Basic Theory of Solid-State NMR
ρ t( ) = ρ 0( )cosωt − iω H , ρ 0( )$%
&'sinωt
4th Winter School on Biomolecular Solid-State NMR, Stowe, VT, Jan. 10-15, 2016
Mei Hong, Department of Chemistry, MIT
Spin Angular Momentum & Magnetic Dipole Moment
2
µ = γ
S
E = −µ iB
γN = gN
e2mp
= gNµN
τ =µ ×BEnergy of a magnetic
moment in a B field: Torque on a magnetic moment in a B field:
τ =µ ×B
τ = d
S dt
#$%
&%⇒ d
µdt
= γµ ×B
Precession of the Magnetic Moment around B
3
For rotational motion:
Recall linear motion : F = dp dt( )
dµx dt = γBzµydµy dt = −γBzµxdµz dt = 0
#
$%
&%
Solution for µ(0) in the x-z plane:
µx t( ) = µx 0( )cosωtµy t( ) = µx 0( )sinωtµz t( ) = µz 0( ) = cnst
"
#$
%$
Bloch eqn:
Larmor frequency:
ω=−γB
Fields, Frequencies, Energies & Hamiltonians
Larmor frequency = transition frequency of nuclear spin energy levels.
ρ t( ) = e−iHt ρ 0( )e+iHt
4
H EH = −
µ ⋅B = −γ
S ⋅B0
E = ω0= −µ ⋅B0
B0
B0 = 18.8 Tesla 1H Larmor frequency: ω0 = -2π 800 MHz
Energy splitting: E = 3.3 µeV
ω0 = −γB0 = 2πν0
Resonance: apply electromagnetic radiation near the Larmor frequency ω0.
Resonance, the Rotating Frame, and RF Pulses
Lab frame Rotating frame, On resonance
Rotating frame, Off resonance
5 ωrf
2π 50 –1000 MHz, ω1
2π 50 – 100 kHz
Resonance: Maximizing Transition Probability
Complete transition (Pab=1) can be achieved even when the applied field strength Vba=ħω1 << ħω0 (~100 kHz vs ~100 MHz).
W << H(0)
6
Pab t( ) =Vba
2
42
sin2ωrtωr
2
=ω=ωab
sin2ωrt
Rabi frequency:
ωr =12
ωba0 −ω( )
2+ Vab
22
Phase of Radiofrequency Pulses
• Change in phase of B1(t) by 90˚ in the lab frame corresponds to a change in direction of B1 in the rotating frame (e.g. from x to y)
• Pulse phase can also change due to frequency change: PMLG = FSLG.
B1Lab t( ) = B1
sinωrf t
00
"
#
$$$
%
&
'''=B12
sinωrf t
−cosωrf t
0
"
#
$$$$
%
&
''''+B12
sinωrf t
cosωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1Lab t( ) = B1
2
sinωrf t
cosωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1rot t( ) =
B12
010
"
#
$$$
%
&
'''
B1Lab t( ) = B1
cosωrf t
00
"
#
$$$
%
&
'''=B12
cosωrf t
sinωrf t
0
"
#
$$$$
%
&
''''+B12
cosωrf t
−sinωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1Lab t( ) = B1
2
cosωrf t
−sinωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1rot t( ) =
B12
100
"
#
$$$
%
&
'''
7
Analyzing Pulses: Spin Echo & the Vector Model
• Precession at chemical shift offsets in the rotating frame;
• Pulse effects: left hand rule; • No quantum mechanics needed.
8
A Central Equation: Local Fields & Frequencies
9
• Local fields from electrons & other nuclei are much weaker than Zeeman fields.
• Local fields’ interactions with spins are described by Hamiltonians.
Truncation of Weak Interactions: Keep B//
Btot =
B0 +
Bloc = B0 + Bloc,//( )2 + Bloc,⊥2
= B0 + Bloc,//( ) 1+Bloc,⊥2
B0 + Bloc,//( )2
= Taylor expansion
B0 + Bloc,//( ) 1+Bloc,⊥2
2 B0 + Bloc,//( )2
<< 1
"
#
$$$$
%
&
''''
≈ B0 + Bloc,//
ωL ≈ −γ B0 + Bloc,//( ) =ω0 −γBloc,//
10
HZeeman = −γB0 I z chemical shielding tensor
HCS = γI ⋅σ ⋅
B0
HD = −µ04π
γ jγk3I j ⋅ rjk rjk( ) I k ⋅ rjk rjk( )− I j ⋅ I k
rjk3
k∑
j∑
electric quadrupole moment
HQ =e Q
2I 2I −1( )I ⋅V ⋅
I
electric field gradient tensor
Nuclear Spin Interactions: Hamiltonians
11 Hamiltonians expressed in frequency units.
HCS = γσ zzLabB0 ⋅ Iz = σ isoω0 +
12δ 3cos2θ −1−η sin2θ cos2φ( )*
+,-
./⋅ Iz
HD,IS = −µ04πγ IγSr3
123cos2θ −1( ) ⋅2 IzSz
HD,II = −µ04π
γ 2
rjk3123cos2θ jk −1( )
k∑
j∑ ⋅ 3Izj Szk -
I j ⋅I k2
34
567
HQ =eQ
2I 2I −1( )VzzLab 3Iz Iz −
I ⋅I2
34
567, where Vzz
Lab =1
2eq 3cos2 θ −1−η sin2 θ cos 2φ( )
8
9
:::::
;
:::::
Truncated Hamiltonians: Keep the Operators that Commute with the Zeeman Interaction
HLocal = ωiso +ωaniso θ ,φ( )$% &'
↓
spatial part, affected by MAS& sample alignment
⋅ Spin Part( )
↓
Affected by rf pulses
12
HZeeman =ω0 ⋅ I z HCS,D,Q
Describing Pulse Sequences For Coupled Spin Systems
The vector model cannot conveniently describe • how dipolar & J interactions change the magnetization; • complex multiple-pulse sequences.
Need to know density operators.
13
• Spin is quantized (shown by the Stern-Gerlach expt). Spin-1/2 nuclei have 2 basis states: and .
• In a suitably chosen basis, the above states are represented by vectors:
• A pure spin state is a linear
superposition of the basis states:
Pure Spin States and Spin Operators
↑ =
∧ 10
#
$%
&
'(, ↓ =
∧ 01
#
$%
&
'(
• are QM observables with eigenvalue eqns:
• follow commutation rules:
• are rep’ed by 2 x 2 matrices (Pauli matrices) for spin-1/2 nuclei:
I z ±z =±
12±z , I x ±x =±
12±x , etc.
I x , I y!"
#$= iIz , I y , I z!
"#$= iI x , I z , I x!
"#$= iI y
I x =12
0 11 0
!
"#
$
%&, I y =
12
0 −ii 0
!
"#
$
%&, I z =
12
1 00 −1
!
"#
$
%&
I x2 = I y
2 = I z2 =
14
1 00 1
!
"#
$
%&=141
Spin operators
Spin states
↑
ψ =C+ ↑ +C− ↓ =∧ C+
C−
&
'((
)
*++
↓
Commutator: A, B!
"#$≡ AB− BA
14
A Mixture of Spins: Density Operator
ρ = pk ψk ψkk∑
• Average value of an observable A:
A ≡ ψ Aψ =Tr ρA( )
15
• At thermal equilibrium:
ρeq =e−H kT
Tr e−H kT( )≈
12I +1
1− HkT
$
%&
'
()
Dropped: commutes with all operators
Trace, sum of diagonal ω0 << kT
• A statistical ensemble of spins can only be described by a density operator:
• The observable in NMR is the magnetization...
ρeq' ≈
12I +1
γB0kT
Iz$
%&
'
() ∝ I z
• Magnetization: the sum of all magnetic dipole moments, • The only non-zero traces are: • The x, y components of the magnetization are non-0 only when ρ contains Ix
and Iy terms:
M = γ
I =
I x
I y
I z
"
#
$$$
%
&
'''= γ
Tr ρ I x( )Tr ρ I y( )Tr ρ I z( )
"
#
$$$
%
&
'''
Tr Ix
2( ), Tr Iy2( ), Tr Iz
2( )
I x ± iI y =Tr cosωt ⋅ I x + sinωt ⋅ I y( ) I x ± iI y( )( ) = cosωt ± isinωt= e±iωt = f t( )
e.g. for ρ = cosωt ⋅ I x + sinωt ⋅ I y ,
When the density operator contains Ix, Iy terms, then x and y magnetization are observed.
M =
µi∑
Time Evolution of the Density Operator
17
• If H and ρ commute, then ρ does not change (e.g. Sz magnetization is not affected by IzSz dipolar coupling).
• If H does not depend on time, then the formal solution is:
• More useful solution: (obtained by Taylor-expanding the above exponential operators)
ρ t( ) = e−iHt ρ 0( )eiHt = U t( ) ρ 0( )U−1 t( ), where U t( ) = e−iHt is the propagator.
ρ t( ) = ρ 0( )cosωt +
H , ρ 0( )#$
%&
iωsinωt, provided H , H , ρ 0( )#
$%&
#$
%&=ω
2ρ 0( )
id ψdt
= H ψ
ψ 0( ) ⇒ H
ψ t( )
Schrödinger eqn dρdt
= −i H , ρ#$
%&
von Neumann eqn ρ 0( ) ⇒
dipolar couplings, etc
rf pulses, chem shifts ρ t( )
Evolution of ρ Under 2-Spin Dipolar Coupling
18
Let ρ 0( ) = I x , H IS =ωIS2 I zSz
ρ t( ) = ρ 0( )cosωt +H , ρ 0( )#$
%&
iωsinωt, provided H , H , ρ 0( )#
$%&
#$
%&=ω
2ρ 0( )
H , ρ 0( )#$
%&= ωIS2 I zSz , I x#
$%&=ωIS2 I z , I x#
$%&Sz =ωIS ⋅2iI ySz
H , H , ρ 0( )#$
%&
#$
%&= ωIS2 I zSz ,ωIS ⋅2iI ySz#
$%&=ωIS
2 4i IzSz , I ySz#$
%&=ωIS
2 4i Iz , I y#$
%&Sz2
=ωIS2 i −iI x( ) =ωIS
2 I x
Thus, H , H , ρ 0( )"
#$%
"#
$%=ω
2ρ 0( ) is satisfied, with ω =ωIS
ρ 0( ) ⇒HIS
ρ t( )?
ρ t( ) = I x cosωt +2 I ySz sinωt
Detected as x-magn. unobservable.
HCS = σ isoω0 +12δ 3cos2θ −1−η sin2θ cos2φ( )(
)*+
,-⋅ I z
HD,IS = −µ04πγ IγSr3
123cos2θ −1( ) ⋅2 I zSz
HD,II = −µ04π
γ 2
rjk3123cos2θ jk −1( )
k∑
j∑ ⋅ 3I z
jSzk −I j ⋅I k( )
HQ =eQeq
2I 2I −1( )123cos2θ −1−η sin2θ cos2φ( ) 3I z Iz −
I ⋅I( )
The Spatial Part of Nuclear Spin Hamiltonians
HLocal = ωiso +ωaniso θ ,φ( )$% &'spatial part, affected by MAS
& sample alignment
⋅ Spin Part( )
Affected by rf pulses
19
Chemical Shift Tensor & the Principal Axis System
20
HCS = γσ zz
LabB0 ⋅ I z = ?
σ isoω0 +12δ 3cos2θ −1−η sin2θ cos2φ( )*
+,-
./⋅ I z
HCS = γI ⋅σ ⋅B0 = γ Ix Iy Iz( )
σ xxLab σ xy
Lab σ xzLab
σ yxLab σ yy
Lab σ yzLab
σ zxLab σ yz
Lab σ zzLab
$
%
&&&&&
'
(
)))))
00B0
$
%
&&&
'
(
)))
=truncation
γ Izσ zzLFB0
= γ IzB0 0 0 1( )... ... ...... ... ...... ... σ zz
LF
$
%
&&&
'
(
)))
001
$
%
&&&
'
(
)))
bilinear product, invariant with frame change
= γ IzB0 ⋅b0T σ b0( )
PAS
Choose the principal axis system (PAS), where the tensor is diagonal
σ PAS =
σ xxPAS 0 0
0 σ yyPAS 0
0 0 σ zzPAS
"
#
$$$$
%
&
''''
= cosφ sinθ , sinφ sinθ , cosθ( )
ωxxPAS 0 0
0 ωyyPAS 0
0 0 ωzzPAS
$
%
&&&&
'
(
))))
cosφ sinθsinφ sinθcosθ
$
%
&&&
'
(
)))
b0PAS
Iz
Chemical Shift Anisotropy & Asymmetry Principal values: ω ii
PAS =ω0σ iiPAS
= ωxxPAS cos2 φ sin2θ +ωyy
PAS sin2 φ sin2θ +ωzzPAS cos2θ( ) ⋅ Iz
HCS = γ IzB0 ⋅
b0T σ b0( )
PAS
Isotropic shift : ωiso ≡13ωxx +ωyy +ωzz( )
Anisotropy parameter : δ ≡ωzzPAS −ωiso;
Asymmetry parameter : η ≡ωyy −ωxx
ωzz −ωiso
ω θ ,φ;δ,η( ) = δ
23cos2θ −1−η sin2θ cos2φ( )+ωiso
Directional cosines
Chemical shift anisotropy can also be expressed in terms of directional cosines.
• (θ, φ): molecular orientations in the sample: powder, oriented samples, single crystal, etc.
• (δ, η): electronic environment around the nuclear spin → structure info.
21
3 principal values: directly read off from maximum and step positions.
P θ( ) dθ = S ω( ) dω
⇒ S ω( ) = 1
6δ ω + δ / 2
For η=0,
Static Powder Spectra Spectra S(ω(θ, φ)) reflect orientation distribution P(θ, φ) of the molecules.
ω θ( ) = 1
2δ 3cos2 θ−1( )+ωiso
22
Schmidt-Rohr & Spiess, 1995.
Orientation dependence allows the measurement of: • Helix orientation in lipid bilayers using oriented samples; • Changes in bond orientation due to motion; • Torsion angles, i.e. relative orientation of molecular segments.
NMR Frequencies are Orientation-Dependent
ω θ,φ( ) = 1
2δ 3cos2 θ−1−ηsin2 θcos2φ( )+ωiso
23
Magic-Angle Spinning:
b0Rotor =
cosωrt sinθmsinωrt sinθmcosθm
#
$
%%%
&
'
(((
= cosωrt sinθm , sinωrt sinθm , cosθm( )
ωxxrotor ωxy
rotor ωxzrotor
ωyxrotor ωyy
rotor ωyzrotor
ωzxrotor ωzy
rotor ωzzrotor
#
$
%%%%
&
'
((((
cosωrt sinθmsinωrt sinθmcosθm
#
$
%%%
&
'
(((
=ωiso +123cos2θm −1( ) ωzz
rotor −ωiso( )− ωyyrotor −ωxx
rotor( )sin2θm cos2ωrt
+ωxyrotor sin2θm sin2ωrt +ωxz
rotor sin2θm cosωrt +ωyzrotor sin2θm cosωrt
ωCS θ ,φ t( )( ) =ω0 ⋅
b0T σ b0( )
Rotor
θm = 54.7˚:
123cos2θm −1( ) = 0
HLocal t( ) =ω t( ) ⋅ Spin Part( )
θm ≠ 54.7˚: Off MAS, SAS, DAS...24
Periodic Time Signal Under MAS
f t( ) = exp i ωCS (t' )dt '
0
t
∫#
$%
&
'(= eiωisot exp i ωCSA (t' )dt '
0
t
∫#
$%
&
'(• At all times,
time signal:
At t = ntr = n ⋅2π ωr , ωCS θ ,φ( ) =ωiso ⇒ f tr( ) = eiωisot
Rotation echoes
• Onto 60 kHz – 100 kHz MAS: ωr >>ωijrotor
ωCS θ ,ωrt( ) =ωiso + 0 − ωyyrotor −ωxx
rotor( )sin2θm cos2ωrt +ωxyrotor sin2θm sin2ωrt + ...
ωCS (t' )dt '
0
t
∫ =ωisot + ciji, j∑
ωijrotor
ωrcosnijωrt → f t( ) ≈ eiωisot
25
26
Dipolar Coupling Tensor
DPAS =
DxxPAS 0 0
0 DyyPAS 0
0 0 DzzPAS
!
"
####
$
%
&&&&
=
−ωd 2 0 0
0 −ωd 2 0
0 0 ωd
!
"
###
$
%
&&&
ωIS θ;δ( ) = 1
2δ 3cos2θ −1( ), where δ = − µ0
4πγ IγSr3
• Uniaxial (η = 0) along the internuclear vector. • Isotropic component = 0.
Common dipolar coupling constants (δ): • 1H-1H in CH2 groups: ~1.8 Å, ~32 kHz (x 1.5 for homonuclear) • 13C-1H single bond: 1.1 Å, ~23 kHz • 15N-1H single bond: 1.05 Å, ~10 kHz • 13C-13C single bond: 1.54 Å (aliphatic), ~3.1 kHz; 1.4 Å (aromatic), ~4.2 kHz • 13C-15N bonds in protein backbones: 1.5 Å, 900 Hz – 1 kHz
27
Voltage: U = −dφdt
= −µ0dMx t( )dt
dA∫= −µ0ωLM0 cosωLt dA∫
Bx t( ) = µ0Mx t( ) = µ0M0 sinωLt
Magnetic flux: φ t( ) = B t( )dA∫
$
%&&
'&&
The precessing M induces a voltage in the rf coil, which is detected.
NMR signal ∝ωLM0 cosωLt
• NMR signals are measured in the rotating frame:
• The spectrum S(ω) is the “weight” of the signal eiωt with a given ω in the total f(t):
f t( ) = 12π
S ω( )eiωt dω−∞
∞∫
f t( ) = f0 cos ωL −ωrf( )t + isin ωL −ωrf( )t#$
%&= f0e
i ωL−ωrf( )t
Detecting the NMR Time Signal
SN=ωLM0 cosωLt
ωL ∝ γ
5 2B03 2
T
• The spectrum S(ω) is the “weight” of the signal eiωt with a given ω in the total f(t):
• The spectrum is calculated from the time signal by Fourier transformation:
From Time Signals to Frequency Spectra: FT
S ω( ) = 12π
f t( )e−iωt dt−∞
∞∫
f t( ) = 12π
S ω( )eiωt dω−∞
∞∫
Many useful Fourier theorems to relate the shape of f(t) & S(ω): • Inverse width: dwell time → 1/spectral width, Δν1/2 → 1/T2 • Convolution • Integral theorem • Symmetry theorem • Sampling theorem
28
Acknowledgement • Schmidt-Rohr and Spiess, “Multidimensional solid-state NMR and Polymers”
• Cohen-Tannoudji, Diu and Laloe, “Quantum Mechanics”
• Former & current students of various NMR spectroscopy courses & tutorials.
Further elucidation of: • Decoupling & recoupling local fields under MAS: Jaroniec, Mueller, Griffin • Orientation dependence of NMR frequencies: Opella, Cross • Orientational changes due to motion: Hong, Reif, McDermott • Phase cycling of rf pulse sequences: Hong • Coherence transfer for nD correlation: Ladizhansky, McDermott • NMR detection sensitivity: Griffin, Reif • Sampling in Fourier transformation: Rovnyak • Quadrupolar nuclei: Grandinetti
29