .04833,
Ss Ø .01588, Sg Ø 4.833, Sf -> 28.37, n Ø 2.42<;
U238Th = 8r Ø .0191, nd Ø .04833, Ss Ø .4301,
Sg Ø .13194, Sf -> 0, n Ø 0<;
U235frac = .0072;
SigSNatUTh =
;
2
Moderation.nb
HU235frac Ss ê. U235ThL + HH1 - U235fracL Ss ê. U238ThL;
SigGNatUTh = HU235frac Sg ê. U235ThL +
HH1 - U235fracL Sg ê. U238ThL;
SigFNatUTh = HU235frac Sf ê. U235ThL +
HH1 - U235fracL Sf ê. U238ThL;
kinfNatUTh = HHn U235frac Sf ê. U235ThL +
Hn H1 - U235fracL Sf ê. U238ThLL ê
HSigGNatUTh + SigFNatUThL;
Print@" The thermal k-factor for an infinite
body of natural uranium is ", kinfNatUThD;
H* Fast neutron values *L
C12data = 8r Ø .00160, nd Ø .08023,
Ss -> .3811, Sg Ø .0002728 , Sf Ø 0, n Ø 0<;
U235Fast = 8r Ø .01886, nd Ø .04833, Ss Ø .328644,
Sg Ø .0120825, Sf -> .06766, n Ø 2.6<;
U238Fast = 8r Ø .0191, nd Ø .04833, Ss Ø .33347,
Sg Ø .007732, Sf -> .004591, n Ø 2.6<;
SigSNatUFast = HU235frac Ss ê. U235FastL +
HH1 - U235fracL Ss ê. U238FastL;
SigGNatUFast = HU235frac Sg ê. U235FastL +
HH1 - U235fracL Sg ê. U238FastL;
SigFNatUFast = HU235frac Sf ê. U235FastL +
HH1 - U235fracL Sf ê. U238FastL;
kinfNatUFast = HHn U235frac Sf ê. U235FastL +
Hn H1 - U235fracL Sf ê. U238FastLL ê
HSigGNatUFast + SigFNatUFastL;
Print@" The fast k-factor for an infinite body
of natural uranium is ", kinfNatUFastD;
The thermal k-factor for an
infinite body of natural uranium is 1.33581
The fast k-factor for an
infinite body of natural uranium is 1.02411
(b) The average energy of a neutron created in a fission event is about 2 MeV. What
fraction of the neutron's energy is lost in an elastic collision with a 12C nucleus where
the average of the cosine of the scattering angle is 32A ?
(c) How many collisions does it take to slow the neutron down to thermal energy (~1
eV)?
(b) The average energy of a neutron created in a fission event is about 2 MeV. What
Moderation.nb
fraction of the neutron's energy is lost in an elastic collision with a 12C nucleus
where3
the average of the cosine of the scattering angle is 32A ?
(c) How many collisions does it take to slow the neutron down to thermal energy (~1
eV)?
H* ModFac is the moderation factor =
fractional energy loss in
an elastic scattering collision
EF = Energy of the fast neutrons H2.5 MeVL
ETh = Energy when thermal
cross sections are applicable H1 eVL
NumScatt = number of elastic scatterings
EThêEF = ModFac^NumScatt --Ø
NumScatt =
Log@EThêEFDêLog@1-ModFacD *L
Avalue = 12; EF = 2. µ 10 ^ 6; ETh = 1.;
ModFac = N@2 Avalue ê H1 + AvalueL ^ 2D;
NumScatt = Floor@Log@ETh ê EFD ê Log@1 - ModFacDD;
Print@" For scattering from Carbon the fraction
of neutron energy lost is ", ModFacD;
Print@" Thus, ", NumScatt, " scatterings are required
to bring the neutron to thermal energies"D;
For scattering from Carbon the
fraction of neutron energy lost is 0.142012
Thus, 94 scatterings are required
to bring the neutron to thermal energies
(d) Now mix in graphite so that the number densities fractions are x of 12C and (1-x)
uranium (natural). Calculate the aggregate macroscopic cross sections for this mixture
using the fast neutron values. From these, calculate the probability, p(x), that a neutron survives to reach thermal energies. Plot the survival probability as a function of x.
(e) Now, that the neutrons are slowed down (with probability, p), recalculate the macroscopic cross sections for the x mixture using the thermal values and calculate the
critical factor, k(x), given by
n Sf
n Sf
k HxL = p
+ H1 - pL
.
Sa Thermal
Sa Fast
What is the optimal mixing fraction yielding the greatest value for k?
4
Moderation.nb
What is the optimal mixing fraction yielding the greatest value for k?
Clear@CmixD
SigSMixFast =
HCmix Ss ê. C12dataL + H1 - CmixL SigSNatUFast;
SigGMixFast = HCmix Sg ê. C12dataL +
H1 - CmixL SigGNatUFast;
SigFMixFast = H1 - CmixL SigFNatUFast;
Prob =
SigSMixFast
SigGMixFast + SigSMixFast + SigFMixFast
H1 - ProbL n SigFMixFast
NumScatt
;
ê. U235Fast;
SigGMixFast + SigFMixFast
SigSMixTh = HCmix Ss ê. C12dataL + H1 - CmixL SigSNatUTh;
SigGMixTh = HCmix Sg ê. C12dataL + H1 - CmixL SigGNatUTh;
SigFMixTh = H1 - CmixL SigFNatUTh;
Prob n SigFMixTh
kMixTh =
ê. U235Th;
SigGMixTh + SigFMixTh
kinfNatU = kMixTh + kMixFast;
kMixFast =
Plot@kinfNatU ê. Cmix Ø x, 8x, 0, 1<D
CmixMax = Cmix ê.
Flatten@FindRoot@∂Cmix kinfNatU, 8Cmix, 0.98`<DD;
kCmix = kinfNatU ê. Cmix Ø CmixMax;
kMixFast ê. Cmix Ø CmixMax;
kMixTh ê. Cmix Ø CmixMax;
1
MFPFast =
ê.
SigSMixFast + SigGMixFast + SigFMixFast
Cmix Ø CmixMax;
1
MFPTh =
ê.
SigSMixTh + SigGMixTh + SigFMixTh
Cmix Ø CmixMax;
1
AbsLenFast =
ê. Cmix Ø CmixMax;
SigGMixFast + SigFMixFast
Moderation.nb
AbsLenTh =
LFast =
1
SigGMixTh + SigFMixTh
AbsLenFast MFPFast
3
ê. Cmix Ø CmixMax;
ê. Cmix Ø CmixMax;
ê. Cmix Ø CmixMax;
3
LMix = Prob LTh + H1 - ProbL LFast ê. Cmix Ø CmixMax;
Print@" The probability of surviving to
thermal speeds is ", Prob ê. Cmix Ø CmixMaxD
Print@" The optimal mix of carbon is ",
CmixMax, " yielding k = ", kCmixD
Print@" Absorption lengths: AbsHFastL = ",
AbsLenFast, "
AbsHThermalL = ", AbsLenTh, " cm"D
Print@" Mean Free Path lengths: MFPHFastL = ",
MFPFast, "
MFPHThermalL = ", MFPTh, " cm"D
Print@" Diffusion lengths: LHFastL = ",
LFast, "
LHThermalL = ", LTh,
" cm \!\H\*OverscriptBox@\HL\L, \H_\LD\L = ", LMixD
AbsLenTh MFPTh
LTh =
1.20
1.15
1.10
1.05
1.00
0.95
0.2
0.4
0.6
0.8
1.0
5
6
Moderation.nb
The probability of surviving to thermal speeds is
0.834869
The optimal mix of carbon is
0.963601 yielding k = 1.20202
Absorption lengths: AbsHFastL =
1371.58
AbsHThermalL = 72.8205 cm
Mean Free Path lengths: MFPHFastL =
2.63093
MFPHThermalL = 2.52202 cm
Diffusion lengths: LHFastL = 34.6821
ê
LHThermalL = 7.82421 cm L = 12.2593
ü Monte Carlo simulation
(f) Develop a Monte Carlo simulation for this reactor configuration and calculate the
neutron multiplication factor, k, for your optimal mix.
Moderation.nb
ü Set up the simulation by defining the time step in terms of the
velocity and group cross sections
Clear@vcm, vboost, dPs, dPg, dPf, dPa, dP, dt, vmagD
H* This block can only be
executed after the previous section *L
Fastdata = 8r Ø 0.01886, nd Ø 0.04833,
Ss Ø SigSMixFast, Sg Ø SigGMixFast,
Sf Ø SigFMixFast, n Ø 2.6< ê. Cmix Ø CmixMax
Thermaldata = 8r Ø 0.01886, nd Ø 0.04833,
Ss Ø SigSMixTh, Sg Ø SigGMixTh,
Sf Ø SigFMixTh, n Ø 2.42< ê. Cmix Ø CmixMax
data = 8Fastdata, Thermaldata<;
muavg@A_D = 2 ê H3 AL; Avalue = 12;
vboost@vmag_D = vmag ê H1 + AvalueL ;
H* boost velocity connecting CM and Lab frames *L
vcm@vmag_D = Avalue vmag ê H1 + AvalueL ;
H* velocity of neutron in CM frame *L
EFast = 2.5 µ 10 ^ 6; vF = 100 Sqrt@2 He EFastL ê mnD ê. Cons;
ETh = 1.0; vTh = 100 Sqrt@2 He EThL ê mnD ê. Cons;
vroom = 100 vavg@TroomD ê. Cons;
dtofv@v_, ig_D := 1 ê H10 v HSs + Sg + SfLL ê. data@@igDD;
dtTh = dtofv@vTh, 2D; dtF = dtofv@vF, 1D;
dsF = vF dtF; dsTh = vTh dtTh;
dPsF = Ss dsF ê. Fastdata; dPsTh = Ss dsTh ê. Thermaldata;
dPfF = Sf dsF ê. Fastdata; dPfTh = Sf dsTh ê. Thermaldata;
dPgF = Sg dsF ê. Fastdata; dPgTh = Sg dsTh ê. Thermaldata;
dPaF = dPfF + dPgF; dPaTh = dPfTh + dPgTh;
dPF = dPsF + dPaF; dPTh = dPsTh + dPaTh;
Print@" Fast time step = ", dtF,
" sec, and average speed of, v = ", vF,
" cmês, distance per step = ", dsF, " cm"D
Print@" Thermal time step = ", dtTh,
, vTh,
7
8
Moderation.nb
" sec, and average speed of, v = ", vTh,
" cmês, distance per step = ", dsTh, " cm"D
Print@" The probabilities of each
possible event in one time step:"D
Print@"
Fast values:
scatter = ", dPsF,
" fission = ", dPfF, "
capture =", dPgFD;
Print@"
Thermal values: scatter = ", dPsTh,
" fission = ", dPfTh, "
capture =", dPgThD
8r Ø 0.01886, nd Ø 0.04833, Ss Ø 0.379365,
Sg Ø 0.000545448, Sf Ø 0.000183637, n Ø 2.6<
8r Ø 0.01886, nd Ø 0.04833, Ss Ø 0.382775,
Sg Ø 0.00629738, Sf Ø 0.00743502, n Ø 2.42<
Fast time step = 1.203 µ 10-10
sec, and average speed of, v = 2.18697 µ 109
cmês, distance per step = 0.263093 cm
Thermal time step = 1.82337 µ 10-7
sec, and average speed of, v = 1.38316 µ 106
cmês, distance per step = 0.252202 cm
The probabilities of
each possible event in one time step:
Fast values:
scatter = 0.0998082
fission = 0.0000483135
capture =0.000143503
Thermal values: scatter = 0.0965367
fission = 0.00187513
capture =0.00158821
Moderation.nb
ü Run the simulation:
Nexp = number of "experiments"
Nneutron= number of neutrons per experiment
TimingBRAbsAvgTable = 8<; kTable = 8<; NScattTable = 8<;
ElossTable = 8<; Rmax = 4 LMix; NRbins = 40;
Rmax
dR =
; NumDen = Table@0, 8i, 1, NRbins + 1<D;
NRbins
SsGroup = 8Ss ê. dataP1T, Ss ê. dataP2T<;
SgGroup = 8Sg ê. dataP1T, Sg ê. dataP2T<;
SfGroup = 8Sf ê. dataP1T, Sf ê. dataP2T<;
nGroup = 8n ê. dataP1T, n ê. dataP2T<;
Nexp = 10; Nneutrons = 200; NThTotal = 0;
0. DoBNFiss = 0; RAbsTable = 8<; NScattToTh = 0;
NTh = 0; NFastScattTotal = 0; Eloss = 0;
DoBForB8ig = 1, v = 80, 0, vF<; vmag = vF;
dt = dtofv@vmag, igD; r = 80, 0, 0<;
iStep = 1; iStop = -1; NFastScatt = 0<,
iStop < 0 && iStep < 105, iStep ++,
:ran = RandomReal@D; ds = vmag dtofv@vmag, igD;
dPs = ds SsGroupPigT; dPg = ds SgGroupPigT;
dPf = ds SfGroupPigT; r = r + v dt;
IfBran < dPs + dPg + dPfH* does it interact? *L,
H*YES, it interacts*L:IfBran < dPs
H* Is the interaction elastic? *L,
H*YES, elastic *L:vmagsave = vmag;
thcm = p RandomReal@D; phicm = 2 p RandomReal@D;
v = vcm@vmagD 8Sin@thcmD Cos@phicmD,
Sin@thcmD Sin@phicmD, Cos@thcmD< +
vboost@vD; vmag =
v.v ; IfBvmag § vTh
H* Is the new speed thermal? *L,
9
10
Moderation.nb
H* Is the new speed thermal? *L,
H* yes, change to group 2,
and keep speed constant from now on *L
vTh v
:ig = 2; v =
>, H* no, accumulate
vmag
fractional energy loss *L:Eloss =
Eloss + 1 -
vmag2
; NFastScatt ++>F;>,
vmagsave2
H* NO, inelastic *L8iStop = 1;
If@ran < dPs + dPfH* Was the inelastic
event fission? *L, H* Yes, accumulate
fission neutrons generated *LNFiss =
NFiss + nGroupPigTH* no, do nothing *LD<
FH* End elastic scattering IF *L
>FH* End interaction IF *L;
>FH* End For *L;
If@ig ã 2,
NScattToTh = NScattToTh + NFastScatt; NTh ++D;
NFastScattTotal = NFastScattTotal + NFastScatt;
rmag =
r.r ; AppendTo@RAbsTable, rmagD;
rmag
rindex = FloorB
F + 1; If@rindex > NRbins,
dR
NumDenPNRbins + 1T ++, NumDenPrindexT ++D;,
NFiss
8j, 1, Nneutrons<F; AppendToBkTable,
F;
Nneutrons
LenRabs = Length@RAbsTableD;
AppendToBRAbsAvgTable,
RAbsTablePiT
⁄Nneutrons
i=1
Nneutrons
NScattToTh
AppendToBNScattTable, NB
FF;
NTh
;
F;
Moderation.nb
AppendToBElossTable, NB
Eloss
NFastScattTotal
FF;
NThTotal = NThTotal + NTh;, 8iexp, 1, Nexp<F;
⁄iexp=1 ElossTablePiexpT
Nexp
ElossAvg =
8137.41, 0.141855<
Nexp
F
11
12
Moderation.nb
ü Calculate averages and deviations and plot the flux density
H* k Avergage *L
Lenk = Length@kTableD;
kAvg = N@Sum@kTable@@iDD, 8i, 1, Lenk<D ê LenkD;
Delk =
Sqrt@Sum@HkTable@@iDD - kAvgL ^ 2, 8i, 1, Lenk<D ê LenkD;
H* Average absorption radius *L
LenR = Length@RAbsAvgTableD;
RAbsAvg =
N@Sum@RAbsAvgTable@@iDD, 8i, 1, LenR<D ê LenRD;
DRAbs = Sqrt@N@Sum@HRAbsAvgTable@@iDD ^ 2 - RAbsAvg ^ 2L,
8i, 1, LenR<D ê LenRDD;
H* Average number of
scatterings to reach thermal speed *L
LenNScatt = Length@NScattTableD;
NScattAvg =
Sum@NScattTable@@iDD, 8i, 1, LenNScatt<D ê LenNScatt;
ProbSurviveToThermal = N@NThTotal ê Nneutrons ê NexpD;
H* Print Monte Carlo results *L
Print@" Neutron multiplication
constant
kHMonte CarloL = ", kAvg,
" +ê- ", Delk, " kHanalyticL = ", kCmixD
Print@" Average radius value at absorption = ",
RAbsAvg, " +ê_ ", DRAbs, " cm"D
Print@" Average number of scatterings needed
to reach thermal speeds = ", NScattAvgD
Print@" Survival probability HMCL = ",
ProbSurviveToThermal, " HanalyticL =",
Prob ê. Cmix -> CmixMaxD
Moderation.nb
13
Neutron multiplication constant
kHMonte CarloL =
1.18256 +ê- 0.0921539 kHanalyticL = 1.20202
Average radius value at absorption =
8.88485 +ê_ 0.310202 cm
Average number of scatterings
needed to reach thermal speeds = 92.3824
Survival probability HMCL = 0.8345 HanalyticL =0.834869
norm = ‚ NumDenPiT;
NRbins
i=1
DenNormed = TableBNB
NumDenPiT
norm
IShellVol@iR_D = dR3 4 p iR2;M
PhiMCTable = TableB
x
Phi@x_D =
‰- LD
x
F, 8i, 1, NRbins<F;
DenNormedPiT
ShellVol@iD
, 8i, 1, NRbins<F;
; Inorm = 4 p IntegrateAPhi@xD x2,
8x, 0, ¶<, Assumptions Ø Re@LDD > 0E;M
ê. LD Ø LMix; ; PhiMax =
norm
Max@PhiMCTableD;
PhiAnalyticTable =
Table@Phi0@Hi - 0.5`L dRD, 8i, 1, NRbins<D;
horizaxis = Style@"r HcmL", FontFamily Ø "Tahoma",
FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
vertaxis = Style@"j", FontFamily Ø "Tahoma",
FontColor Ø Blue, FontWeight Ø Bold, FontSize Ø 12D;
plotname = Style@"Neutron Flux", FontFamily Ø "Tahoma",
FontColor Ø Black, FontWeight Ø Bold, FontSize Ø 14D;
p1 = ListPlot@PhiMCTable, DisplayFunction Ø Identity,
PlotStyle Ø 8RGBColor@1, 0, 0D, PointSize@0.015`D<,
Frame Ø True, GridLines Ø Automatic, PlotLabel Ø
plotname, FrameLabel Ø 8horizaxis, vertaxis<,
,
,
Phi0@x_D =
Phi@xD
Moderation.nb
plotname, FrameLabel Ø 8horizaxis, vertaxis<,
ImageSize Ø 400, Background Ø LightOrange,
PlotRange Ø 8-.01 PhiMax, 1.2 PhiMax<D;
p2 = ListPlot@PhiAnalyticTable, Joined Ø True,
DisplayFunction Ø Identity,
PlotStyle Ø 8Black, Thickness@0.005`D<, Frame Ø True,
GridLines Ø Automatic, PlotLabel Ø plotname,
FrameLabel Ø 8horizaxis, vertaxis<,
ImageSize Ø 400, Background Ø LightOrange,
PlotRange Ø 8-.01 PhiMax, 1.2 PhiMax<D;
Show@p1, p2, DisplayFunction Ø $DisplayFunctionD
Neutron Flux
0.0005
0.0004
0.0003
j
14
0.0002
0.0001
0.0000
0
10
r HcmL
20
30
40