Vij D.R. Handbook of Applied Solid State Spectroscopy

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1. Nuclear Magnetic Resonance Spectroscopy

where it is difficult to distinguish the different isotropic chemical shifts.

Another problem for obtaining spectra from solid samples for rare nuclei (low

natural abundance) with small gyromagnetic ratio, such as

C, is associated

with the low sensitivity related to two problems: small NMR signal and long

spin-lattice relaxation times [8]. In order to circumvent these problems three

methods were proposed: dipolar (homo and heteronuclear) decoupling and

magic-angle spinning (MAS) for improving the resolution of the spectra, and

cross-polarization for increasing the sensitivity.

1.5.1 Dipolar Decoupling

The dipolar decoupling technique is used to suppress the magnetic dipolar

interaction between nuclear spins. When observing rare

C nuclei, the

homonuclear dipolar interaction (

C-

C) is negligible due to their small

natural abundance and the only important dipolar interaction comes from the

abundant

H nuclei. To suppress the

C-

H dipolar interaction, the method

called heteronuclear decoupling is used and was proposed by Sarles and Cotts

in 1958 [13]. It consists in the continuous wave (CW) irradiation with rf at the

H resonance frequency while observing the

C signal. In order to effectively

suppress the

C-

H dipolar interaction, it is necessary to apply rf

B fields

satisfying the condition

1dip

ȖȞB !' , where

dip

' is the line width due to the

heteronuclear dipolar interaction. In the cases of residual line widths arising

from insufficient proton decoupling power, a simple two-pulse phase

modulation (TPPM) scheme greatly improves the quality of the spectra [14].

In the case of homonuclear dipolar interaction, normally

H-

H dipolar

interaction, the method employed is called homonuclear decoupling. There

are several ways to suppress this interaction and the most common are those

based on multiple pulses, such as WaHuHa [15], MREV-8 [16, 17], and BR-

24 [18], or on the Lee-Goldburg method [19].

1.5.2 Magic-Angle Spinning (MAS)

Because this method is of fundamental importance for the advanced

techniques to be presented later in this chapter, the magic-angle spinning

technique will now be discussed in more detail. In 1959 Andrew et al. [20]

and Lowe [21] proposed, independently, this method to suppress the magnetic

dipolar interaction. In 1962, Andrew and Eades [22] showed that MAS could

also be applied to eliminate other anisotropic interactions. To introduce this

method, let us take as an example the magnetic dipolar interaction between

the nuclei

C and

H. The dipolar magnetic field produced by

H along the

static magnetic field on the

C site is given by



 

dip

µ3cosș 1

Br|,

where r is the modulus of the distance vector connecting both nuclei, and

ș is

1.5 High Resolution Solid State NMR Methods

the angle between r

and

. The term





3cos ș 1 describes the anisotropy

of the dipolar interaction. If this term were equal to zero, the dipolar

interaction would be eliminated. One way to do this is to choose

șș 54.74 |

, where

ș is called the magic-angle. Of course, it is

impossible to put all the spin pairs aligned along this specific orientation,

except in very special cases. The method proposed to reach this condition

consists in spinning the sample around the magic-angle orientation with a

rotation frequency

dip

!' . In this situation, all the internuclear vectors r

will be, on the average, along the magic direction resulting in an average

dipolar field







dip m

µ3cosș 10

Br |. The same procedure can be

applied for suppressing other nuclear spin interactions but, in the general case,

it is necessary to use the orientation of the principal axis of the interaction

tensor, and the rotation frequency should be

Ȟ,

!' where

' is the line-

broadening introduced by the specific interaction. In the case of

C-

dipolar interaction,

dip

' is in the range of 1 kHz to 100 kHz. For

chemical shift anisotropy, since this interaction depends on the intensity of the

magnetic field, choosing, for example, B

| 10 T, 'Q

csa

is in the range of 1

around 25 kHz. In this situation, it would be relatively easy to suppress the

C chemical shift anisotropy from the

C spectra, but it would be difficult to

suppress the

C-

H dipolar interaction. For suppressing the

C-

H dipolar

interaction, it is necessary to apply both heteronuclear decoupling and MAS.

In the cases where it is impossible to spin the sample with

!' , spinning

sidebands will be observed in the spectrum. This topic will be discussed in

detail after equation (1.69).

Let us now analyze MAS using the general secular Hamiltonian [12]:



sec

0,0 0,0 2,0 2,0

0,0 2,0

33cosș 11

3 Ĳį Șsin școs 2 .

222

HCTRTR

CT CT



ªº



  I

«»

¬¼

(1.62)

In order to introduce the effects of spinning the sample around the magic-

angle [12], we have to express the

lm,

tensor in a coordinate system

associated to the rotor, Figure 1.1, where its components will be denoted by

l,m

Ȟ .

In Figure 1.1 three reference systems can be observed: the laboratory

system (

,,xyz), the rotor system (RS) (

RS RS RS

,,xyz), and the temporary

system (1, 2, 3), the latter representing the transition between laboratory and

rotor systems. To connect the laboratory and rotor systems we have to employ

the Wigner rotation equation:

kHz to 10 kHz. For usual magic-angle spinning systems, the maximum Q is

1. Nuclear Magnetic Resonance Spectroscopy



,', ,'

,Ȥ, Ȟ

lm m m lm



<)

(1.63)

Because

sec

is invariant under rotations around

, the Euler angle < ,

can be chosen equal to zero. The angle between the rotor spinning axis

and

, F will be chosen later as the magic-angle. The angle ) represents the

instantaneous phase

Ȧ t , during the rotation of the sample with frequency

around z

Figure 1.1 Laboratory, temporary, and rotor coordinate systems.

Taking all these assumptions, the secular Hamiltonian can be written as [12]









sec 2

0,0 0,0 2,0 ',0 r 2, '

0,0 0,0 2,0 2,0

ȦȦ 2Ȧ 2Ȧ

2,0 2,1 2, 1 2,2 2, 2

Ȟ 0, Ȥ,ȦȞ

Ȟ 3cos Ȥ 1 Ȟ

sin2ȤȞ Ȟ sin ȤȞ Ȟ .

rr r r

it it it it

HCT CT t

CT CT

CT e e e e







ªº

 

«»

¬¼

ªº



«»

¬¼

(1.64)

Now,

l,m

Ȟ can be diagonalized by a new Wigner rotation to its principal

axes system (PAS), where its components will be denoted by

,lm

U and written

in the following way:



ȞĮ,ȕ, Ȗȡ

mm lm

ml 



(1.65)

The angles

Į , ȕ , and Ȗ are the Euler angles connecting PAS and RS.

Using the equation above we have [12]







sec 2

0,0 0,0 2,0

2,0

ȡį3cos Ȥ 1

1 Ș 3

3cos ȕ 1sinȕ cos 2Ȗįȟ,

22 2

HCT CT

CT t

 

ªº

u 

«»

¬¼

(1.66)

where



1r1r2 r2 r

ȟ cos Ȧ sin Ȧ cos 2Ȧ sin 2Ȧt C tS tC tS t  

(1.67)

and

1.5 High Resolution Solid State NMR Methods



sin 2Ȥ sinȕ cosȕȘcos 2Ȗ 3cosĮȘsin 2Ȗ sin Į

C ª º

¬¼

(1.68)



sin 2Ȥ sinȕ cosȕ 3 Școs 2Ȗ sin ĮȘsin 2Ȗcos Į

S ª º

¬¼



22 2

13Ș

sin Ȥ sin ȕ cos 2Ȗ 1cosȕ

222

½

ªº



®¾

«»

¬¼

¯¿



22 2

13Ș

sin Ȥ sin ȕ cos 2Ȗ 1cosȕ sin2ĮȘcosȕsin2Ȗ cos 2Į .

222

½

ªº

  

®¾

«»

¬¼

¯¿

From equation (1.66) it can be observed that when the sample is spun

around the magic-angle (

Ȥș ) with frequency

Z , then



sec

0,0 0,0 2,0

ȡįȟ

HCT CTt 

(1.69)

The second term of equation (1.69) will be suppressed only if

Ȟ!' . If

this condition is not satisfied, the spectrum will present spinning sidebands.

Before discussing in detail the spinning sidebands, let us illustrate how they

are formed using the

C-

H dipolar interaction to exemplify this

phenomenon. Normally, each individual internuclear vector

is tilted by an

angle

İ relatively to the

z -axis, Figure 1.2a. When spinning the sample

around the magic-angle, the orientation





șȦ

t of the internuclear vector

relative to

will oscillate within the interval





șİ,șİ. Since the

dipolar magnetic field on the

C site depends on





șȦt as





3cos șȦ 1t



it will also oscillate. Therefore, the resonance frequency of each

C will

periodically oscillate with a period

2ʌȦt , returning to the original

frequency every complete turn.

Figure 1.2 a) Sample rotation with the internuclear vector aligned along a direction different

from the rotation axis (z

). b) Schematic representation of magnetization dephasing due to

MAS and rotational echo.

cos 2Į  Școsȕsin2Ȗ sin2Į

1. Nuclear Magnetic Resonance Spectroscopy

In the real case of an amorphous or polycrystalline sample, there will be

C-

H pairs whose internuclear vectors will make all the possible İ angles

with respect to

z -axis. Hence, during the rotation period

t , there will be an

interval of resonance frequencies for each individual

C-

H pair

mín máx

Ȧ : Ȧ ,Ȧ . After the

C nuclei excitation (Figure 1.2b) there will be a

coherent transverse magnetization consisting of the vector sum of all

individual

C magnetic moments, which under MAS will evolve with

frequencies that will oscillate within the specific interval

mín máx

Ȧ : Ȧ ,Ȧ . At

t , the

C magnetic moments will spread out in the transverse plane

resulting in a strong decay in the NMR signal. However, at

t all the

magnetic moments will have the same resonance frequencies they had after

the excitation, resulting in the refocalization of the spins, as in a spin-echo

experiment. This echo effect will occur every complete rotation period,

resulting in a train of echoes called rotational echoes.

When

t is longer than the transverse relaxation time

2ʌȦT ', several

rotational echoes will show up in the detected NMR signal separated by

t .

The respective NMR spectrum, obtained after the Fourier transformation of

this train of rotational echoes, will result not only in the spectral lines

expected for each isotropic chemical shift of the

C belonging to the

molecule, but also in several replicas of them. These replicas are named

spinning sidebands and are equally separated from the real isotropic chemical

shift frequencies by multiples of the spinning frequency

. When

tT , the

rotational echoes disappear and the corresponding spectrum will contain only

the isotropic chemical shift frequencies. This, together with the magic-angle

condition, ensures the suppression of the anisotropic components of the

secular Hamiltonian.

Knowing the nature of the rotational echoes, we now return to the analysis

of MAS using the secular Hamiltonian under the magic-angle condition [12],

now only for the case of chemical shift anisotropy:











sec

cs 0 0

Ȗı Ȗįȟ

HBIBtI  == .

(1.70)

In this case, each individual

C nucleus i will present a chemical shift

resonance frequency given by





ȦȦıȦįȟ

iiii

t 

(1.71)

and will evolve in the transverse plane with the following phase:

    

000

ȦȦıȦįȟ

ttt

ii i ii

t t dt dt t dtI 

cc c cc

³³³

(1.72)

One important property of equation (1.72) is that the second integral

vanishes at each rotation period, which means the total accumulated phase

due to the anisotropic internal interactions is zero for one rotation period

(Figure 1.2b).

1.5 High Resolution Solid State NMR Methods

  

aniso aniso

Ȧ 0

ttdtI

(1.73)

This condition can be perturbed by external interactions, like rf pulses or

molecular rotations, making the MAS spectrum susceptible to spin

manipulation or motion effects.

1.5.3 Cross-Polarization (CP)

As already discussed, it is experimentally difficult to obtain spectra from solid

samples for rare nuclei (low natural abundance) or with small gyromagnetic

ratios, such as

C. The low sensitivity is associated to two problems: very

small NMR signal and long spin-lattice relaxation times

T [5, 8]. In order to

circumvent these problems the cross-polarization method was proposed [23].

This method is based on the transference of polarization from the abundant

spins, with short spin-lattice relaxation times

T , to the rare

C nuclei. After

this polarization transfer, the rare nucleus signal intensity is increased by a

factor

abundant rare

ȖȖ when compared with the excitation with a ʌ 2 pulse. In the

case of

C and

H this factor is about 4. Despite the fact that the NMR

experiment is performed for the

C nuclei, the repetition rate for signal

averaging is determined by the short

T , because the

C magnetization is

now defined by the

H nuclei. In order to establish an efficient contact

between

H and

C nuclei for the polarization transfer, it is necessary to

satisfy the Hartmann-Hahn condition

H1H C1C

ȖȖBB . This means that the rf

fields have to be applied simultaneously in such a way that the precession

frequencies of both nuclei around the respective resonant rf fields are the

same. As a result, a resonance exchange of energy between

C and

H can

readily occur through a mutual spin flip mechanism [8].

1.5.4 The CP-MAS Experiment

The combination of heteronuclear decoupling, MAS, and CP techniques in

only one experiment was proposed in 1976 by Schaefer and Stejskal [24] and

marked the birth of high resolution solid state NMR spectroscopy for rare

nuclei. Figure 1.3 shows the procedure to perform this clever experiment.

To summarize the experiment, one applies S/2 pulse to

H, which is

followed by a change of 90q in the phase of the

H rf field relative to the first

pulse leading to the spin-locking of the abundant spin system. Still under the

spin-locking condition, which was used to lower the

H spin temperature

much below the lattice temperature, the Hartmann-Hahn condition is

established, and this situation is kept until the polarization transfer from

H to

C spins is complete. After that, the observation of the

C FID is performed

under heteronuclear decoupling. In order to increase the signal-to-noise ratio

1. Nuclear Magnetic Resonance Spectroscopy

C FID, the CP-MAS experiment is repeated as many times as it is

necessary for the total

H spin-lattice relaxation between acquisitions to occur.

MAS is continuously applied during the experiment.

Figure 1.3 CP-MAS pulse sequence. The sample is under MAS during the entire experiment.

1.5.5 NMR Spectra

It can be observed that the secular Hamiltonians obtained above (equation

(1.28)) are independent of rotations around the static magnetic field

BBz

since the Hamiltonians are invariant under rotations around the principal axis

of the laboratory frame. However, the same Hamiltonians depend, in general,

on both T and I. T



is the angle between the principal axis of the PAS and the

principal axis of the laboratory frame (which corresponds to the direction of

the static magnetic field). I is the rotation angle around the principal axis of

the PAS and is important only when

Ș 0z

. Therefore, it is fundamental that

one be acquainted with the orientation of the interaction tensor relative to a

molecular frame so one may analyze the Hamiltonians and the corresponding

spectra.

The first term of the secular Hamiltonian is independent of the orientation

of the PAS relative to the laboratory frame and, due to this, is called the

isotropic term. This term is different from zero for the chemical shift

interaction and equal to zero for dipolar and quadrupolar interactions. The

second term, which depends both on



T and I, is known as the anisotropic

term of the interaction, and contains all the necessary information for studying

dynamics and conformation in solid materials.

The measurement of the tensor parameters

Ĳ , į , and Ș , as well as the

orientation angles T and I is one of the main purposes of NMR spectroscopy.

There are several important factors governing the NMR experiment to access

these parameters:

x the relative intensity of the internal Hamiltonians compared to the

Zeeman Hamiltonian;

x the relative intensities of the internal Hamiltonians;

x the manner in which the internal Hamiltonians manifest themselves in

NMR spectra;

1.5 High Resolution Solid State NMR Methods

x the natural (molecular motions) or intentional (advanced NMR methods)

averaging of the internal interactions in both ordinary

and/or spin

spaces.

Keeping in mind all this information, we have several ways of

obtaining

the NMR spectra:

x High resolution NMR in liquids:

Due to the fast Brownian molecular

motion, the anisotropic terms of the internal Hamiltonians are averaged out

and the spectra are composed of narrow lines and defined only by the

isotropic terms.

x Wide-line NMR in amorphous or polycrystalline solids

: Due to the

anisotropic components of the internal Hamiltonians, which contribute

differently for each tensor orientation, the spectra are very broad due to the

superposition of the NMR lines (powder pattern spectra, Figure 1.4).

x High resolution NMR in single crystals

: Despite the contribution of the

anisotropic components, the spectra are again composed of narrow lines

because there is only one orientation relative to the external static field for

all the tensors.

x High resolution NMR in amorphous or polycrystalline solids:

Due to the

utilization of advanced solid state NMR methods, such as hetero- or

homonuclear decoupling or magic-angle spinning, the anisotropic terms of

the internal Hamiltonians are averaged out and the spectra are composed

of narrow lines and defined only by the isotropic terms.

Figure 1.4 Powder-pattern spectra: Chemical shift under a) cubic, b) axial





Ș 0 , and c) non-

axial



Ș 0z symmetries; d) dipolar interaction between two spins 1/2 distant from each other

of fixed distance

r ; and e), f) quadrupolar interaction for spins 1 and 3/2.

1. Nuclear Magnetic Resonance Spectroscopy

1.6 PRINCIPLES OF TWO-DIMENSIONAL

SPECTROSCOPY

There is a wide variety of NMR methods based on the time acquisition (t

) of

a series of FIDs, where each one is acquired with a different value of a time-

dependent variable evolving during a second period of time (t

) before

detection [9]. This has been termed two-dimensional (2D) spectroscopy and

was first suggested in 1971 by Jenner [25] and implemented in 1976 by Aue,

Bartholdi, and Ernst [26]. Most of these experiments include an additional

time interval before FID detection, the mixing time (t

) [27]. The general

pulse sequences for obtaining 2D spectra are illustrated in Figure 1.5. In both

cases, a and b, the preparation period involves, for example, the generation of

transverse magnetization through a S/2 pulse or cross-polarization procedure.

During the evolution period t

, the magnetization evolves under a specific

spin interaction, and during the FID detection period t

, the magnetization is

observed evolving under the action of the same or another spin interaction. To

detect the 2D time signal S(t

), taking into account the magnetization

evolution during t

and t

, a series of experiments must be performed with a

systematic variation of the evolution time t

, by adding increments 't

and

acquiring the corresponding FIDs during t

. These FIDs acquired as a function

of t

will have amplitudes or phases modulated by the interaction acting

during the evolution period. As a consequence, S(t

, t

) will contain the

information of the frequencies defined by the interactions independently

acting during t

and t

. The 2D Fourier transform of S(t

, t

), namely,

S(Z

, Z

), will be the probability map of finding a frequency Z

in the first

period and a frequency Z

in the detection period [7]. In order to perform the

double Fourier transformation and get 2D spectra without phase distortions,

some special procedures have to be performed, for example: the off-resonance

technique [7], the time-proportional-phase incrementation method (TPPI)

[28], and hypercomplex data acquisition [29]. More specific interpretations of

the spectral intensity will be given for each 2D experiment described in this

chapter [7].

Figure 1.5 General timing of two-dimensional NMR experiments.

In the mixing time period, which is not mandatory for all 2D experiments,

one can employ suitable pulses for effecting coherence transfer, monitor

cross-relaxation, or chemical exchange, etc. [30].

1.7 Molecular Dynamics and Local Molecular Conformation in Solid Materials

1.7 MOLECULAR DYNAMICS AND LOCAL

MOLECULAR CONFORMATION IN SOLID

MATERIALS

Molecular dynamics and local conformation have important effects on

mechanical, transport, and optical properties of polymers [31], activity of

proteins [32], stability of pharmaceuticals [33], transport properties in zeolites

[34], behavior of amorphous materials near the glass transition [35,36], ion

transport in organic and inorganic ionic conductors [37–39], and other

structural properties of organic and inorganic systems [40, 41]. Solid state

NMR provides powerful techniques for elucidating details of segmental

dynamics and local conformation in solid materials [7]. NMR methods allow

the study of dynamics occurring in a wide range of frequencies with a high

degree of detail. Fast dynamics (~100 MHz) can be characterized to some

extent by NMR relaxation time measurements [6]. Dynamics occurring within

the kHz frequency scale can be studied by line shape analysis [42] or dipolar-

chemical shift correlation methods [43]. Slow dynamics (1 Hz – 1 kHz) can

be studied by the so-called exchange NMR experiments, where relatively slow

segmental reorientations are observed in terms of changes of orientation-

dependent NMR frequencies [7]. Concerning structural studies, NMR also

provides a series of methods capable of producing reliable measurements of

torsion angles between localized sites, short and intermediate range structure,

as well as interatomic distances [44–48]. In the following sections some of

these methods and their applications for studying molecular dynamics and

local molecular conformation will be discussed. Because NMR relaxation

methods have been extensively reviewed and there are many specialized

books that address this topic [6, 49], they will not be discussed in this chapter.

We will concentrate our attention on the NMR methods capable of yielding

amorphous solids

(mostly organic solids).

1.7.1 Line Shape Analysis

The analysis of the behavior of the NMR line shapes as a function of

temperature is one of the most traditional procedures for probing molecular

dynamics using solid state NMR. To provide a better understanding of this

statement, let us remember that only under static conditions the NMR

Hamiltonians can be written as in equation (1.28), which results in the NMR

powder patterns shown in Figure 1.4. In other words, if there is molecular

motion, regular powder patterns will only be observed if its correlation time W

is much longer than the inverse of the anisotropy parameter of the spin

interaction, i.e.,

>> 2S/G (slow exchange regime). At larger motional rates,

i.e., smaller W

, (W

~ 2S/G, the intermediate exchange regime) the NMR

spectrum is not simply defined by the static anisotropic NMR frequency,

information about molecular dynamics and local structure of