Basic Theory of Solid-State NMR
Klaus Schmidt-Rohr
Dept. of Chemistry, Iowa State University, Ames, IA 50011
w = -g B
^
[H, r^(0)]
r(t) = r(0) coswt + iw sinwt
^
F=wt
^
Outline
- NMR without serious quantum mechanics
- Angular momentum (spin) and magnetic dipole moment
- Precession around B, with w = - g B
- Fields, frequencies, energies, Hamiltonians
- The rotating frame
- Radio-frequency pulses
- A simple pulse sequence: Hahn spin echo
- Truncation of weak fields
- Truncation in the interaction frame
- Local fields and interaction tensors: - Chemical shift - Dipolar couplings
- Orientation dependence of NMR frequencies
- PAS - Powder spectrum (for h = 0)
- Frequency w(t) in MAS
- Dipolar decoupling
- Time signals, Fourier transformation
- 2D NMR
- Basic spin quantum mechanics for NMR
- The state of the spin system: density operator
- Spin evolution: The von Neumann equation and its solutions
- Oscillating solutions: - Precession
- Hˆ , rˆ 0 = 0: invariant spin state
- Heteronuclear spin-1/2 dipolar evolution
- Pulses and rotations in spin space
- Propagators and pulse sequences: - Spin echoes
- Homonuclear couplings - Spin exchange - Multiple-quantum coherences
2
Angular Momentum (Spin) and Magnetic Dipole Moment
3

Precession around B, with w = - g B





Torque t = m x B (1) Angular force = torque = t = dS/dt = d(angular momentum)/dt (2)


Force
=
F
=
dp/dt
= d(momentum)/dt


Linked by m = g S (3)
dSx/dt = g Sy Bz




(1), (2), (3)  dS/dt = g S x B

dSy/dt = -g Sx Bz
for B along z
dSz/dt = 0

Sx(t) = Sx(0) cos(wt)
Sy(t) = Sx(0) sin(wt) w = - g B
Sz(t) = Sz(0) = constant
For S(0) in the x-z plane:
r
r
M = N spin m




Precession of S or m around B with frequency w = - g B
4
A Central Equation of NMR
B
5
Fields,

B
Example:
Supercon magnet
B0 = 14 tesla
Frequencies,

Energies,
w = - g |B|
E=ħw=hn
w = 2p n
 .
- B
1H
Larmor frequency
w0 = -2p 600 MHz
E= m
Energy splitting
E = 2.5 meV
Hamiltonians
^
^
“H = E”
^
^ 

.
H=-m B=-g
^ 

S.B
Zeeman interaction
^
^ .
^
H
=
g
S
B
=
w
S
0
z
0
L z
- In NMR, the strength of nuclear interactions (local fields, energies, Hamiltonians)
is usually given as frequencies (in Hz)
- This corresponds to a line width or frequency shift in the NMR spectrum
The Rotating Frame and Radio-Frequency Irradiation
7
Radio-Frequency Pulses
- In the laboratory frame: B1(t) = 2 B1 cos(wrf t) field oscillates with wrf = 2p 150 MHz
- In the rotating frame: Magnetization precesses around the static B1 field,
with w1 = -g B1 = -75 kHz
- Pulse flip angle f = w1 tP = -g B1 tP  “area under the pulse”
8
Phase and Direction of Radio-Frequency Pulses
- Change in phase of B1(t) (e.g. by 90o) in the lab frame corresponds to a
change in direction of B1 in the rotating frame (e.g. x to y)
- Pulse phase change Dw t occurs due to frequency change Dw: PMLG / FSLG
9
A Simple Pulse Sequence: Hahn Echo
In the rotating frame
10
A Central Equation of NMR
- NMR frequency measures B field strength, with < 0.1 ppm resolution
11
Truncation of Weak Fields: Keep the Parallel Component

wL = - g |Btot|

|Btot|
=


|B0 + Bloc|
=
B 0  B loc, z   B loc, 
2
2
2
B loc ,
= B 0  B loc ,z  1 
2
B

B



0
loc ,z
The Magic Angle



2
B loc ,
1

 B 0  B loc ,z 1 
2 

2 B 0  B loc ,z  

= B0 +
B loc ,
1

B loc ,
Bloc,z
2 B 0  B loc ,z 

<<1
wL ≈ - g (B0 + Bloc,z) = w0 - g Bloc,z

Magic angle: Bloc,z = 0
12
Truncation in the Interaction Frame:
Oscillating Components Are Averaged
- Alternative view, which matches quantum-mechanical truncation
- View local fields from the
rotating frame (“interaction frame”)
- Average the rapidly oscillating components to zero
- The time-independent components
are the truncated fields:

Keep field component parallel to B0: Bloc,z
13
Chemical Shift


- Slight shielding of nucleus from B0 by electron clouds: Local field BCS
- Linear in B0 (“ppm”):

|BCS|


|B0|


but not necessarily parallel: BCS = - s B0

with the chemical-shift tensor s


- B0 along z: BCS = -
 ...

sB0 =  ...

 ...
- After truncation, retain

...
...
...
s xz  0 
 s xz B 0 
  

s yz  0  =  s yz B 0 
  

s zz  B 0   s zz B 0 
in the lab frame (“LF”)
BCS,z = - szzLF B0
- Chemical-shift frequency
wCS = - g BCS,z = - szzLF w0
14
Dipolar Field and Dipolar Tensor
- Electrodynamics: The magnetic field, at a pointr, of a dipolem at (0,0,0) is:
(Griffiths, eq.(3.104))
r
B dip =


- m along z, & truncation by B0: B dip ,z =
0
r  
m = 0
 
m

r
- Dipolar tensor D: B dip =

- Dipolar frequency


LF
with D
=
m0
4p
m0
4p
r
3( m z rz /r ) rz - m z
2
4p
m0
m0
r r 2 r r
3( m  r /r ) r - m
r
r r 2 r r
3( m  r /r ) r - m
4p
r
3
3(r /r) r /r -  
r
3
=
3
=
=
m0
4p
t
D
m0
4p
3
3 cos  3 - 1
2
m
r
3
r
m
3 cos   cos   -  
r
3
wdip = -g Bdip,z = -g m DzzLF
15
NMR Frequency: zz-Element of a Coupling Tensor
- Chemical-shift frequency
wCS = - g BCS,z = - szzLFw0
- Dipolar frequency
wdip = -g Bdip,z = -g m DzzLF
- Both frequencies are proportional to the zz-element of a coupling tensor
(symmetric 3x3 matrix) in the lab frame.
- We can derive many properties of these frequencies by
considering the lab-frame zz-element wzz

of a generalized frequency matrix w
wzz =0
LF
0
 w LF
xx
 LF
1 w yx
 w LF
 zx
w xy
LF
w
LF
yy
w yz
LF
LF
w xz  0  r T

B
LF  
w yz  0  = 0
B0
LF  
w zz  1 
t
w
r
B0
B0
r
T
= b0
t
w
Unit vector along
r
r
T
b0 = b0

t
w
r
r
B0
b 0 :=
B0
r PAS r
b 0 = b 0T
 
t
w
r
b0

B0

RF

- We’ll evaluate this vector – matrix – vector product
-matrix is diagonal)

- in the principal-axes system (w
to get w,f)
- in the rotor frame to get w(t) in MAS
16
Principal Axes System

- Like the coordinates (B0x, B0y, B0z) of a vector B0,

the elements w (incl. wzz) of the tensor w depend on the coordinate system.
 w PAS
xx

 0
 0


w is diagonal =
0 

0 
PAS
w zz 
0
w yy
PAS
0
in the Principal Axes System (PAS)
- “Principal values” wxxPAS etc.

PAS
- Dipolar coupling: wzzPAS = , wxx
= wPAS
yy = -/2
- Calculate the frequency “in the PAS”:
r
T
b0
wCS = wzz = 
LF
= w xx
PAS
=w
PAS

xx
b0,x
PAS

2
t
w
 w yy
PAS
yy

b 0,y
PAS

2
 w zz
cos  2  w
2
PAS
PAS
zz
b0,z
PAS
PAS
0  b 0,x 
 PAS 
0  b 0,y 
PAS
PAS
w zz  b 0,z 
0

PAS
cos 1  w
2
 w PAS
xx
r PAS

PAS
PAS
PAS
= b 0,x , b 0,y , b 0,z  0
b0
 0

w yy
PAS
0

2
cos  3
2
with cosn
 cos f sin  
r


PAS
b 0 = sin f sin 




 cos  
 
= b . ePAS
0
n

= w xx cos f sin   w yy sin f sin   w zz cos 
PAS
2
2
PAS
2
2
PAS
2
17
Orientation Dependence of NMR Frequencies
Separate orientation-independent isotropic shift wiso = ( w xx  w yy  w zz )/3
and the chemical-shift / dipolar anisotropy wCSA(,f) = wCS(,f) - wiso
PAS
Define w z = w zzPAS - w iso Anisotropy parameter  = wz
PAS
(w of largest magnitude)

wx+ wy+ wz = 0
PAS
Biaxiality (or asymmetry) parameter h =
wy -wx
wz

wCSA = w xxPAS cos 2 f
= w xx
PAS


2
PAS
2
- w iso cos f sin   w yy
2
2
PAS
2
PAS
2
= - 1  h 
2
wz
2
= w z cos  2

w CSA =

2
2
2
2
2
wz
2
2
sin  - h
2
wz
2
cos

PAS
2
wz
2
- w iso cos 
sin f sin   w z cos 
2
2
using h wz = wy- wx = - wz- 2wx
2
cos f sin  - 1 - h 
2
- w iso
2
- w iso sin f sin   w zz
= w x cos f sin   w y sin f sin   w z cos 


sin   w yy sin f sin   w zz cos 
2
2
f - sin f sin 
2
2
using cos2f + sin2f = 1
using cos2 + sin2 = 1,
cos2f - sin2f = cos2f
3cos  - 1 - h sin  cos2 f 
2
2
18
(h = 0)
Utilize NMR orientation dependence to measure
(1) Bond orientation in oriented samples: Helix orientation, etc.
(2) Changes in segmental orientation: Dynamics
(3) Relative orientation of segments: Torsion angles
19
Powder Spectra
- Spectrum of all orientations: Frequency w,f)
contributes with intensity P(,f) = sin/2p
- Analytical:
Simplest case: h = 0, w) =  (3cos2 -1)/2
“Differential conservation of the integral”
S(w) |dw | = P() |d |
S(w) = P((w)) |d/dw |
 S (w ) =
1
6
w   /2
- Numerical: Sample ,f) on a unit sphere,
add P(,f) into the calculated spectrum at w,f)

- The three principal values wx, wy, wz can be read off from maximum and step positions:
20
Magic-Angle Spinning
____
zPAS
_
MAS with wr >> : Evolution under time-averaged frequency wCS
m
w CS = w
Proof of the
central equality:


t
t
t 
A BC =


LF
zz
=
1
t

t
0
w
LF
zz
(t' ) dt ' =
 
 

 
A zz  
t
t
t
so A  B  C

1
t

t
0
t
w (t ' ) dt '
 
 

 
B zz  

zz

LF
zz
 
 
=
 
C zz  
t LF zz element
= w zz
of average tensor




A zz  B zz  C zz 
= A zz  B zz  C zz

- The average tensor wt =
1
t

t
_0
t
w (t ' ) dt ' is also a 3x3 matrix.

- It has uniaxial symmetry,
h = 0, around the rotor axis.
- The angle between the unique principal axis (rotor axis) and B0 is m,

so
t LF 

2
2
2
w CS = w zz = 3 cos  m - 1 - h sin  m cos 2 g   w iso = 3 cos  m - 1  w iso
2
2
- When 3cos2m-1 = 0 (magic angle m = 54.74o), the frequency is equal to the isotropic shift.

21
Magic Zero, Periodic Signal, Spinning Sidebands
Slow MAS:
The uniaxiality argument still holds for completed rotations
LF
kt r
w C SA ( kt r ) =
1
kt r

w C SA (t ) dt =
0
1
kt r
tr = rotation period
 kt r

t

2
 w (t ) dt 
=
3cos
m - 1 = 0

2


 0
 zz


tr
This contains the “magic zero” of MAS:
w
CSA
( t ) dt = 0
0
_
 After full rotation periods, the anisotropic frequency is refocused, w(ktr) = wiso
Rotational echoes
 The MAS time signal has a
periodic component, with period tr
-- The Fourier
transform of a signal with period tr

has intensity at the “harmonics” N x 2p/tr:
Spinning sidebands spaced by wr = 2p/tr
22
Instantaneous Frequency in Magic-Angle Spinning
The instantaneous chem.-shift frequency under MAS is
r
T
w CS (t) = b 0

t
w
r
b0

RF
with
 cos w r t sin  m 
r RF 

b 0 = sin w r t sin  m =




cos  m


evaluated in the rotor frame RF, where the
B0-field rotates with the spinning frequency wr


 cos w r t
2
sin w r t
3 
 1/2





3cos2m-1 = 0
cos2m= 1/3
sin2m= 2/3
The vector-matrix-vector product gives wCS(t) as
2
~
~
~
~
wCS(t) = wiso+ C1 cos(wrt) + S1 sin(wrt) + C2 cos(2wrt) + S2 sin(2wrt) = S Gm eimwrt
m=-2
with coefficients that depend on  and h, and on the orientation , , g of the PAS in the rotor
- Verify the “magic zero”
tr
w
CSA
( t ) dt = 0
with wr tr = 2p
0
- Signal with constant chemical shift:
s(t) = cos(wCS t) + i sin(wCSt) = exp(iwCSt)

- MAS signal

exp  i

- Signal with varying chemical shift:
s(t) = exp(iwCS(t1) Dt) exp(iwCS(t2) Dt) exp(iwCS(t3) Dt)
t

0


i w iso t
w CS ( t' ) dt ' = e
exp  i


t
w
0
CSA

(t ' )dt '

23
Heteronuclear Dipolar Decoupling
- Dipolar coupling produces line broadening
- 1H-X dipolar coupling can be truncated by a strong B1 field:
Heteronuclear dipolar decoupling
- Note: Presence or absence of 1H magnetization is irrelevant here
24
Time Signal, Spectrum, Fourier Transformation
- NMR signal from oscillating transverse component of the magnetization
(induces a voltage V = - Nturns d(magnetic flux)/dt in the radio-frequency coil)
- NMR is effectively measured in the rotating frame
s(t) = s0 (cos(wL-wrf)t + i sin(wL-wrf)t) = s0 eiwt
with w = wL- wrf
- The spectrum S(w) is the “weight” of the signal eiwt with a given w in the total s(t):
s(t) = ∫-∞∞ S(w) eiwt dw
“Fourier superposition”
- The spectrum can be calculated from s(t) according to
S(w) = (2p)-1∫-∞∞ s(t) e-iwt dt
“Fourier transformation”
- Many useful Fourier theorems: “inverse-width”, “zero-point - integral”, “shift”,
“convolution”, “sampling”, etc. relate properties of (or operations on) s(t) and S(w).
25
26
Basic Spin Quantum Mechanics for NMR
Quantum mechanics is needed to treat
(i) homonuclear coupled spins
(ii) multi-spin coherences (including DQ/ZQ coherences)
(iii) quadrupolar coupling
(iv) heteronuclear-coupling multiplets
Translation between classical picture and quantum mechanics:
Magnetization, state of the spin system: Density operator r^
^
B-fields / interactions / couplings: Terms in the Hamiltonian H
Parallel fields: Commuting Hamiltonians
^
NMR conventions: - Write H^ for standard H/ħ
(frequency units)
- Write S^x/y/z for standard S^x/y/z/ħ (unitless)
^

^. 
^
^


^
^
- Write m for standard m/ħ (retain H = - m B and m = g S )
27
Fields,

B
Example:
Supercon magnet
B0 = 14 tesla
Frequencies,

Energies,
w = - g |B|
E=ħw=hn
w = 2p n
 .
- B
1H
Larmor frequency
w0 = -2p 600 MHz
E= m
Energy splitting
E = 2.5 meV
Hamiltonians
^
^
“H = E”
^
^ 

.
H=-m B=-g
^ 

S.B
Zeeman interaction
^
^ .
^
H
=
g
S
B
=
w
S
0
z
0
L z
- In NMR, the strength of nuclear interactions (local fields, energies, Hamiltonians)
is usually given as frequencies (in Hz)
- This corresponds to a line width or frequency shift in the NMR spectrum
State of the Spin System: The Density Operator
- Pure spin state can be described by |y
- State of a system of >1018 spins can be described by a density operator r^ = Sn pn |yn yn|
- Combines quantum mechanics (“|y ”) and statistical mechanics (“pn”)
^
- In thermal equilibrium, pn = exp(-En/(kT))/Z (Boltzmann),
r^eq= Z1 exp(-H/(kT))

^
- The density matrix r is the matrix representation of r
in the energy basis:

rnm = yn| r^ |ym 2 x 2 matrix for spin-1/2 (“Pauli matrices”)
- The average value of an observable A in quantum mechanics is the
^
expectation value A of its Hermitian operator A
^
- Pure state: A = y| A |y
^
- Mixed state: Statistically weighted combination of expectation values yn|A|yn:



A = Sn pn yn| A^ |yn = … = Sn Sm Anm rnm = Tr(A 
r) = Tr(r
A)
Tr = ‘trace’ = sum of diagonal elements


x-magnetization ~ Sx = Tr(Sx 
r)


Note: Tr(Sx2) = Tr(Sy2) = Tr(Sz2) ≠ 0, all other traces are 0
Combine x- and y-components into complex NMR signal



 
s(t) ~ Sx + i Sy = Tr({Sx + i Sy} r) = “Tr(S+ r)”
^
^
If r^ = cosF Sx + sinF Sy, then s(t) ~ cosF + i sinF =exp(iF)
29
The Equilibrium Spin State in the High-T Approximation
H nm
At high T,
<< 1

kT
req =
^
1
Z
 - Hˆ 
exp 

 kT 

Hˆ 
1 
 1
kT 
1
2S
If the Zeeman interaction (B0) is the largest term in the Hamiltonian,

g B 0 Sˆ z 
req  2 S  1 1  kT 



^
1
^
The first term (“1”) can be dropped: (i) It does not change since [H, 1] = 0 and



(ii) gives no signal: Tr(Sx/y/z 1) = Tr(Sx/y/z) = 0

The reduced density operator r^ red = r^ - 1/(2S+1) in NMR
gives the same Sx/y/z as the full one.
It is written as r^ without index red.
^
req = gB0 Sz corresponds to z-magnetization: Spins point along B0.
^
kT (2S+1)
^ ^
Szeq = Tr( gB0 Sz Sz) = g B 0 S ( S  1)
kT (2S+1)
kT
3
bigger z-magnetization for
larger g, B0, S, & lower T
30
Spin-State Evolution: The von Neumann Equation
- A pure spin state |y(t) evolves according to the Schrödinger equation i
dy
= Hˆ y
dt
^
- Analogously, H also drives the time evolution of the density operator r^(t) in the
von Neumann equation:
d rˆ
dt

= -i Hˆ , rˆ

^ ^
^^ ^^
(Commutator [A, B] := AB – BA )

^
- Three types of solutions r^ (t) (for time-independent H):
^
(i) Oscillation/precession:

[H, r^(0)]
^
^
r(t) = r(0) coswt + iw sinwt
^
^
Requires [H, [H, r^ (0)]] = w2 r^ (0)
(ii) Propagator-based:
r^ (t) = e-i Ht r^(0) ei Ht
^
^
(iii) Multiple-quantum coherences: rnm(t) = e-i(En-Em)t rnm(0)
(Verify by inserting each proposed solution into the von Neumann equation)
31
Precession around


B0
 
- Evolution in B0 field: H0 = - m . B0 = - g Sz B0 = w0 Sz (we’ll drop the operator “ ^ “
- Initial magnetization along x: r(0) = Sx, precesses around B0:
from now on)
(i) Oscillation/precession: r(t) = r(0) coswt + [H, r(0)]/(iw) sinwt
if [H, [H, r(0)]] = w2 r(0)
To evaluate [H, r(0)] and [H, [H, r(0)]], we use the
fundamental commutation relations between spin component operators:
[Sx, Sy] = i Sz
By cyclic relabeling of x-, y-, z-axes:
so
[Sy, Sz] = i Sx
[Sz, Sx] = i Sy
0
[H, r(0)] = [w0 Sz, Sx] = w0 [Sz, Sx] = w0 i Sy
[H, [H, r(0)]] = w02 [Sz, i Sy ] = w02 i2 (-Sx) = w02 Sx
= w02 r(0) as required
r(t) = Sx cosw0t + Sy sinw0t

Precession around B0
0
32
Heteronuclear Dipolar Coupling to a Spin-1/2
HIS = wIS 2 Iz Sz
- Initial magnetization along x: r(0) = Sx
(i) r(t) = r(0) coswt + [H, r(0)]/(iw) sinwt
if [H, [H, r(0)]] = w2 r(0)
[H, r(0)] = [wIS 2 Iz Sz, Sx] = wIS 2 Iz [Sz, Sx] = wIS 2 Iz i Sy = iwIS Sy 2 Iz
[H, [H, r(0)]] = wIS2 4 Iz2 [Sz, i Sy ] = wIS2 i2 4 Iz2 (-Sx) = wIS2 4 Iz2 Sx
To evaluate Iz2, we use a Pauli matrix, which represents Iz for spin-1/2:
 1
t
2
Iz = 

 0

0
-
1
2





t 2  14
Iz = 
0
0 1 1
= 
1
4 0
4
0 1
1
=
1
=

1 4
4
[H, [H, r(0)]] = wIS2 4 ¼ Sx = wIS2 Sx = wIS2 r(0) as required

r(t) = Sx coswISt + Sy 2 Iz sinwISt
Oscillating magnetization
33
Invariant Spin States
(i) Oscillation/precession: r(t) = r(0) coswt + [H, r(0)]/(iw) sinwt
[H, r(0)] = 0:
When
if [H, [H, r(0)]] = w2 r(0)
[H, [H, r(0)]] = 0 = 02 r(0): w = 0
Then
r(t) = r(0) cos0t + 0/(iw) sin0t = r(0)
Classical analogue: B-field
parallel to the magnetization
The spin system remains unchanged in its original state,
does not evolve under this Hamiltonian.
Examples:
(a) Equilibrium z-magnetization under the truncated homonuclear dipolar coupling:
r(0) = Sn Iz,n ~ w0 Sn Iz,n = H0 (Zeeman interaction);
[HII, r(0)] ~ [HII, H0] = 0 (truncated HII commutes with H0)
(but nonequilibrium r(0) = Sn pn Iz,n evolves: spin diffusion)
 
(b) J-coupling HJ = J IA. IB of equivalent spins (e.g. in H2O):
Initial x-magnetization of both spins, r(0) = Ix,A+ Ix,B
 
[IA. IB, Ix,A+ Ix,B] = 0,
no J-evolution/splitting for J-coupled 1H in H2O
34
Radio-Frequency Pulses and Spin Rotations
 
Evolution under an x-pulse: H1,x = - m . B1 = - g Sx B1 = w1 Sx
Initial magnetization along z: r(0) = Sz, precesses (nutates) around B1 (along x-axis):
(i) Nutation:
r(t) = Sz cosw1t + [Sx, Sz]/i sinw1t = Sz cosw1t - Sy sinw1t
since [H, [H, r(0)]] = w12 [Sx, [Sx, Sz]] = w12 Sz = w12 r(0), as required
Set this equal to
(ii) Propagator-based: r(t) = e-iw1Sxt Sz eiw1Sxt
Get:
f = - p/2 = -g B1 tP
e-i f Sx Sz ei f Sx = Sz cosf - Sy sinf
Conclusion: e-i f Sx Sz ei f Sx = Sz rotated by f around the x-axis
This applies generally.
Example:
e-i p/2 Sy Sx ei p/2 Sy = “Sx rotated by p/2 around y-axis” = - Sz
e-i p/2 Sy Sx ei p/2 Sy = Sx cos p/2 - Sz sin p/2 = - Sz
35
Propagators: Hahn Echo
Redefine t = 0:
Equilibrium (b):
(b): r(0) = Sz
After p/2-pulse (c):
(c): r(0’) = Sy
After free evolution (d):
r(0) = Sz
r(t90) = ei p/2 Sx Sz e-i p/2 Sx = Sy
r(t90+t) = e-iHCSt Sy eiHCSt
(d): r(0”) = r(t90+t) After p-pulse (f): r(t90+t+t180) = ei p -Sy e-iHCSt Sy eiHCSt e-i p -Sy
= eip-Sy e-iwCSSzt ei p Sy e-i p Sy Sy ei p Sy e-i p Sy eiwCSSzt e-ip-Sy
=
e+iwCSSzt
Sy
e-iwCSSzt
After 2nd free evolution (g): r(t90+t+t180+t) = e-iwCSSzt e+iwCSSzt Sy e-iwCSSzt eiwCSSzt = Sy
Echo condition: “r(2t) = r(0)”
Note: We were able to solve this problem even though the Hamiltonian is (stepwise) time-dependent.
36
Spherical Tensors
- Instead of the coupling matrices and Cartesian product operators used here,
one can use irreducible spherical tensors (for the spatial part)
and irreducible tensor operators (for the spin part).
For instance, for the full homonuclear dipolar coupling
rˆ t rˆ
ˆ
H =I D J =
2
 (- 1)
m
D
m
2
-m
0
Tˆ2 where D 2 =
3
2
0
D zz , Tˆ2 =
1
6
rˆ rˆ
ˆ
ˆ
(3 I z J z - I  J )
1
2
(D xx - D yy )  iD xy
m = -2
1
2
D 2 = m(D zx  iD zy ), D 2 =
- The transformation of a spherical tensor under rotation is particularly simple:
- In the rotating frame (around z) and B0-truncation:
˜ -m
- m im w t
Tˆ2 = Tˆ2 e 0
t - average

0
Tˆ2
- MAS: wzzLF ~ D20 in the rotor frame (rotating around an axis tilted from z by m):
2
D˜ =
0
2
D
m'
2
W
m '0
2
( m ) e
- im ' w r t
t - average

D 2 W 2 ( m ) = D 2
0
00
0
1
2
3 cos
2
 m - 1
m '= - 2
37
Homonuclear Dipolar Couplings of Two Spins-1/2
 
- HII = wII (3 Iz Jz - I . J)
 
- Initial magnetization along x:
- No effect from
 
 
I.J
r(0) = Ix + Jx
= IxJx+ IyJy+ IzJz
 
[I . J, Ix + Jx] = 0, [I . J, I J] = 0
 Evolution under HII’ = wII 3 Iz Jz
r(t) = r(0) coswt + [H, r(0)]/(iw) sinwt
A general commutator:
[I . J, Ix + Jx] = [IxJx+ IyJy+ IzJz, Ix+ Jx] =
0 + [IyJy, Ix+ Jx] +[IzJz, Ix+ Jx] =
Jy(-iIz) + Iy(-iJz) + Jz iIy + Iz iJy= 0
A commutator for spins-1/2 :
[I J, I J] = (I I) (J J) - (I I) (J J) = 0
(use Pauli matrices)
t
1 0
Ix = 
2 1
1 t
1 0
,
I
=
 y

0
2i
-i 

0
if [H, [H, r(0)]] = w2 r(0)

[H, r(0)] = [wII 3 Iz Jz, Ix + Jx] = 3wII {Iz [Jz, Jx] + Jz [Iz, Ix]}= 3wII {Iz iJy+ Jz iIy}
[H, [H, r(0)]] = 9 wII2 [Iz Jz, Iz iJy+ Jz iIy] = 9 wII2 (Iz2 Jx + Ix Jz2)
To evaluate Iz2 and Jz2, again use the Pauli matrix result: Iz2 = ¼ 1I, Jz2 = ¼ 1J
 [H, [H, r(0)]] = 9 wII2 (¼ Jx + Ix ¼) = (3/2 wII)2 (Ix + Jx) = (3/2 wII)2 r(0)
r(t) = (Ix + Jx) cos 3/2wIIt + 3(Iy Jz + Jy Iz) sin 3/2wIIt Oscillating x-magnetization
38
Homonuclear Exchange of Spin-1/2 z-Magnetization
 
- HII = wII (3 Iz Jz- I . J) = wII (2 Iz Jz- (Ix Jx + Iy Jy))
- Nonequilibrium z-magnetization: r(0) = Iz – Jz = 2 IzZQ
Commutators for spins-1/2 :
[Iz Jz, I J] = 0
[Iy Jy, Iz Jx] = 0,
[Iy Jy, Iy Jx] = -¼ i Jz
- No effect from Iz Jz: [Iz Jz, Iz - Jz] = 0, [Iz Jz, I J] = 0
 Evolution under HII’ = - wII (Ix Jx + Iy Jy) = - wII IxZQ
- [H, r(0)] = [-wII IxZQ, 2 IzZQ] = -wII 2 [IxZQ, IzZQ] = 2wII iIyZQ
IxZQ = Ix Jx + Iy Jy [IxZQ, IyZQ] = iIzZQ
IyZQ = Iy Jx - Ix Jy [IyZQ, IzZQ] = iIxZQ
IzZQ = ½(Iz – Jz)
[IzZQ, IxZQ] = iIyZQ
[H, [H, r(0)]] = [-wII IxZQ, 2wII iIyZQ ] = wII2 2 IzZQ = wII2 r(0), as required
r(t) = (Iz – Jz) coswIIt + (Iy Jx - Ix Jy) sinwIIt
Iz ~ coswIIt, Jz ~ - coswIIt : I-J spin exchange
- Alternative notation using I± := Ix ± iIy: Ix Jx + Iy Jy = (I+ J- + I- J+)/2
I+ raises mI, J- lowers mJ: I+ J- can be interpreted as producing a “spin flip-flop”
39
Multiple-Quantum Coherences
- An |n|-quantum coherence rotates n times faster around z than magnetization does:
e-iwtIz TLn eiwtIz = TLn ei nwt
Note: wIz = w Sn Iz,n
- Linear combinations for ±n are also called |n|-quantum coherences
- z-magnetization etc. (diagonal elements of the density matrix) are not coherences
Examples: -- Ix and Iy are single-quantum coherences: [w(Iz + Jz), Ix] = 1w i Iy
-- IxZQ = Ix Jx + Iy Jy and IyZQ = Iy Jx - Ix Jy are zero-quantum coherences
(but IzZQ = ½(Iz – Jz) is not)
since [w(Iz + Jz), IZQ] = 0, no evolution
-- IxDQ = Ix Jx – Iy Jy and IyDQ = Iy Jx + Ix Jy are double-quantum coherences
(but IzDQ = ½(Iz + Jz) is not)
since [2w ½(Iz + Jz), IxDQ] = 2w i IyDQ
- Under H = wI Iz + wJ Jz, DQ coherence evolves with wI + wJ, ZQ coherence with wI  wJ
- A pulse phase shift corresponds to a rotation around z:
Iy = e-ip/2Iz Ix eip/2Iz
ei p/2Iy
T22 ei p/2Iy
= e-ip/2Iz ei p/2Ix eip/2Iz T22 e-ip/2Iz ei p/2Ix eip/2Iz
= e-ip/2Iz ei p/2Ix T22 e2ip/2 ei p/2Ix eip/2Iz = e-ip/2Iz ei p/2Ix (-T22) ei p/2Ix eip/2Iz
- This enables phase cycling to select n-quantum coherences
40
- NMR without serious quantum mechanics
- Angular momentum (spin) and magnetic dipole moment
- Precession around B, with w = - g B
- Fields, frequencies, energies, Hamiltonians
- The rotating frame
- Radio-frequency pulses
- A simple pulse sequence: Hahn spin echo
- Truncation of weak fields: Keeping only the parallel component
- Truncation in the interaction frame: Oscillating components are averaged
- Local fields and interaction tensors: - Chemical shift - Dipolar couplings
- Orientation dependence of NMR frequencies
- PAS - Powder spectrum (for h = 0)
- Frequency w(t) in MAS
- Dipolar decoupling
- Time signals, Fourier transformation
- 2D NMR
- Basic spin quantum mechanics for NMR
- The state of the spin system: density operator
- Spin evolution: The von Neumann equation and its solutions
- Oscillating solutions: - Precession
- Hˆ , rˆ 0 = 0: invariant spin state
- Heteronuclear spin-1/2 dipolar evolution
- Pulses and rotations in spin space
- Propagators and pulse sequences: - Spin echoes
- Homonuclear couplings - Spin exchange - Multiple-quantum coherences
41