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• 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.

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.

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.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

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

H(t) = h If(t) (1)

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

H(t) = H0Iz + h If(t) (2)

[H(t),H(t)] 6= 0

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

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

H(t) = h If(t) (1)

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

H(t) = H0Iz + h If(t) (2)

[H(t),H(t)] 6= 0

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

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

H(t) = h If(t) (1)

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

H(t) = H0Iz + h If(t) (2)

[H(t),H(t)] 6= 0

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

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

P(k) =||H(+)||2

( + E E)2 + 142(3)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

P(k) =(2

)Re

0

exp(it 12

t)|H()||U(t)H(+)U(t)|) dt

(4)

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

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

P (k) =

pP(k) =(2

)Re

0

exp(it 12

t)(H()H(+)(t))av dt (5)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

Blume developed a solution for the lineshape:

P (w) =2

(2I0 + 1)

m1m0,m1m0

I1m1|H()|I0m0ab

paI0m0I1m1a|L1|I0m0m1bI0m0|H(+)|I1m1 (6)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

Propogator has a following form (s = i):

L = s1W ij

V j Fj (7)

L = s1W ij

(Vj Fj Fj Vj) (8)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

Propogator has a following form (s = i):

L = s1W ij

V j Fj (7)

L = s1W ij

(Vj Fj Fj Vj) (8)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Theory Overview

For easier implementation we can also rewritte as:

P (s) =2

(2I0 + 1)H()m1m0H(+)m0m1 [sabm1m1m0m0

(a|W |b)m0m0m1m1

i(a|F |a)ab[I0m0|Vj |I0m0m0m0 I1m1|Vj |I1m1m1m1 ]

1

(9)

Oleksandr Kazakov Stochastic Theory of Lineshape

• Problem

Algorithm to generate arrays values and perform where index i andj for Ai,j are composed of 3 subindices A(m0,m1,a),(m0,m1,b) in asimplest case having only nuclear spin.

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

1I ,b)

=

[sabm1m1m0m0 (a|W |b)m0m0m1m1 i(a|F |a)ab[I0m0|Vj |I0m0m0m0 I1m1|Vj |I1m

1m1m1 ]

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?

Oleksandr Kazakov Stochastic Theory of Lineshape

• Where to go?

1 Matlab

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

2 Java

Supports Multidimensional arrays implementation since 1995

Oleksandr Kazakov Stochastic Theory of Lineshape

• Where to go?

1 Matlab

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

2 Java

Supports Multidimensional arrays implementation since 1995

Oleksandr Kazakov Stochastic Theory of Lineshape

• Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m0][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][m0][m1]Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m0][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][m0][m1]Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m0][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][m0][m1]Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

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

Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m0][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][m0][m1]Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• Algorithm Development in Stages

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

1]

Cast 6-D array into 2-D: A[a][b][m0][m1][m0][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][m0][m1]Go back to the equation 6 and do the summation.

Plot the results.

Oleksandr Kazakov Stochastic Theory of Lineshape

• 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 11,1/2, 1/2 010 21, 1/2,1/2 001 32, 1/2, 1/2 100 42, 1/2, 1/2 101 52,1/2,1/2 111 6

Oleksandr Kazakov Stochastic Theory of Lineshape

• 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

• Code Sample

Casting 6-D array into 2-D:

for(int k=0; k

• 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

• 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;

}

Oleksandr Kazakov Stochastic Theory of Lineshape

• 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;

}

Oleksandr Kazakov Stochastic Theory of Lineshape

• Oleksandr Kazakov Stochastic Theory of Lineshape

• 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 zdirection

Oleksandr Kazakov Stochastic Theory of Lineshape

• Results

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

Oleksandr Kazakov Stochastic Theory of Lineshape

• Results

W = 0.1

Oleksandr Kazakov Stochastic Theory of Lineshape

• Results

W = 0.3

Oleksandr Kazakov Stochastic Theory of Lineshape

• Re

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