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 magnetic
resonance.
2 Algorithm approach on assignment and inversion of Liouville
Matrix.
3 STOL Ver. 1.0 for simulating stochastic lineshapes.

Theory Overview
Hamiltonian for a nucleus in a randomly varying magnetic ﬁeld
h(x, y, z):
H(t) = γh · If(t) (1)

Theory Overview
h(x, y, z):
H(t) = γh · If(t) (1)
What will happen if we add a ﬁxed magnetic ﬁeld H0 along
positive x?
H(t) = γH0Iz + γh · If(t) (2)

Theory Overview
h(x, y, z):
H(t) = γh · If(t) (1)
What will happen if we add a ﬁxed magnetic ﬁeld H0 along
positive x?
H(t) = γH0Iz + γh · If(t) (2)
[H(t), H(t) ] = 0

Theory Overview
Probability of the emission of a photon with wave vector k and
frequency ω by the system from state |λ to its ﬁnal state |α :
Pλα(k) =
| α|H(+)|λ |2
(ω + Eα − Eλ)2 + 1
4Γ2
(3)

Theory Overview
Pλα(k) =
2
Γ
Re
∞
0
exp(iωt −
1
2
tΓ) 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
2
tΓ)( 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
|I0m0m1b I0m0|H(+)
|I1m1 (6)

Theory Overview
Propogator has a following form (s = iω):
L = s1 − W − i
j
V ×
j Fj (7)

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(−)
δm1m0 H(+)
δm0m1
[sδabδm1m1
δm0m0
− (a|W|b)δm0m0
δm1m1
− i(a|F|a)δab[ I0m0|Vj|I0m0 δm0m0
− I1m1|Vj|I1m1 δm1m1
]−1
(9)

Problem
Algorithm to generate arrays values and perform where index i and
j for Ai,j are composed of 3 subindices A(m0,m1,a),(m0,m1,b) in a
simplest case having only nuclear spin.
Ai,j = A(m0S,m1S,m0I ,m1I ,a),(m0S,m1S,m0I ,m1I ,b) =
[sδabδm1m1
δm0m0
− (a|W|b)δm0m0
δm1m1
− i(a|F|a)δab[ I0m0|Vj|I0m0 δm0m0
− I1m1|Vj|I1m1 δm1m1
]−1
(10)
It is possible to do assignment of Liouville Matrix by hand up to
size 8 × 8 but what if its size expands to 256 × 256 or
1 · 106 × 1 · 106?

Where to go?
1 Matlab
Multidimensional arrays(N-D) implementation appeared in
recent Matlab 2014b
2 Java
Supports Multidimensional arrays implementation since 1995

Algorithm Development in Stages
Generate 6-D array: A[a][b][m0][m1][m0][m1]

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

A [i][j] → A[a][b][m0][m1][m0][m1]
Go back to the equation 6 and do the summation.

A [i][j] → A[a][b][m0][m1][m0][m1]
Go back to the equation 6 and do the summation.
Plot the results.

Encoding Table Example
A simple example of encription of 6 diﬀerent combinations between
3 indecies that are used to cast back 6-D array from
2-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 after
converting into 2-D array. Elements being converted and saved
into 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;
}

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 that
in 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 that
in 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 Hamiltonian
was introduced of the form:
H = HIz + h · If(t) (11)
Where ﬁxed magnetic ﬁeld set along z−direction

Results
NMR line shape for spin-1
2 nucleus in a fixed magnetic field
along 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
along z axis and ﬂuctuating ﬁeld along z axis

Results
along z axis and ﬂuctuating ﬁeld along y axis. Jump rate
W = 0.1and100

Plotting 3-D graphs using JZY3D OpenGL library
Figure: NMR line shape for spin- 1
2
nucleus in a fixed magnetic field along z axis and fluctuating field along z
axis. 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 is
keeping track of coordinate frames in which various quantaties are
deﬁned. To unravel such comlications concept of irreducable
tensor operator(ISTO) should be intorduced:
T
(J,M)
i → T
(J,M)
f =
J
M =−J
T
(J,M )
i DJ
M ,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 an
applied magnetic ﬁeld, an eﬀective Hamiltonian using a so-called g
tensor is used. Note that Hamiltonian has the following form:
Hzm = µBH · g · S
and thus Zeeman magnetic tensor and spin operators can be
written in principal axis frame as following:
F
[2](P)
±2(g) = −
1
2
βe(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
3
βe(gzz −
1
2
(gxx + gyy) A
[2](P)
0(g) = −
2
3
SzHz
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) = (Dl
m,2(ΩL→P ) + Dl
m,−2(ΩL→P ))∗F
[2](P)
2(g) + (Dl
m,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
2
sin(β)cos(β)F
2(P)
0(g) )
F
[2],(L)
0(g) =
3
2
sin2
(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)
2
cos(2γ) + icos(β)sin(2γ)]F
2(P)
2(g) +
3
8
sin2
(β)F
2(P)
0(g) )

Zeeman Splitting
Now we all set to construct Hamiltonian where all quantities are
evaluated 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 between
the EPR active electron and neighboring nuclei. Most general
Hamiltonian has the following form:
Hhf = I · A · S
Hyperﬁne tensor
and spin operators can be written in principal axis frame as following:
F
[2](P)
±2(hf) = −
1
2
βe(Axx − Ayy) A
[2](P)
±2(hf) = −
1
2
S±I±
F
[2](P)
±1(hf) = 0 A
[2](P)
±1(hf) = ±
1
2
(S±Iz + SzI±)
F
[2](P)
0(hf) = −
2
3
βe(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 Low
Field(H=8)

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

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

Results
W = 0.1, H = 100

Results
W = 0.01, H = 2 · 105

Conclusion:
Algorithm for populating and inverting stochastic luneshape
have been developed and succesfully tested.
Developed algorithm is universal not only to stochastic indices
but spin values can be varied as well.
Ongoing Work:
Eﬃciency of the computationg have to be boosted and
currently code have been transferred to the Matlab.
Extending possible number of stochastic and quantum states
for coupled S − I spins.

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