Harry Pfeifer's NMR experiment in 1951Harry Pfeifer's NMR experiment in 1951Harry Pfeifer's NMR experiment in 1951Harry Pfeifer's NMR experiment in 1951
H. Pfeifer: Über den Pendelrückkoppel-empfänger und die Beobachtungen von magnetischen Kernresonanzen, About pendulum feedback receiver and observation of magnetic resonances, Diplomarbeit, Universität Leipzig, 1952
NMR of quadrupolar nucleiNMR of quadrupolar nucleiNMR of quadrupolar nucleiNMR of quadrupolar nuclei
WEB of Science refers to approximately 10 000 papers per year, which concern NMR investigations.
35% of these works refer to 1H, 25% to 13C, 8% to 31P, 8% to 15N, 4% to 29Si and 2% to 19F as one of the nuclei under study. In all these nuclei, we have the nuclear spin I = ½.
If we look at nuclei with a quadruple moment and half-integer spin I > ½, we find the nuclei 27Al in 3% of all the NMR papers and 1% for each of the nuclei 11B, 7Li, 23Na and 51V.
For even numbered spin, only the I = 1 nuclei are frequently encountered: 2H in 4% and 14N and 6Li in 0.5% of all NMR papers.
Dieter Freude, Institut für Experimentelle Physik I der Universität Leipzig METU-Center Workshop on Solid State NMR, 30 and 31 October 2007
Solid-state NMRSolid-state NMRof quadrupolar nuclei of quadrupolar nuclei
with half-integer spinwith half-integer spin
Solid-state NMRSolid-state NMRof quadrupolar nuclei of quadrupolar nuclei
with half-integer spinwith half-integer spin
Introduction to quadrupolar line broadening in solid-state NMR spectra of nuclei with half-integer spin
Excitation problems for solid-state NMR
Double-rotation (DOR) and Multiple-quantum solid-state (MQMAS) NMR of quadrupolar nuclei with half-integer spin
HistoryHistoryHistoryHistory
Classic techniques: single crystal and broad line NMR
Encanced rf power allows pulse excitation and echo observation in broad line NMR
1980, MAS reduces the powder line broadening of the central transition by 1/4
1984, nutation techniques improve the quadrupolar resolution
SATRAS reduces quadrupolar line broadening for powders
1988, DOR removes line broadening for powders
1988, DAS removes line broadening for powders
1995, MQMAS removes line broadening for powders, less laborious than DOR/DAS
MQMAS techniques are improved in sensitivity and resolution
1998, MQMAS is combined with DOR, Carr-Purcell, cross-pol., REDOR, VAS
2000, STMAS cobines high single-quantum sensitivity wit high resolution
2004, SPAM (soft pulse added mixing) increases sensitivity
Zeeman splitting of energy levelsZeeman splitting of energy levelsZeeman splitting of energy levelsZeeman splitting of energy levelsNeglecting quadrupole interaction, we have the pure Zeeman splitting
Em = m B0
As an example, we consider the Zeeman splitting of a I = 5/2-nucleus:
Double-headed arrows showsingle-quantum up to quintuple-quantum transitions.
m
5/2
5/2
3/21/21/2 3/2
The right-hand side demonstrates level populations, which van be changed, e. g., by double frequency sweep (DFS).
Parameters of quadrupole interactionParameters of quadrupole interactionParameters of quadrupole interactionParameters of quadrupole interactionThe quadrupole coupling constant Cqcc is commonly defined as
h
qQeC
2
qcc
For the quadrupole frequency, Q, different definitions exist in the literature. We use
122
3
122
3 qcc2
Q
II
C
hII
qQe
where h denotes Planck's constant. Elements of the traceless tensor of the electric field gradient V are given in the principal axis system. The ZZ-component is VZZ = eq, where e denotes the elementary charge and the value q alone has no physical meaning in SI units. Q is the quadrupole moment. eQ is called the electric quadrupole moment.
where I denotes the nuclear spin. The asymmetry parameter is in the range 0 1. With the convention |VZZ| |VYY| |VXX| we obtain
ZZ
YYXX
V
VV
Euler's angles andEuler's angles andthe angle-dependent quadrupole frequency the angle-dependent quadrupole frequency ''QQ
Euler's angles andEuler's angles andthe angle-dependent quadrupole frequency the angle-dependent quadrupole frequency ''QQ
A positive rotation to a frame (x, y, z) about the Euler angles includes the rotation about the original z axis, the rotation about the obtained y' axis, and the rotation about the final z" (identical with z''') axis.
x
y
y', y''
x'
x''
z, z'
z'', z'''
x'''
y'''
2cossin
22
1cos3 22
Quadrupole shift of theQuadrupole shift of the Larmor frequency Larmor frequencyQuadrupole shift of theQuadrupole shift of the Larmor frequency Larmor frequency
Assuming resonance offset and chemical shift to be zero, the quadrupole shift is given as m,m' = L. Conventions m,m+1 and m,m for single-quantum transitions and symmetric transitions, respectively, assign the central transition 1/2,+1/2 to m = 1/2. The first-order quadrupole shift becomes
2
1Q1, mmm
We see that there is no quadrupole shift for the central transition in first-order perturbation theory. Symmetric satellites appear around the central transition.
But for all transitions a second-order quadrupole shift exist. It is for the central transition
CBAII
24
L
2Q
2/1,2/1 coscos4
31
6
static MAS
A 27
8
9
42
3
822 2 cos cos
21
16
7
82
7
4822 2 cos cos
B 15
4
1
22 2
3
422 2 2 cos cos
9
8
1
122
7
2422 2 2 cos cos
C 3
8
1
3
1
42
3
822 2 2 cos cos
5
16
1
82
7
4822 2 cos cos
Note that the second-order quadrupole shift and broadening in powder spectra is proportional to (Q/L)2 , if expressed in ppm.
Quadrupolar shift for single crystalsQuadrupolar shift for single crystalsQuadrupolar shift for single crystalsQuadrupolar shift for single crystals
2
1Q1, mmm
+5/2+3/2 +3/2+1/2 +1/2–1/2 –1/2 –3/2 –3/2 –5/2
L
Zeeman
first-order
second-order
,6
131
3
171
30
313191
30
L
2Q
2
L
2Q
1,
fmmII
mmIImm
A useful different form of the equationA useful different form of the equationA useful different form of the equationA useful different form of the equationSecond-order quadrupole shift under MAS conditions can be written in a different form as
Table taken from Gan 2001: Ratios of expansion coefficients between satellite (m* - ½ m* + ½) and central transition (m* = 0).
CBAmmII
mmIImm
24
L
2Q
2
L
2Q
1,
coscos6
131
3
171
30
313191
30
2cos48
352cos
8
35
16
105 22A
2cos24
352cos5
12
5
8
45 222 B
2cos48
352cos
8
5
3
1
16
9 222 C
I = 9/2, m* = ±4
rank 0 rank 4
I = 3/2, m* = ±1 2 8/9
I = 5/2, m* = ±1 1/8 7/24
I = 5/2, m* = ±2 7/2 11/6
I = 7/2, m* = ±1 2/5 28/45
I = 7/2, m* = ±2 7/5 23/45
I = 7/2, m* = ±3 -22/5 12/5
I = 9/2, m* = ±1 5/8 55/72
I = 9/2, m* = ±2 1/2 1/18
I = 9/2, m* = ±3 19/8 9/8
5 50/18
rank 0
rank 4
Ratios on the left are the base of MAS satellite (SATRAS) and satellite transition (STMAS) spectroscopy.
Quadrupole line shapes for half-integeger spin Quadrupole line shapes for half-integeger spin II > ½ > ½
first-order, cut central transition second-order, central transition onlyfirst-order, cut central transition second-order, central transition only
Quadrupole line shapes for half-integeger spin Quadrupole line shapes for half-integeger spin II > ½ > ½
first-order, cut central transition second-order, central transition onlyfirst-order, cut central transition second-order, central transition only
169 16
9 329 1 0 0 5
6 149 4
21
L
Q2
L161
34
I I
= 0
= 0.5
= 1
MAS static
L
Q
3 2 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3
= 0 = 1
I = 3/2
I = 5/2
I = 7/2
= 0 = 1
I = 3/2
I = 5/2
I = 7/2
Q
L
43
116 L
2Q
L
II
All presented simulated line shapes are slightly Gaussian broadened,
in order to avoid singularities.L is the Larmor frequency.spectral range: Q(2I 1) or 3 Cqcc/ 2I
Excitation, a broad line problemExcitation, a broad line problemExcitation, a broad line problemExcitation, a broad line problem
Basic formula for the frequency spectrum of a rectangular pulse with the duration and the carrier frequency 0 with = 0:
sind2cos
1 2/
2/
ttf
We have a maximum f () = 1 for = 0 and the first nodes in the frequency spectrum occur at = 1/. The spectral energy density is proportional to the square of the rf field strength given above. If we define the usable bandwidth of excitation 1/2 in analogy to electronics as full width at half maximum of energy density, we obtain the bandwidth of excitation
886.0
2/1
It should be noted here that also the quality factor of the probe, Q = / probe, limits the bandwidth of
excitation independently from the applied rf field strength or pulse duration. A superposition of the free induction decay (FID) of the NMR signals (liquid sample excited by a very short pulse) for some equidistant values of the resonance offset (without retuning the probe) shows easily the bandwidth probe of the probe.
Excitation profile of a rectangular pulseExcitation profile of a rectangular pulseExcitation profile of a rectangular pulseExcitation profile of a rectangular pulse
5 4 3 2 1 0 1 2 3 4 5
/ MHz
We denote the frequency offset by Positive and negative values of are symmetric with respect to the4 carrier frequency 0 of the spectrometer. The rectangular pulse of the duration has the frequency spectrum (voltage)
The figure describes a pulse duration = 1 µs. The first zero-crossings are shifted by 1 MHz with respect to the carrier frequency.
2/
2/sindcos
2/
2/
ttf
Solid-state NMR spectrometer use pulse durations in the range = 1 10 µs. Respectively, we have single-pulse excitation widths of 886 – 88.6 kHz.
The full width at half maximum of the frequency spectrum correspond to a power decay to half of the maximum value or a voltage decay by 3 dB or by 0.707.
886.0
2/1
For example, NOESY and stimulated
echo require 3 pulses. Than we have
n
k
Tkf1
cos212/
2/sin
Tf
cos212/
2/sin
Excitation profile ofExcitation profile of 2n + 1 pulses 2n + 1 pulsesExcitation profile ofExcitation profile of 2n + 1 pulses 2n + 1 pulses
The figure on the left side corresponds to a pulse duration = 1 µs and a symmetric pulse distance of 10 µs. Correspondingly, the first zero-crossings are shifted by 100 kHz with respect to the carrier frequency. The beat minima are shifted by 1 MHz.
5 0 5
/ MHz
0,5 0,1 0 0,1 0,5 / MHz
21 subsequent 1-µs-pulses with 10 µs spacing21 subsequent 1-µs-pulses with 10 µs spacing21 subsequent 1-µs-pulses with 10 µs spacing21 subsequent 1-µs-pulses with 10 µs spacing
10
1
cos212/
2/sin
k
Tkf
The central excitation has a fwhm of 4 kHz, whereas we have had 30 kHz for three pulses.
The excitation bandwith of the center band decreases with increasing number of pulses.
0,5 0 0,5
/ MHz
21 subsequent 1-µs-pulses with 1 ms spacing21 subsequent 1-µs-pulses with 1 ms spacing21 subsequent 1-µs-pulses with 1 ms spacing21 subsequent 1-µs-pulses with 1 ms spacing
The excitation band width of the center band decreases with increasing number of pulses and increasing pulse distance. Therefore, the excitation profile should be simulated in critical cases.
1 0 1 / kHz
40 Hz
Selective excitation, a quadrupole problemSelective excitation, a quadrupole problemSelective excitation, a quadrupole problemSelective excitation, a quadrupole problemThe intensity of the free induction decay G(t = 0) after the pulse with the radio frequency field strength rf and the duration is for nonselective excitation of all transitions m m + 1
The equation above gives also the relative intensities of all transitions, e. g. 12/30, 9/35 and 4/21 for the central lines in the case of nonselective excitation of the I = 3/2, 5/2 and 7/2 nuclei, respectively. The selective excitation of a single transition can be described by
Comparison of both equations reveals that the maximum observed intensity is reduced by
rf
selectivenon1, sin
1212
1130
III
mmIIG mm
rf
selective1, 11sin
1212
1130
mmIIIII
mmIIG mm
11 mmII , but, the effective nutation frequency is enhanced by the same value.
For the central transition, m = 1/2, we obtain
rfeffrf 2
1
I
This is very important and should be discussed in detail on the table.
Highly resolved spectra of quadrupolar nucleiHighly resolved spectra of quadrupolar nucleiHighly resolved spectra of quadrupolar nucleiHighly resolved spectra of quadrupolar nuclei
Resolution of signals having relatively small chemical shift differences
Determination of quadrupole parameters for resolved signals
Improvement of the sensitivity
Simulated line shape of a central transition with an anisotropy factor = 0.2 and
slight Gaussian broadening;
the static NMR spectrum without MAS,
the MAS NMR spectrum for rot > static linewidth,
the rotation-synchronized MQMAS NMR spectrum.
DOR spectrum looks like MQMAS spectrum, but many spinning side bands appear.
Motivation:
MQMAS, the MQMAS, the zz-filter experiment-filter experimentMQMAS, the MQMAS, the zz-filter experiment-filter experiment
,4cos702cos3601812960
1017136
360
3143
2
40,4
240,2
40,0
2
L
22Q
L
222Q
2/,2/
dddpIIp
pIIppp
t
tpp
11
2
2 ,27136
1017136tpIRt
II
pIIpt
Election of coherence pathway is obtained by corresponding phase cycles.
p
5 4 3 2 1 012345
two rectangular pulses + z-filter
t1 t t2
p = 1 is the single-quantum central transition. p = 3 and 5 refer to triple- and quintuple- quantum transitions, respectively. d(4) denote Wiegner's reduced matrices of rank 4.
Averaging of the anisotropic (rank 4) contributions in the echo after the time t2:
The phase development for symmetric single and multiple-quantum transitions is described by
Where we go?Where we go? Where we go?Where we go?
The pulse sequence has to be further prolonged, in order to include "cyclops" and to get real and imaginary part of the signal (after Fourier-transform in F2-direction) for the 2D-Fourier-transform in F1-direction.
The transformation (spin density operator) of a coherence of sp at rotation around z-direction by the angle is described by
Now we consider a quintuple-quantum experiment and set the phases of the first and third pulses and the receiver phase to 0°. But the phase of the second pulse is incremented in steps of (360/5)° starting from 0°. After five steps, we have five times accumulated the coherences p = +5 and p = 5, and averaged (to zero) other coherences; unfortunately, except the coherences p = 0 which were also five times accumulated. In order to quench the coherences p = 0, we increment ten times by (360/10)° and use an alternating receiver phases 0° and 180°. The signal for p = 0 is quenched after two scans with alternating receiver phases. After 10 scans we remain with the coherences p = 5 only.
The preparation pulse creates coherences in all orders p 0. Of course, their intensity is drastically decreasing with the number of the order. How can we go a selected pathway?
A radio frequency pulse with the phase transfers at the time t coherences from the order p to the order p+. In the following, tand p- means before pulse, t+ and p+ means after pulse. Then we have
ppFp z iexp
pptPtP pp iexp1
Improved coherence transfer and Improved coherence transfer and whole-echo-split-whole-echo-split-tt11 MQMAS techniques MQMAS techniques
Improved coherence transfer and Improved coherence transfer and whole-echo-split-whole-echo-split-tt11 MQMAS techniques MQMAS techniques
3Q MAS 3Q DFS 3Q FAM II
+3 +2 +1
p = 0 1 2 3
131
12t
131
12t
131
12t
131
19t
131
19t
131
19t
2t
2t
2t
For 5QMAS split-t1-factors are 12/37 and
25/37.Echo occurs at Echo occurs at tt 2==. The evolution time . The evolution time tt11 is split is split
between MQ and 1Q coherences. DFS means between MQ and 1Q coherences. DFS means double frequency sweepdouble frequency sweep, FAM denotes , FAM denotes fast fast amplitude modulation.amplitude modulation.
Kentgens introduced 1999 the double frequency sweeps in static, MAS and MQMAS NMR experiments on the basis of an amplitude modulation to the carrier frequency.
The double frequency sweeps (DFS's) are generated by a programmed time-dependent amplitude modulation of the rf which causes two sidebands that are adiabatically swept from a start frequency to a final frequency during the pulsing.
What is an adiabatic frequency sweep?
Adiabatic sweep for Adiabatic sweep for II = 5/2 = 5/2Adiabatic sweep for Adiabatic sweep for II = 5/2 = 5/2
Single crystal, no sample rotation, Q’ = 300 kHz, z-axis of EFG and B0 coincide. The energy (m) is a function of the resonance offset in the rotating system (E = offset m h). In the frequency-stepped adiabatic half-passage (FSAHP) the spin system is far off-resonance at the beginning of the irradiation.
The frequency is then stepped through the region of resonances slowly enough, that the density operator can follow the Hamiltonian. Switching off the rf power at the Larmor frequency creates a single-quantum coherence like a /2 pulse applied to a spin-1/2 system. A full passage would be comparable with a nonselective pulse. The figure inset shows the level repulsion at the crossing of the -1/2 and +1/2 levels. There is an energy gap of h rf between the upper and lower branch
600 400 200 0 400 200 600
0.5
0
0.5
1
resonance offset / kHz
ener
gy
h
/ MH
z
2/3 2/3 2/5 2/5
2/1 2/1
2/1 2/1
m = 1 m = 3 m = 2
Optimization of pulse lengthsOptimization of pulse lengthsOptimization of pulse lengthsOptimization of pulse lengths
MQMAS NMR pulse sequences require the adjustment of pulse lengths. It depends on the nuclear spin, on the order of multiple-quantum coherences, on the rotation frequency, on the nutation frequency and on the quadrupole parameters of the species under study.
Pulse optimization by SIMPSON (Bak, Rasmussen, Nielsen) features an excellent agreement with the results of pulse optimization by NMR experiments. Both, SIMPSON optimization and the experiment show that the pulse optimization should focus on those species in the sample which have the largest quadrupole coupling constant. This leads to the lowest distortions of the quantitative character of the spectrum.
Result of a SIMPSON-simulation of the first two pulses of a 3QMAS-split-t1 experiment for one of the 27Al signals in the spectrum of the zeolite AlPO4-14 which has a quadrupole coupling constant of 4.08 MHz.
2727Al 3QMAS NMR study of AlPOAl 3QMAS NMR study of AlPO44-14 -14 2727Al 3QMAS NMR study of AlPOAl 3QMAS NMR study of AlPO44-14 -14
40 30 20 10 0
40
30
20
10
0
1/ ppm
2/ ppm
position 1
position 2
position 3
position 5
AlPO4-14, 27Al 3QMAS spectrum (split-t1-whole-echo, DFS pulse) measured at 17.6 T with a rotation frequency of 30 kHz.
The parameters CS, iso = 1.3 ppm, Cqcc = 2.57 MHz, = 0.7 for aluminum nuclei at position 1, CS, iso = 42.9 ppm, Cqcc = 1.74 MHz, = 0.63, for aluminum nuclei at position 2, CS, iso = 43.5 ppm, Cqcc = 4.08 MHz, = 0.82, for aluminum nuclei at position 3, CS, iso = 27.1 ppm, Cqcc = 5.58 MHz, = 0.97, for aluminum nuclei at position 5, CS, iso = 1.3 ppm, Cqcc = 2.57 MHz, = 0.7 were taken from Fernandez et al.
1177O 3QMAS NMR O 3QMAS NMR and MAS NMR spectra of zeolite Na-Aand MAS NMR spectra of zeolite Na-A1177O 3QMAS NMR O 3QMAS NMR and MAS NMR spectra of zeolite Na-Aand MAS NMR spectra of zeolite Na-A
Two signals without shoulders are resolved in the isotropic projection of the 3Q MAS spectrum. This corresponds to three different SiOAl bond angles, which can be determined from the X-ray data of the hydrated zeolite Na‑A (Si/Al = 1). But three slices from the 2D spectrum were taken, see right side, since the existence of three sites was proved by the DOR spectrum. The deconvolution of the MAS spectrum (bottom right) uses the quadrupole parameters obtained by a simulation of the three anisotropic slices of the MQ MAS spectrum and gives finally the real intensities of three signals. The sheared 3Q MAS spectrum is presented with anisotropic projection on the top and isotropic projection on the side on the left hand side.
Na-A
2 / ppm
60.0 50.0 40.0 30.0 20.0 10.0
MQMASiso / ppm
64
56
48
40
32
24
01020304050 2 / ppm60
017345168 / ppm
MQMAS iso = 35.9 ppm
= 45.0 ppm
= 46.4 ppm
3
1 2
Double rotation (DOR)Double rotation (DOR)Double rotation (DOR)Double rotation (DOR)
Double rotation was introduced by Samoson et al. in 1989. It averages the anisotropic contribution of the second-order quadrupole shift by fast sample spinning around the magic-angle (54.74°) and an angle of 30,56° or 70,12° in addition. This technique is based on an excellent fine mechanic and a computer controlled pneumatic unit.
,271363cos30cos3528
9
4cos702cos3601812960
4
31
90
3
24
40,4
240,2
40,0
2
L
2Q
L
22Q
QanisoQiso2/,2/
II
ddd
IIpppp
30.56° or 70.12°
145/966
arccos
= goes to zero for
DOR setupDOR setupDOR setupDOR setup
νouter = 12 kHz, νinner = 510 kHz
Iy/Ix has to be adjusted, in order to have J parallel to Z.
2
1
2
1
y
z
Z
L
Jy
2 sin1
Jz (1 +2 cos1)
B0
DOR NMR spectra of the zeolite Na-ADOR NMR spectra of the zeolite Na-ADOR NMR spectra of the zeolite Na-ADOR NMR spectra of the zeolite Na-A
Three signals, one peak with shoulder and another well-resolved peak can be found in the DOR spectrum of zeolite Na-A in the field of 17.6 T. This corresponds to three different SiOAl bond angles. Intensities were obtained n a direct fit of the center line of the DOR spectrum under the assumption of equal envelope line shapes for the spinning sidebands of all species. The intensities ratio of ca.1:1:2 for sites O-1, O-2, O-3, respectively, is in good agreement with the relative occurrence of the SiOAl bond angles in the X-ray data.
* ** *
*
20 0204060 /ppm
*
*
***
11.7 T
20 0204060 /ppm
17.6 T
MAS, MQMAS or DOR?MAS, MQMAS or DOR?MAS, MQMAS or DOR?MAS, MQMAS or DOR?
Signal-to-noise ratios of AlPO4-14 and andalusite spectra. The ratios of the experimentally obtained spectra were recalculated, in order to base all values to the same acquisition time of 10 h. The acquisition time is the product of repetition time and number of scans for MAS and DOR, whereas for the 2D experiments the number of experiments is included.
The numbers listed in the table show the large differences between the techniques. As example, we have for AlPO4-14 at 9.4 T the values 4740 and 56 for MAS and 5QMAS, respectively. It means that we need the (4740/56)2-fold acquisition time, in order to get an identical signal-to-noise ratio of the 5QMAS DFS spectrum compared to the MAS spectrum. It is well-known that 5QMAS takes the four-fold acquisition time as 3QMAS. The high signal-to-noise ratio of the DOR experiments compared with the 3QMAS techniques is remarkable. From this point of view, the combination of DOR with MAS and simulation of the MAS spectra seems to be the most effective procedure. Particularly for amorphous materials or other samples with a large distribution of isotropic values of the chemical shift, MQMAS experiments should be favored.
MAS DOR 3QMAS DFS 5QMAS DFS
AlPO4-14 at 17.6 T 4222 2791 149 75
AlPO4-14 at 9.4 T 4740 2782 114 56
andalusite at 17.6 T 555 615 197
andalusite at 9.4 T 569 635 129
I acknowledge support from
Özlen ErdemHorst Ernst
Johanna KanellopoulosBernd Knorr
Thomas Loeser Dieter Michel
Lutz MoschkowitzUlf Pingel
Ekaterina RomanovaDagmar Prager
Daniel Prochnow Ago Samoson
Denis Schneider
Deutsche ForschungsgemeinschaftMax-Buchner-Stiftung