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