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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) = 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
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
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)
Oleksandr Kazakov Stochastic Theory of Lineshape
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(+)†
Oleksandr Kazakov Stochastic Theory of Lineshape
Theory Overview
P (k) =∑λα
pλPλα(k) =(2
Γ
)Re
∫ ∞0
exp(iωt− 1
2tΓ)(〈H(−)H(+)(t)〉)av dt (5)
Oleksandr Kazakov Stochastic Theory of Lineshape
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)
Oleksandr Kazakov Stochastic Theory of Lineshape
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)
Oleksandr Kazakov Stochastic Theory of Lineshape
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)
Oleksandr Kazakov Stochastic Theory of Lineshape
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)
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),(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?
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][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.
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
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 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
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<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;
Oleksandr Kazakov Stochastic Theory of Lineshape
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;
Oleksandr Kazakov Stochastic Theory of Lineshape
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;
Oleksandr Kazakov Stochastic Theory of Lineshape
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 z−direction
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
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 100
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 z 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
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 100
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 y axis. Jump rateW = 0.1and100
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 0.1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 100
Oleksandr Kazakov Stochastic Theory of Lineshape
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
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
)
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
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) )
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
Abstracting from Blume simplified two-state model
Oleksandr Kazakov Stochastic Theory of Lineshape
Octahedron
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 0.1
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 0.5
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Zeeman Splitting Lineshape
W = 100
Oleksandr Kazakov Stochastic Theory of Lineshape
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+)
)
Oleksandr Kazakov Stochastic Theory of Lineshape
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
Oleksandr Kazakov Stochastic Theory of Lineshape
Slides with both Hyperfine and Zeeman coupling S-I 1/2 at LowField(H=8)
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
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 100
Oleksandr Kazakov Stochastic Theory of Lineshape
Slides with both Hyperfine and Zeeman coupling S(1/2)-I(1/2) athigh field.H = 12 · 103
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
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 100
Oleksandr Kazakov Stochastic Theory of Lineshape
Hyperfine and Zeeman splitting for spins S(1/2)-I(1) coupling
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
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 0.1, H = 100
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 0.3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 1
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 3
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 10
Oleksandr Kazakov Stochastic Theory of Lineshape
Results
W = 0.01, H = 2 · 105
Oleksandr Kazakov Stochastic Theory of Lineshape
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.
Oleksandr Kazakov Stochastic Theory of Lineshape