Basic Theory of Solid-State NMR - … · 5th Winter School on Biomolecular Solid-State NMR, Stowe,...
Transcript of Basic Theory of Solid-State NMR - … · 5th Winter School on Biomolecular Solid-State NMR, Stowe,...
Basic Theory of Solid-State NMR
ρ t( ) = ρ 0( )cosωt − iω H , ρ 0( )$%
&'sinωt
5th Winter School on Biomolecular Solid-State NMR, Stowe, VT, Jan. 7-12, 2018
Mei Hong, Department of Chemistry, MIT
Magnetic Dipole Moment From Nuclear Spins
2
µ = γ
S γ N = gN
e2mp
E = −µ iB
τ =µ ×BEnergy of a magnetic
moment in a B field: Torque on a magnetic moment in a B field:
τ =µ ×B
τ =
dSdt
=1γd µdt
$
%&
'&⇒ dµdt
= γµ ×B
Precession of the Magnetic Moment around B
3
For rotational motion:
Recall linear motion : F = dp dt( )
Solution for µ(0) in the x-z plane:
µx t( ) = µx 0( )cosωtµy t( ) = µx 0( )sinωtµz t( ) = µz 0( ) = cnst
"
#$
%$
!µ ×!B =
i!
j!
k!
µx µy µz0 0 B0
=!i µyB0 −
!jµxB0 +
!k ⋅0
Bloch eqn
dµx dtdµy dt
dµz dt
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟= γ
µyB0−µxB00
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
Larmor frequency
Fields, Frequencies, Energies & Hamiltonians
Precession frequency = transition frequency
ρ t( ) = e−iHt ρ 0( )e+iHt
4
H EH = −
µ ⋅B = −γ
S ⋅B0
E = − µ ⋅B0
ΔE = ω0 B0
B0 = 18.8 Tesla 1H Larmor frequency: ω0 = -2π 800 MHz
Energy splitting
ω0 = −γB0 = 2πν0
Resonance: apply electromagnetic radiation near the Larmor frequency ω0.
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)
5
Pab t( ) =Vba
2
42
sin2ωrtωr
2
=ω=ωab
sin2ωrt
Rabi frequency:
ωr =12
ωba0 −ω( )
2+ Vab
22
Resonance, the Rotating Frame, and RF Pulses
Lab frame Rotating frame, On resonance
Rotating frame, Off resonance
6 ωrf
2π 50 –1000 MHz, ω1
2π 50 – 100 kHz
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)
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
B1Lab t( ) = B1
sinωrf t
00
"
#
$$$
%
&
'''=B12
sinωrf t
−cosωrf t
0
"
#
$$$$
%
&
''''+B12
sinωrf t
cosωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1Lab t( ) = B1
2
sinωrf t
cosωrf t
0
"
#
$$$$
%
&
'''' ⇒
B1rot t( ) =
B12
010
"
#
$$$
%
&
'''
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 the Zeeman field …
ωL = −γ ⋅!B
Resonance frequency
Magnetic field at the nucleus
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 = −γ I zB0
11 Hamiltonians expressed in frequency units.
chemical shielding tensor
HCS = γ!I ⋅σ↗ ⋅
!B0
HD = − µ04π
γ jγ k3!I j ⋅ !rjk rjk( ) !I k ⋅ !rjk rjk( )− !I j ⋅ !I k
rjk3
k∑
j∑
electric quadrupole moment
HQ = e ↗Q2I 2I −1( )"
#I ⋅V↘ ⋅
#I
electric field gradient tensor
Local Field Interactions: Hamiltonians
HCS = γσ zzLabB0 ⋅ Iz = σ isoω0 +
12δ 3cos2θ −1−η sin2θ cos2φ( )⎡
⎣⎢⎤⎦⎥⋅ Iz
HD,IS = − µ04π!γ Iγ Sr3
123cos2θ −1( ) ⋅2 IzSz
HD,II = − µ04π!
γ 2
rjk3123cos2θ jk −1( )
k∑
j∑ ⋅ 3Izj Szk -
"I j ⋅"I k( )
HQ = eQ2I 2I −1( )!Vzz
Lab 3Iz Iz −"I ⋅"I( ), where Vzz
Lab =1
2eq 3cos2 θ − 1− η sin2 θ cos 2φ( )
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
Understanding Pulse Sequences for Coupled Spins
The vector model cannot conveniently describe • how dipolar & J interactions change the magnetization; • complex multiple-pulse sequences.
14
Need statistical and quantum mechanics: density operators.
• Spin angular momentum is quantized, as shown by the Stern-Gerlach expt.
• Spin-1/2 nuclei have 2 basis states:
Spin States Represented by Vectors
↑ =
∧ 10
⎛⎝⎜
⎞⎠⎟, ↓ =
∧ 01
⎛⎝⎜
⎞⎠⎟
↑ and ↓
ψ =C+ ↑ +C− ↓ =∧ C+
C−
&
'((
)
*++
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• In a suitably chosen basis, these 2 states are represented by vectors:
• A pure spin state is a linear superposition of the basis states:
• Observables are QM (Hermitian) operators with eigenvalue eqns:
A, B⎡⎣ ⎤⎦ ≡ AB − BA
I x , I y!"
#$= iIz , I y , I z!
"#$= iI x , I z , I x!
"#$= iI y
Observables Represented by Matrices
• Spin operators follow commutation relations:
• Spin-1/2 operators are represented by 2 x 2 Pauli matrices:
I z ±z = ± 1
2±z , I x ±x = ± 1
2±x , I y ±y = ± 1
2±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 = 1
41 00 1
⎛⎝⎜
⎞⎠⎟= 141
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Commutator
Our Samples: an Ensemble of Spins • Expectation (average) value of an observable A:
• The observable in NMR is the magnetization...
• 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
ρeq' ≈
12I +1
γB0kT
Iz$
%&
'
() ∝ I z
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For pure states: = C+* C−
*( ) A++ A+−A−+ A−−
⎛
⎝⎜
⎞
⎠⎟
C+
C−
⎛
⎝⎜
⎞
⎠⎟ =
C+*C+A++ + C−
*C−A−−
+C+*C−A+− + C−
*C+A−+
A ≡ ψ Aψ
ρ ≡ pk ψ k ψ kk∑ = pk
Ck+
Ck−
⎛
⎝⎜
⎞
⎠⎟ Ck+
* Ck−*( )
k
∑ =C+C+
* C+C−*
C−C+* C−C−
*
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
• A statistical ensemble of spins is described by a density operator:
For mixtures: A = pk ψ k Aψ k = ∑ Tr ρ A( )
• Magnetization: the sum of all magnetic dipole moments:
M = γ
I =
I x
I y
I z
"
#
$$$
%
&
'''= γ
Tr ρ I x( )Tr ρ I y( )Tr ρ I z( )
"
#
$$$
%
&
'''
18
I x =
12
0 11 0
⎛⎝⎜
⎞⎠⎟, I y =
12
0 −ii 0
⎛⎝⎜
⎞⎠⎟, I z =
12
1 00 −1
⎛⎝⎜
⎞⎠⎟, I x
2 = I y2 = I z
2 = 14
1 00 1
⎛⎝⎜
⎞⎠⎟= 141
Tr Ix
2( ), Tr Iy2( ), Tr Iz
2( )• The only non-zero traces are:
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 ,
• So the x, y magnetization are non-0 only when ρ contains Ix and Iy terms.
Time Evolution of the Density Operator
19
• If H and ρ commute, then ρ does not change in time. e.g. Sz magnetization is not affected by IzSz dipolar coupling.
id ψdt
= H ψ
ψ 0( ) H⎯ →⎯ ψ t( )
Schrödinger eqn dρdt
= −i H , ρ#$
%&
von Neumann eqn ρ 0( ) rf pulses, chem shifts
dipolar couplings⎯ →⎯⎯⎯⎯⎯⎯ ρ t( )
ρ t( ) = e−iHt ρ 0( )eiHt = U t( ) ρ 0( )U−1 t( ), where U t( ) = e−iHt .
• If H is time-independent, then the formal solution to the von Neumann eqn is:
ρ t( ) = ρ 0( )cosωt +
H , ρ 0( )#$
%&
iωsinωt, provided H , H , ρ 0( )#
$%&
#$
%&=ω
2ρ 0( )
• More useful solution: (by Taylor-expansion of the exponential operators)
Evolution of ρ Under 2-Spin Dipolar Coupling
Let ρ 0( ) = I x , H IS =ωIS2 I zSz
ρ t( ) = ρ 0( )cosωt +
H , ρ 0( )⎡⎣ ⎤⎦iω
sinωt, provided H , H , ρ 0( )⎡⎣ ⎤⎦⎡⎣
⎤⎦ =ω
2ρ 0( )
Thus, H , H , ρ 0( )⎡⎣ ⎤⎦⎡
⎣⎤⎦ =ω IS
2 ρ 0( )
ρ 0( ) ⇒HIS
ρ t( )?
ρ t( ) = I x cosωt +2 I ySz sinωt
Detected as x-magn. unobservable.
20
H , ρ 0( )⎡⎣ ⎤⎦ = ω IS2 I zSz , I x⎡⎣ ⎤⎦ =ω IS2 I z , I x⎡⎣ ⎤⎦ Sz =ω IS ⋅2iI ySz
H , H , ρ 0( )⎡⎣ ⎤⎦⎡⎣
⎤⎦ = ω IS2 I zSz ,ω IS ⋅2iI ySz⎡⎣ ⎤⎦ =ω IS
2 4i Iz , I y⎡⎣ ⎤⎦ Sz2 =ω IS
2 i −iI x( ) =ω IS2 I x
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
21
Chemical Shift Tensor & the Principal Axis System
HCS = γ
!I ⋅"σ ⋅!B0 = ... σ isoω0 +
12δ 3cos2θ −1−η sin2θ cos2φ( )⎡
⎣⎢⎤⎦⎥⋅ I z
HCS = γ!I ⋅"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 zzLabB0
= γ IzB0 0 0 1( )... ... ...... ... ...... ... σ zz
Lab
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
001
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
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
#
$
%%%%
&
'
((((
22
= γ IzB0 ⋅
!b0T "σ !b0( )PAS
bilinear, invariant with coordinate change
= 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 : ω iso ≡13ω xx +ω yy +ω zz( )
Anisotropy parameter : δ ≡ω zzPAS −ω iso
Asymmetry parameter : η ≡ω yy −ω xx
ω zz −ω iso
Directional cosines
Chemical shift anisotropy can also be expressed in terms of directional cosines.
• (θ, ϕ): reflect orientations of the molecules in the sample: powder, oriented, single crystal, etc.
• (δ, η): reflect electronic environment at the nuclear spin –> structure info.
23 ω θ ,φ;δ ,η( ) = δ
23cos2θ −1−η sin2θ cos2φ( ) +ω iso
Orientation dependence allows the measurement of: • Helix orientation in lipid bilayers; • Changes in bond orientation due to motion; • Torsion angles, i.e. relative orientation of molecular segments.
NMR Frequencies are Orientation-Dependent
24
ω θ ,φ;δ ,η( ) = δ
23cos2θ −1−η sin2θ cos2φ( ) +ω iso
3 principal values: directly read off from maximum and step positions.
P θ( ) dθ = S ω( ) dω , and P θ( ) = sinθ
⇒ S ω( ) = P θ( ) dθdω
= 1
6δ ω + δ 2( )( )1 2
For η=0,
Static Powder Spectra Spectra S(ω) reflect orientation distribution P(θ, ϕ) of the molecules.
ω θ( ) = 1
2δ 3cos2 θ−1( )+ωiso
25
Schmidt-Rohr & Spiess, 1995.
b0Rotor =
sinθm cosωrtsinθm sinωrtcosθm
#
$
%%%
&
'
(((
= sinθm cosω rt, sinθm sinω rt, cosθm( )ω xxrotor ω xy
rotor ω xzrotor
ω yxrotor ω yy
rotor ω yzrotor
ω zxrotor ω zy
rotor ω zzrotor
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
sinθm cosω rtsinθm sinω rtcosθm
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
ωCS θ ,φ t( )( ) =ω0 ⋅
b0T σ b0( )
Rotor
Spinning: HLocal t( ) =ω t( ) ⋅ Spin Part( )
θm = 54.7˚: 1
23cos2θm −1( ) = 0 θm ≠ 54.7˚: OMAS, SAS, DAS...
26
=ω iso +123cos2θm −1( ) ω zz
rotor −ω iso( )− ω yyrotor −ω xx
rotor( )sin2θm cos2ω rt
+ω xyrotor sin2θm sin2ω rt +ω xz
rotor sin2θm cosω rt +ω yzrotor sin2θm cosω rt
Lab frame Rotor frame
Periodic Time Signal Under MAS
f t( ) = ei ωCS (t' )dt '0
t
∫= eiω isote
i ωCSA (t' )dt '0
t
∫• At all times:
At t = ntr = n ⋅2π ωr , ωCS θ ,φ( ) =ωiso ⇒ f tr( ) = eiωisot
Rotation echoes
ωCS θ ,ωrt( ) =ωiso + 0 − ωyyrotor −ωxx
rotor( )sin2θm cos2ωrt +ωxyrotor sin2θm sin2ωrt + ...
• At 60 – 100 kHz MAS: ωr >>ωijrotor
ωCS (t' )dt '
0
t
∫ =ωisot + ciji, j∑
ωijrotor
ωrcosnijωrt → f t( ) ≈ eiωisot
27
28
Dipolar Coupling Tensor
DPAS =
DxxPAS 0 0
0 DyyPAS 0
0 0 DzzPAS
!
"
####
$
%
&&&&
=
−δ 2 0 0
0 −δ 2 0
0 0 δ
!
"
###
$
%
&&&
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 (includes 1.5 x 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, 0.9 – 1.0 kHz
29
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.
Voltage: U = −dφdt
= −µ0dMx t( )dt
dA∫= −µ0ωLM0 cosωLt dA∫
• NMR signals are measured in the rotating frame:
f t( ) = f0 cos ωL −ωrf( )t + isin ωL −ωrf( )t#$
%&= f0e
i ωL−ωrf( )t
Detecting the NMR Time Signal
M0 = N γ( )2 B0
14kT ΔE∝γ, µ∝γ
SN∝γ 5 2B0
3 2
T NMR signal ∝ωLM0 cosωLt
SN=ωLM0 cosωLt
ωL
• The spectrum S(ω) is the weighting factors of eiωt for a given ω in the total f(t):
• The spectrum is the Fourier transform of the time signal.
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 relate the shapes of f(t) and S(ω): • Inverse width: dwell time = 1/spectral width, Δν1/2 = 1/πT2 • Convolution theorem • Integral theorem • Symmetry theorem • Sampling theorem
30