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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