some topics of solid-state nuclear magnetic resonance

Dieter Freude, Institut für Experimentelle Physik I der Universität Leipzig Workshop in the Ibnu Sina Institute for Fundamental Science Studies

at the Universiti Teknologi Malaysia, 12-16 May 2008

Some topics of Some topics of solid-state nuclear magnetic resonancesolid-state nuclear magnetic resonance

Some topics of Some topics of solid-state nuclear magnetic resonancesolid-state nuclear magnetic resonance

http://images.google.de/imgres?imgurl=http://www.cibklutm.com/images/UTM.gif&imgrefurl=http://www.cibklutm.com/&h=280&w=280&sz=40&hl=de&start=1&um=1&tbnid=lzwk0acuoma3pM:&tbnh=114&tbnw=114&prev=/images%3Fq%3Dutm%2B%252B%2Bmalaysia%26svnum%3D10%26um%3D1%26hl%3Dde%26rlz%3D1T4GGIH_deDE240TR247%26sa%3DN

Harry Pfeifer's NMR-Experiment 1951 in LeipzigHarry Pfeifer's NMR-Experiment 1951 in LeipzigHarry Pfeifer's NMR-Experiment 1951 in LeipzigHarry Pfeifer's NMR-Experiment 1951 in Leipzig

H. Pfeifer: About the pendulum feedback receiver and the obsevation of nuclear magnetic resonances, diploma thesis, Universität Leipzig, 1952

How works NMR: a nuclear spin How works NMR: a nuclear spin I I = 1/2 in an magnetic field = 1/2 in an magnetic field BB00How works NMR: a nuclear spin How works NMR: a nuclear spin I I = 1/2 in an magnetic field = 1/2 in an magnetic field BB00 B0, z

y

x

L

B0, z

y

x

L

Many atomic nuclei have a spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is

The component in the direction of an applied field is

Lz = Iz m = ½ for I = 1/2.

.)1( IIL

Atomic nuclei carry an electric charge. In nuclei with a spin, the rotation creates a circular current which produces a magnetic moment µ.

An external homogenous magnetic field B results in a torque T = µ B with a related energy of E = µ·B.

The gyromagnetic (actually magnetogyric) ratio is defined by

µ = L.

The z component of the nuclear magnetic moment is

µz = Lz = Iz m .

The energy for I = 1/2 is split into 2 Zeeman levels

Em = µz B0 = mB0 = B0/2 = L/2.Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet.

Larmor frequencyLarmor frequencyLarmor frequencyLarmor frequency

Joseph Larmor described in 1897 the precession of electron orbital magnetization in an external magnetic field.

Classical model: the torque T acting on a magnetic dipole is defined as the time derivative of the angular momentum L. We get

By setting this equal to T = µ B , we see that

The summation of all nuclear dipoles in the unit volume gives us the magnetization. For a magnetization that has not aligned itself parallel to the external magnetic field, it is necessary to solve the following equation of motion:

.dd1

dd

ttμL

T

.dd

Bμμ

t

.dd

BMM t

B0, z

M

y

x L

We define B (0, 0, B0) and choose M(t 0) |M| (sin, 0, cos). Then we obtain

Mx |M| sin cosLt, My |M| sin sinLt, Mz |M| cos with L = B0.

The rotation vector is thus opposed to B0 for positive values of . The Larmor frequency

is most commonly given as an equation of magnitudes: L = B0 or.

2 0L B

Some of the 130 NMR isotopesSome of the 130 NMR isotopesSome of the 130 NMR isotopesSome of the 130 NMR isotopes

nucleus natural abundance

/%

spin quadrupole moment Q/fm2

gyromagnetic ratio

/107 T s

-frequency 100 MHz

(1H)

rel. sensitivity at natural

abundance

1H 99.985 1/2 26.7522128 100.000000 1.000 2H 0.015 1 0.2860 4.10662791 15.350609 1.45 10 6 6Li 7.5 1 0.0808 3.9371709 14.716106 6.31 10 4 7Li 92.5 3/2 4.01 10.3977013 38.863790 0.272 11B 80.1 3/2 4.059 8.5847044 32.083974 0.132 13C 1.10 1/2 6.728284 25.145020 1.76 10 4 14N 99.634 1 2.044 1.9337792 7.226330 1.01 10 3 15N 0.366 1/2 2.71261804 10.136784 3.85 10 6 17O 0.038 5/2 2.558 3.62808 13.556430 1.08 0 5 19F 100 1/2 25.18148 94.094008 0.834

23Na 100 3/2 10.4 7.0808493 26.451921 9.25 10 2 27AI 100 5/2 14.66 6.9762715 26.056890 0.21 29Si 4.67 1/2 5.3190 19.867187 3.69 10 4 31P 100 1/2 10.8394 40.480742 6.63 10 2 51V 99.750 7/2 5.2 7.0455117 26.302963 0.38

R. K. Harris, E. D. Becker, S. M. C. de Menezes, R. Goodfellow, P. Granger:NMR nomenclature: Nuclear spin properties and conventions for chemical shifts -

IUPAC recommendations 2001, Pure Appl. Chem. 73 (2001) 1795-1818

Chemical shift of the NMRChemical shift of the NMRChemical shift of the NMRChemical shift of the NMR

H+

external magnetic field B0

shielded magnetic

fieldB0(1)

OH

electronshell

We fragment hypothetically a water molecule into hydrogen cation plus hydroxyl anion. Now the 1H in the cation has no electron shell, but the 1H in the hydroxyl anion is shielded (against the external magnetic field) by the electron shell. Two signals with a distance of about 35 ppm appear in the (hypothetical) 1H NMR spectrum.

Chemical shift rangeChemical shift rangeof some nucleiof some nucleiChemical shift rangeChemical shift rangeof some nucleiof some nuclei

Ranges of the chemical shifts of a few nuclei and the reference substances, relative to which shifts are related.

1, 2H TMS

6, 7Li 1M LiCl

11B BF3O(C2H5)2

13C MS = (CH3)4Si

14, 15N NH4+

19F CFCl3

23Na 1M NaCl

27Al [Al(H2O)6]3+

29Si TMS = (CH3)4Si

31P 85% H3PO4

51V VOCl3

1000 100 10 0 10 100 1000/ ppm

129, 131Xe XeOF4

An NMR spectrum is not shown as a

function of the frequency

= ( / 2) B0(1), but rather on

a ppm-scale of the chemical shift

= 106 (ref ) /L, where the

reference sample is defined by UPAC,

e. g. TMS (tetramethylsilane) for 1H, 2H, 13C, and 29Si NMR.

OffsetOffsetOffsetOffset

The basic frequency is 400.13 MHz for 1H (AVANCE 400). For other nuclei, the value with 6 characters after the decimal point is given in tables.

Having a correct adjusting of the external magnetic field and no offset,the transmitter frequency equals the basic frequency, and we obtain the position of 0 ppm exact in the middle of the spectral range.

An offset can be obtained by using the commands O1 in channel F1 or O2 in Channel F2 and setting the values 0 Hz. Then the transmitter (carrier) frequency equals the sum of basic frequency and offset, e.g. in channel F1 we have SFO1 = BF1 + O1. But we obtain the position of the offset frequency exact in the middle of the spectral range, since the referencing of the scale in Hz or ppm is performed with respect to the basic frequency. With other words, changing the offset does not change the ppm-value of an NMR signal. Note that the transmitter frequency offset is not identical with the signal offset.

However, the ppm-value of an NMR signal can be changed by using the command SR. For solid-state NMR the use of SR is not recommended. Set always SR = 0 and reset SR = 0 after changing the nucleus (special feature of xWinNMR), but check the value "Field" on the hardware panel.

Shimming and adjusting the magnetic field with PDMSShimming and adjusting the magnetic field with PDMSShimming and adjusting the magnetic field with PDMSShimming and adjusting the magnetic field with PDMS

Polydimethylsiloxane (PDMS, [-Si(CH3)2-O-]n) has a chemical shift of 0.07 ppm.

It should be run at rot = 35 kHz. Keep the sample permanent in a rotor.

Shimming and field adjusting must be performed for each probe. It should be done

again, if the probe has been repaired or a stupid person changed the data.

Always read the proper shim file for the probe and check the value "Field".

MAS shimming is much more simple than shimming a liquid-state spectrometer.

For shimming and adjusting use the gs command (go setup) and call bsmsdisp.Shimming target is either a very long mono-exponential decay of the free induction or a maximum Lorenz-shaped signal in the frequency domain (frq).Bruker MAS probes have usually the rotor in the y-z-plane. Thus x = 0.Shim first z and y alternating, if necessary then x2-y2 and xy. Keep 0 for other. Lorenz-shape is often not achieved. A 50% narrow signal about 10 Hz broad plus a 50% signal 40 Hz broad are sufficient.

Only after shimming (or reading the shim file) the "Field"-value has to be adjusted (or checked). Use an offset of 0.07 ppm (28 Hz for AVANCE 400). Change "Field" until the signal is in resonance. "Field" amounts 7200 with 4.shim at 25 April 2008. 87 Field units correspond to 400 Hz 1H-shift or to 1 ppm for all nuclei.

Fourier transform NMRFourier transform NMRFourier transform NMRFourier transform NMR

History

Continuous wave (cw) spectrometers with stationary rf input (omit dotted box) are

rarely applied these days. To prevent saturation, cw spectrometers use a weak rf

input of a few µT at constant frequency with variable magnetic field or vice-versa.

Pulse spectrometers use radio frequency pulses. In the study of liquids,

rf induction in the mT range is sufficient, whereas in the study of solids,

maximal rf induction and minimal pulse width is desired, for example:

12 mT with an pulse length /2 of 1 µs is necessary to rotate the magnetization

of 1H-nuclei from the z-direction into the x-y-plane.

rf synthesizer pulse generator

transmitter receiver processor

B0

rf coil

coil of the superconducting magnet

Free induction and Fourier transformFree induction and Fourier transformFree induction and Fourier transformFree induction and Fourier transform

1 2 3 4 0

10 20 30 40 50 60 70 0

t/ms

t/s

0 1 2 3 4 5 / ppm

The figure shows at left the free induction decay (FID) as a function of time and at right the Fourier transformed 1H NMR spectrum of alcohol in fully deuterated water. The individual spikes above are expanded by a factor of 10. The singlet comes from the OH groups, which exchange with the hydrogen nuclei of the solvent and therefore show no splitting. The quartet is caused by the CH2 groups, and the triplet corresponds to the CH3 group of the ethanol. The splitting is caused by J-coupling between 1H nuclei of neighborhood groups via electrons.

Jean Baptiste Joseph Fourier’s transformationJean Baptiste Joseph Fourier’s transformationJean Baptiste Joseph Fourier’s transformationJean Baptiste Joseph Fourier’s transformationFourier’s original form from 1822 was conceived to describe the spatial distribution of temperature by infinite sums. In spectroscopy, it is mainly used to transform signals from the time domain into the frequency domain and vice-versa. The symmetric form of the Fourier transform is written

diexp2

1tftg

tttgf diexp2

1

Note that exp(i0t) = cos0t + i sin0t.

Let us now consider the function g(t) = exp(t / Td + i0t).

The Fourier transform is

2

d2

0

d0d2

d2

0

d

d0

12i

1

1

2i2

i

T

TT

T

T

T

f

Setting 0 = 0, we see that the Fourier transform of a mono-exponential decay

gives f() = 1 / (1 + 2Td2). In this case there is no imaginary part of the frequency domain. But

usually we have real part und imaginary part in the time and frequency domain as well.

Lorentz line shapeLorentz line shapeLorentz line shapeLorentz line shape

A mono-exponential decay of the free induction corresponds to

G(t) = exp(t/T2),

where T2 denotes the

transverse relaxation time.

The Fourier-transform gives fLorentz = const. 1 / (1 + x2) with x = ( 0)T2,

see red line. The "full width at half maximum" (fwhm) in frequency units is

.1

22/1 T

21/2=2/T2=1/2

0

fLorentz

1

1/2

Advantage of FT NMR compared to cw NMRAdvantage of FT NMR compared to cw NMRAdvantage of FT NMR compared to cw NMRAdvantage of FT NMR compared to cw NMR

1 2 3 4 0

10 20 30 40 50 60 70 0

t/ms

t/s

0 1 2 3 4 5 / ppm

Supposed, we would measure the signal in the frequency domain by cw NMR. Then we would measure a signal only in the narrow intervals. The time we spend for measuring the broad intervals is lost.

But for very broad signals, with other words for a very short FID, the dead time of the pulse spectrometer causes a disadvantage compared to the cw technique.

Sampling theoremSampling theoremSampling theoremSampling theorem

The sampling theory named after Harry Nyquist tells us that for the unique identification of a cosine function, at least two measurements must be taken per oscillation period. For the duration of the sampling of a measurement value (dwell time) we then get < 1/(2) or in other words, the sampling rate has to be at least twice the oscillation frequency to be measured. If the sampling rate is exactly double or less, we get, after Fourier transformation, mirror symmetric replicates or aliasing. These are mirrored into the unique spectral range from 1/(2) from outside. If the sampling rate is much higher than twice the frequency of the sampled signal, no advantage is gained in non-noisy signals. This so-called oversampling, however, simplifies the determination of signals buried in noise.

dwell time t = 1 ms t /ms 15 0

Frequency domain of the Fourier transformation:

0 Hz 250 Hz 500 Hz

5 10

The figure shows measurements with a dwell time of 1 ms. The dashed line with a frequency of 250 Hz has 4 measured values per period (double oversampling). The dotted 500 Hz line contains only 2 measured values per period and, after a Fourier transformation, would appear on both edges of the frequency range from 0 to 500 Hz, since it is indiscernible from 0 Hz (all points pass through a line). The straight line for 0 Hz yields the same result. The 1 kHz line contains only one measured point per period and would be mirrored in to both edges of the measurement range.






0 Hz 250 Hz 500 Hz

5 10

Fast Fourier transformFast Fourier transformFast Fourier transformFast Fourier transform

The numerical calculation is based on a a certain number of measured points, e. g. 1024 points, if size (SI) and time domain (TD) were set to 1K. The Integral (that means summation) has to be performed over the same number of points. Therefore, the calculation effort is increasing with the square of the size. Fast Fourier Transform (FFT) is a calculation procedure that reduces the effort in such a way that it is only proportional to the size.

We have always for digital Fourier transforma dwell time (DW) of t and a

correspondent sweep width (SWH) = 1 / 2t or SWH = 1 / 2DW.

Starting from the sweep width in ppm (SW = SWH/ SFO1, the latter given in MHz), the dwell time is calculated corresponding to this relation.

In opposite to the analog acquisition mode, an oversampling is used for the digital acquisition mode.

The size (SI) of the spectrum determines the number of points in the frequency domain. We have SI / 2 points in the real part and the same number in the imaginary part of the spectrum. Therefore, the frequency resolution ( denotes the distance between two points on the scale) equals

= / ½ SI = DW-1 / SI = SWH / SI or ppm = SW / SI.The time domain (TD) can be much shorter than SI.

Basics for synthesis and detection of frequenciesBasics for synthesis and detection of frequenciesBasics for synthesis and detection of frequenciesBasics for synthesis and detection of frequencies

Addition theorem

coscoscoscos2

A phase sensitive detector (also known as a lock-in amplifier) is a type of amplifier that can extract a signal with a known carrier wave from noisy environment. It multiplies the signal voltage with an reference voltage coming from the same origin (look-in).

tttt 212121 2cos2cos2cos2cos2

Operational amplifier + RC unit

R

C

Uout

Uref

Usignal

ssUsRCU RCt

RCt

d2sin sig

1

1 refrefout

Uref

Quadrature detectionQuadrature detectionQuadrature detectionQuadrature detection

After the preamplifier the same signal goes to two independent phase sensitive detectors having reference signals with the same frequency but a 90° shifted phase. Their output is the input of (two units of) the analogue-to-digital conversion (ADC). The ADC's create digital values each time interval of the dwell time DW. One value goes to the real part of the time domain, the next to the imaginary part and so on. It means that we have time intervals of 2DW between the measuring values in one part.

The Fourier transform needs always real part and imaginary part of the spectrum. In addition it is based on the proper adjustment for analogue acquisition. It means the amplification of the both branches should be identical and the phase difference of the two references should amount exactly 90°. Otherwise we obtain a sharp peak exact in the middle of the sweep range and mirror signals with respect to the middle. No problems for fully digitalized acquisition.

The difference between solid-state and liquid NMR,The difference between solid-state and liquid NMR,e. g. the lineshape of watere. g. the lineshape of water

The difference between solid-state and liquid NMR,The difference between solid-state and liquid NMR,e. g. the lineshape of watere. g. the lineshape of water

10 20 30 400

/ kHz

-30 -20 -10-40

0.1 0.2 0.3 0.40

/ Hz

-0.3 -0.2 -0.1-0.4

solid water (ice)

liquid water

Line broadening effects in solid-state NMRLine broadening effects in solid-state NMRLine broadening effects in solid-state NMRLine broadening effects in solid-state NMR

chemical shift anisotropy

distribution of the isotropic value of the chemical shift

dipolar interactions

first-order quadrupole interactions

second-order quadrupole interaction

inhomogeneities of the magnetic susceptibility

Chemical shift anisotropyChemical shift anisotropyChemical shift anisotropyChemical shift anisotropy

0

csa

xx yy

zz

xx yy

zz

csa

zz xx yy

1

2

csa

zz xx yy

1

2

zz yy xx zz yy xx

yy xx 0 zz

yy xx 0 zz

csa

asymmetry factoranisotropy

= 0 0

Distribution of isotropic values of the chemical shiftDistribution of isotropic values of the chemical shiftDistribution of isotropic values of the chemical shiftDistribution of isotropic values of the chemical shift

No common model exists for this very important broadening effect.

Dipolar broadening of a two-spin systemDipolar broadening of a two-spin systemDipolar broadening of a two-spin systemDipolar broadening of a two-spin system

= II,IS (3 cos2 - 1)

0

DDWW I 1

2

44

3 03

II

2I

II r

42

03

IS

SIIS r

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

Fast rotation (160 kHz) of the sample about an axis oriented at 54.7° (magic-angle) with respect to the static magnetic field removes all broadening effects with an angular dependency of

o7.543

1cosarc

That means chemical shift anisotropy,dipolar interactions,first-order quadrupole interactions, and inhomogeneities of the magnetic susceptibility.

It results an enhancement in spectral resolution by line narrowing also for soft matter studies.

High-resolution solid-state MAS NMRHigh-resolution solid-state MAS NMRHigh-resolution solid-state MAS NMRHigh-resolution solid-state MAS NMR

2

1cos3 2

rotor with samplein the rf coil zr

rot

θ

gradient coils forMAS PFG NMR

B0

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

Effective field and Rabi frequencyEffective field and Rabi frequencyEffective field and Rabi frequencyEffective field and Rabi frequency

We get into the so-called "rotating" coordinate system, which rotates with the angular frequency around the z axis of the laboratory coordinate system. The radio frequency field is applied to the coil (including the sample) in the x‑direction of the laboratory coordinate system with the frequency and amplitude 2Brf. This linear polarized field can be described by two circular polarized fields which rotate with the frequency in the positive and negative sense around the z-axis. From that we get an x-component Brf in the rotating coordinate system. The external magnetic field is in the rotating coordinate system replaced by the resonance offset (L ) / .

The effective working field in the rotating coordinate system is a vector addition of the rf field and the offset

Beff = (Brf, 0, B0 /).

The nutation of the macroscopic magnetization corresponds to a rotation in the rotating coordinate system. If the offset is small compared to Brf, the so-called nutation frequency or Rabi frequency is rf = Brf or

.2 rfrf B

Longitudinal relaxation time Longitudinal relaxation time TT11Longitudinal relaxation time Longitudinal relaxation time TT11

All degrees of freedom of the system except for the spin (e.g. nuclear oscillations,

rotations, translations, external fields) are called the lattice. Setting thermal

equilibrium with this lattice can be done only through induced emission. The

fluctuating fields in the material always have a finite frequency component at the

Larmor frequency (though possibly extremely small), so that energy from the spin

system can be passed to the lattice. The time development of the setting of

equilibrium can be described after either switching on the external field B0 at time

t 0 (difficult to do in practice) with,1 1

0

T

t

enn

T1 is the longitudinal or spin-lattice relaxation time an n0 denotes the difference in

the occupation numbers in the thermal equilibrium. Longitudinal relaxation time

because the magnetization orients itself parallel to the external magnetic field.

T1 depends upon the transition probability P as

1/T1 = 2P 2B½,+½ wL.

TT1 1 determination by IRdetermination by IRTT1 1 determination by IRdetermination by IR

The inversion recovery (IR) by -/2

1210Tenn

By setting the parentheses equal to zero, we get 0 T1 ln2 as the passage of

zero.

0

Line width and Line width and TT22Line width and Line width and TT22

A pure exponential decay of the free induction (or of the envelope of the echo, see next page) corresponds to

G(t) = exp(t/T2).

The Fourier-transform gives fLorentz = const. 1 / (1 + x2) with x = ( 0)T2,

see red line. The "full width at half maximum" (fwhm) in frequency units is

.1

22/1 T

Note that no second moment exists for a Lorentian line shape. Thus, an exact Lorentian line shape should not be observed in physics.

Gaussian line shape has the relaxation function G(t) = exp(t2 M2 / 2) and a line

form fGaussian = exp (2/2M2), blue dotted line above, where M2 denotes the

second moment. A relaxation time can be defined by T22 = 2 / M2. Then we get

21/2=2/T2=1/2

0

fLorentz

1

1/2

( ) ( ) ( ) .Hz/×12.74ln

Hz/=s/

2=s/ 2

2/1

22

2/122

2-2

≈

TM

TT2 2 and and TT22**TT2 2 and and TT22**

( ) 2

2

e= TG

( ) 2e= Tt

tG

Correlation time Correlation time cc, relaxation times , relaxation times TT11 and and TT22Correlation time Correlation time cc, relaxation times , relaxation times TT11 and and TT22

tftfG

c

GG

exp0

2

L

2

L06

24

1 21

8

1

2

4

1

5

11

c

c

c

cII

rT

2

L

2

L06

24

2 21

2

1

53

4

1

5

11

c

c

c

cc

II

rT

T1

T2

ln T1,2

1/T

T1 min

T2 rigid

The relaxation times T1 and T2 as a function of the reciprocal absolute temperature

1/T for a two spin system with one correlation time. Their temperature dependency

can be described by c 0 exp(Ea/kT).

It thus holds that T1 T2 1/c when Lc « 1 and T1 L2 c when Lc » 1.

T1 has a minimum of at Lc 0,612 or Lc 0,1.

Rotating coordinate system and the offsetRotating coordinate system and the offsetRotating coordinate system and the offsetRotating coordinate system and the offset

For the case of a static external magnetic field B0 pointing in z-

direction and the application of a rf field Bx(t) = 2Brf cos(t) in x-

direction we have for the Hamilitonian operator of the external interactions in the laboratory sytem (LAB)

H0 + Hrf = LIz + 2rf cos(t)Ix,

where L = 2L = B0 denotes the Larmor frequency, and the

nutation frequency rf is defined as rf = Brf.The transformation from the laboratory frame to the frame rotating with gives, by neglecting the part that oscillates with the twice radio frequency,

H0 i + Hrf i = Iz +

rf Ix,

where = L denotes the resonance offset

and the subscript i stays for the interaction representation.

B0

M

x

y

z

B0

M x’

y

z

Magnetization phases develop in this interaction representation in the rotating coordinate system like = rf or = t.

Quadratur detection yields value and sign of .

Bloch Bloch equation and stationary solutionsequation and stationary solutions Bloch Bloch equation and stationary solutionsequation and stationary solutions

We define Beff (Brf, 0, B0 /) and introduce the Bloch equation:

1

0

2

effd

d

T

MM

T

MM

tzx zyyx eee

BMM

Stationary solutions to the Bloch equations are attained for dM/dt 0:

.

1

1

,21

,21

0

212rf

222

2

L

22

2

L

rf0rf

212rf

222

2

L

2

rf0rf

212rf

222

2

L

22L

MTTBT

TM

HMBTTBT

TM

HMBTTBT

TM

z

y

x

Hahn echoHahn echoHahn echoHahn echo

B0

M

x

y

z B0

M x

y

z B0

x

y

z

5 4

1 2

3

B0

x

y

z

1 2

5 4

3

B0

M x

y

z

/2 pulse FID, pulsearound the dephasing around the rephasing echo y-axis x-magnetization x-axis x-magnetization

(r,t) = (r)·t (r,t) = (r,) + (r)·(t )

11H MAS NMR spectra, TRAPDORH MAS NMR spectra, TRAPDOR11H MAS NMR spectra, TRAPDORH MAS NMR spectra, TRAPDOR

H-ZSM-5 activated at 550 °C

420246 8 10 / ppm

20 468 10 / ppm

4

4.2 ppm 2.9 ppm2.9 ppm

2.2 ppm

1.7 ppm

2.2 ppm1.7 ppm2.9 ppm2.9 ppm

with dephasing

without dephasing

difference spectra

2

Without and with dipolar dephasing by 27Al high power irradiation and difference spectra are shown from the top to the bottom. The spectra show signals of SiOH groups at framework defects, SiOHAl bridging hydroxyl groups, AlOH group.

H-ZSM-5 activated at 900 °C

4.2 ppm

4.2 ppm

4.2 ppm

11H MAS NMR of porous materialsH MAS NMR of porous materials11H MAS NMR of porous materialsH MAS NMR of porous materials

4 2 0 2 4 6 7 5 ppm

3 1 1 2

Bridging OH groups in small channels and cages of zeolites SiOHAl

Disturbed bridging OH groups in zeolite H-ZSM-5 and H-Beta SiOH

Bridging OH groups in large channels and cages of zeolites SiOHAl

Cation OH groups located in sodalite cages of zeolite Y and in channels of ZSM-5 which are involved in hydrogen bonds

CaOH, AlOH, LaOH OH groups bonded to extra-framework aluminium species which are located in cavities or channels and which are involved in hydrogen bonds

AlOH Silanol groups at the external surface or at framework defects

SiOH

Metal or cation OH groups in large cavities or at the outer surface of particles MeOH

1313C MAS NMR and cross polarizationC MAS NMR and cross polarization1313C MAS NMR and cross polarizationC MAS NMR and cross polarization

Cross polarization

Hartman-Hahn condition under MAS

Decoupling

2929Si MAS NMR spectrum of silicalite-1Si MAS NMR spectrum of silicalite-12929Si MAS NMR spectrum of silicalite-1Si MAS NMR spectrum of silicalite-1

SiO2 framework consisting of 24 crystallographic different silicon sites per unit cell (Fyfe 1987).

2929Si MAS NMRSi MAS NMR2929Si MAS NMRSi MAS NMR

130 110 90 70 60 80 ppm

100 120

Si(1 Zn)

Si(2 Zn)

zincosilicate-type zeolites VP-7, VPI-9 Q4

alkali and alkaline earth

silicates

Q0

Q2

Q1

Q4

Si(1 Al)

Si(0 Al)

Si(2 Al)

Si(3 Al)

Si(4 Al)

Si(3Si, 1OH)

aluminosilicate-type zeolites

Q3

Q4

Q3

2929Si MAS NMR shift and Si-O-Si bond angle Si MAS NMR shift and Si-O-Si bond angle 2929Si MAS NMR shift and Si-O-Si bond angle Si MAS NMR shift and Si-O-Si bond angle Considering the Q4 coordination alone, we find a spread of 37 ppm for zeolites in the previous figure. The isotropic chemical shift of the 29Si NMR signal depends in addition on the four Si-O bonding lengths and/or on the four Si-O-Si angles i, which occur between neighboring tetrahedra. Correlations between the chemical shift and the arithmetical mean of the four bonding angles i are best described in terms of

The parameter describes the s-character of the oxygen bond, which is considered to be an s-p hybrid orbital. For sp3-, sp2- and sp-hybridization with their respective bonding angles = arccos(1/3) 109.47°, = 120°, = 180°, the values = 1/4, 1/3 and 1/2 are obtained, respectively. The most exact NMR data were published by Fyfe et al. for an aluminum-free zeolite ZSM-5. The spectrum of the low temperature phase consisting of signals due to the 24 averaged Si-O-Si angles between 147.0° and 158.8° (29Si NMR linewidths of 5 kHz) yielded the equation for the chemical shift

1coscos

44.216.287ppm Take away message from this page:

Si-O-Si bond angle variations by a distortion of the short-range-order in a crystalline material broaden the 29Si MAS NMR signal of the material.

Determination of the Si/Al ratio by Determination of the Si/Al ratio by 2929Si MAS NMRSi MAS NMRDetermination of the Si/Al ratio by Determination of the Si/Al ratio by 2929Si MAS NMRSi MAS NMR

For Si/Al = 1 the Q4 coordination represents a SiO4 tetrahedron that is surrounded by four AlO4-tetrahedra, whereas for a very high Si/Al ratio the SiO4 tetrahedron is surrounded mainly by SiO4-tetrahedra. For zeolites of faujasite type the Si/Al-ratio goes from one (low silica X type) to very high values for the siliceous faujasite. Referred to the siliceous faujasite, the replacement of a silicon atom by an aluminum atom in the next coordination sphere causes an additional chemical shift of about 5 ppm, compared with the change from Si(0Al) with n = 4 to Si(4Al) with n = 0 in the previous figure. This gives the opportunity to determine the Si/Al ratio of the framework of crystalline aluminosilicate materials directly from the relative intensities In (in %) of the (up to five) 29Si MAS NMR signals by means of the equation

4

0

400Al

Si

nnnI

Take-away message from this page:

Framework Si/Al ratio can be determined by 29SiMAS NMR. The problem is that the signals for n = 04 are commonly not well-resolved and a signal of SiOH (Q3) at about 103 ppm is often superimposed to the signal for n = 1.

2727Al MAS NMRAl MAS NMR2727Al MAS NMRAl MAS NMR

0 10 20 30 40 50 60 70 80 90 100 10 110 120 ppm

aluminates

aluminosilicates

aluminoborates

aluminophosphates

aluminates

aluminosilicates

aluminoborates

aluminophosphates

aluminates

aluminosilicates

aluminoborates

aluminophosphates

aluminosilicates

3-f

old

co

ord.

4-f

old

co

ordi

nate

d

5-f

old

co

ordi

nate

d

6-f

old

co

ordi

nate

d

20

2727Al MAS NMR shift and Al-O-T bond angleAl MAS NMR shift and Al-O-T bond angle2727Al MAS NMR shift and Al-O-T bond angleAl MAS NMR shift and Al-O-T bond angle

Aluminum signals of porous inorganic materials were found in the range -20 ppm to 120 ppm referring to Al(H2O)6

3+. The influence of the second coordination sphere can be demonstrated for tetrahedrally coordinated aluminum atoms: In hydrated samples the isotropic chemical shift of the 27Al resonance occurs at 7580 ppm for aluminum sodalite (four aluminum atoms in the second coordination sphere), at 60 ppm for faujasite (four silicon atoms in the second coordination sphere) and at 40 ppm for AlPO4-5 (four phosphorous atoms in the second coordination sphere).

In addition, the isotropic chemical shift of the AlO4 tetrahedra is a function of the mean of the four Al‑O‑T angles (T = Al, Si, P). Their correlation is usually given as

/ppm = -c1 + c2.

c1 was found to be 0.61 for the Al-O-P angles in AlPO4 by Müller et al. and 0.50 for the Si-O-Al angles in crystalline aluminosilicates by Lippmaa et al. Weller et al. determined c1-values of 0.22 for Al-O-Al angles in pure aluminate-sodalites and of 0.72 for Si-O-Al angles in sodalites with a Si/Al ratio of one.

Aluminum has a nuclear spin I = 5/2, and the central transition is broadened by second-order quadrupolar interaction. This broadening is (expressed in ppm) reciprocal to the square of the external magnetic field. Line narrowing can in principle be achieved by double rotation or multiple-quantum procedures.

/

2277Al MAS NMR spectra Al MAS NMR spectra of a hydrothermally treated zeolite ZSM-5of a hydrothermally treated zeolite ZSM-5

2277Al MAS NMR spectra Al MAS NMR spectra of a hydrothermally treated zeolite ZSM-5of a hydrothermally treated zeolite ZSM-5

L = 195 MHz

Rot = 15 kHz

/ ppm

60 40 20 0 20 40 60 80 100

L = 130 MHz

Rot = 10 kHz

four-fold coordinated

five-fold coordinated

six-fold coordinated

Take-away message:

A signal narrowing by MQMAS or DOR is not possible, if the line broadening is dominated by distributions of the chemical shifts which are caused by short-range-order distortions of the zeolite framework.

some topics of solid-state nuclear magnetic resonance

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external magnetic field

g iz g

magnetic dipole

h nmr spectrum

nuclear magnetic moment

nuclear spin properties

nmr nomenclature

si nmr