Nuclear Angular Correlation: Evaluation of Experimental Data

Pavla Sretrova, Vit Prochazka, Jan Riha
Palacky University in Olomouc, Faculty of Science, Department of Experimental Physics

Content

Mathematica at Department of Experimental Physics

Basic theory of Nuclear Angular Correlation

Simulations of TDPAC

Measurement of TDPAC

Random Events

Conclusion

Mathematica at Department of Experimental Physics

Mathematica is a powerful tool for universities

teaching

Mathematical Software in Natural Sciences - usage of Mathematica in daily life of student

Seminar in Mathematics for Physicists - interactive study materials for basic course of higher mathematics

research

calculations (e.g. calculation of scattering coefficients in the spherical harmonics basis to describe the effect of texture in nuclear resonant scattering experiments)

simulations (e.g. crystallization of materials placed in an external magnetic field)

data processing (fitting experimental data from TDPAC experiments)

Basic theory of Nuclear Angular Correlation

Intro to Nuclear Angular Correlation

Nuclear spectroskopic method ⟶ a studied material is built-in into a radioactive matrice (probe)

Atomic nucleus interacts with electrons and other nuclei via quadrupole and magnetic moments ⟶ hyperfine interactions cause splitting of nuclear levels

Physical chemical properties of matter

(^57)Co as a research probe

Radioactive nucleus disintegrates to its daughter nucleus (^57)Fe in the upper excited state ((2^nd)) (Overscript[⟶, (Subscript[γ, 1])]) the lower excited state with short lifetime (Subscript[τ, N]) ((1^st)) (Overscript[⟶, (Subscript[γ, 2])]) the ground state (G)

Picture 1: Scheme of gamma cascade after disintegration of nucleus.             Picture 2: Splitting of upper and lower excited states and ground state.

2 gamma photons of energies (Subscript[E, (Subscript[γ, 1])])=122 keV and (Subscript[E, (Subscript[γ, 2])])=14.4 keV emitted from ONE nucleus ⟶ ANGULAR DEPENDENCE

There is a correlation of two successive photons (Subscript[γ, 1])and (Subscript[γ, 2]) as a function of the angle θ between the propagation directions (Subscript[(Overscript[k, →]), 1]) and (Subscript[(Overscript[k, →]), 2]) of these radiations by one nucleus.

Picture 3: Detectors configuration at angular correlations measurement.                     Picture 4: Number of events vs. time delay between (Subscript[γ, 1]) and (Subscript[γ, 2]).

Time delay of successive photons in combination with splitting nuclear states cause modulation of exponential function (e^(-(t/τ))) by characteristic oscilations.

The oscillations are theoretically described by correlation function W(θ,t).

Dependence on hyperfine parameters, mainly on magnetic field (Overscript[B, →]) and the largest component of the electric field gradient tensor (Subscript[V, zz]).

Description of TDPAC

The general correlation function can be derived from quantum mechanics and has the following form

W(θ,t)∝exp(((-t)/(Subscript[τ, N])))(∑_k)(Subscript[A, kk])(Subscript[G, kk])(t)(Subscript[P, k])(cosθ)

Exponential attenuation (e^(-(t/(Subscript[τ, N])))).

Anisotropy coeficient (Subscript[A, kk]) - constant for given photon cascade .

(Subscript[P, k])(cos θ) is Legendre polynomial of (k^th) order with argument of angle θ.

Perturbation function for powder (Subscript[G, kk])(t) ⟶ contains all information on the interaction of the lower excited state of nuclei with their environment.

(Subscript[G, kk])(t)=(1/(2k+1))(∑_(N=-k)^k)(Subscript[G, kk]^NN)(t)

“Single crystal” perturbation function

(Subscript[G, (Subscript[k, 1])(Subscript[k, 2])]^((Subscript[N, 1])(Subscript[N, 2])))(t)=(∑_((Subscript[m, a])(Subscript[m, b])))(((-1))^(2I+(Subscript[m, a])+(Subscript[m, b])))(Sqrt[(2(Subscript[k, 1])+1)(2(Subscript[k, 2])+1)])·(({{I}, {(Subscript[m, a'])}})({{I}, {-(Subscript[m, a])}})({{(Subscript[k, 1])}, {N}}))·(({{I}, {(Subscript[m, b'])}})({{I}, {-(Subscript[m, b])}})({{(Subscript[k, 2])}, {N}}))
·(∑_(mm'))<(Subscript[m, b])|m(SuperStar[>])<(Subscript[m, a])|m><(Subscript[m, b'])|m'><(Subscript[m, a'])|m'(SuperStar[>])·exp{((-i)/ℏ)((Subscript[E, m])-(Subscript[E, m']))t}

The case of (^57)Fe

(^57)Co (Overscript[⟶, EC]) upper excited state of (^57)Fe with spin (SuperMinus[((5/2))]) (Overscript[⟶, (Subscript[γ, 1])]) lower excited state of (^57)Fe with  spin (SuperMinus[((3/2))]) (Overscript[⟶, (Subscript[γ, 2])]) ground state of (^57)Fe  spin (SuperMinus[((1/2))])

Three-level system with γ-γ cascade

Formula (1), (2) and (3) transform to the following forms

W(θ,t)=exp(((-t)/(97.8·(10^(-9)))))[1+(1/20)(Subscript[G, 22])(t)((3/2)(cos^2)(θ)-(1/2))],

where powder perturbation function (Subscript[G, 22])(t):

(Subscript[G, 22])(t)=(1/5)+(∑_(N=-2)^2)(Subscript[G, 22]^NN)(t),

where single perturbation function (Subscript[G, 22]^NN)(t) depends on type of hyperfine interactions.

(Subscript[G, (Subscript[k, 1])(Subscript[k, 2])]^((Subscript[N, 1])(Subscript[N, 2])))(t)=(∑_((Subscript[m, a])=-3/2)^(3/2))(∑_((Subscript[m, b])=-3/2)^(3/2))(((-1))^(2*(3/2)+(Subscript[m, a])+(Subscript[m, b])))(Sqrt[(2(Subscript[k, 1])+1)(2(Subscript[k, 2])+1)])·(({{I}, {(Subscript[m, a'])}})({{I}, {-(Subscript[m, a])}})({{(Subscript[k, 1])}, {(Subscript[N, 1])}}))·(({{I}, {(Subscript[m, b'])}})({{I}, {-(Subscript[m, b])}})({{(Subscript[k, 2])}, {(Subscript[N, 2])}}))
·(∑_(m=-3/2)^(3/2))(∑_((m^(,))=-3/2)^(3/2))<(Subscript[m, b])|m(SuperStar[>])<(Subscript[m, a])|m><(Subscript[m, b'])|m'><(Subscript[m, a'])|m'(SuperStar[>])·exp{((-i)/ℏ)((Subscript[E, m])-(Subscript[E, m']))t}

Procedure of perturbation and correlation functions calculation:

TreeGraph[{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1400,EdgeStyle->Arrowheads[.025]]

Simulations of TDPAC

Calculation of perturbation and correlation function for (^57)Fe

1. Hamiltonian

TreeGraph[{Style[1,Lighter[Blue]],2,3,4,5},{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1300,EdgeStyle->Arrowheads[.025]]

Finding of general Hamiltonian for lower excited state with spin (SuperMinus[((3/2))])

Hamiltonian of magnetic interactions

HM=(3/2)γ*ℏ*B{{-Cos[α],((Sqrt[3])/3)(E^(I β))Sin[α],0,0},{((Sqrt[3])/3)(E^(-I β))Sin[α],-(1/3)Cos[α],(2/3)(E^(I β))Sin[α],0},{0,(2/3)(E^(-I β))Sin[α],(1/3)Cos[α],((Sqrt[3])/3)(E^(I β))Sin[α]},{0,0,((Sqrt[3])/3)(E^(-I β))Sin[α],Cos[α]}};
MatrixForm[HM]

Hamiltonian of electric interactions

HQ=(1/4)e Q V{{1,0,(η/(Sqrt[3])),0},{0,-1,0,(η/(Sqrt[3]))},{(η/(Sqrt[3])),0,-1,0},{0,(η/(Sqrt[3])),0,1}};
MatrixForm[HQ]

Final Hamiltonian of mixed state

H=HM+HQ;
MatrixForm[H]

2. Auxiliary functions

TreeGraph[{Style[2,Lighter[Blue]],2,3,4,5},{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1300,EdgeStyle->Arrowheads[.025]]

Modifying of built-in functions such as Eigenvectors and ThreeJSymbol, because more conditions were required.

eigvec2[a_,b_]:=If[1≤ b≤ Length[eigvec],eigvec[[a,b]],0] (**)
mindex[m_]:=If [m==3/2,mindex[m]=1,If[m==1/2,mindex[m]=2,If[m==-1/2,mindex[m]=3,If[m==-3/2,mindex[m]=4,0]]]](*function for transition from one state to another*)
j3[ma_,n_,k_]:=If[(Abs[-(-ma+n)]≤ 3/2)&&Abs[n]≤ k,ThreeJSymbol[{(3/2),-(-ma+n)},{(3/2),-ma},{k,n}],0]

3. Single crystal perturbation function

TreeGraph[{Style[3,Lighter[Blue]],2,3,4,5},{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1300,EdgeStyle->Arrowheads[.025]]

Rewriting of formula (6) into code.

GNNkk[N1_,N2_,k1_,k2_,t_]:=(∑_(ma=-(3/2))^(3/2))(∑_(mb=-(3/2))^(3/2))(((-1))^(2*(3/2)+ma+mb))(Sqrt[(2k1+1)((2k2+1))])j3[ma,N1,k1]*j3[mb,N2,k2](∑_(o=1)^4)(∑_(p=1)^4)Conjugate[eigvec2[o,mindex[mb]]]*eigvec2[o,mindex[ma]]*eigvec2[p,mindex[mb-N2]]*Conjugate[eigvec2[p,mindex[ma-N1]]](E^((-(I/( ℏ))(eigval[[o]]-eigval[[p]])t)))

4. Powder perturbation function

TreeGraph[{Style[4,Lighter[Blue]],2,3,4,5},{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1300,EdgeStyle->Arrowheads[.025]]

Rewriting of formula (5) into code.

Gkk[k_,t_]:=FullSimplify[ComplexExpand[(1/(2k+1)) (∑_(n=-k)^k)GNNkk[n,n,k,k,t]]]

5. Correlation function

TreeGraph[{Style[5,Lighter[Blue]],2,3,4,5},{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->24]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->24]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->24]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->24]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->24]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.75,ImageSize->1300,EdgeStyle->Arrowheads[.025]]

Rewriting of formula (4) into code.

W[θ_,t_,Gkk_]:=((1+(1/20)*Gkk*((3/2)LegendreP[2,Cos[θ]]-(1/2))))(*without attenuation factor (E^(-(t/(Subscript[τ, N]))))*)

Perturbation and correlation functions simulations

Pure quadrupole interaction

Manipulate[(
dataQ=(Hobec=H/.{B->0,η->eta,ℏ -> 1.054*10^-34,e-> 1.602*10^-19,γ-> -4.96*10^6,Q-> 0.16*10^-28,V->v*6.005*10^21,α->0,β->0};
eigval=Eigenvalues[Hobec];
eigvec=Eigenvectors[Hobec];
Gkk[2,t]/.ℏ -> 1.054*10^-34);
Grid[{
Table[Style[i,Bold,FontFamily->"Arial",FontSize->36],{i,{"Perturbation function","Correlation function"}}],{"(Subscript[G, 22])(t)="dataQ,"W(90°,t)="W[Pi/2,t,dataQ]  },{" ","W(180°,t)="W[Pi,t,dataQ],SpanFromLeft},{Plot[{dataQ},{t,0,0.0000003},Ticks-> {{{2.5*10^-8,25},{5*10^-8,50},{7.5*10^-8,75},{10*10^-8,100},{12.5*10^-8,125}},Automatic},
Frame-> True,Axes->False,FrameLabel->Table[Style[i,Black,FontSize->30,FontFamily->"Arial"],{i,{"t [ns]","(Subscript[G, 22])(t)"}}],FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->Blue],
Plot[{W[Pi/2,t,dataQ],W[Pi,t,dataQ]},{t,0,0.0000003},Frame-> True,FrameLabel->Table[Style[i,Black,FontSize->30,FontFamily->"Arial"],{i,{"t [ns]","exp[t/(Subscript[τ, N])] W(θ,t)"}}],FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[Table[Style[i,FontFamily->"Arial",FontSize->30],{i,{"θ = 90°","θ = 180°"}}],{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{eta,0.25,"η"},Range[0,1,.25],RadioButton},{{v,3,"(Subscript[V, zz]) [mm/s]"},Range[0,5,.5],RadioButton},LabelStyle->36]

Pure magnetic interaction

Manipulate[(
dataM=(Hobec=H/.{B->b,η-> 0,ℏ -> 1.054*10^-34,e-> 1.602*10^-19,γ-> -4.96*10^6,Q-> 0*10^-28,V->0,α->alpha,β->beta};
eigval=Eigenvalues[Hobec];
eigvec=Eigenvectors[Hobec];
Gkk[2,t]/.ℏ -> 1.054*10^-34);
Grid[{Table[Style[i,Bold,FontFamily->"Arial",FontSize->36],{i,{"Perturbation function","Correlation function"}}]
,{"(Subscript[G, 22])(t)="dataM,"W(90°,t)="W[Pi/2,t,dataM]  },(*{" ","W(180°,t)="W[Pi,t,dataM],SpanFromLeft},*){Plot[{dataM},{t,0,0.0000003},Ticks-> {{{2.5*10^-8,25},{5*10^-8,50},{7.5*10^-8,75},{10*10^-8,100},{12.5*10^-8,125}},Automatic},
Frame-> True,Axes->False,FrameLabel->Table[Style[i,Black,FontSize->30,FontFamily->"Arial"],{i,{"t [ns]","(Subscript[G, 22])(t)"}}],FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->Blue],
Plot[{W[Pi/2,t,dataM],W[Pi,t,dataM]},{t,0,0.0000003},Frame-> True,FrameLabel->Table[Style[i,Black,FontSize->30,FontFamily->"Arial"],{i,{"t [ns]","exp[t/(Subscript[τ, N])] W(θ,t)"}}],FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[Table[Style[i,FontFamily->"Arial",FontSize->30],{i,{"θ = 90°","θ = 180°"}}],{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{b,30,"B [T]"},Range[0,50,3],RadioButton},{{alpha,(π/6),"α"},Range[0,Pi,Pi/6],RadioButton},{{beta,(π/3),"β"},Range[0,2Pi,Pi/6],RadioButton},LabelStyle->36]

Combined electric quadrupole and magnetic dipole interactions

Manipulate[(
mixed=(Hobec=H/.{B->b,η-> eta,ℏ -> 1.054*10^-34,e-> 1.602*10^-19,γ-> -4.96*10^6,Q-> 0.16*10^-28,V->(v*6.005*10^21),α->alpha,β->beta};
eigval=Eigenvalues[Hobec];
eigvec=Eigenvectors[Hobec];
Gkk[2,t]/.ℏ -> 1.054*10^-34);
Grid[{{Style["Perturbation function",Bold,FontFamily->"Arial",FontSize->36],Style["Correlation function",Bold,FontFamily->"Arial",FontSize->36]},{"(Subscript[G, 22])(t)="mixed,"W(90°,t)="W[Pi/2,t,mixed]  },(*{" ","W(180°,t)="W[Pi,t,mixed],SpanFromLeft},*)
{Plot[{mixed},{t,0,0.0000003},Ticks-> {{{2.5*10^-8,25},{5*10^-8,50},{7.5*10^-8,75},{10*10^-8,100},{12.5*10^-8,125}},Automatic},
Frame-> True,Axes->False,FrameLabel->{Style["t [ns]",Black,FontSize->30,FontFamily->"Arial"],Style["(Subscript[G, 22])(t)",Black,FontSize->30,FontFamily->"Arial"]},FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->Blue],
Plot[{W[Pi/2,t,mixed],W[Pi,t,mixed]},{t,0,0.0000003},Frame-> True,FrameLabel->{Style["t [ns]",Black,FontSize->30,FontFamily->"Arial"],Style["exp[t/(Subscript[τ, N])] W(θ,t)",Black,FontSize->30,FontFamily->"Arial"]},FrameTicks->{{Automatic,None},{Table[{i*10^-8,i*10},{i,0,100,10}],None}},FrameTicksStyle->Directive[Black,30],ImageSize->700,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[{Style["θ = 90°",FontFamily->"Arial",FontSize->30],Style["θ = 180°",FontFamily->"Arial",FontSize->30]},{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{eta,0.25,"η"},Range[0,1,.25],RadioButton},{{v,3,"(Subscript[V, zz]) [mm/s]"},Range[0,5,.5],RadioButton},{{b,30,"B [T]"},Range[0,50,3],RadioButton},{{alpha,0,"α"},Range[0,Pi,Pi/6],RadioButton},{{beta,0,"β"},Range[0,2Pi,Pi/6],RadioButton},LabelStyle->36]

Measurement of TDPAC

Experimental setup

a built-in sample into a radioactive matrice

scintillation detectors with high time resolution (10 ns or better)

constant fraction discriminators

time to amplitude convertors

multi channel analyzers

PC with software Maestro

Picture 5: Schema of TDPAC experimental arrangement.

Data

Time delayed correlation histogram

fitting function

(e^(-((((t-(Subscript[t, 0])))^2)/(σ^2))))⊗[H(t-(Subscript[t, 0]))*(A (e^(-(t/(Subscript[τ, random]))))+B (e^(-(t/(Subscript[τ, N])))))]

Gaussian peak  (e^(-((((t-(Subscript[t, 0])))^2)/(σ^2))))

Heaviside step function H(t-(Subscript[t, 0]))

“Slow” exponential function A (e^(-(t/(Subscript[τ, random])))) ⟶ Random events (pairing of photons from different nuclei)

“Fast” exponential function B (e^(-(t/(Subscript[τ, N])))) ⟶ TDPAC ((Subscript[τ, N]) is lifetime of the first excited state of (^57)Fe)

The first approach: fitting only by fast exponential function

There are problems with our measuring system

Random events measurement (slow exponential function)

Random events

Probability of photon (Subscript[γ, 2]) detection in certain time interval (T = 2 ms) from detection of photon (Subscript[γ, 1])

The time between specific pulses ((Subscript[γ, 1]) and (Subscript[γ, 2])) from different nuclei is completely unpredictable

The time intervals do converge to a well-defined average τ≈(T/N)

An average rate of pulses r=(1/τ)

The time delayed histogram of random coincidence between (Subscript[γ, 1]) (122 keV) and (Subscript[γ, 2]) (14.4 keV) pulses is describe by following formula

(Subscript[I, 1])(t)=r (e^(-r t))

Data

Import and drawing experimental data

SetDirectory[NotebookDirectory[]];
d=Import["october_2019/16_10_random_coincideence_2milions_ns.Spe","List"];(*import of data file*)
dat=d[[13+5;;1023-16]];(*selection of data*)
data=Table[{i*1953,dat[[i]]/Max[dat]},{i,1,Length[dat]}];(*data normalization*)
dataListPlot=ListPlot[Tooltip[data],
PlotStyle->Black,PlotLegends->{"random coincidence,\ntime window 2 ms"},
AxesLabel->{"t [ms]","I(t)"},LabelStyle->Directive[30,Black],
Ticks->{Table[{i*10^6,i},{i,.25,2,.25}],Automatic},
PlotRange-> {{-5*10^4,2*10^6},Automatic},
ImageSize->900](*plotting of normalized data*)

Data fitting

fit=FindFit[data,(A/τ )Exp[-(t/(τ))],{A,{τ,2.38*10^6}},t];(*fit of normalized data*)
fitPlot=Plot [A/τ  Exp[-(t/(τ))]/.fit,{t,-5*10^4,2*10^6},
PlotStyle->Directive[Red,Thickness[.005]],
AxesLabel->{"t [ms]","I(t)"},
LabelStyle->Directive[30,Black],
PlotLegends->{StringJoin["fit ","τ = ",ToString[(τ/.fit)*10^-6]," ms"]},
ImageSize->900];(*plotting of fitting function*)
Show[dataListPlot,fitPlot]

The average rate of pulses r

N[1/((τ/.fit)*10^-9)]//Round[#,1.]&(*counts per second*)

Number of 14.4 keV photons per second measured by CFD : 422

Theoretical number of 14.4 keV photons per second calculated from total activity formula of radioactive sample: 420

q[r_]:=Exp[-r t*10^-9]
Plot[{A/τ  Exp[-(t/(τ))]/.fit,q[422],q[420]},{t,-5*10^4,2*10^6},
PlotStyle->Table[Directive[i,Thickness[.005]],{i,{Red,Purple,Darker[Green]}}],
AxesLabel->{"t [ms]","I(t)"},
LabelStyle->Directive[30,Black],
PlotLegends->{StringJoin["(Subscript[τ, FIT]) = ",ToString[(τ/.fit)*10^-6]," ms\n(Subscript[r, FIT]) = ",ToString[N[1/((τ/.fit)*10^-9)]//Round[#,1.]&],"cnt/s"],StringJoin["(Subscript[τ, CFD]) = ",ToString[N[1/422]*10^3]," ms\n(Subscript[r, FIT]) = 422 cnt/s"],StringJoin["(Subscript[τ, calculated]) = ",ToString[N[1/420]*10^3]," ms\n(Subscript[r, calculated]) = 420 cnt/s"]},
ImageSize->900]

Conclusion

Mathematica is a powerful tool for universities ⟶ usage in teaching and research

Calculations, simulations, evaluations of experimental data

TDPAC is a high resolution spectroscopic method

study of influence of hyperfine interaction on a shape of correlation funcion

Random events

background signal in TDPAC experiments

Thank you for your attention