Principles of High Resolution Solid State Nuclear Magnetic Resonance Spins manipulation as much as you can

Thibault Charpentier CEA / IRAMIS / SIS2M - UMR CEA-CNRS 3299 91191 Gif-sur-Yvette cedex, France

2`eme ´ecole de RMN du GERM 18th -23th , Carg`ese (Corse)

Content (Part I)

Introduction

NMR Interactions

Basic (static) NMR Techniques

Magic Angle Spinning

Isotropic versus Anisotropic NMR in the solid state CO

Cα

300

280

260

240

220

200

180

160 13

140 120 100 C Chemical shift (ppm)

80

60

40

20

0

-20

-40

40

20

0

-20

-40

Cα

CO

300

280

260

240

220

200

180

160 13

13

C NMR spectra of Glycine

140 120 100 C Chemical shift (ppm)

80

60

NMR in Solid phase: powder spectrum

z  B0

y x

In liquid phase (isotropic media), the Brownian motions average out the anisotropy of NMR interactions.

Solid State NMR today

Thanks to major advances in the last two decades, ssNMR Methodologies (pulse sequences) have now reached the level of complexity (and powefullness) of liquid state NMR. Adapted from Supporting Information, JACS 2010, 132:2393

Solid State NMR today: main components

I

Magic Angle Spinning (MAS, fast-MAS, ultra-fast MAS)

I

Cross-Polarization (correlation spectroscopy)

I

Decoupling (heteronuclear and homonuclear)

I

Recoupling (correlation spectroscopy, Distance measurements/restraints)

I

Through bond (J) or through space (D) correlation

I

Spin diffusion (1 H)

I

Anisotropic information (static or recoupling/MAS, dynamics)

I

DFT computation of NMR shifts

(Nuclear) Spin Interactions

 B 0 NMR

EPR Electrons

Nuclear Spins I

 B loc Nuclear Spins S

 t  B 1 NMR

Phonons MicroWaves, ....

Adapted, M. Mehring, NMR-Basic Principles and Progress, Vol. 11, High Resolution NMR in Solids, 2nd ed., (Springer-Verlag, New York, 1983).

The Zeeman Interaction Coupling of nuclear spins with external magnetic fields.

~ 0 = B0~z • static magnetic field B ~ 0 = −γ~B0 · Iz = ~ω0 Iz H0 = −γ~~I B Larmor Frequency ω0 = −γB0

~ 1 (t) = B1 {~x cos(ωt) + ~y sin(ωt)} • RF magnetic field B HRF

~ 1 (t) = −γ~B1 {cos (ωt) Ix + sin (ωt) Iy } = −γ~~I · B  ~ω1  −iωt I+ e + I− e +iωt = −γ~B1 I+ e −iωt + I− e +iωt = 2

Nutation Frequency ω1 = −γB1 Note: NMR convention H ⇒ H/~, e −iEt/~ ⇒ e −iEt

Rotating frame as a first introduction to AHT . . . The problem in time-dependent: H(t)

= ω0 Iz +


ω1 I+ e −iωt + I− e +iωt 2

The purpose of the rotating frame (or interaction frame) is to make the problem time-independent (thus solvable). We seek for a transformation that removes the dominant part of H. To achieve this we rotate the frame around Iz U0 (t) = exp {−iωIz t} = Rz (ωt)

Rotation of angle φ = ωt around Iz.

How does H(t) change ? e H(t)

d = U0† (t)HU0 (t) − iU0† (t) U0 (t) = U0† (t)HU0 (t) − ωIz dt   −iωt ˜ = ω0 Iz + ω1 I+ (t)e + ˜I− (t)e +iωt − ωIz

Similar transformations are needed for understanding the spin dynamics underlying most of NMR pulse sequences (recoupling,decoupling, MAS, rotational resonance, rotary resonance).

Rotating frame as a first introduction to AHT . . . ˜Iα (t) = exp {+iωIz t} Iα exp {−iωIz t} = U † (t)Iα U0 (t) rotating frame 0 ˜I+ (t) = I+ exp {+iωt} ˜I− (t) = I− exp {−iωt} ˜Iz (t) = Iz In the rotating frame e H(t)

  = ω0 Iz + ω1 ˜I+ (t)e −iωt + ˜I− (t)e +iωt − ωIz ω1 (I+ + I− ) = ∆Iz + ω1 Ix Q.E.D = (ω0 − ω) Iz + 2

The problem is now time-independent: precession of the spin around Ix . The issue in decoupling/recoupling is to find the effective H1 around which the spins precess ! That is to find the proper U0 (t) to obtain a e time-independent H. Similarly, for all interactions, we only keep time-independent part in the rotating frame. The effect of time-dependent parts is negligible to first order: they are non-resonant. Q.E.D: Quod erat demonstrandum.

The Chemical Shift Anisotropy (CSA) CSA originates from the magnetic shielding of the external magnetic field ~ ind = −σ B ~0 through the induced magnetic field: B H HCSA

  ~0 + B ~ ind = −γ~I · (1 − σ) · B ~0 = −γ~I · B ~ 0 = −ω0~I · σ · ~z = γ~I · σ · B

I

~ ind ∦ B ~ 0) σ is a tensor (matrix) (B

I

~ 0 is Secular approximation: In High Field, only the component k B effective. HCSA = −ω0 σzz Iz ( σxz and σyz : higher order effects )

I

Only the symmetric part is active (to first order) in NMR σzz

(s)



HCSA

σxx = γ~I · σyx σzx

σxy σyy σzy

   σxz 0 σyz  ·  0  ≈ −ω0 σzz Iz σzz B0

Anisotropy of NMR interactions . . . NMR spectra of a single molecule (or crystal)

B0

C

B0

O C

B0 C

O O

 zz

 yy

 xx

Anisotropy of NMR interactions . . . NMR spectrum (CSA) of a powder

 B 0

z

    , 





The powder average

y

x

Z S(t) = 

=90 °

=0 °

sin θdθdφ

× exp {−iν(θ, φ)t}

Digression: How to get the FID/Spectrum from H ? Initial state (after 90 pulse) ρ(0) = Ix . We detect (convention) I− so that the detector operator is I+ . The FID is  S(t) = Tr I+ e −iHt Ix e +iHt P UsingPthe definition of Tr {A} = m hm| A |mi and inserting 1 = m |mi hm|, we obtain S(t)

=

X

2

|hm + 1| I+ |mi| e −iωm,m+1 t

m

with the one quantum transition frequencies ωm,m+1

m1

= hm + 1| H |m + 1i − hm| H |mi = Em+1 − Em 2

and the intensities |hm + 1| I+ |mi| . ex. 1: Derive those equations ex. 2: What are the equations in the case of a non-diagonal Hamiltonian H = X † ΛX

E m1

m1,m m

Em

The Chemical Shift Anisotropy (CSA) Principal values (eigenvalues), σZZ , σYY , σXX of σ

ZZ



σzz (θ, φ)

= σXX sin2 θ cos2 φ + σYY sin2 θ sin2 φ + σZZ cos2 θ   3 cos2 θ − 1 η 2 + sin θ cos 2φ = σiso + ∆ 2 2

 B 0 YY

XX



I

σiso : isotropic chemical shielding

I

∆ : chemical shielding anisotropy

I

η : asymmetry parameter

Isotropic Chemical shift (ppm)     ref
σiso − σiso ν − ν ref ref =− δiso = ≈ − σiso − σiso ref ν ref 1 − σiso Do not forget: CSA (in Hz) scales with B0 : −ω0 σzz !

The (Shielding) Chemical Shift Anisotropy NMR static lineshape

σiso

σiso

∆0

η=0

σyy σxx

σzz

η = 0.5

σyy σxx

σzz ∆

∆

η=1

200

100

0 ppm

-100

-200

200

100

0 ppm

-100

-200

The Dipolar Hamiltonian IS HD

µ0 ~γI γS = 3 4π rIS

  ~I · ~S − 3 (~S · ~rIS )(~I · ~rIS ) = ~I · D · ~S 2 rIS

No isotropic shift Tr {D} = 0 , IS HD = C D {A + B + C + D + E + F} A  , 

z

A

 B 0

S

 , 

D

 , 

y x

B C

 ,

r IS 

 ,

A

A



I

B flip-flop

A

C,D (1Q)

 , 

E,F (2Q)

C,D (1Q)

 , C,D (1Q)

C,D (1Q)

 ,

E F


= Iz Sz 3 cos2 θ − 1
1 = − (I+ S− + I− S+ ) 3 cos2 θ − 1 4
3 = − (I+ Sz + Iz S+ ) cos θ sin θe −iφ 2
3 = − (I− Sz + Iz S− ) cos θ sin θe +iφ 2
3 = − (I+ S+ ) sin2 θe −2iφ 4
3 = − (I− S− ) sin2 θe +2iφ 4

The Heteronuclear Dipolar Interaction IS HD =


µ0 ~γI γS 3 cos2 θ − 1 Iz Sz = CISD × 3 |{z} 4π rIS constant

 3 cos2 θ − 1 × 2Iz Sz | {z } 2 | {z } SpinTerm



Orientation

h

(S)

i

(I )

ω0 Sz + ω0 Iz , I+ S− + I− S+ 6= 0: Flip-Flop does not conserve Zeeman Energy

CISD (

 kHz) =

γI γS γH2

 ×

122 (˚ A)

3 rIS

C − H = 1˚ A ⇒ CISD = 30.7 kHz

IS Note: for 3 cos2 θ − 1 = 0, HD = 0.

Dipolar Interactions provide a direct access to interatomic distances νD ∝

1 3 rIS

The Heteronuclear Dipolar Powder Spectrum IS HD = CISD |{z}



3 cos2 θ − 1 2

 2Iz Sz

d

d

EmI ,mS ωmS =+1/2 o

θ = 90

ωmS =−1/2

Energy levels |mI , mS i
= d 3 cos2 θ − 1 hmI , mS | Iz Sz |mI , mS i
= d 3 cos2 θ − 1 /2
= −d 3 cos2 θ − 1 /2 Dipolar Splitting: d = CISD

θ=0

d

 ,

 , 

 ,

 , 

o

-d/2

The Homonuclear Dipolar Interaction II HD =

µ0 ~γI2 (A + B) = CIID 4π rij3



3 cos2 θ − 1 2

  1 i j I+ I− + I−i I+j 2Izi Izj − 2

I+i I−j + I−i I+j : flip-flop terms I

Efficient mechanism magnetization exchange among (like) nuclear spins (spin diffusion)

I

many-body physics ⇒ large width of 1 H static NMR spectra

I

[ω0 (Izi + Izj ), I+i I−j + I−i I+j ] = 0

Dipolar Interactions provide a direct access to interatomic distances. IS Note: for 3 cos2 θ − 1 = 0, HD = 0.

The Homonuclear Dipolar Powder Spectrum II Hamiltonian HD in the    1 2 d 3 cos θ − 1  II 0 HD = 0 2 2 0

3d/2

o

θ = 90

 ,

o

3d/2

 , s -3d/4

0 −1 −1 0

 , 

 , a θ=0

|mi , mj i basis 0 −1 −1 0

 0 |++i 0  |−+i 0  |+−i −1 |−−i

The Homonuclear Dipolar Powder Spectrum |si =

1 √ {|−+i + |+−i} 2

1 |ai = √ {|−+i − |+−i} 2

1 II ˜D H =d



3 cos2 θ − 1 2



√0  2 j i ˜I+ + ˜I+ =   0 0

ν(|++i−|si) =


3d 3 cos2 θ − 1 4



2

0  0 0

0 −1 0 0

0 0 0 0

 0 |++i 0  |si 0  |ai 1 |−−i 2

 0 0 0 |++i 0 0 0  |si  |ai 0 0 0 √ 2 0 0 |−−i

ν(|si−|−−i) = −


3d 3 cos2 θ − 1 4

Indirect J couplings

Small Interaction but, thanks to high-resolution (see homonuclear decoupling , MAS), can be measured now and used in NMR to provide through bond information. It possesses an anisotropic part which is neglected because of its small strength (incorporated into the Dipolar interaction). Only the isotropic part is retained. HJIS = 2π~I · JIS · ~J ≈ 2πJIS~I · ~S For unlike spins, HJIS = 2πJIS Iz Sz (See liquid NMR)

NMR interactions in general



~ (A) B loc

A 1 σ Q D J

Axx ~ = Ayx = AX Azx ~ X ~ ~ RF B0 ,B ~0 -B ~I ~S ~S

Axy Ayy Azy

   Axz Xx Ayz  · Xy  Azz Xz

~ HA = −γ~ × ~I · A · X

Interaction Zeeman interaction Magnetic Shielding Quadrupolar interaction ( I ≥ 1 ) Dipolar interaction (spatial) Indirect J couplings (through bond)

Calculation DFT DFT structure DFT

NMR Interactions: The Principal Axes System (PAS) I z  B 0

A ZZ

A XX

x

 AXX 0 0 AYY 0  A= 0 0 0 AZZ I Isotropic : Aiso = 1 Tr [A] 3 

AYY

I

Anisotropy : δA = AZZ − Aiso

y

I

Asymmetry : ηA = (AYY − AXX ) /δA with 0 ≤ ηA ≤ 1

Diagonalization of A (3D rotation) provides the principal values (Aα,α ) and its orientation with respect to the reference frame (crystallographic or molecular frame). Convention: |AZZ − Aiso | ≥ |AXX − Aiso | ≥ |AYY − Aiso |

AZZ

AYY Aiso

A XX

NMR Interactions: The Principal Axes System (PAS) II

APAS

= Aiso 1   1 − 2 (1 + ηA ) 0 0 + δA  0 − 21 (1 − ηA ) 0 0 0 1

A

= X−1 A APAS XA

XA

= R(αA , βA , γA ) =

exp(−iαA Iz )

× exp(−iβA Iy ) × exp(−iγA Iz )

(x, y , z): reference frame axis (αA , βA , γA ): Euler angles A NMR interaction is characterized by 6 parameters: Aiso , δA , ηA , αA , βA , γA . For a single interaction, (αA , βA , γA ) does not affect the powder lineshape but can be determined for single crystal NMR.

Single Crystal NMR

T. Vosegaard, PhD Thesis, November 1998

Single Crystal NMR

Single Crystal NMR allows the determination of the six parameters

Multiple Interactions: CSA + Dipolar Cα

CO

300

280

260

240

220

200

180

160 13

140 120 100 C Chemical shift (ppm)

80

60

40

20

0

-20

-40

40

20

0

-20

-40

Cα

CO

300

280

260

240

220

200

180

160 13

13

140 120 100 C Chemical shift (ppm)

80

60

C NMR spectra of Glycine (bottom) and U-13 C-Glycine Sensitivity to the relative orientation of the interaction.

NMR interactions: Tensorial Formalism

Tensorial formalism allows describing rotations in both coordinate (orientation) and spin space. λ

H (Ω) = C

λ

m=+2 X

λ (−1)m R2,−m (Ω) T2,m

m=−2

λ = CSA, Dipolar, Quadrupolar, . . . I

C λ : constant

I

λ R2,−m (Ω) : orientation of the PAS in the LAB frame

I

T2,m Spin operators (combination of Iαi Iβj operators)

NMR interactions: Tensorial Formalism Rotation of coordinates (orientation). λ R2,−m (Ω)

I

Ω : orientation = Euler angles (α, β, γ) Rz (α)Ry (β)Rz (γ) = e −iαIz e −iβIy e −iγIz

I

λ ⇒ Rotation from the PAS ρλ2,n to the laboratory frame R2,m .

λ R2,m (Ω) =

n=+2 X n=−2

I

ρλ2,n = ( η2λ , 0,

q

ηλ 3 2 δλ , 0, 2 )

2 ρλ2,n Dm,n (Ω)

NMR interactions: Euler Angles

Wigner Matrix Elements: 2 2 Dm,n (Ω) = e −imα dm,n (β)e −imγ 2 dm,n (β) Reduced Wigner Matrix (Second order polynomial of (cos β, sinβ))

Rz (α)Ry (β)Rz (γ) = e −iαIz e −iβIy e −iγIz 

cos α cos β cos γ  − sin α sin γ  − cos α cos β sin γ   − sin α cos γ cos α cos β

sin α cos β cos γ + cos α sin γ − sin α cos β sin γ + cos α cos γ sin α sin β

− sin β cos γ



  − sin β sin γ    cos β

NMR interactions: Tensorial Formalism Rotation of spin operators T2,m I

T2,m : tensorial spin tensors [Iz , T2,m ] = mT2,m Rz† (φ)T2,m Rz (φ) = e −imφ T2,m Ry† (φ)T2,m Ry (φ) =

n=+2 X

2 T2,n dn,m (φ)

n=−2

R † (Ω)T2,m R(Ω) =

n=+2 X

2 T2,n Dn,m (Ω)

n=−2

I

T2,0 is invariant under Zeeman interaction: λ Hλ (Ω) ≈ C λ R2,0 (Ω) T2,0

Basic NMR technique: Cross-Polarization & Decoupling 1

H

Cross-Polarization 90

CP

Decoupling

Magnetization Transfer X CP

CP I

Signal enhancement (from 1 H Magnetization)

I

CP from nuclei with shorter T1

I

Mechanism for heteronuclear correlation

Principles of (Proton) Decoupling Principles: observed spins see rotating spins to average out heteronuclear dipolar interactions Physics of proton-decoupling is a difficult topic. Optimization of proton is still a subject of (hot) research. Simple modulation such as continuous wave (CW) is often not sufficient, introduction of an additional modulation is required. A popular scheme is the Two Point Phase Modulation (TPPM). Continuous Wave (CW) decoupling 1

Decoupling

Two Point Phase Modulation (TPPM) 2

2



−

z

z

B1 x

=15 °

1

y

B1 x

− B1

y

Principles of (Proton) Decoupling H = δIz +

X

(i)

ωD Iz Sz(i) + HSS

i

|

{z

Broadening

}

Apply continuous RF field on S spins so that I spins see rotating S spins. H = δIz +

X

(i)

(S)

ωD Iz Sz(i) + HSS + HRF (t)

i † ˜ In the toggling frame, S(t) = URF (t)SURF (t)

˜ = δIz + H

X

n o (i) ˜ SS ωD Iz S˜ (i) (t) + H

i

˜ SS Decoupling is efficient if S˜ (i) (t) = 0, fast enough with respect to H (S) (ω1 > kHSS k) ˜ = δIz + H

n o  (i) (i) ˜ SS ωD  Iz S˜ (t) + H i  X

Cross-Polarization: Spin Locking Spin Locking

The nuclear magnetization can be locked applying an RF magnetic field in the same direction. Its decay is similar to longitudinal relaxation but in the rotating frame T1ρ (rotating-frame relaxation)

exp −/T 1  90Y

X z

z

B1

x

y

B1

x

y

Cross-Polarization: Principles

Double-rotating frame: Laboratory frame I 0

 ≠ I

Rotating frame

S 0

I

S I

0

S

0

I

1

 I B1I = S B S1

S S

1

Magnetization Transfert

Cross-Polarization: How does it work ? Two spins system H

IS = ω1I Ix + ω1S Sx + 2ωD Iz Sz

90◦ around y : xz → zx ˜ = Ry† (π/2)HRy (π/2) H IS ˜ = ω1I Iz + ω1S Sz + 2ωD H Ix Sx Working with |mI , msi  I  (ω1 + ω1S ) 0 0 ωD |+, +i = |1i I S   1 0 (ω ω 0 − ω ) D 1 1  |+, −i = |2i ˜=  H  |−, +i = |3i 0 ωD −(ω1I − ω1S ) 0 2 I S ωD 0 0 −(ω1 + ω1 ) |−, −i = |4i Fictitious Spin operators: Double Quantum (DQ) subspace Iα14 , Zero Quantum (ZQ) Subspace Iα23

˜ = ω1I + ω1S Iz34 + ωD Ix34 + ω1I − ω1S Iz23 + ωD Ix23 H

Cross-Polarization: How does it work ?
For ω1I + ω1S  ωD , ω1I = ω1S = ω1 ˜ ≈ ω1 Iz14 + ωD Ix23 H Flip-Flop in the ZQ subspace leads to magnetization exchange.  I  (ω1 + ω1S ) 0 0 0 |+, +i = |1i   |+, −i = |2i 1 0 ω 0 0 D  ˜≈  H  |−, +i = |3i 0 ωD 0 0 2 0 0 0 −(ω1I + ω1S ) |−, −i = |4i Or in the original basis set: I

ω1 Iz14 = ω1 (Sz + Iz ): Conservation of the total magnetization: the spin lock !

I

ωD Ix23 = ωD (S+ I− + S− I+ ): Magnetization exchange between Iz and Sz

Cross-Polarization: Two spins system Starting from ρ(0) = βIx (no magnetization on S), one arrives at ρ∞ = β2 Ix + β2 Sx . The magnetizations have equilibrated towards a common spin temperature β2

Cross-Polarization: Thermodynamics limits For multiple spins system, general equation of the CP dynamics (I ⇒ S) is      !       γI τ τ 1 MS (τ ) = exp − I MIeq − exp − IS  γS 1−λ T1ρ T  |  |{z} | {z CP }   {z } Gain!

λ=

IS TCP I T1ρ

Gain

Loss

6 IS : fast/slow CP dynamics depending on the spatial TCP ∝ 1/rIS proximity of nuclei I. CP provides a means to probe I surroudings 1

T IS

Abundant

TCP

S spins Diluted

I

S

T1

T 1 ≈∞

0,8 Magnetization

I spins

Important: CPMAS spectra are not quantitative ! CP-build-up curves are needed.

T1ρ

0,6

0,4

0,2

Lattice

0 0

1000

2000 3000 CP Contact Time

4000

Cross-Polarization: Ramp-CP II CP with HD : HH broadening. 1

13

Glycine H- C CPMAS - B0=11.75T - Hartmann-Hahn condition

I

S

I

S

S

 IID 1I

1I

S1

1I

1S

 IID

S1

CP efficiency (arb. units)

I

26

24

22 20 18 13 1 νRF( C) [dB] - νRF( H) = 10 dB

Ramp-CP: Enhancing Cross Polarization Efficiency, reducing sensitivity to HH mismatch 1

H

Cross-Polarization 90

CP

Decoupling

Magnetization Transfer X RAMP-CP

CP

I

Shape can be optimized

I

Adiabatic passage also possible

16

14

Cross Polarization: HETeronuclear CORrelation 1

H

90 Which Dec During CP2 ?

CP1 C

Decoupling

t1

13

CP1

CP2

N

15

CP2

t2

I

CP provides an efficient (and general purpose) mechanism for heteronuclar correlation through space (dipolar couplings)

I

τCP allows one to control the space length probed

I

Back polarization from X (13 C,15 N,...) to 1 H is possible (but resolution...)

The Magic Angle

 3 cos2 θ − 1 η 2 = δiso ω0 + δσ ω0 + sin θ cos 2φ × Iz 2 2   2 2 µ0 ~ γI γS 3 cos θIS − 1 = × 2Iz Sz 3 4π rIS 2    
µ0 ~2 γI2 3 cos2 θ12 − 1 1 1 2 1 2 1 2 = × 2Iz Iz − I I + I− I+ 3 4π r12 2 2 +− 

HCSA IS HD II HD

Magic Angle θm : 3 cos2 θm − 1 = 0 Origin: Zeeman truncation (or secular averaging) [Iz , T20 ] = 0

⇒

λ λ Hλ (Ω) = R20 (Ω)T20

No more valid for second-order interaction (see Quadrupolar nuclei)

Magic Angle Sample Spinning: Principles Introducing a coherent sample motion

Spinning the Sample around the Magic Angle θm

Static Sample

z

z

 B 0

m

 B 0

m



y x

y x

Interaction along θ

Effective interaction along θm !

Magic Angle Sample Spinning 13

U- C-Glycine B0=11.75T TPPM @ 100 kHz δiso

Cα CO

νR=0 kHz

* νR = 2 kHz

*

νROT

*

*

*

*

*

*

Spinning Sideband

* *

*

* *

*

**

*

νROT

νR = 15 kHz 300

250

200 13

150 100 C chemical shift (ppm)

50

0

-50

Magic Angle Sample Spinning: Principles Dealing with NMR frequency with time-dependent orientation.  B PAS 0 FRAME Magic angle θm and spinning frequency m  ,  ,  ω  R R R ROT = 2πνrot . AZZ

ROTOR FRAME



ROT

, m , 0

ν(t) = ν(θ(t), φ(t)) = νiso + R20 (Ω(t))

 AXX

LABORATORY FRAME

AYY

Two frames must be introduced. y

x

(αR , βR , γR ) ⇒ (ωROT , θm , 0) | {z } | {z } PAS−ROTOR

ROTOR−LAB

With the help of Wigner matrix algebra (easy !), X 2 2 2 Dm,0 (θ(t), φ(t), 0) = Dm,p (αR , βR , γR )Dp,0 (ωROT , θm , 0) p

Magic Angle Sample Spinning: NMR frequencies • In a static sample: R2,0 (Ω) =

X

2 ρ2,n Dn,0 (Ω)

n

• Under MAS: R2,0 (Ω(t)) =

X

2 2 ρ2,n Dn,p (αR , βR , γR )Dp,0 (ωROT , θm , 0)

n,p 2 2 with Dp,0 (ωROT , θm , 0) = e −ipωROT t dp,0 (θm ), one obtains

˜ 2,0 (t) R2,0 (Ω(t)) = R 2,0 + R with P −ipωROT t ˜ R2,0 (t) = p6=0 R2,p (αR , βR , γR )e R 2,0 ∝

2 d0,0 (θm )

=

• Powder average on (αR , βR , γR ).

1 2

R τR 0

˜ 2,0 (t)dτR = 0 R

p = −2, −1,
A0, +1, +2
3 cos θm − 1 = 0 2

NMR signal under MAS I • NMR signal with a time-dependent frequency ω

S(t) = exp{−iωt} = exp{−iψ(t)} with ψ(t) = ωt • Under MAS, time-dependent frequency ω(t), the NMR signal is Z

t

Z

t

ω(u)du

ω(u)du} = exp{−iψ(t)} with ψ(t) =

S(t) = exp{−i

0

0

• With ω(t) = ωiso + ω e (t), ω e (t)

=

X

ωp e −ipωR t

p

e decreases as ψ(t)

ωp ωR

e = ψ(t)

X ωp  e −ipωR t − 1 i pωR p

(ωp : anisotropy). ωR > ωp ⇒ small SSBs.

NMR signal under MAS II

• NMR signal with a time-independent frequency ω e S(t) = exp{−iωiso t} × exp{−i ψ(t)} e R ) = ψ(0) e =0 ω e (t) is periodic ⇒ ψ(τ e −i ψ(t) = e

X

Ip e −ipωR t

SSBs manifold

p

exp{−iωiso t} position

Magic Angle Sample Spinning 13

U- C-Glycine B0=11.75T TPPM @ 100 kHz

νROT = 2 kHz

300

250

200 13

150 100 C chemical shift (ppm)

50

0

-50

•: NOTE: CSA Spinning Sideband manifold increases with B0 . Higher spinning frequency required at higher field !

Setting the Magic Angle:

81

Br in KBr

1

Rotational Echoes

Signal (arb. units)

0,5 0

100 200 300 400 500 600 700 800 900 1000 Time (µs)

isotropic component 0

-0,5 1000

0

1000

2000

3000

4000

500

0 ppm

5000 6000 Time (µs)

-500

7000

8000

-1000

9000

10000

I

KBr is the standard sample to setup the Magic Angle

I

Optimize the SSB rot. Echoes

MAS NMR apparatus CPMAS PROBE STATOR

B. Meier, Cargese 2008.

ROTORS

diam. (mm) 4 3.2 2.5 1.3

Spin rate (kHz) 15 24 35 67

SSBs/CSA manipulation: Magic Angle Turning (MAT)

T=NTROT

π

π

π

π

π t1

t1

t1 +1 0 -1 0

T/6

T/3

T/2

2T/3

5T/6

SMAT (t1 , t2 ) = e −iωiso t1 × SMAS (t2 )

T

SSBs manipulation: Magic Angle Turning (MAT) MAS vR = 10 kHz

100

50

0

-50 P Chemical Shift ( ppm )

-100

-150

-200

31

-50 P Chemical Shift ( ppm )

-100

-150

-200

31

MAS vR = 5 kHz

100

50

0

-50

Isotropic dimension ( ppm )

MAT vR = 5 kHz -40

-30

-20

-10

0

10

50

0

-50

-100

-150

-200

SSBs/CSA manipulation: Phase Alternated Spinning Sideband (PASS) N x TROT

π

π

π

π

π

θ1 (+2π)

θ2 (+4π)

θ3 (+6π)

θ4 (+8π)

θ5 (+10π)

+1 0 -1 0 0

2π (+12π)

Θ Pitch (degrees)

60 120 180 240 300 360

0

60

120 180 240 Pulse position (degrees)

SPASS (t) = e −iωiso t

X p

300

360

Ip exp −ipωR t × e −ipΘ

SSBs/CSA manipulation: Phase Alternated Spinning Sideband (PASS) PASS

-1 0 +1

260

240

220

200

180

160

140 120 100 80 60 C chemical shift (ppm)

13

40

20

0

-20

-40

SSBs manipulation: Magic Angle Turning (MAT)

I

Even overlapping spinning sidebands can be resolved

I

IS Remember: CSA SSBs are narrow (as well HD )

I

What about homonuclear dipolar interactions ?

Magic Angle Sample Spinning: Homonuclear Dipolar Interaction . . .

* νROT= 15 kHz

*

νROT= 10 kHz

*

* *

* *

*

*

*

* * * νROT= 8 kHz

*

*

*

νROT= 5 kHz

200

Glycine

150

100

50 0 -50 1 H chemical shift (ppm)

-100

-150

-200

Magic Angle Sample Spinning: Homonuclear Dipolar Interaction . . .

12

10

8

6

4

2

-4 0 -2 -6 H chemical shift (ppm)

-8

-10

-12

-14

-16

-18

1

I

Unresolved spectra: Why ?

I

II MAS (frequency) provides unsufficient averaging: ωR < kHD k

I

Influence of three-spins terms: flip-flop mechanism is still active !

NMR Interactions under MAS: general considerations • Static (First-Order Terms)(a) H(Ω) = R2,0 (Ω)T20 • Under MAS H(t) =

X

Hp (ΩR )e −ipωR t with H0 (ΩR ) = 0

p

• CSA Iz, heteronuclear Iz Sz or isolated homonuclear spin pairs (heterogeneous system)  Z t  0 [H(t), H(t )] = 0 ⇒ U(t, 0) = exp −i H(u)du (narrow SSBs) 0

• homonuclear spin pairs (spin diffusion) (homogeneous)  Z t  0 b [H(t), H(t )] 6= 0 ⇒ U(t, 0) = T exp −i H(u)du (broad SSBs) 0

b Dyson Time-Ordering Operator T ⇒ Average Hamiltonian Theory, Floquet Theory, Effective Hamiltonian (a): Second Order Terms see quadrupolar interaction

Homogeneous NMR Interactions under MAS: Theoretical Framework P Hp (ΩR )e −ipωR t . Floquet Theory(a) says:
U(t, 0) = P(t) exp −iHt P † (0)

H(t) is periodic: H(t) =

p

I

P(t) is periodic ( P(τR ) = P(t)): spinning sidebands

I

H is the smooth compoment: width of the bands (residual homo. dip. int.)

Perturbation Expansion in 1/ωR yields Z τR (1) H = H(t)dt = H0

(= 0 : MAS) (=AHT)

0

H(2)

=

1 X [H−p , Hp ] 2 p>0 pωR

(6= 0 : even under MAS)

Hm two spins terms ⇒ H(2) three spins terms ! 1 H requires (very) large spinning frequency for enhancing the resolution. (a): For AHT/Magnus Expansion, see Tutorial on decoupling

Cross Polarizarion under MAS Generalized HH conditions under MAS ωI = ωS ± nνR

13

1

νRF( C) = νRF( H) - N * νROT CO Cα

N=0 N=1 N=2 N=3

30

28

26

24 22 20 13 1 νRF( C) [dB] ( νRF( H) = 8 dB )

18

16

14

Cross Polarization under MAS 13

1

13

U- C- Glucose - H- C CPMAS - B0= 11.75T - vROT= 12.5 kHz

Signal Intensity (arb. units)

1,5

D

C

1

D

H H

TCP

HH HH

H H

0,5

H C

H

TCP -t/TCP

1- (1-α) e

-αe

-3t/2TCP

cos(bt)

H H

0

0

200

400 600 CP contact Time (µs)

800

1000

ω1 (1 H) = ω1 (13 C) + ωR : recoupling of (first-neighbors) 1 H-13C dip. int.

Cross Polarization under MAS Hamiltonian under CPMAS (=static case but with ωD (t)): ˜ H(t) =



   ω1I + ω1S Iz34 + ωD (t)Ix34 + ω1I − ω1S Iz23 + ωD (t)Ix23

˜ 14 (t) + H ˜ 23 (t) = H P with ωD (t) = p ωD,p e −ipωR t (p=-2,-1,1,2) I

ωD,0 = 0: NO CP at the Hartmann Hahn condition !

I

BUT ωD (t) modulated at 2ωR and ωR

If ω1I − ω1S = nωR , a resonance phenomenon appears, selectively in the 2 ↔ 3 subspace (Zero Quantum System).

(ZQ) Cross Polarization under MAS 2 I

S

1−1  B RF

Introduce the rotating frame with respect to the transition 2 ↔ 3 (i.e. the resonant transition !) a the resonance frequency nωR

p R  D t 

3 R ˜ 23 H (t) =

+



 ω1I − ω1S − nωR Iz23 + ωD,+n I−23 + ωD,−n I+23

non-resonant terms

If ω1I − ω1S − nωR = 0:   0 ωD,+n R ˜ = ωD,+n S+ I− + ωD,−n S− I+ H23 (t) = ωD,−n 0 Flip-Flop between S and I : magnetization exchange (CPMAS works!)

(DQ) Cross Polarization under MAS 1 I

S

1 −1  B RF

Introduce the rotating frame with respect to the transition 1 ↔ 4 (i.e. the resonant transition !) a the resonance frequency nωR

p R  D t 

4 R ˜ 14 H (t) =

+



 ω1I + ω1S − nωR Iz14 + ωD,+n I−14 + ωD,−n I+14

non-resonant terms

If ω1I + ω1S − nωR = 0:   0 ωD,+n R ˜ H14 (t) = = ωD,+n S+ I+ + ωD,−n S− I− ωD,−n 0 Double Quantum Flip between S and I : magnetization exchange (CPMAS works!) Note: in (23) subspace ω1I − ω1S = −2ω1S + nωR < 0, negative signal with respect to the ZQ CPMAS !

Rotary resonances in MAS

Conclusion: Under MAS, when applied RF fields are resonant with the spinning frequency, this introduces new resonances that can lead to recoupling of interaction What about decoupling ...

Decoupling under MAS Avoid rotary resonance conditions, in second frame, interaction are on-resonance so recoupling can occurs CO

Cα

CW

4 νR

4 νR

2 νR

2 νR

1 νR

1 νR

νR/2

νR/2

0 νR

0 νR

200

190

180 170 160 C Chemical shift (ppm)

13

150

60

CW

50 40 30 13 C Chemical shift (ppm)

20

Decoupling under MAS CO

Cα

100 kHz (νR = 15 kHz)

100 kHz (νR = 15 kHz)

CW CW

TPPM

TPPM 180

178 176 174 13 C Chemical shift (ppm)

172

50

45 40 C Chemical shift (ppm)

13

35

J coupling under MAS: MAS-J-INADEQUATE Thanks to the MAS averaging of anisotropic interactions, J coupling can be used in ssNMR (allmost) as in liquid NMR.

I

Double Quantum Spectroscopy

I

Refocused Scheme (in Solids)

I

Such Experiments are mainly limited by T2 (Coherence life-time, Spin Echo)

J coupling under MAS: MAS-J-INADEQUATE An Application to 31 P-31 P homonuclear correlation spectroscopy in ceramic (CaO)0.4 -(Na2 O)0.1 -(P2 05 )-0.5