Oral Exam Slides

Stochastic Theory of Lineshape

Oleksandr Kazakov

January 28, 2015

Oleksandr Kazakov Stochastic Theory of Lineshape

Outline

1 Treatment of motional and exchange narrowing in magneticresonance.

2 Algorithm approach on assignment and inversion of LiouvilleMatrix.

3 STOL Ver. 1.0 for simulating stochastic lineshapes.


Outline





Outline





Theory Overview

Hamiltonian for a nucleus in a randomly varying magnetic fieldh(x, y, z):

H(t) = γh · If(t) (1)

What will happen if we add a fixed magnetic field H0 alongpositive x?

H(t) = γH0Iz + γh · If(t) (2)

[H(t),H(t)′] 6= 0


Theory Overview


H(t) = γh · If(t) (1)


H(t) = γH0Iz + γh · If(t) (2)

[H(t),H(t)′] 6= 0


Theory Overview


H(t) = γh · If(t) (1)


H(t) = γH0Iz + γh · If(t) (2)

[H(t),H(t)′] 6= 0


Theory Overview

Probability of the emission of a photon with wave vector k andfrequency ω by the system from state |λ〉 to its final state |α〉:

Pλα(k) =|〈α|H(+)|λ〉|2

(ω + Eα − Eλ)2 + 14Γ2

(3)


Theory Overview

Pλα(k) =(2

Γ

)Re

∫ ∞0

exp(iωt− 1

2tΓ)〈λ|H(−)|α〉〈α|U†(t)H(+)U(t)|λ〉) dt

(4)

Where U(t) = exp(−iHt) and H(−) = H(+)†


Theory Overview

P (k) =∑λα

pλPλα(k) =(2

Γ

)Re

∫ ∞0

exp(iωt− 1

2tΓ)(〈H(−)H(+)(t)〉)av dt (5)


Theory Overview

Blume developed a solution for the lineshape:

P (w) =2

Γ(2I0 + 1)

∑m1m0,m1′m0′

〈I1m1|H(−)|I0m0〉∑ab

pa〈I0m0I1m1a|L−1|I0m′0m′1b〉〈I0m′0|H(+)|I1m′1〉 (6)


Theory Overview

Propogator has a following form (s = iω):

L = s1−W − i∑j

V ×j Fj (7)

L = s1−W − i∑j

(Vj ⊗ Fj − Fj ⊗ Vj) (8)


Theory Overview

Propogator has a following form (s = iω):

L = s1−W − i∑j

V ×j Fj (7)

L = s1−W − i∑j

(Vj ⊗ Fj − Fj ⊗ Vj) (8)


Theory Overview

For easier implementation we can also rewritte as:

P (s) =2

Γ(2I0 + 1)H(−)δm1m0H(+)δm′

0m′1

[sδabδm1m′1δm0m′

0

− (a|W |b)δm0m′0δm1m1′

− i(a|F |a)δab[〈I0m0|Vj |I0m′0〉δm0m′0−〈I1m1|Vj |I1m′1〉δm1m′

1]−1

(9)


Problem

Algorithm to generate arrays values and perform where index i andj for Ai,j are composed of 3 subindices A(m0,m1,a),(m′

0,m′1,b)

in asimplest case having only nuclear spin.

Ai,j = A(m0S ,m1S ,m0I ,m1I ,a),(m0S ,m′1S ,m

′0I ,m

′1I ,b)

=

[sδabδm1m′1δm0m′

0− (a|W |b)δm0m′

0δm1m′

1

− i(a|F |a)δab[〈I0m0|Vj |I0m′0〉δm0m′0−〈I1m1|Vj |I1m′1〉δm1m′

1]−1

(10)

It is possible to do assignment of Liouville Matrix by hand up tosize 8× 8 but what if its size expands to 256× 256 or1 · 106 × 1 · 106?


Where to go?

1 Matlab

Multidimensional arrays(N-D) implementation appeared inrecent Matlab 2014b

2 Java

Supports Multidimensional arrays implementation since 1995


Where to go?

1 Matlab

Multidimensional arrays(N-D) implementation appeared inrecent Matlab 2014b

2 Java

Supports Multidimensional arrays implementation since 1995



Algorithm Development in Stages

Generate 6-D array: A[a][b][m0][m1][m′0][m

′1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m′0][m

′1]→ A′[i][j]

Perform 2-D array inversion: A′[i][j] = A′[i][j]−1

Cast 2-D array back into 6-D array:A′[i][j]→ A[a][b][m0][m1][m

′0][m

′1]

Go back to the equation 6 and do the summation.

Plot the results.




′1]


′1]→ A′[i][j]



′0][m

′1]


Plot the results.




′1]


′1]→ A′[i][j]



′0][m

′1]


Plot the results.




′1]


′1]→ A′[i][j]



′0][m

′1]


Plot the results.




′1]


′1]→ A′[i][j]



′0][m

′1]


Plot the results.




′1]


′1]→ A′[i][j]



′0][m

′1]


Plot the results.


Encoding Table Example

A simple example of encription of 6 different combinations between3 indecies that are used to cast back 6-D array from2-D(A[a][:][m0][m1][:][:]→ A′[i][:]).

a,m0,m1 Output Count

1, 1/2, 1/2 → 000 1

1,−1/2, 1/2 → 010 2

1, 1/2,−1/2 → 001 3

2, 1/2, 1/2 → 100 4

2, 1/2, 1/2 → 101 5

2,−1/2,−1/2 → 111 6


Code Sample

Java Method to store reference for the elements of 6-D array afterconverting into 2-D array. Elements being converted and savedinto string format.

public static String[] genind(int m_0,int m_1, int a1 )

int x=1;

String Y[] = new String[max];

for(int k=0;k<m_0;k++)

for(int l=0;l<m_1;l++)

for(int a=0;a<a1;a++)

Y[x-1]=String.format("%03d",a+10*k+100*l);

x++;

return Y;


Code Sample

Casting 6-D array into 2-D:

for(int k=0; k<m_0;k++)

for(int l=0; l<m_1; l++)

for(int m=0; m<m_01; m++)

for(int n=0;n<m_11; n++)

for(int a=0; a<a1; a++)

for(int b=0; b<b1;b++)

tr[index(k,l,a,Y)][index(m,n,b,Y)]=matrix[a][b][k][l][m][n];

return tr;


Code Sample

Converting from 2-D array back to 6-D array:

public static Object[][][][][][] matrixturn(Complex

tr[][], String Y[])

for(int k=0; k<max;k++)

for(int l=0; l<max; l++)

hn[a1indback(k,Y)][b1indback(l,Y)][m0indback(k,Y)]...

[m1indback(k,Y)][m01indback(l,Y)][m11indback(l,Y)]=tr[k][l];

return hamatrixnew;


Code Sample

Method for the extraction of element reference in 2-D array to thatin 6-D:

public static int m0indexback( int k, String Y[])

int m0=0;

int f=Integer.parseInt(Y[k]);

m0=(int) Math.floor((f/100));

return m0;

public static int m1indexback( int k,String Y[])

int m1=0;

int ff=Integer.parseInt(Y[k]);

m1=(int) Math.floor((ff-Math.floor(ff/100)*100)/10);

return m1;


Code Sample

Method for the extraction of element reference in 2-D array to thatin 6-D to perform summation:

public static int a1indexback(int k,String Y[])

int fff=Integer.parseInt(Y[k]);

int a1=0;

a1= (int) (Math.floor(fff-Math.floor(fff/100)*100)-...

Math.floor((fff-Math.floor(fff/100)*100)/10)*10);

return a1;



Results

To illustrate that algorithm works Nuclear Zeeman Hamiltonianwas introduced of the form:

H = HIz + h · If(t) (11)

Where fixed magnetic field set along z−direction


Results

NMR line shape for spin-12 nucleus in a fixed magnetic fieldalong z axis and fluctuating field along x axis


Results

W = 0.1


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 100


Results

NMR line shape for spin-12 nucleus in a fixed magnetic fieldalong z axis and fluctuating field along z axis


Results

W = 0.1


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 100


Results

NMR line shape for spin-12 nucleus in a fixed magnetic fieldalong z axis and fluctuating field along y axis. Jump rateW = 0.1and100


Results

W = 0.1


Results

W = 1


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 100



Plotting 3-D graphs using JZY3D OpenGL library

Figure: NMR line shape for spin- 12

nucleus in a fixed magnetic field along z axis and fluctuating field along zaxis. Frequency along x-axis, fluctuating field along y-axis and intensity along z-axis. Transition rateW = 0.1


ISTO Approach

Most important aspect of calculating matrix elements of Lioville iskeeping track of coordinate frames in which various quantaties aredefined. To unravel such comlications concept of irreducabletensor operator(ISTO) should be intorduced:

T(J,M)i → T

(J,M)f =

J∑M ′=−J

T(J,M ′)i DJM ′,M (Ξi→f )

Also spin Hamiltonian is a rank-zero tensor:

H(Ω) =∑µ,m,l

F (l,−m)ν,µ A(l,m)

ν,µ


Zeeman Splitting

To account for the anisotropy of the Zeeman response to anapplied magnetic field, an effective Hamiltonian using a so-called gtensor is used. Note that Hamiltonian has the following form:

Hzm = µBH · g · S

and thus Zeeman magnetic tensor and spin operators can bewritten in principal axis frame as following:

F[2](P )±2(g) = −1

2βe(gxx − gyy) A

[2](P )±2(g) = 0

F[2](P )±1(g) = 0 A

[2](P )±1(g) = 0

F[2](P )0(g) = −

√2

3βe(gzz −

1

2(gxx + gyy) A

[2](P )0(g) = −

√2

3SzHz

F[0](P )0(g) =

1√3

(gxx + gyy + gzz A[0](P )0(g) =

1√3

)


Coordinate Frames Transformation

F[l](L)m(g) = (Dlm,2(ΩL→P ) +Dlm,−2(ΩL→P ))∗F

[2](P )2(g) + (Dlm,0(ΩL→P ))∗F

[2](P )0(g)

F tensor components should satisfy identity:

(F[l]q )∗ = (−1)2+qF

[l]−q


Coordinate Frames Transformation(Explicit)

F[2],(L)1(g) = −(F

[2],(L)−1(g) )∗ = −e−iα([sin(β)cos(β)cos(2γ)− isin(β)sin(2γ)]F

2(P )2(g) −

√3

2sin(β)cos(β)F

2(P )0(g) )

F[2],(L)0(g) =

√3

2sin2(β)cos(2γ)F

2(P )2(g) +

1

2(3cos2(β)− 1)F

2(P )0(g)

F[2],(L)2(g) = (F

[2],(L)−2(g) )∗ = ei2α([

1 + cos2(β)

2cos(2γ) + icos(β)sin(2γ)]F

2(P )2(g) +

√3

8sin2(β)F

2(P )0(g) )


Zeeman Splitting

Now we all set to construct Hamiltonian where all quantities areevaluated in the lab frame:

Hzm = A[2]0 F

[2]0 +A

[0]0 F

[0]0


Abstracting from Blume simplified two-state model


Octahedron


Zeeman Splitting Lineshape

W = 0.1



W = 0.5



W = 1



W = 3



W = 10



W = 100


In EPR, it is important to consider hyperfine interaction betweenthe EPR active electron and neighboring nuclei. Most generalHamiltonian has the following form:

Hhf = I ·A · S

Hyperfine tensorand spin operators can be written in principal axis frame as following:

F[2](P )±2(hf) = −1

2βe(Axx −Ayy) A

[2](P )±2(hf) = −1

2S±I±

F[2](P )±1(hf) = 0 A

[2](P )±1(hf) = ±1

2(S±Iz + SzI±)

F[2](P )0(hf) = −

√2

3βe(Azz −

1

2(Axx +Ayy)) A

[2](P )0(hf) = −

√2

3

(SzIz −

1

4(S+I− + S−I+)

)F

[0](P )0(hf) =

1√3

(Axx +Ayy +Azz) A[0](P )0(hf) =

1√3

(SzIz −

1

4(S+I− + S−I+)

)


In the same manner as Zeeman Hamiltonian we can rewritte:

Hhf = A[2]0 F

[2]0 +A

[0]0 F

[0]0 +A

[2]1 (F

[2]1 )∗ +A

[2]−1F

[2]1

so generalizing:Hres = Hzm +Hhf


Slides with both Hyperfine and Zeeman coupling S-I 1/2 at LowField(H=8)


Results

W = 0.1


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 100


Slides with both Hyperfine and Zeeman coupling S(1/2)-I(1/2) athigh field.H = 12 · 103


Results

W = 0.1


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 100


Hyperfine and Zeeman splitting for spins S(1/2)-I(1) coupling


Results

W = 0.1


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 0.1, H = 100


Results

W = 0.3


Results

W = 1


Results

W = 3


Results

W = 10


Results

W = 0.01, H = 2 · 105


Conclusion:

Algorithm for populating and inverting stochastic luneshapehave been developed and succesfully tested.Developed algorithm is universal not only to stochastic indicesbut spin values can be varied as well.

Ongoing Work:

Efficiency of the computationg have to be boosted andcurrently code have been transferred to the Matlab.Extending possible number of stochastic and quantum statesfor coupled S − I spins.


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