Nuclear Angular Correlations:
Perturbation and Correlation Functions

Pavla Šretrová, Vít Procházka, Jan Říha

Palacky University in Olomouc
Faculty of Science
Department of Experimental Physics

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Content

Basic theory of Nuclear Angular Correlation

The case of (^57)Fe

Calculations of perturbation and correlation functions step by step guide

Graphs of (Subscript[G, 22])(t) and W(θ,t)

Experiment simulations

Summery

Basic Theory of Nuclear Angular Correlation

Background

Material research in solids focuses on the phase transformations and chemical reaction.

Nuclear analytical methods  (mainly Mőssbauer spectroscopy)

atomic nuclei as a probe for inspection of electronic configuration (magnetic and charge ordering).

Picture 1: Scheme of Mössbauer spectrometer.

In several cases there is a failure of these techniques -> using other methods for detailed observation in the local arrangement of atoms.

(Time Differential Perturbed) Angular Correlation.

Basic Theory of Nuclear Angular Correlation

Intro to Nuclear Angular Correlation

Atomic nuclei as an investigative probe of solid and liquid structures.

Incorporation of a specific radioactive isotope into the investigated sample.

Specific radiation (α, β, γ) is emitted via radioactive nuclei decay.

Isotope characterization

Mass M

Proton number charge Z

Nucleon number A

Spin (Overscript[I, →])

Quadrupole moment Q

Magnetic moment (Overscript[m, →])

Interaction of a nucleus with electrons via magnetic and quadrupole moments reflecting arrangement of electrons.

Basic Theory of Nuclear Angular Correlation

Random arrangement of nuclei magnetic moments.

Random emission of gamma photons -> isotropic radiation field.

Picture 2: Isotropic radiation field.                                 Picture 3: Anisotropic radiation field.

Orientation of nuclei (anisotropic radiation field) can be caused by extremely low temperatures -> too complicated.

Cascade of two successive gamma photons also creates anisotropic radiation field of nuclei.

Radioactive nucleus disintegrates to its daughter nucleus 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 4: Scheme of gamma cascade after disintegration of nucleus.               Picture 5: Splitting of upper and lower excited states and ground state.

Information of nuclei orientation in the moment of their deexcitation is encoded in direction of emitted photons.

Nuclear interaction with electrons and other nuclei via quadrupole and magnetic moments. These interactions, called hyperfine, cause splitting of nuclear levels.

Basic Theory of Nuclear Angular Correlation

The observation of gamma photon in a given direction selects an ensemble of nuclei oriented in one direction.

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 6: Detectors configuration at angular correlations measurement.                       Picture 7: Number of events vs. time delay between (Subscript[γ, 1]) and (Subscript[γ, 2]).

Time delay of successive photons in combination with splitting nuclear state causes modulation of exponential function by characteristic oscilation.

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]).

Basic Theory of Nuclear Angular Correlation

The 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

We are interested only in radioactive Mössbauer isotope (^57)Co.

(^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}

TreeGraph[{1\[DirectedEdge] 2,2\[DirectedEdge] 3,3\[DirectedEdge] 4,4\[DirectedEdge] 5},VertexLabels->{1->Placed[{Style["Hamiltonian",Bold,FontSize->16]},Center],2->Placed[{Style["Eigenvalues\n+\nEigenstates",Bold,FontSize->16]},Center],3->Placed[{Style["Single crystal\nperturbation\nfunction",Bold,FontSize->16]},Center],4->Placed[{Style["Powder\nperturbation\nfunction",Bold,FontSize->16]},Center],5->Placed[{Style["Correlation\nfunction",Bold,FontSize->16]},Center]},VertexShapeFunction->"Rectangle",VertexSize->.5,ImageSize->1000]

Procedure of perturbation and correlation functions calculation:

Calculations of perturbation and correlation functions step by step guide

1. Hamiltonian

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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

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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

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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

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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

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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]))))*)

Graphs of (Subscript[G, 22])(t) and W(θ,t)

Pure quadrupole interaction

Perturbation function for pure quadrupole interaction

the asymmetry parameter: η = 0.3

the largest component of the electric field gradient tensor: (Subscript[V, zz]) = 3 mm/s

(Subscript[G, 22])(t)=(1/5)+(4/5)cos(ωt),

where ω = eQ(Subscript[V, zz]) (Sqrt[1 + ((η^2)/3)])/2ℏ.

Correlation function

W(θ,t)=(e^(-(t/(97.8·(10^(-9))))))[1+(1/20)((1/5)+(4/5)cos(ωt))((3/2)(cos^2)(θ))]

Graphs

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[{
{Style["Perturbation function",FontFamily->"Arial",FontSize->20],Style["Correlation function",FontFamily->"Arial",FontSize->20]},{"(Subscript[G, kk])="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->{Style["t [ns]",Black,FontSize->20,FontFamily->"Arial"],Style["(Subscript[G, 22])(t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{Automatic,None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->Blue],
Plot[{W[Pi/2,t,dataQ],W[Pi,t,dataQ]},{t,0,0.0000003},Frame-> True,FrameLabel->{Style["t [ns]",Black,FontSize->20,FontFamily->"Arial"],Style["exp[t/(Subscript[τ, N])] W(θ,t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{{0.95,{1,"1.00"},1.05},None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[{Style["θ = 90°",FontFamily->"Arial",FontSize->20],Style["θ = 180°",FontFamily->"Arial",FontSize->20]},{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{eta,0.3,"η"},{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},RadioButton},{{v,3,"(Subscript[V, zz]) [mm/s]"},Range[0,5,.5],RadioButton}]

Pure magnetic interaction

Perturbation function for pure magetic interaction

B = 30 T, α = (π/6), β = (π/3)

(Subscript[G, 22])(t)=(1/5)+(2/5)cos((Subscript[ω, L])t)+(2/5)cos(2(Subscript[ω, L])t),

where (Subscript[ω, L]) = ((g·(Subscript[μ, N])·B)/ℏ).

Correlation function

W(θ,t)=(e^(-(t/(97.8·(10^(-9))))))[1+(1/20)((1/5)+(2/5)cos((Subscript[ω, L])t)+(2/5)cos(2(Subscript[ω, L])t))((3/2)(cos^2)(θ))]

Graphs

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[{
{Style["Perturbation function",FontFamily->"Arial",FontSize->20],Style["Correlation function",FontFamily->"Arial",FontSize->20]},{"(Subscript[G, kk])="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->{Style["t [ns]",Black,FontSize->20,FontFamily->"Arial"],Style["(Subscript[G, 22])(t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{Automatic,None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->Blue],
Plot[{W[Pi/2,t,dataM],W[Pi,t,dataM]},{t,0,0.0000003},Frame-> True,FrameLabel->{Style["t [ns]",Black,FontSize->20,FontFamily->"Arial"],Style["exp[t/(Subscript[τ, N])] W(θ,t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{{0.95,{1,"1.00"},1.05},None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[{Style["θ = 90°",FontFamily->"Arial",FontSize->20],Style["θ = 180°",FontFamily->"Arial",FontSize->20]},{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{b,30,"B [T]"},{0,5,10,15,20,25,30,35,40,45,50},RadioButton},{{alpha,(π/6),"α"},{0,Pi/6,Pi/5,Pi/4,Pi/3,Pi/2,5Pi/6,Pi},RadioButton},{{beta,(π/3),"β"},{0,Pi/6,Pi/5,Pi/4,Pi/3,Pi/2,5Pi/6,3Pi/2,Pi,3Pi/2,2Pi},RadioButton}]

Combined electric quadrupole and magnetic dipole interactions

Graphs

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",FontFamily->"Arial",FontSize->20],Style["Correlation function",FontFamily->"Arial",FontSize->20]},{"(Subscript[G, kk])="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->20,FontFamily->"Arial"],Style["(Subscript[G, 22])(t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{Automatic,None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->Blue],
Plot[{W[Pi/2,t,mixed],W[Pi,t,mixed]},{t,0,0.0000003},Frame-> True,FrameLabel->{Style["t [ns]",Black,FontSize->20,FontFamily->"Arial"],Style["exp[t/(Subscript[τ, N])] W(θ,t)",Black,FontSize->20,FontFamily->"Arial"]},FrameTicks->{{{0.95,{1,"1.00"},1.05},None},{{{0,0},{10*10^-8,100},{20*10^-8,200},{30*10^-8,300},{40*10^-8,400},{50*10^-8,500},{60*10^-8,600}},None}},FrameTicksStyle->Directive[Black,20],ImageSize->500,PlotStyle->{Directive[Black],Directive[Red]},PlotLegends->Placed[{Style["θ = 90°",FontFamily->"Arial",FontSize->20],Style["θ = 180°",FontFamily->"Arial",FontSize->20]},{Right,Top}],PlotRange->{{0,30*10^-8},{0.93,1.08}},ImageSize->500]}}]),{{eta,0.3,"η"},{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1},RadioButton},{{v,3,"(Subscript[V, zz]) [mm/s]"},Range[0,5,.5],RadioButton},{{b,30,"B [T]"},{0,5,10,15,20,25,30,35,40,45,50},RadioButton},{{alpha,0,"α"},{0,Pi/6,Pi/5,Pi/4,Pi/3,Pi/2,5Pi/6,Pi},RadioButton},{{beta,0,"β"},{0,Pi/6,Pi/5,Pi/4,Pi/3,Pi/2,5Pi/6,3Pi/2,Pi,3Pi/2,2Pi},RadioButton}]

Experiment simulations

W(θ,t) vs. Mössbauer spectra

Comparison of Mőssbauer Spectroscopy & Time Differential Perturbed Angular Correlation  (TDPAC)

Similarity of Mössbauer Spectra in compare with TDPAC

Comparable measurement time

({{, }, {, }, {, }})

Picture 8: Comparisson between Mossbauer spectra and correlation functions with the same hyperfine parameters.

Averaging of W(θ, t) throughout certain range of angles

Influence of finite size of detector on correlation function

2.5 cm ⟶ θ ± 14°

5 cm → θ ± 27°

Hyperfine parameters

Grid[{{"(Subscript[V, zz]) [mm/s]","η","B [T]","α [rad]","β [rad]"},{1.5,0.3,10,0,0}},Dividers->{{2-> Gray,3-> Gray,4->Gray,5-> Gray},{2-> Gray}},ItemStyle-> Directive[19]]

(Legended[, (LineLegend["point detector", "2.5 cm", "5.0 cm"])])

Picture 9: Correlation functions simulated for different detector size.

Convolution of W(θ,t) with time resolution of detector

Time resolution of detector impacts the shape of oscillations in W(θ,t)

Worse time resolution causes oscillations more spread

Grid[{{"(Subscript[V, zz]) [mm/s]","η","B [T]","α [rad]","β [rad]"},{1,0,20,0,0}},Dividers->{{2-> Gray,3-> Gray,4->Gray,5-> Gray},{2-> Gray}},ItemStyle-> Directive[19]]

Picture 10: Correlation functions simulated for different detector time resolutions.

Summary

This presentation coved:

Introduction to Nuclear Angular Correlation.

Case of (^57)Co.

(Subscript[G, 22])(t) and W(θ,t) graphs for varying hyperfine parameters.

Simulations of experiments.

Thanks to Mathematica some calculations were made symbolically.

Powerful tool for data processing - simulation and fitting experimental data.

Using all simulations we are able to upgrade experimental setup.

Thank you for your attention!

Refferences

FUNK, Emerson G., Jr., Gamma - Gamma Directional Correlation Measurement in (Se^76) and (Cd^110). Disertation thesis. The University of Michigan, March 1958.

BUTZ, T.: Nuclear Quadrupole Interactions Studies by Time Differential Perturbed Angular Correlations. Z. Naturforsch. 51a, 396-410, 1996.