Principles of High Resolution Solid State Nuclear Magnetic Resonance Manipulating spins without restriction
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)
Homonuclear Decoupling
Recoupling
Notes
Homonuclear Decoupling
I
2013: Ultra-high spinning frequency (JEOL: 110 kHz, o.d. 0.75 mm)
Instead of sample spinning, rotate the nuclear spins !
Homonuclear Decoupling: Lee-Goldburg I.a Principles: High power Off-resonance irradiation
HRF (t) = 2ω1 cos {(ω0 ± ∆)t} Ix
z
In the rotating frame:
0
0± B 0 m
=
1
2
Ie
1 For derivation, see exercices
x
II II H = HD + ω1 Ix + ∆Iz = HD + ωe Ie II k, secular approximation for HII ωe > kHD D in a second rotating frame (ωe Ie ): 3 cos2 θm − 1 II II e +H (t) He = HD,e D,e 2
Homonuclear Decoupling: Lee-Goldburg I.b Principles: High power Off-resonance irradiation
z B0 m
Ie
iso
HCS,e = ωiso cos θIe
x For derivation, see exercices
Spin precession around Ie results in a scaling of the chemical shift:
Homonuclear Decoupling: Lee-Goldburg I.c The full derivation . . . buckle up !
H = HII + ω1 Ix + ∆Iz = HII + ωe Ie Rotation around Iy , angle θ (Ie ⇒ Iz ). Let’s Wigner do the work ! 1 X ij † ij ωD Ry (θ)T20 Ry† (θ)HRy (θ) = Ry (θ) + ωe Iz 2 i6=j ! 1 X ij X ij 2 ωD T2n dn,0 (θ) + ωe Iz = 2 n i6=j
Rotating Frame (again and again ...) e −iωe Iz t ˜ H(t)
=
1 X ij ωD 2 i6=j
! X
ij 2 T2n dn,0 (θ)e −inωe Iz t
n
The secular (time independent) part X ij ij 2 2 ˜ = 1 H ωD T20 d0,0 (θ) = d0,0 (θ) × HII 2 i6=j
=0 with M.A. irr.
Homonuclear Decoupling: What this damned secular approx stands for ?!! II k. Secular approximation holds if ωe > kHD
IID eI
eI
IID Secular approx.
Secular approx.
Homonuclear Decoupling: Lee-Goldburg II The Time-reversal symmetry !
Frequency swichted LG
Phase Modulated LG (on-resonance) −X
−
=2 e =1/e =2 e =1/e =0 =0
=2 e =1/e =2 e =1/e z
2
−x error
Ie
Ie
x
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Phase
X
error
−z
0
Pulse Index
Faster phase than frequency switch Compensates for RF / offset mis-setting Symmetrisation cancels higher-order terms (Time reversal)
Homonuclear Decoupling: High Resolution NMR of proton 2D approach (High Resolution t1 × Poor Resolution t2 )
Windowed pulse sequence ( A = building block )
t2 90
A0°
*
A180°
*
A0°
*
A180°
A0°
* C
window
*
Homonuclear Decoupling: 1H NMR (L-Alanine) Alanine - B0=11.75T - vROT=12.5 kHz
H FSLG dimension (ppm)
-2
1
FSLG Resolution
-4
0 2 4 6 8 10
14
12
10
8
6
4 2 0 -2 MAS dimension (ppm)
MAS Resolution
-4
-6
-8
-10
Homonuclear Decoupling:
13
C NMR (Admantane)
Adamantane - B0=11.75T - vROT=15 kHz
H
H C
H
CW (5kHz)
H
1/2
H
J/3
H C H
H
FSLG 44
42
40
38
36 34 13 Chemical shift (ppm)
32
30
28
⇒ Weak couplings such as J can be recovered !! ⇒ J-base experiments can be performed with 1 H !
26
Homonuclear Decoupling: in silico design Numerical optimization using spin dynamics simulations. This approach has been pioneered with the DUMBO pulse sequence, digitized phase modulation φ(τ ) at constant RF amplitude, including a time-reversal symmetry (φ(1 − τ ) in the second half, τ = t/τm )
0≤τ ≤ φ(τ ) =
1 2
n=K X
continuous
{an cos(2πnτ )
n=0
+bn sin(2πnτ )}
C =m
N
Windowed (w)
m 1 2
≤τ ≤1
φ(τ ) = φ(1 − τ ) + π
* C
E. Salager et al., Chem. Phys. Lett. 469 (2009) 336-341
N
Homonuclear Decoupling Design: Direct spectral optimization → The moss reallistic spin-dynamics simulator is the spectrometer ! Optimization using the spectrometer output.
B. Elena, Chem. Phys. Lett. 398 (2004) 532-538
WAHUHA Discrete rotation around the magic angle
Ie
z y x
A first introduction to Average Hamiltonian Theory Introduction to the toggling frame. The AHT sandwich.
U , 0
H
H R H R
The bracketed spin evolution is equivalent to an evolution under the rotated spin Hamiltonian. U(t, 0) = Rφ† (θ) exp {−iHt} Rφ (θ) n o = exp −iRφ† (θ)HRφ (θ)t = exp {−iHφ (θ)t}
Note: At the end cycle, we have URF = R † R = 1. This is general property of recoupling/decoupling scheme.
WAHUHA: A introduction to Average Hamiltonian Theory H
X
H
zz
H
X
Y zz
zz
H
H=
H
zz
H
xx
2
zz
2
2
H
zz
H
yy
H
xx
Y
H
X
H
X zz
H
H
zz
ij = ωD 3Izi Izj − Ixi Ixj − Iyi Iyj
Hxx
= Ry (90)Hzz Ry† (90)
Hxx
ij = ωD 3Ixi Ixj − Izi Izj − Iyi Iyj
Hyy
= Rx (90)Hzz Rx† (90)
Hyy
ij = ωD 3Iyi Iyj − Ixi Ixj − Izi Izj
zz
yy
H
zz
1 2 H xx 2 H yy 2 H zz 6 H
zz
H
Hzz
Hxx + Hyy + Hzz = 0
Total Evolution is (to first order) H(6τ ) = Hzz τ + Hyy τ + 2Hxx τ + Hyy τ + Hzz τ = 0
WAHUHA: A introduction to Average Hamiltonian Theory
X
Y
2
Y
X
Iz
Iz
Iz
Iz
Iz
2
Iz
Iz
Ix
Iz
Iz
2
Iz
Iy
Ix
Iy
Iz
X
I z=
6 1 I I I 3 x y z
X
Iz
=
1 (Ix + Iy + Iz ) 3
Magnetization precesses around Ie 1 Ie = √ (Ix + Iy + Iz ) 3 Chemical shift scales as: 1 HCS = √ ωiso Ie 3
Homonuclear Decoupling: Pulse sequences overview Numerous scheme have been developped for homonuclear decoupling (and can’t be reviewed here). Most popular are( a) I
Solid Echo based (90 pulses): WHH4, MREV8, BR24, BLEW12, DUMBO.
I
Magic Echo sandwich: TREV8, MSHOT3
I
Lee-Goldburg based: LG, FSLG, PMGLn, wPMGLn
I
Rotor-synchronized: CNνn , RNνn , SAM
Criteria I
Spinning Frequencies regime
I
RF field required
I
Electronics (switch)
S. Paul, P. K. Madhu, J. of the Indian Institute of Science 90 (2010)
Homonuclear Decoupling: High-resolution 3D The HCNA experiment
I 1H
PMLG: High Resolution 1 H t1
I 1 H-13 C
CP +
13
C t2
I 13 Cα -15 N
specific-CP: frequency selective (LG)-CP that selects Cα
I 15 N t3
J. Biomol. NMR, 25 (2003) 217
Recoupling Interactions: Principles
Hλ = C λ × R λ (Ω) × |{z} Tλ | {z } orientation
Decoupling (under MAS)
spins
Recoupling under MAS, synchronized sample/spin rotation
Sample motion (MAS or Brownian) R λ (Ω(t)) = 0
R λ (Ω(t)) = 0, T λ (t) = 0
Spins motion(rotation)
but
T λ (t) = 0
R λ (Ω(t)) × T λ (t) = 0
Recoupling Interactions: Principles Recoupling under MAS (AHT) for the Nuts
MAS modulation R λ (Ω(t)) ≈ cos ωR t Rotor-synchronized spin modulation (RF field) T λ (t) ≈ cos ωR t 1 cos ωR t + 2 2 Average Hamiltonian Theory
Hλ (t) = C λ cos2 ωR t =
λ
H =
Cλ 2
Recoupling Heteronuclear Dipolar Interactions: REDOR 2
R
N R
N R
D t
REDOR : Rotational Echo Double Resonance
HIS = ωD (t)Iz Sz With ω D (t) = 0. With π pulses,
I z S z t
HIS (t) = ωD (t) {Iz Sz } (t) But
D t × I z S z t ≠0
HIS = ωD (t) {Iz Sz } (t) 6= 0
Recoupling Heteronuclear Dipolar Interactions: REDOR
The variation of the signal amplitude with respect to the recoupling time is a dipolar oscillation. Analysis of the latter gives the interatomic distance.
Recoupling Homonuclear Dipolar Interactions: DRAMA Dipolar Recoupling At the Magic Angle
Spin part H
MAS modulation R 4
X
R 2
X
R 4
R 4 H zz t
X
R 2
H yy t
X
R 4 H zz t
R
R 4 H zz t
R 2 X
R 4
H zz t
X
C 1 t
H zz t
Hyy = 0 Hzz = 0
S1 t R H zz t
H yy t
Hyy 6= 0 Hzz 6= 0
H zz t
Hyy = 0 Hzz = 0 C 2 t
ij ωD (t)
=
X
Cn cos(nωR t)
n=1,2
+ Sn sin(nωR t)
S2 t
Hyy = 0 Hzz = 0
Recoupling Homonuclear Dipolar Interactions: DRAMA Dipolar Recoupling At the Magic Angle R 4 H zz t
X
R 2
H yy t
X
0 < t < τR /4
R 4 H zz t
C 1 t
H1 = CDij
Z 0
τ /4
! ij ij C1 (t)dt Tzz = CDij ΛTzz
τR /2 < t < 3τR /4 ij ij Hαα = CDij ωD (t)Tαα
ij Tzz
= 2Izi Izj − Ixi Ixj − Iyi Iyj
ij Txx
= 2Ixi Ixj − Izi Izj − Iyi Iyj
ij Tyy
= 2Iyi Iyj − Ixi Ixj − Izi Izj ij ij ij Tzz + Txx + Tyy =0
ij H2 = CDij (−2Λ)Tyy
3τR /4 < t < τR ij H3 = CDij (Λ)Tzz
H = H1 + H2 + H3
Recoupling Homonuclear Dipolar Interactions: DRAMA Dipolar Recoupling At the Magic Angle R 4
Y
R 2
X
X
R 4
Y
R R 4
X
R 2
X
R 4
6 C D I z I z −I y I y ij
Y
R
j
i
j
Y
R
6 C ijD I ix I xj −I iy I yj
Y
H
i
= H1 + H2 + H3 =
ij ij 2ΛCDij Tzz − Tyy
=
6ΛCDij Izi Izj − Iyi Iyj
R
H Recoupling of the homonuclear dipolar interaction !
= =
6ΛCDij Ixi Ixj − Iyi Iyj 6ΛCDij I+i I+j + I−i I−j
Y
Recoupling Homonuclear Dipolar Interactions: BABA Back to Back
DQ Spin dynamics
Y
R 2
Y X
R 2
X
R
Double Quantum (DQ) Hamiltonian H
Λ
= 6ΛCDij Ixi Ixj − Iyi Iyj = 6ΛCDij I+i I+j + I−i I−j = HDQ Z τ /4 = S1 (t)dt 0
ρ(0) Izi + Izj Izi + Izj
Izi + Izj HDQ −−−→ I+i I+j , I−i I−j HDQ ←−−− I+i I+j , I−i I−j =
H
e −iωi t
H
e −iωj t
H
e −i(ωj +ωj )t
H
e +i(ωj +ωj )t
I+i
−−CS →
I+i
−−CS →
I+i I+j
−−CS →
I−i I−j
−−CS →
Interatomic Distance Measurements 1
H
Distance measurement 90
Dec
CP
Dec
REC X CP
P
Rec
P
P
...
P
P
Homonuclear Dipolar Correlation Experiments 1
H
1
90
H
90
Dec
CP
MIX
X 90
CP
P
AB
P
Dec
CP
Rec
P
...
EXC
X 90
P
90 Rec
CP
P
DQ
P
A
P
...
RCV
SQ
Rec 90
P
AA
2 A
A AB
A B BA
B
2 B
B
BA
BB
B
A
B
A
Homonuclear decoupling + DQ: DQ-CRAMPS
L. Mafra et al. , JMR 196 (2009) 88-91
Homonuclear decoupling + DQ: DQ-CRAMPS
S.P. Brown et al., J. Am. Chem. Soc. 126 (2004) 13230
Rotational Resonance NMR CO
Cα
200
180
160
140
120 13
100 80 C Chemical shift (ppm)
60
40
20
0
40
20
0
Cα CO ∆ = νROT
200
180
160
140
120 13
100 80 C Chemical shift (ppm)
60
1 2 When δiso − δiso = nνR , recoupling of homonuclear dipolar interactions !
Perpespectives: Dipolar Truncation
M.J. Bayro et al., J. Chem. Phys. 130 (2009) 114506
Dipolar truncation: Rotational Resonance NMR Chemical Selectivity in the Polarization curve measurements
1
H
R2 Distance measurement 90
Dec
CP
X CP
90
Selective inversion
m
90
JACS 2003 (125) 2718-2722
Dipolar truncation: Proton Mediated X-X Correlation Proton Spin diffusion correlation
JACS 124 (2002) 9704-9705;
Quantum Mechanics Engineering for NMR I The evolution of the spin systems is fully characterized by the density matrix or operator ρ(t) which obeys the Liouville-von Neummann equation i
d ρ(t) = [H(t), ρ(t)] dt
Its formal solution is given by ρ(t) = U † (t, 0)ρ(0)U(t, 0) where U(t, 0) is the quantum evolution operator (or propagator) Z t b H(u)du U(t, 0) = T exp −i 0
Quantum Mechanics Engineering for NMR II
Th determine its evolution, ρ(t) has to be expanded on a basis set of (suitable) operators: ρ(t) =
X
aα (t)Aα where Aα operator basis
α
For example, for two spins I
Product Basis Iα1 Iβ2 (Ix1 Ix2 , Ix1 Iy2 , . . .)
I
12 ( T 12 , T 12 ) Tensorial basis Tk,m 1,m 2,m
I
Fictitious spin operators Ixij , Iyij , Izij (transition between |ii and |ji levels )
Quantum Mechanics Engineering for NMR III
e −iHt Ae +iHt
=
A + (−it)[H, A] +
... +
(−it)2 [H, [H, A]] 2
(−it)n [H, ..., [H, A] . . .] + . . . {z } n! | n commutators
Simplification with cyclic commutation rules [A, B] = iC e −iφB Ae +iφB
[B, C ] = iA
[C , A] = iB
= A cos φ − i[B, A] sin φ = A cos φ + C sin φ
For example {Ix , Iy , Iz }
{Ix , 2Iy Sz , 2Iz Sz }
{Sy , 2Iz Sz , 2Iz Sx }
Quantum Mechanics Engineering for NMR IV Common situation in NMR H(t) = Hbig (t) + Hsmall (t) What is the effect of Hbig (t) on Hsmall (t) ? Go into the Hbig (t) frame (generalized rotating frame) to take Hbig (t) out. † ρe(t) = Ubig (t, 0)ρ(t)Ubig (t, 0) e H(t)
† † d = Ubig H(t)Ubig −iUbig Ubig {zdt } |
Corriolis / Gauge term
=
† Ubig
esmall (t) (H(t) − Hbig (t)) Ubig = H
ee e e small + H (t) H(t) ≈H small
e small = 1 H τbig
Z
τbig
esmall (u)du H 0
Reduced Wigner matrix and irreducible Tensor
Reduced Wigner matrix and irreducible Tensor
