principles of high resolution solid state nuclear magnetic...
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Principles of High Resolution Solid StateNuclear Magnetic ResonanceManipulating spins without restriction
Thibault Charpentier
CEA / IRAMIS / SIS2M - UMR CEA-CNRS 329991191 Gif-sur-Yvette cedex, France
2eme ecole de RMN du GERM 18th-23th, Cargese (Corse)
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
00±
B0
z
x
m
1
=1
2
I e
HRF (t) = 2ω1 cos (ω0 ±∆)t Ix
In the rotating frame:
H = HIID + ω1Ix + ∆Iz = HII
D + ωe Ie
ωe > ‖HIID‖, secular approximation for HII
D
in a second rotating frame (ωe Ie):
He = HIID,e
(3 cos2 θm − 1
2
)+
HIID,e(t)
For derivation, see exercices
Homonuclear Decoupling: Lee-Goldburg I.b
Principles: High power Off-resonance irradiation
B0z
x
m
iso
I eSpin precession around Ie results in ascaling of the chemical shift:
HCS ,e = ωiso cos θIe
For derivation, see exercices
Homonuclear Decoupling: Lee-Goldburg I.cThe full derivation . . . buckle up !
H = HII + ω1Ix + ∆Iz = HII + ωe Ie
Rotation around Iy , angle θ (Ie ⇒ Iz). Let’s Wigner do the work !
R†y (θ)HRy (θ) =1
2
∑i 6=j
ωijDR†y (θ)T ij
20Ry (θ) + ωe Iz
=1
2
∑i 6=j
ωijD
(∑n
T ij2nd
2n,0(θ)
)+ ωe Iz
Rotating Frame (again and again ...) e−iωe Iz t
H(t) =1
2
∑i 6=j
ωijD
(∑n
T ij2nd
2n,0(θ)e
−inωe Iz t
)
The secular (time independent) part
H =1
2
∑i 6=j
ωijD
(T ij
20d20,0(θ)
)= d2
0,0(θ)×HII =0 with M.A. irr.
Homonuclear Decoupling: What this damned secularapprox stands for ?!!
Secular approximation holds if ωe > ‖HIID‖.
eI
DII
DII
eI
Secular approx. Secular approx.
Homonuclear Decoupling: Lee-Goldburg IIThe Time-reversal symmetry !
Frequency swichted LG
z
x
I e
−z
−x
I e error
error
X −X
−
=2e=1 /e=2e=1 /e
Phase Modulated LG (on-resonance)
=0 =0=2e=1 /e =2e=1 /e
3210 45678 3210 45678
Pulse Index
Ph
ase
2
0
Faster phase than frequency switchCompensates for RF / offset mis-settingSymmetrisation 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 )
90
t2
A0°
A180°
A0°
A180°
A0°
C
**
**
*window
Homonuclear Decoupling: 1H NMR (L-Alanine)
-10-8-6-4-202468101214MAS dimension (ppm)
-4
-2
0
2
4
6
8
10
1 H F
SLG
dim
ensi
on (p
pm)
Alanine - B0=11.75T - vROT=12.5 kHz
MAS Resolution
FSLG
Res
olut
ion
Homonuclear Decoupling: 13C NMR (Admantane)
2628303234363840424413 Chemical shift (ppm)
CW (5kHz)
FSLG
J/31/2
C
H
H
C
H
H
H
H
H
H
Adamantane - B0=11.75T - vROT=15 kHz
⇒ Weak couplings such as J can be recovered !!⇒ J-base experiments can be performed with 1H !
Homonuclear Decoupling: in silico designNumerical optimization using spin dynamics simulations. This approachhas been pioneered with the DUMBO pulse sequence, digitized phasemodulation φ(τ) at constant RF amplitude, including a time-reversalsymmetry (φ(1− τ) in the second half, τ = t/τm)
0 ≤ τ ≤ 12
φ(τ) =n=K∑n=0
an cos(2πnτ)
+bn sin(2πnτ)
12 ≤ τ ≤ 1
φ(τ) = φ(1− τ) + π
Windowed (w)
C=m
continuous
N
CN
*
m
E. Salager et al., Chem. Phys. Lett. 469 (2009) 336-341
Homonuclear Decoupling Design: Direct spectraloptimization
→ The moss reallistic spin-dynamics simulator is the spectrometer !Optimization using the spectrometer output.
B. Elena, Chem. Phys. Lett. 398 (2004) 532-538
A first introduction to Average Hamiltonian Theory
Introduction to the toggling frame. The AHT sandwich.
H
U ,0
H
RH R
The bracketed spin evolution is equivalentto an evolution under the rotated spinHamiltonian.
U(t, 0) = R†φ(θ) exp −iHtRφ(θ)
= exp−iR†
φ(θ)HRφ(θ)t
= exp −iHφ(θ)t
Note: At the end cycle, we have URF = R†R = 1. This is generalproperty of recoupling/decoupling scheme.
WAHUHA: A introduction to Average Hamiltonian Theory
Y YX X2
H zz H zz H zz H zz H zz
X X2
H zz H zz H xx H zz H zz
2
H zz H yy H xx H yy H zz
H =1
62H xx2H yy2H zz
H
Hzz = ωijD
(3I i
z Ijz − I i
x Ijx − I i
y I jy
)Hxx = Ry (90)HzzR
†y (90)
Hxx = ωijD
(3I i
x Ijx − I i
z Ijz − I i
y I jy
)Hyy = Rx(90)HzzR
†x(90)
Hyy = ωijD
(3I i
y I jy − I i
x Ijx − I i
z Ijz
)Hxx +Hyy +Hzz = 0
Total Evolution is (to first order)
H(6τ) = Hzzτ +Hyyτ + 2Hxxτ +Hyyτ +Hzzτ= 0
WAHUHA: A introduction to Average Hamiltonian Theory
Y YX X2
I z I z I z I z I z
X X2
I z I z I x I z I z
2
I z I y I x I y I z
6
I z=13
I xI yI z
I z =1
3(Ix + Iy + Iz)
Magnetization precesses around Ie
Ie =1√3
(Ix + Iy + Iz)
Chemical shift scales as:
HCS =1√3ωiso Ie
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 , RNν
n , 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 3DThe HCNA experiment
I 1H PMLG: High Resolution 1H t1
I 1H-13C CP + 13C t2
I 13Cα-15N specific-CP: frequency selective(LG)-CP that selects Cα
I 15N t3J. Biomol. NMR, 25 (2003) 217
Recoupling Interactions: Principles
Hλ = Cλ × Rλ(Ω)︸ ︷︷ ︸orientation
× Tλ︸︷︷︸spins
Decoupling (under MAS)
Sample motion (MAS or Brownian)
Rλ(Ω(t)) = 0
Spins motion(rotation)
Tλ(t) = 0
Recoupling under MAS,synchronized sample/spin rotation
Rλ(Ω(t)) = 0,Tλ(t) = 0
but
Rλ(Ω(t))× Tλ(t) = 0
Recoupling Interactions: PrinciplesRecoupling under MAS (AHT) for the Nuts
MAS modulation
Rλ (Ω(t)) ≈ cos ωRt
Rotor-synchronized spin modulation (RF field)
Tλ(t) ≈ cos ωRt
Hλ(t) = Cλ cos2 ωRt =1
2+
cos ωRt
2
Average Hamiltonian Theory
Hλ=
Cλ
2
Recoupling Heteronuclear Dipolar Interactions: REDOR
2
N R N R
R
Dt
I z Sz t
Dt × I z Sz t ≠0
REDOR :Rotational Echo Double Resonance
HIS = ωD(t)IzSz
With ωD(t) = 0. With π pulses,
HIS(t) = ωD(t) IzSz (t)
But
HIS = ωD(t) IzSz (t) 6= 0
Recoupling Heteronuclear Dipolar Interactions: REDOR
The variation of the signal amplitudewith respect to the recoupling time isa dipolar oscillation. Analysis of thelatter gives the interatomic distance.
Recoupling Homonuclear Dipolar Interactions: DRAMADipolar Recoupling At the Magic Angle
Spin part
R
XR2 X
R4
R4
R
X
R2
X
R4
R4
H zz t H zz t H zz t
H zz t H yy t H zz t
ωijD(t) =
∑n=1,2
Cn cos(nωRt)
+ Sn sin(nωRt)
MAS modulation H
XH zz t
R2
XH yy t H zz t
R4
R4
C1t Hyy 6= 0Hzz 6= 0
S1t Hyy = 0Hzz = 0
C2t
Hyy = 0Hzz = 0
S2t Hyy = 0Hzz = 0
Recoupling Homonuclear Dipolar Interactions: DRAMADipolar Recoupling At the Magic Angle
XH zz t
R2
XH yy t H zz t
R4
R4
C1t
Hαα = C ijDωij
D(t)T ijαα
T ijzz = 2I i
z Ijz − I i
x Ijx − I i
y I jy
T ijxx = 2I i
x Ijx − I i
z Ijz − I i
y I jy
T ijyy = 2I i
y I jy − I i
x Ijx − I i
z Ijz
T ijzz + T ij
xx + T ijyy = 0
0 < t < τR/4
H1 = C ijD
(∫ τ/4
0C1(t)dt
)T ij
zz = C ijDΛT ij
zz
τR/2 < t < 3τR/4
H2 = C ijD(−2Λ)T ij
yy
3τR/4 < t < τR
H3 = C ijD(Λ)T ij
zz
H = H1 +H2 +H3
Recoupling Homonuclear Dipolar Interactions: DRAMADipolar Recoupling At the Magic Angle
R
XR2 X
R4
R4
H = H1 +H2 +H3
= 2ΛC ijD
(T ij
zz − T ijyy
)= 6ΛC ij
D
(I iz I
jz − I i
y Ijy
)Recoupling of the homonuclear dipolar
interaction !
R
XR2 X
R4
R4Y Y
R
Y Y6CDij I z
i I zj−I y
i I yj
R
Y Y6CDij I x
i I xj−I y
i I yj
H = 6ΛC ijD
(I ix I
jx − I i
y Ijy
)= 6ΛC ij
D
(I i+I j
+ + I i−I j−
)
Double Quantum (DQ) Hamiltonian
Recoupling Homonuclear Dipolar Interactions: BABABack to Back
R
XR2 XY Y
R2
Double Quantum (DQ) Hamiltonian
H = 6ΛC ijD
(I ix I
jx − I i
y Ijy
)= 6ΛC ij
D
(I i+I j
+ + I i−I j−
)= HDQ
Λ =
∫ τ/4
0
S1(t)dt
DQ Spin dynamics
ρ(0) = I iz + I j
z
I iz + I j
z
HDQ−−−→(I i+I j
+, I i−I j−
)I iz + I j
z
HDQ←−−−(I i+I j
+, I i−I j−
)
I i+
HCS−−→ e−iωi t
I i+
HCS−−→ e−iωj t
I i+I j
+HCS−−→ e−i(ωj+ωj )t
I i−I j−
HCS−−→ e+i(ωj+ωj )t
Homonuclear Dipolar Correlation Experiments
90
CP
1H
X MIX
CP Dec
Rec
P P P...P P
90 90
AB
BA
A
B A
B
A
B
90
CP
1H
X EXC
CP Dec
Rec
P P P...
90 90Rec
RCVDQ SQ
AB
BA
2A
B A
2B
AA
BB
AB
Rotational Resonance NMR
02040608010012014016018020013C Chemical shift (ppm)
CO
Cα
02040608010012014016018020013C Chemical shift (ppm)
COCα
∆ = νROT
When δ1iso − δ2
iso = nνR , recoupling of homonuclear dipolar interactions !
Dipolar truncation: Rotational Resonance NMR
Chemical Selectivity in the Polarization curve measurements
90
CP
R2 Distance measurement1H
X
CP Dec
Selectiveinversion90 90m
JACS 2003 (125) 2718-2722
Dipolar truncation: Proton Mediated X-X CorrelationProton Spin diffusion correlation
JACS 124 (2002) 9704-9705;
Quantum Mechanics Engineering for NMR I
The evolution of the spin systems is fully characterized by thedensity matrix or operator ρ(t) which obeys the Liouville-vonNeummann equation
id
dtρ(t) = [H(t), ρ(t)]
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)
U(t, 0) = T exp
−i
∫ t
0H(u)du
Quantum Mechanics Engineering for NMR II
Th determine its evolution, ρ(t) has to be expanded on a basis setof (suitable) operators:
ρ(t) =∑α
aα(t)Aα where Aα operator basis
For example, for two spins
I Product Basis I 1αI 2
β (I 1x I 2
x , I 1x I 2
y , . . .)
I Tensorial basis T 12k,m ( T 12
1,m, T 122,m)
I Fictitious spin operators I ijx , I ij
y , I ijz
(transition between |i〉 and |j〉 levels )
Quantum Mechanics Engineering for NMR III
e−iHtAe+iHt = A + (−it)[H,A] +(−it)2
2[H, [H,A]]
. . . +(−it)n
n![H, ..., [H,A] . . .]︸ ︷︷ ︸
n commutators
+ . . .
Simplification with cyclic commutation rules
[A,B] = iC [B,C ] = iA [C ,A] = iB
e−iφBAe+iφB = A cos φ− i [B,A] sinφ = A cos φ + C sin φ
For example
Ix , Iy , Iz Ix , 2IySz , 2IzSz Sy , 2IzSz , 2IzSx
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.
ρ(t) = U†big (t, 0)ρ(t)Ubig (t, 0)
H(t) = U†bigH(t)Ubig −iU†big
d
dtUbig︸ ︷︷ ︸
Corriolis / Gauge term
= U†big (H(t)−Hbig (t))Ubig = Hsmall(t)
H(t) ≈ Hsmall +
˜Hsmall(t) Hsmall =1
τbig
∫ τbig
0
Hsmall(u)du
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