PHGN590
Introduction to Nuclear Reactor Physics
Temperature Dependence of Resonance Capture
J. A. McNeil
Physics Department
Colorado School of Mines
4/2009
Packages to be Used
Data
Cons = 8kB Ø 1.3806505 µ 10 ^ -23 , Troom Ø 293.15, e -> 1.60217653 µ 10 ^ -19,
mn Ø 1.674929 µ 10 ^ -27, clight Ø 2.99792458 µ 10 ^ 8, mnMeV Ø 939.56536<;
Eth@T_D = kB T ê e ê. Cons; H* Thermal energy in eV *L
D2Odata = 8r Ø .001105, nd -> .03323, Ss -> .4519, Sg Ø 4.42 µ 10 ^ -5 , Sf Ø 0, n Ø 0<;
C12data = 8r Ø .00160, nd Ø .08023, Ss -> .3811, Sg Ø .0002728 , Sf Ø 0, n Ø 0<;
H* Thermal neutron values *L
Nadata = 8r Ø .00097, nd Ø .02541, Ss Ø .08131, Sg Ø .01347 , Sf Ø 0, n Ø 0<;
U235data = 8r Ø .01886, nd Ø .04833, Ss Ø .01588, Sg Ø 4.833, Sf -> 28.37, n Ø 2.42<;
U238data = 8r Ø .0191, nd Ø .04833, Ss Ø .4301, Sg Ø .13194, Sf -> 0, n Ø 0<;
Pu239data = 8r Ø .0196, nd Ø .04938, Ss Ø .3902, Sg Ø 13.27, Sf -> 36.66 , n Ø 2.98<;
H* Fast neutron values *L
Nadata = 8r Ø .00097, nd Ø .02541, Ss Ø .083853, Sg Ø .000020328 , Sf Ø 0, n Ø 0<;
U235data = 8r Ø .01886, nd Ø .04833, Ss Ø .328644, Sg Ø .0120825, Sf -> .06766, n Ø 2.6<;
U238data = 8r Ø .0191, nd Ø .04833, Ss Ø .33347, Sg Ø .007732, Sf -> .004591, n Ø 2.6<;
Pu239data = 8r Ø .0196, nd Ø .04938, Ss Ø .33578, Sg Ø .0128388, Sf -> .091353, n Ø 2.98<;
2
TemperatureDependence.nb
Temperature dependence - effective cross section from thermal
averaged rates
ü Lorentzian resonance in cm frame
Clear@En, E0, G, v, v0, vL, vA, a, b, c, A, aD
Lorentzian@En_, E0_, G_D = a ^ 2 G ê H2 p L ê HH a ^ 2 En - E0L ^ 2 + G ^ 2 ê 4 L 2 ê H1 + 2 ArcTan@2 E0 ê GD ê pL
Integrate@Lorentzian@En, E0, GD, 8En, 0, +Infinity<,
Assumptions Ø 8Im@E0D ã 0, Im@GD ã 0, Re@GD > 0, Re@a ^ 2D > 0, Im@a ^ 2D ã 0<D
a2 G
p JI-E0 + En a2 M +
2
G2
4
N 1+
2 ArcTanB
2 E0
G
F
p
1
ü In the lab frame the target is moving with thermal velocity, vA, and the neutron is moving
with velocity, vL.
The neutron center of mass velocity is vcm = a (vL - vA), a = A/(A+1).
Thus, Ecm = a^2 ( EL + EA/A - 2 Sqrt[EL EA/A] cos(q) ). Integrating over angles gives:
Clear@PhiAvg, a, a, b, c, A, EL, EA, E0, GD
abc = 8a Ø a ^ 2 H EL + EA ê AL, b Ø 2 a ^ 2 Sqrt@EL EA ê AD, c Ø G ê 2 <;
PhiAvg@EL_, EA_, A_D = Simplify@
2 a ^ 2 ê H1 + 2 ArcTan@2 E0 ê GD ê pL s0 G ê H2 p L H1 ê 2L Integrate@1 ê HHa - b m - E0L ^ 2 + c ^ 2L, 8m, -1,
1<, Assumptions Ø 8Im@aD ã 0, Im@E0D ã 0, Im@bD ã 0, Im@cD ã 0, Re@cD > 0, Re@bD > 0<D ê. abcD
EA EL
EA+A EL-2 A
2 E0-
A
A
s0 ArcTanB
G
2
EA EL
A
a2
EA EL
EA+A EL+2 A
F + ArcTanB
Ip + 2 ArcTanA
2 -E0+
A
A
G
a2
F
2 E0
EM
G
ü Multiply by the Maxwell-Boltzmann distribution to get the thermal
average integrand
fMaxwell@EA_, kT_D = 2 ê Hp ^ H1 ê 2L HkTL ^ H3 ê 2LL Exp@-EA ê kTD;
Integrate@fMaxwell@EA, kTD Sqrt@EAD, 8EA, 0, Infinity<, Assumptions Ø 8Re@kTD > 0<D
1
TemperatureDependence.nb
integrand@EL_, EA_, A_, kT_D = Simplify@Sqrt@EAD fMaxwell@EA, kTD PhiAvg@EL, EA, ADD
EA EL
EA+A EL-2 A
2 E0-
‰
EA
kT
G
kT3ê2
a2
EA EL
EA+A EL+2 A
F + ArcTanB
A
EA s0 ArcTanB
EA EL
A
A
p Ip + 2 ArcTanA
2 -E0+
A
A
G
a2
F
2 E0
EM
G
ü Lorentzian folded with the Maxwell-Boltzmann distribution
ü Target and resonance data
Avalue = 12.; H* Target mass *L
E0value = 5.;H* location of resonance in eV *L
Gvalue = .1; H* width of resonance in eV *L
sub = 8s0 Ø 1, A Ø Avalue, E0 Ø E0value, G Ø Gvalue, a Ø Avalue ê HAvalue + 1L<
Sigma0@EL_D = s0 Lorentzian@EL, E0, GD ê. sub;
8s0 Ø 1, A Ø 12., E0 Ø 5., G Ø 0.1, a Ø 0.923077<
Tvalue = 300; kTvalue = Eth@TvalueD;
ELmin = Abs@HE0value - 15 GvalueL ê a ^ 2 ê. subD; ELmax = HE0value + 15 GvalueL ê a ^ 2 ê. sub ;
sTh@EL_, T_D := Abs@ NIntegrate@ integrand@EL, EA, Avalue, Eth@TDD ê. sub, 8EA, 0, Infinity<DD;
p1 = Plot@sTh@EL, TvalueD, 8EL , ELmin, ELmax<, PlotRange Ø All, PlotStyle Ø RGBColor@1, 0, 0DD;
p2 = Plot@Sigma0@ELD, 8EL, ELmin, ELmax<, PlotRange Ø All, PlotStyle Ø RGBColor@0, 0, 1DD;
Show@p1, p2D
sum = 0.; Npts = 400; dEL = HELmax - ELminL ê Npts;
Do@sum = sum + sTh@ELmin + dEL i, TvalueD, 8i, 1, Npts<D;
Print@" Normalization of thermalized cross section = ", sum dELD
Print@" Normalization of bare cross section = ", NIntegrate@Sigma0@ELD, 8EL, ELmin, ELmax<DD
5
4
3
2
1
5.0
5.5
6.0
6.5
7.0
7.5
Normalization of thermalized cross section = 0.981735
Normalization of bare cross section = 0.981913
3
4
TemperatureDependence.nb
ü Lorentzian fit
Enmin = HE0value - GvalueL ê a ^ 2 ê. sub; Enmax = HE0value + GvalueL ê a ^ 2 ê. sub;
Npts = 100; dEn = HEnmax - EnminL ê Npts;
sThtable = Table@8Enmin + i dEn, 1 ê sTh@Enmin + Hi - 1L dEn, TvalueD<, 8i, 1, Npts<D;
fitfun = Fit@sThtable, 81, x, x ^ 2<, xD;
Normfit = a ^ 4 ê fitfun@@3DD ê. sub ê. x Ø 1;
E0fit = - Normfit fitfun@@2DD ê H2 a ^ 2L ê. sub ê. x Ø 1;
Gfit = 2 Sqrt@Normfit fitfun@@1DD - E0fit^ 2D;
Print@" Overall Norm: ", Normfit, " HfitL ", a ^ 2 G ê H2 pL ê. sub, " HT=0L"D
Print@" Resonance energy: ", E0fit, " HfitL ", E0value, " HT=0L"D
Print@" Width: ", Gfit, " HfitL ", Gvalue, " HT=0L"D
sThfit@x_D = Normfit ê HHa ^ 2 x - E0fitL ^ 2 + Gfit^ 2 ê 4L ê. sub;
Enmin = Abs@HE0value - 10 GvalueL ê a ^ 2 ê. subD; Enmax = HE0value + 10 GvalueL ê a ^ 2 ê. sub ;
p1 = Plot@sThfit@EnD, 8En, Enmin, Enmax<, PlotRange Ø All, PlotStyle Ø RGBColor@1, 0, 0DD;
p2 = Plot@sTh@En, TvalueD, 8En, Enmin, Enmax<, PlotStyle Ø RGBColor@0, 0, 1DD;
p3 = Plot@Sigma0@EnD, 8En, Enmin, Enmax<, PlotRange Ø All, PlotStyle Ø RGBColor@0, 1, 0DD;
Show@8p1, p2, p3<D
H*ListPlot@LtableD*L
Overall Norm: 0.0468978 HfitL
0.0152975 HT=0L
Resonance energy: 5.00156 HfitL 5. HT=0L
Width: 0.234878 HfitL 0.1 HT=0L
6
5
4
3
2
1
5.0
5.5
6.0
Note: The actual resonance shape (blue) appears to drop off faster than the Lorentzian fit (red). Let's try a Gaussian form.
TemperatureDependence.nb
ü Gaussian fit
Enmin = HE0value - 1. GvalueL ê a ^ 2 ê. sub; Enmax = HE0value + 1. GvalueL ê a ^ 2 ê. sub;
Npts = 100; dEn = HEnmax - EnminL ê Npts;
LogsThtable = Table@8Enmin + Hi - 1L dEn, Log@sTh@Enmin + Hi - 1L dEn, TvalueDD<, 8i, 1, Npts<D;
fitfun = Fit@LogsThtable, 81, x, x ^ 2<, xD;
dE2fit = -1 ê fitfun@@3DD ê. sub ê. x Ø 1;
E0fit = fitfun@@2DD dE2fit ê H2 L ê. sub ê. x Ø 1;
sThfitNorm = Exp@fitfun@@1DD + E0fit^ 2 ê dE2fitD
sThfitNorm = 2 s0 ê Sqrt@p dE2fitD ê H1 + Erf@E0 ê Ha ^ 2 Sqrt@dE2fitDLDL ê. sub
H* analytic normalization *L
sThfit@En_D = sThfitNorm Exp@-HEn - E0 ê a ^ 2L ^ 2 ê dE2fitD ê. sub;
Enmin = Abs@HE0value - 10 GvalueL ê a ^ 2 ê. subD; Enmax = HE0value + 10 GvalueL ê a ^ 2 ê. sub;
p1 = Plot@sThfit@EnD, 8En, Enmin, Enmax<, PlotRange Ø All, PlotStyle Ø RGBColor@1, 0, 0DD;
p2 = Plot@sTh@En, TvalueD, 8En, Enmin, Enmax<, PlotRange Ø All, PlotStyle Ø RGBColor@0, 0, 1DD;
p3 = Plot@Sigma0@EnD, 8En, Enmin, Enmax<, PlotRange Ø All, PlotStyle Ø RGBColor@0, 1, 0DD;
Show@p1, p2, p3D
3.34596
4.05149
6
5
4
3
2
1
5.0
5.5
6.0
The Gaussian fit (red) has a higher peak, but drops more quickly than the Maxwell-Boltzmann folded Lorentzian (blue). The
original Lorentzian (green) is shown for comparison.
5
6
TemperatureDependence.nb
ü Examine temperature dependence of the Gaussian width
H* Gaussian fit *L
T0 = 175; dT = 40; NTemps = 30; DEtable = Table@0, 8iT, 1, NTemps<D;
Do@
8Tvalue = T0 + iT dT; kTvalue = Eth@TvalueD;
Enmin = HE0value - 1. GvalueL ê a ^ 2 ê. sub; Enmax = HE0value + 1. GvalueL ê a ^ 2 ê. sub ;
Npts = 100; dEn = HEnmax - EnminL ê Npts;
LogsThtable = Table@8Enmin + Hi - 1L dEn, Log@sTh@Enmin + Hi - 1L dEn, TvalueDD<, 8i, 1, Npts<D;
fitfun = Fit@LogsThtable, 81, x, x ^ 2<, xD;
dE2fit = -1 ê fitfun@@3DD ê. sub ê. x Ø 1;
DEtable@@iTDD = Sqrt@dE2fitD<, 8iT, 1, NTemps<D;
DE2plot = Table@8HT0 + iT dTL, DEtable@@iTDD ^ 2<, 8iT, 1, NTemps<D;
ListPlot@DE2plotD
DE2ofT@T_D = Fit@DE2plot, 81, T<, TD
0.06
0.05
0.04
0.03
200
400
600
0.00705232 + 0.0000418745 T
800
1000
1200
1400
TemperatureDependence.nb
H* Plot temperature dependence of Gaussian line shape *L
Clear@x, T, A, a, En, E0D
Enmin = Abs@HE0value - 20 GvalueL ê a ^ 2 ê. subD; Enmax = HE0value + 20 GvalueL ê a ^ 2 ê. sub ;
sub = 8s0 Ø 1, A Ø Avalue, E0 Ø E0value, G Ø Gvalue, a Ø Avalue ê HAvalue + 1L<;
sfit@En_, E0_, T_D := If@HEn - E0 ê a ^ 2L ^ 2 ê DE2ofT@TD > 100, 0, 2 s0 ê Sqrt@p DE2ofT@TDD ê
H1 + Erf@E0 ê Ha ^ 2 Sqrt@DE2ofT@TDDLDL Exp@-HEn - E0 ê a ^ 2L ^ 2 ê DE2ofT@TDDD ê. sub;
Eshift = 1;
p0 = Plot@sfit@En - Eshift, E0value, 300D,
8En, Enmin, Enmax + Eshift<, PlotRange Ø All, PlotStyle Ø RGBColor@1, 0, 0DD;
p1 = Plot@sfit@En, E0value, 1200D, 8En, Enmin, Enmax + Eshift<,
PlotRange Ø All, PlotStyle Ø RGBColor@1, 0, 0DD;
p2 = Plot@sTh@En - Eshift, 300D, 8En, Enmin, Enmax + Eshift<,
PlotRange Ø All, PlotStyle Ø RGBColor@0, 0, 1DD;
p3 = Plot@sTh@En, 1200D, 8En, Enmin, Enmax + Eshift<,
PlotRange Ø All, PlotStyle Ø RGBColor@0, 0, 1DD;
Show@
p0,
p1,
p2,
p3D
4
3
2
1
5
6
7
8
Integrate@sfit@En, E0value, 300D, 8En, 0, Infinity<D
sum = 0.; Npts = 1000; dEL = HE0 + 20 GL ê a ^ 2 ê Npts ê. sub;
Do@sum = sum + sTh@dEL i, TvalueD, 8i, 1, Npts<D;
sum dEL
1.
0.99197
7
8
TemperatureDependence.nb
Temperature dependent absorption cross section
ü Set up parameters for Monte Carlo simulation
Clear@speed, T, E0, i, Lfit, vmag, vboost, vD
AModerator = 12; vboost@v_D = v ê H1 + AModeratorL ; vcm@vmag_D = AModerator vmag ê H1 + AModeratorL ;
Ss = Ss ê. C12data; Sg = Sg ê. C12data; Sf = Sf ê. C12data; sResonanceRatio = 10 000;
Sg0 = sResonanceRatio Sg; H* ratio = resonance height ê sHscatteringL *L
KE@speed_D = H1 ê 2L mnMeV 10 ^ 6 H speed ê H100 clightLL ^ 2 ê. Cons;
H* Returns KE in eV as a function of speed in cmêsec *L
vofEn@En_D = Sqrt@2 En ê HmnMeV 10 ^ 6LD H100 clightL ê. Cons;
H* Returns speed in cmêsec as a function of KE in eV *L
NRes = 1; E0List = Table@i E0value, 8i, 1, NRes<D;
Sgfun@speed_, T_D = Sg + Sg0 Sum@sfit@KE@speedD, E0List@@iDD, TD, 8i, 1, NRes<D;
dtofv@speed_, T_D = 1 ê H5 speed HSs + Sgfun@speed, TD + SfLL;
Estart = .5 µ 10 ^ 2;
Plot@Sgfun@speed, 1200D, 8speed, vofEn@Eth@12 000DD, vofEn@EstartD<, PlotRange Ø AllD
6
5
4
3
2
1
4 µ 106
6 µ 106
8 µ 106
ü Monte Carlo simulation of survival probability as a function of temperature
SurvivalTable = 8<; LifetimeTable = 8<;
TemperatureDependence.nb
Timing@
Estart = 2. µ 10 ^ 6; vTh = vofEn@Eth@12 000DDH* speed at 1 eV = 12000 K *L;
Nexp = 10; Nneutrons = 10 000;
Tstart = 100; dT = 200;
Do@
Tvalue = Tstart + HiT - 1L dT; NThTable = 8<; tTable = 8<;
Do@H* Do Nexp number of experiments to get
statistics on the number of neutrons that reach thermal energies *L
NTh = 0;
Do@
For@8vmag = vofEn@EstartD; v = 80, 0, vmag<; r = 80, 0, 0<; iStep = 1; iStop = -1; ig = 1; t = 0<,
iStop < 0 && iStep < 10 ^ 6, iStep++,
8dt = dtofv@vmag, TvalueD; ds = vmag dt;
dPs = ds Ss; dPg = ds Sgfun@vmag, TvalueD;
r = r + v dt; t = t + dt; ran = Random@D;
H* Does it scatter ? *L
If@ran < dPs + dPg,
8H* If yes, was it an elastic scatter event? *L
If@ran < dPs,
8vmagsave = vmag; thcm = Pi Random@D; phicm = 2 Pi Random@D;
v = vcm@vmagD 8Sin@thcmD Cos@phicmD, Sin@thcmD Sin@phicmD, Cos@thcmD< + vboost@vD ;
vmag = Sqrt@v.vD; If@vmag < vTh, iStop = 1; NTh ++D
<, iStop = 1DH* ... if not elastic then an absorption event. *L
<DH* end If Scatter*L
<DH* end For *L
, 8j, 1, Nneutrons<D; H* End number of neutrons loop *L
AppendTo@NThTable, NThD; AppendTo@tTable, tD;
, 8iexp, 1, Nexp<D; H* End number of experiments loop *L
Survival = N@Sum@NThTable@@iDD, 8i, 1, Nexp<D ê HNexp NneutronsLD;
Lifetime = N@Sum@tTable@@iDD, 8i, 1, Nexp<D ê HNexp NneutronsLD;
Print@" T = ", Tvalue, " Survival = ", 100 Survival, " %, Mean lifetime = ", LifetimeD;
AppendTo@SurvivalTable, 8Tvalue, Survival<D;
AppendTo@LifetimeTable, 8Tvalue, Lifetime<D, 8iT, 1, 10<D;
DH* End Timing *L
T = 100 Survival = 56.107 %, Mean lifetime = 1.95205 µ 10-9
T = 300 Survival = 47.772 %, Mean lifetime = 1.63387 µ 10-9
T = 500 Survival = 42.293 %, Mean lifetime = 1.34111 µ 10-9
T = 700 Survival = 37.922 %, Mean lifetime = 1.59115 µ 10-9
T = 900 Survival = 34.452 %, Mean lifetime = 1.44815 µ 10-9
T = 1100 Survival = 31.3 %, Mean lifetime = 1.41613 µ 10-9
T = 1300 Survival = 28.827 %, Mean lifetime = 1.18957 µ 10-9
T = 1500 Survival = 26.64 %, Mean lifetime = 1.08567 µ 10-9
T = 1700 Survival = 24.616 %, Mean lifetime = 1.15688 µ 10-9
T = 1900 Survival = 22.735 %, Mean lifetime = 1.14442 µ 10-9
9
10
TemperatureDependence.nb
822 898.1, Null<
SurvivalTable
88300, 0.8809<, 8600, 0.8832<, 8100, 0.5266<, 8300, 0.4567<, 8500, 0.4237<, 8700, 0.39<,
8900, 0.3522<, 81100, 0.3412<, 81300, 0.3263<, 81500, 0.3059<, 81700, 0.2872<, 81900, 0.2715<,
8100, 0.56107<, 8300, 0.47772<, 8500, 0.42293<, 8700, 0.37922<, 8900, 0.34452<,
81100, 0.313<, 81300, 0.28827<, 81500, 0.2664<, 81700, 0.24616<, 81900, 0.22735<<
H* 1 resonances at E0 = 10 eV, G=.2 eV, A=50, peakêbackground ratio = 10000 *L
SurvivalProb =
88100, 0.56107`<, 8300, 0.47772`<, 8500, 0.42293`<, 8700, 0.37922`<, 8900, 0.34452`<,
81100, 0.313`<, 81300, 0.28827`<, 81500, 0.2664`<, 81700, 0.24616`<, 81900, 0.22735`<<;
ListPlot@SurvivalProb, PlotRange Ø 880, 2000<, 8.2, .6<<D
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0
500
1000
1500
2000
0
500
1000
1500
2000
H* 2 resonances at E0 = 10 eV and 20 eV, G=.2 eV, A=50, peakêbackground ratio = 10000 *L
SurvivalProb = 88100, 0.5266`<, 8300, 0.4567`<, 8500, 0.4237`<, 8700, 0.39`<, 8900, 0.3522`<,
81100, 0.3412`<, 81300, 0.3263`<, 81500, 0.3059`<, 81700, 0.2872`<, 81900, 0.2715`<<;
ListPlot@SurvivalProb, PlotRange Ø 880, 2000<, 8.2, .6<<D
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
TemperatureDependence.nb
H* 1 resonance at E0value = 3 eV, G=.01 eV, ratio = 2000 *L
SurvivalProb = 88100, .3985<, 8300, .3095<, 8600, .2375<,
8900, .2205<, 81200, .2025<, 81500, .182<, 81800, .1705<, 82100, .1705<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.1, .4<<D
88100, 0.3985<, 8300, 0.3095<, 8600, 0.2375<, 8900, 0.2205<,
81200, 0.2025<, 81500, 0.182<, 81800, 0.1705<, 82100, 0.1705<<
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0
500
1000
1500
2000
H* 1 resonance at E0value = 5 eV, ratio = 2000 *L
SurvivalProb = 88100, .597<, 8300, .4865<, 8600, .442<,
8900, .424<, 81200, .406<, 81500, .3855<, 81800, .348<, 82100, .345<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.3, .6<<D
88100, 0.597<, 8300, 0.4865<, 8600, 0.442<, 8900, 0.424<,
81200, 0.406<, 81500, 0.3855<, 81800, 0.348<, 82100, 0.345<<
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0
500
1000
1500
2000
11
12
TemperatureDependence.nb
H* 1 resonance at E0value = 8 eV, ratio = 2000 *L
SurvivalProb = 88100, .712<, 8300, .651<, 8600, .614<,
8900, .582<, 81200, .567<, 81500, .554<, 81800, .535<, 82100, .532<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.5, .8<<D
88100, 0.712<, 8300, 0.651<, 8600, 0.614<, 8900, 0.582<,
81200, 0.567<, 81500, 0.554<, 81800, 0.535<, 82100, 0.532<<
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0
500
1000
1500
2000
H* 1 resonance at E0value = 10 eV, ratio = 2000 *L
SurvivalProb = 88100, .759<, 8300, .732<, 8600, .67<,
8900, .653<, 81200, .634<, 81500, .628<, 81800, .609<, 82100, .615<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.5, .8<<D
88100, 0.759<, 8300, 0.732<, 8600, 0.67<, 8900, 0.653<,
81200, 0.634<, 81500, 0.628<, 81800, 0.609<, 82100, 0.615<<
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0
500
1000
1500
2000
TemperatureDependence.nb
H* 1 resonance at E0value = 20 eV, ratio = 2000 *L
SurvivalProb = 88100, .883<, 8300, .852<, 8600, .8285<,
8900, .8104<, 81200, .8046<, 81500, .7913<, 81800, .7839<, 82100, .7833<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.7, .9<<D
88100, 0.883<, 8300, 0.852<, 8600, 0.8285<, 8900, 0.8104<,
81200, 0.8046<, 81500, 0.7913<, 81800, 0.7839<, 82100, 0.7833<<
0.90
0.85
0.80
0.75
0.70
0
500
1000
1500
2000
H* 1 resonance at E0value = 100 eV, ratio = 2000 *L
SurvivalProb = 88100, .974<, 8300, .963<, 8600, .970<,
8900, .957<, 81200, .954<, 81500, .935<, 81800, .966<, 82100, .944<<
ListPlot@SurvivalProb, PlotRange Ø 880, 2200<, 8.9, 1.0<<D
88100, 0.974<, 8300, 0.963<, 8600, 0.97<, 8900, 0.957<,
81200, 0.954<, 81500, 0.935<, 81800, 0.966<, 82100, 0.944<<
1.00
0.98
0.96
0.94
0.92
0.90
0
500
1000
1500
2000
13
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TemperatureDependence.nb
H* 10 resonances at i^2 E0value H10 eVL , ratio = 2000 *L
SurvivalProb = 8820, .927<, 8100, .907<, 8300, .893<, 8900, .890<,
81500, .895<, 82100, .892<, 82700, .892<, 83300, .867<, 83900, .889<<
ListPlot@SurvivalProb, PlotRange Ø 8.8, 1<D
8820, 0.927<, 8100, 0.907<, 8300, 0.893<, 8900, 0.89<,
81500, 0.895<, 82100, 0.892<, 82700, 0.892<, 83300, 0.867<, 83900, 0.889<<
1.00
0.95
0.90
0.85
0
1000
2000
Resonance (ala Stacey)
3000