An introduction to Solid State NMR and its Interactions

An introduction to
Solid State NMR
and its Interactions
From tensor to NMR spectra
CECAM Tutorial – September 2009
Calculation of Solid-State NMR Parameters
Using the GIPAW Method
Thibault Charpentier - CEA Saclay
thibault.charpentier@cea.fr

Nuclear Magnetic Resonance
Spin
Magnetic Field
The Rabi oscillation
N = N ħ I
Rabi oscillations in quantum dots
Source: http://iramis.cea.fr/drecam/spec/
Pres/Quantro/Qsite/projects/qip.htm

The Zeeman effect
I= 3 2
B 0 ≠0
∣m=3/2 〉
E=− N . B 0
0
ħ H=− N ħ I . B 0 =ħ 0 I Z
B 0 =0
∣m=1/2 〉
The Larmor frequency
0
0
∣m=−1/2 〉
0 =− N B 0
0
∣m=−3/2 〉
0 =− N
2 B 0
NMR spectrum of an isolated nucleus
m=∓1

The spin operators
I= 3 2
I =I X
I Y
I Z
I Z
∣m 〉=m∣m 〉
I ∣m 〉= I I1−mm1∣m1 〉
I −
∣m 〉= I I1−mm−1∣m−1 〉
0 0 0
0 1/2 0 0
I Z =3/2
0 0 −1/2 0
0 0 0 −3/2
I X = 1 2 I I −
I Y = 1 2 I −i I −
3/2 0 0
I =0
0 0 2 0
0 0 0 3/2
0 0 0 0
=
0 0 0 0
0
I
3/2 0 0 0
−
0 2 0 0
0 0 3/2

The Larmor precession
B 0
z
The classical approach to Larmor precession
N
d N
y
d t = N N t ∧ B 0
0
x
The quantum approach to Larmor precession
i ħ d 〈 I 〉
d t
=i ħ d dt 〈t ∣ I ∣ t〉=〈t ∣ [ H , I ] ∣ t〉=i ħ N
〈 I 〉∧ B 0

The nuclear magnetization
In NMR, the only direct observable is
the macroscopic magnetization
M=∑ i
i
But NMR spectroscopist know how to play with
more complex objects...
z
B 0
At equilibrium,
N up
N down
∝exp − E
k B
T
y
y
the magnetization is
along the magnetic field
x
x
M z
≠0
Boltzman populations
M x
=M y
=0

Sensitivity of NMR
The Curie Law (high temperature limit)
M 0
= 0
B 0
=N I
2 ħ 2 I I1
3 k T
B 0
The quantum approach
H=ħ 0 I Z
M 0 =N I 〈 N 〉=N I N ħTr 〚 I eq 〛 eq ∝exp − H
In the high temperature limit
k B
T
eq ∝− 0
k B T I Z
M eq =N I × N ħ× 0
k T ×Tr 〚 I Z
2
〛
Description of NMR experiments in term of I
operators

The Resonance Phenomenum
B 0
z
M 0
Nutation
Application of a tranverse RF
magnetic field at the Larmor
frequency produces the precession
of the nuclear magnetization around
the RF field (nutation) in the socalled
rotating frame
H t= 0
I Z
H RF
t
x T
Pulse Angle
y T
= 1
w
w
H RF
t =− N
2 B 1
I x
cos RF
t =2 1
I x
cos RF
t
d 〈 I 〉
dt
LAB
=〈 I 〉∧
− 0
z2 N
B 1
cos RF
t x
Time Dependent
Rotating frameat RF
around B 0
:x ,y ,z LAB
⇒x ,y , z T
d 〈 I 〉
dt
T
=〈 I 〉∧
{ RF
− 0
}z− 1
x
T
Time Independent

Pulsed Fourier Transform NMR
Typical Timing of a Solid State NMR experiment
Thermal
Equilibrium
&
Relaxation
RF Pulse
p
Free Precession Decay
ms-hours s
s-ms
Macroscopic Magnetization Motion
M p =M 0 x
M 0=M 0 z
M t p =M 0 ×x cos 0 ty sin 0 t
The complex NMR signal Fourier Transform
st∝M 0 exp −i 0 t
0

Pulsed Fourier Transform NMR
Bloch equations
z
dM x
dt
=− M x
T 2
i M y
B 0
dM y
dt
=− M y
T 2
−i M x
y
dM z
dt
=− M z −M eq
T 1
Density operator (quantum state)
x
y
Coherent state
M x
≠0 M y
≠0
t =a x
t I x
a y
t I y
a z
t I z
M
t ∝Tr 〚 I
. t 〛
x
I z
: magnetization ; I
, I −
single quantum coherence
The only directly observable entity

Pulsed Fourier Transform NMR
Free Precession Decay
Signal Processing
(Apodization)
Fourier Transform
(phase correction)
NMR Spectrum

The NMR signal in theory
st=Tr 〚 I exp −i H t I x exp i H t 〛
Initial state:
I x
System evolves under:
H
One quantum transitions are observed:
I
st∝∑ m
∣〈m1∣I ∣m 〉∣ 2 ×exp −i m, m1 t
The NMR frequencies
m, m1 =〈m1∣ H∣m1〉−〈m∣ H∣m〉
The contribution of each transition
to the signal amplitude is (isotropic !!)
∣〈m1∣I ∣m 〉∣ 2
Hamiltonian of NMR interactions are needed H
DFT calculations provide H !

NMR Interactions A complex situation...
H=− N
ħ I⋅ B ext
B local = H Z
H inter.
B 0,
B RF
t
EPR
Electrons
B loc
External fields to
manipulate
the system:
B 0,
B RF
t
NMR
Nuclear Spins I
Nuclear Spins S
NMR
Internal fields:
B loc
inter.
Phonons
inter.
MicroWaves, ....
H= H Z
H CS
H J
H D
H Q
...
0
Chemistry:

B loc
= A X =
NMR Interactions
.
are tensors
A xx
A xy
A xz x
A yx
A yy
A yz
X y
A zx
A zy
A zz
X z
X
H=− N
ħ×I . A . X
Second-rank Tensor
A=1 , X=B 0,
B RF
Zeeman Interaction
A= , X=B 0
A=D , X=S
A=J , X=S
A=Q , X=I
Magnetic shielding
Dipolar Magnetic couplings
(through space)
Indirect Magnetic couplings
(through bond)
Quadrupolar Interaction
(electric couplings)

NMR Interactions: PAS
Diagonalization of A yields the Principal Axis System (PAS)
A xx
A xy
A xz
A A=
.
A XX
0 0
yx
A yy
A yz A
A zx
A zy
A zz=X −1 0 A YY
0 X A
=X A
0 0 A ZZ. −1 . A PAS
. X A
Principal Axes labeling:
A iso
= 1 3 Tr 〚 A〛
∣A ZZ
−A iso ∣≥∣A XX
− A iso ∣≥∣A YY
−A iso ∣
A ZZ
A YY
A iso
A XX

NMR Interactions: PAS
Introducing a convenient representation for encoding
this orientational dependence
1
A PAS
=A iso
1 A−1/2
1
0 0
0 −1/2 1− 0
0 0
Isotropic shift:
A iso
=1/3Tr 〚 A〛
Strength of the anisotropy:
Symmetry of the anisotropy
(asymmetry parameter):
A
=A ZZ
−A iso
A
= A XX
−A YY / A

NMR Interactions: PAS
Relative orientation of the PAS with respect to a reference
frame (crystallographic axes, laboratory frame, ...)
Factorization of X A
provides the three Euler angles
A
, A
, A
X A
=R A
, A
, A
=exp −i A
I z ⋅exp −i A
I y ⋅exp −i A
I z
(x,y,z) is the PAS
of the reference frame B 0
A ZZ
In NMR the position of a line
(single crystal) is dependent
upon the six parameters:
A iso
, A
, A
, A
, A
, A
A XX
A YY

The secular approximation
High Field NMR:
B 0
≫ B loc
Perturbation Expansion of the NMR interactions
H Z
≫ H inter.
H inter.
≈ H 1 CS
H 1 Q
H 2 Q
H 1 D
H 1 J
...
Keeping terms invariant under rotation around B 0
B 0
A XX
A zz
A YY
A ZZ

A general representation of the
NMR interactions
Powerful Tensorial approach to derive all formula !
m2
H
=C
∑ m=−2
−1 m R 2,−m
×T 2m
Euler angles
= , ,
Spatial dependence
R =∑ 2,m n
2, n
2
D n , m
NMR parameters
Spin Operator dependence
[I z
,T 2, m
]=mT 2,m
2,±2
=
2 , 2,±1
T 2, m
(first order ) secular approximation easy : m=0!
H
1 =C R 2,0 ×T 20
=0 ,
2,0 =
3 2
T II 2,0 =
1 3 I 2 Z
−I I1
6

The Chemical Shift Tensor
Absolute chemical shielding (GIPAW output)
Isotropic chemical shift
=−− REF
1
iso
=− iso
− REF
exp
exp iso
ppm=10 6 iso
− REF Hz
×
REF
Hz
Chemical Shift Anisotropy (CSA)
,
z PAS
B 0
H CSA
= CS
, I Z=
2 3 R 2,0 I Z
CS
,= 0 XX
sin 2 cos 2 YY
sin 2 sin 2 ZZ
cos 2
CS
,= 0
iso
0
2
{3cos 2 −1
sin 2 cos2}
y PAS
x PAS

NMR powder spectrum
S powder
t=∫sin d d exp−i ,t⇒ TF ⇒ S powder

The quadrupole Interaction
I1/2
Electric coupling between the nuclear quadrupole moment Q
and local electric field gradient V(r nuc
) at the nucleus
Quadrupolar Coupling Constant ( ~ MHz )
C Q
= eQ h V ZZ
Quadrupolar asymmetry parameter
First order
C Q
H 1 Q
=
6I2I−1 ×R Q ×T
20
20
Second order (complex...)
H Q 2 = 1 0
C Q
Q
= V −V XX YY
V iso
=0
V ZZ
T 2,0=
1 3 I 2 Z
−I I1
6
6I2I−1 × ∑ l=0,2,4
A l ∑ k=−l ,l
B l k
k
D l , 0
A l = f I Z 3 , I Z
Second Order Quadrupolar Induced Shift !
Q 2 =A 0 B 0 0
Isotropic shift

Quadrupolar nuclei
Powder Quadrupolar Static spectra

x LAB
z LAB
The Dipolar Interaction
H D
= ħ {
I S
I .S− 3
I⋅r
3 2 IS
S⋅ r } IS
=I⋅D⋅S
r IS
r IS
B 0
S
=0
r
IS
I
D PAS = −2 ħ I S
3
y r IS
LAB
− 1
0 0
2
0 − 1 0
2
0 0 1
Homonuclear
Heteronuclear
H 1 D
= ħ I S
3
r IS
H 1 D
= ħ I S
3
r IS
×R D IS
20
×T 20
×R D 20
× 3 2 I S Z Z

Indirect J coupling
Small interaction (~Hz), anisotropic effects (so far) neglected
H J
=I J iso
1J ani S≈J iso
I⋅S
Like spins
Unlike spins
I S
J iso
≥∣ iso
− ∣ iso
I S
J iso
≤∣ iso
− ∣ iso
H J
=J iso
I⋅S
H J
=J iso
I Z
S Z
IS
IS 2
IS 3
0
I
0
I
0
I

Multiple interactions
Lineshape is also dependant upon the relative orientation
of CSA PAS with respect to quadrupolar PAS (or vice-versa)
In the crystallographic axes frame (GIPAW calculation)
X ,c
=R ,c
, ,c
, ,c
X Q ,c
=R Q ,c
, Q ,c
, Q ,c
X ,c
X −1 Q ,c
=R ,Q
, ,Q
, ,Q
B 0
V ZZ
ZZ
XX
V XX
V YY
YY

High Resolution NMR
Brownian Motion in Liquids
H t= H iso
H ani
t
H ani
t=0
Only isotropic contributions !
iso
, J iso
In rigid lattices, broad lines
iso
, J iso
+
CSA
, CSA
,C Q
, Q
, CSA ,Q
, CSA ,Q
, CSA ,Q

Magic Angle Spinning Sample
B 0
R
D 20
Second Rank
∝3cos 2 −1
M
A ZZ
A YY
A iso
A XX
3 cos 2 M
−1=0
A ZZ
A XX
A YY
B 0
M A ZZ
M
=0
A ZZ
A ZZ
t =0
A ZZ
A ZZ
t
A ZZ
t

MAS at work
I=1/2
+1/2 -1/2
Static spectrum
CSA + Dipolar ( 13 C- 13 C )
Low speed MAS spectrum
Spinning sidebands
ROT
High speed MAS spectrum
ROT
Proton decoupling is necessary (sample rotation + spin rotation ! )

MAS at work
23 Na
I=3/2
NaAlH 4
+1/2 -1/2
+3/2 +1/2
-3/2 -1/2
B 0
= 7.05 T
v ROT
=10 kHz

NaAlH 4
MAS at work
27 Al
I=5/2
+1/2 -1/2
Central transition
Satellite transitions
B 0
= 7.05 T
v ROT
=10 kHz

The MAS NMR signal in theory
st∝∑ m
∣〈m1∣I ∣m 〉∣ 2 t
×exp−i∫ 0
m, m1 udu
Time-dependent transitions frequencies:
m, m1 =〈m1∣ H t∣m1〉−〈m∣ H t∣m〉
Euler angles of the PAS in the rotor fixed frame
m, m1
,, = 0
, ∑ m≠0
m
, exp {−i m ROT
t}
∫ d exp {−i∫ 0
t
m ,m1 udu}=e −i 0 t , ×∑ k
∣I k ,∣ 2 e −i k ROT t
0 ,: MAS lineshape
I k ,: spinning sidebands pattern

The NMR Laboratory
torturing probe...
MAS probe
MAS
Rotor
Superconducting magnet

NMR parameters: DFT vs Experiment
Quadrupolar parameters (I>1/2) (MHz)
C Q
, Q
Isotropic Chemical shift (ppm)
iso
Chemical shift anisotropy (ppm)
Relative orientation
of CSA PAS in Quad. PAS
(Three Euler angles)
CSA
, CSA
CSA ,Q
, CSA ,Q
, CSA ,Q
Isotropic J couplings (1-3 bonds) 1 J Si−O
2 J Si−O−Si
NMR provides methods for measurements
of Dipolar interactions (only the structure is needed)