CHAPTER 1

Nuclear Magnetic Resonance (NMR) has proved itself to be a powerful, precise

and sensitive method for the non-destructive study of matter. i n solids, for example,

making use of appropriate nuclei, one can probe into a variety of interactions to obtain

Information regarding the structure, internal motions, phase transitions etc.. Basic

princlples of NMR are discussed in dotail, among others, by ~ n d r e w ' , slichter2 and

Abragam 3.

As this thesis is concerned with NMR In diamagnetic insulating solids, the theoretical background relevant to the investigations described in the later chapters

of thls thesls is briefly reviewed in this chapter. An introduction to tho basic concepts

of resonance and relaxation are given in section 1.1 . Section 1 ,2 . describes the nature

of the dipolar interaction and its effects on NMR signal shapos. A brief Introduction

to spin lattice relaxation is given in section 1.3. Spin-rotation Interaction in solids Is

discussed in section 1.4. Modification of tho line shape due to tunneling at low

temperatures is described in section 1.5.

1.1. RESONANCE AN11 RELAXATION

Many atomic nuclei which possess an .intrinsic non zero spin angular

momentum and also a magnetic moment 11 = y.t\ I whero ! is the spin angular momentum, y is the magnetogyric ratio and fi is the Dirac constant. If an assembly

of non-interacting nuclear spins aro subjected t o a magnetic field, the lnteraction

hamiltonlan can be written as

I f the magnetic field is in the Z-direction i.e., H = Ho k

then,

This lnteractlon results in (21 + 1) enerDy lovels, corresponding to the (21 + 1)

eigen values of the operator Ir , with onergies given by E = y .IS H e m where

m = - 1 , - I + 1 , ....... + I

For the case of I - 112 ( e.g. 'H,"F etc.,) we have m - t 112, and

In a macroscopic sample containing such nuclei, the interaction of the spins with

one another and with the other degrees of freedom (referred to as the lattice ) brings

!he spin system into thermal equilibrium among themselves and wlth the surround.

ings. The populations of the Zeeman levels are then governed by the Boltzmann distribution at the temperature of the lattice. i.e., the lower energy levei Is more

populated than the higher energy levei. The application of a radio-frequency (rf) field

H i cos w t in a direction perpendicular to Ho,and with a frequency wo = y HO causes

transitions between adjacent Zeeman levels (selection rule A m - .c 1 ) . The transition probability due to the rf field is proportional t o I c m I m >I* Hence the

excess population in the lower level results in a net absorption o f rf energy by the spin

system with the resonance condition,

For observing the absorption signal contlnuously, the strength of the rf field

should be small so that the rate at which the spin system absorbs rf energy should be

small compared t o the rate at which the spin system can exchange its excess energy

to the lattice. Otherwise 'saturation' occurs resulting in a fall in the signal intensity.

This process by which the spin-system transfers its energy t o the lattice is called the

spin-lattice relaxation and the time constant governing this process is called the

spin-lattice relaxation time (Ti). T i can also be defined awthe time required for the

unmagnetized specimen to acquire the total magnetization after the magnetic field is

applied. The spin-spin relaxation time Tn characterizes the process which destroys

any magnetization that may be present in a plane perpendicular t o Ha. This is due to

the mutual interaction taking place among the spins themselves and causes the spin

system t o attain an internal thermal equilibrium (common spin temperature as different from lattice temperature). The spin-spin interactions endow the NMR line

with a width, shape and structure.

Considering a spin 112 system in the steady state, i f no and n denote the

equilibrium population differences between the two Zeeman levels at time intervals

1-0 and t - t l respectlvely, then they are related by the equation

where W is the probability of transitions induced by the rf field. The two

population differences are equal (i.e.,n =no), when 2WT1 < 1 . We know that W a HI,

where HI is the amplitude of the rf field. Therefore one has to maintain sufficiently

low rf levels t o avoid saturation. In practice, for diamagnetic solids, TI can be as

small as milliseconds and as large as hours and it is temperature dependent.

1.2. DIPOLAR INTERAC'I'ION

1.2.1, Dipolar Hamiltorlinn

The NMR line shape for a system of spin 112 nuclei in diamagnetic solids is

essentlally governed by the nuclear dipole- dipole Interactions. This Interaction gives

rise to a local fleld HIOC of the order of one gauss for a typical interspin distance r = 2 A

and !c = los2= ergslgauss for protons.

The dipolar interaction& can be treated as a perturbation on the Zeeman dp

hamiltonian. The total hamiltonian is

The Zeeman harniltonian is 8SZ = y f i Ho Is where the summation runs over all F ' 0 the spins in the solid, with the energy levels glven by Em =+I wo = -y RHoM

j where M = j m is an eigen value of IZ = 2 j I t

Therefore wo = yHo

In general the dipolar hamiltonian for N spiris has the form

Considering a pair of spirrs dcliolod by I , arid ik , if 0 atid 9 aro Ilio polar

coordinates of the vector rik dcscribirig llrcir relalivo positions, llio Z axis b o i n ~ parallel to the applied ficld (Fig. 1 . I . ) , djh con be wriltoi as

Y Ikhz [=,Ik- 3{ IIzCos3 + 1/2 Sinb (I,+= - i~ + 1 1 - e +id ),

T - ic ~k Ik-Case+ - 1/2 sine ( T k + e + Ik? +i"} ]

Where A = I Ikz (1-3~0s~") 1:

Here I + and I. aro tho laddcr operators, C' and E' sro the t h e Hcrmilian

conjugates of C and E rospectivcly.

Fig. 9.9 Diagram illustrating the frae rotation of a pair of identical spins about a common a x i s .

In perturbation calculation of the nuclear energy levels only the terms A and B of equation (1.9a) which commute with Ez will contribute in l irst order. Therefore

only the 'truncated' part of # is used. dp

Tho terms C.D,E and F which p rod~~cc ! vory wcnk rosponsos at I , J = 0 and t t j = 2111(-,

are neglected in the first order. OC . gives rise to different local fields Hloc at the dp

sites of different spins. This results in a spread of resonance frequency, resulting In

the broadening or spiltting of the NMR absorption line, Through Its dependence on

the Inter-spin space variables the dlpolar broadening is a very useful source of information regarding structure and dynamics In sollds.

pake4 was the first t o study the effect of dipolar interaction on NMR signal line

shapes for a pair of spins. Later Andrew e l als studied the problem for three spins (at

the vertices of an equilateral triangle) and ~ u t o w s k ~ ~ , Bersohan and ~ u t o w s k ~ ' and

loth et ale analyzed the problem for a group of four spins arranged in a tetrahedral

configuration. I t was found that line shape analysis for systems conlalnlng groups

wlth more spins was complicated. However i t was possible t o go beyond a qualitative interpretation by using Van Vleck's method of moments9.

1.2.2. The Method of Mot~rcrits

The method of moments developed by Van Vleck is very useful In that they can be calculated from first principles without having to find the eigen states of the total

hamiltonian. For a resonance curve described by a normailzed shape function f(w) with a maximum at a frequency wo , the n'h moment with respect t o the polnt wo Is

deflned as .

If f(w) is symmetrical wlth respect to wo and is an even function, all odd

moments vanish. A knowledge of the moments gives Information on the shape of the

resonance curve and in particular o n the rate at which It fails down to zero in the

wings, far away from wo. In principle the line shape function f(w) can be described

either by a Gaussian curve (1 -12) or Lorentzlan (1 -13)

Here A* is the second moment of the curve about the centre and d is the half

width at half intensity.

The experimental line shape corresponds very well t o the Gaussian shape at low

temperatures (rigid lattice case), whereas at high temperatures i t would be Lorentzian

(the case of rapid motion). i n the intermodlate temperatures the experimental line

shape can be best described by a weighted superimposition of these two line shapes.

It should be noted that the total second moment of a line does not change, but only

the observed second moment changes. In view of the equation (1.1 1 ) the second

moment Mp of a NMR line can be written as

*2 =su2 t ( u ) do ; M , = s h 2 f ( h ) dh (1 .14 )

o is taken as the central frequency.

The second moment of an absorption line is simply the mean square line width.

The second moment of an NMR line is completely analogous to the moment of inertia of an object with the same shape of the line''. Since moments of a curve are

proportional to the derivatives at the origin of its fourier transform ,

aD

f ( o ) = AS G ( t l . Cos wt.dt 0

where f(w ) is the normalized shape function, A is the normalization constant and

G(t) is an even function defined as

G ( t ) = (2/nA) S f ( w ) Cos ot .dw ( 1 . 1 6 )

G ( t ) = Trace (JHM(t) , Ax)

G ( t ) = Trace ( e iiXt/h

j

(1.17)

6

1, which a 1 The total hamlltonian is 0( - 91, + RAP,

w h e r d represents the truncated part I.e., only A and B terms are taken as they dp

commute with TC, . Hence G(t) can be rewritten as

Further .-

e i t A e i t = ~ ~ c o s o ~ t + r s i n w,t Y

s i n ,t ~r (,iRtip,t hX e-iRdpt 5 )

The second term vanishes since a rotation of the spins through 180' around, say, leaves and Ax unchanged but turns& into 'Av.

dp Y

Hence G(t) is replaced by Gl(t) called the reduced auto- correlation functlon

In turn the various moments of the absorption line can be got from Gi(t) using

the relation, a

Here Gl(t) is expanded in powers of t, the coefficients of the expansion being

traces of operators which are polynomial functions of g and A( at, X

Therefore, we obtain

1

Using only the truncated part of Xdp (1.10), and evaluation of the traces In the above expression In terms of the coordinates of the nuclei, the expression( 1.22)ieads

to the Van Vieck expression for the second moment for a system of identical spins

as

and for a system containing unlike spins S, the contribution to the second moment

of the nuclei of spin I, is

The total second moment of the resonance line of spins I Is given by adding the second moment contribution from equations (1.23) and ( 1 . zr , ) . For a polycrystailine

systam, rho isotropic avuri lgo of (1 .3c0s2 0 ) yiolds o lac lor of 415. Considering Ilrc

contr ibut ions from ldenricai spitrs olonc, vrc hnvc

However experimonlal ly thc .sccot~d momcnt M2 of tho absorpt ion l ines oro obtaincd by numcricnl i l ~ l cg ra l ion of l l l c ob tc rvcd first dorivalivcs using I l l0

expression,

wl\orc h IHO-H', I l ~ o doviirtlotr Iron, IIN! ccnlro and f4(h) is llro hmplltudo of l l lo

derivative signal at h . T'lre intcgrnl is cv t l l i~ntcd ns n surrr over a largo nurnbcr of values

for t i , obtained by d i v i d i r i ~ t l ic expcrirnclltil l spcclrum into a largo riumbcr of equally

spaccd segments. A coti ipi l tcr prnrJr:llrl trncd for this purposc is givclr in

Appendix A.

I t is vdell known that, in niarly solids vlltcrc t l ic nioleculos have syrntnctric Oroups

l iko NH3, CH3, NHq, N(CH3).1 c t c . , l l ~ c r o i s co~~s idornb lw r i l o l i o l ~ of ~ l ~ o l c c u l o s or sub

molecular groups about prcfcrrcd nxcs. S t rc l~ a ~ ~ l o l c c u l a r rnoliorr cal l l lnvc profound

cffoct on tho l ine strnpo and I l lc obscrvctl s c c o ~ ~ t l 1110lllc11t. I I 1110 spil ls arc in rclatlvc

motion l h c n tho local dipolar f icld c x p o r i c ~ ~ c c d by o riuclcus duc to its ncighbours

l l u c t u ~ t c s in l imo and il 1119 IIIOICCUICI~ IIIUI~OII i s fastor l l ln t i Iho l ine width of 1110

resonancel ine, then narrov~ir ig of t t ~ c rcsor1:irtcc lilro occurs. It can also bo oxprcsscd

as

v ~ h c r c tt lc bar indicalcs crrr nvcrogc lal\crl ovcr all l l l c spins of tlrc sonlplc and3 is

tlrc correlation frcqucncy o l tlrc f l uc t i~n t io~rs in t i ic local liclcl arid (YAH ) i s tlro square

root of the aecond moment which is a measure of the line broadening. The presence

of motion modifies equation (1.18) as

The calculation of the average quantity on the right hand side of the above

equation is in general quite compiicated. We consider a simple situation where the two spin j and k are rotating about a fixed axis ON making an angle 0' with HO and

an angle yjk with r jk . Bjk is the angle between rjk and HO (Fig. 1.1 .). In such a

case rjk is constant and Ujk change with time. Using the addition theorem of

spherical harmonics,

For a polycrystaliine material the above equation may be averaged over @'giving

a value of 415.

4 2 2 - 3 y h I ( I+1 ) lk '2 (powder) - (1.31)

A case of special interest is when yjk = n I 2 i.e., when the axis of rotation Is

perpendicular to all the inter nucleus vectors. 1 hen,

1 '2 (powder)

r > k ( r:k

Thus under this condition the second moment reduces by a factor of four

compared to the rigid lattice case. in the case of isotropic reorientations i.e., when

the axis of rotation is changing rapidly such that all the values of Vjk are equally probable.

n 2 c (1-3cor2e*) > = < J (1-~COS 3.1 s i n e a e > =o ( 1 . 3 3 )

0

Consequently, the contribution to Mz from interactions within the rotating group

vanishes. I n the case of rotational motion the centre of rotation of each rotati'ng

group, remains constant whereas it varies in the case of translational motion of the spins. I n the latter case not only the intramolecular M2 but the intermolecular Mp also

tend to average to zero giving rise to a sharp line as in liquids. Earlier It was

mentioned that molecular groups such as CH3, NH3, NH4, N(CH3)4 exhibit rotation

and reorientation. The crystal field provides a potential barrier Vo hindering such

motions. Each group exists in different distinct but energetically equivalent equlllbrium posltions separated by the barrler height VO. These rigid rotors move in

potential wells consisting of N equally spaced minima in a 360' rotation, permitted

by the N fold symmetry axis of the group. (Fig.l.2.shows such a potential barrier).

Near the bottom of the weil, torsional oscillator states exist. Approaching the top

of the barrler, the energy levels begin to resemble those of a free rotor and approach

them asymptotically at energies well above the height of the barrier. Bolh the potential barrier and thermal energy determine the nature of the molecular motion. I f the

energy of the reorienting group is greater than V O , then relatively free rotation takes

place. For energies less than Vo, the group may be excited from one torsional state

to another in the same weil. Alternatively a typical reorientationai Jump occurs when

the molecular group is excited from its torsional ground state to a free rotor state, undergoes a ciassical rotation ( as a rigid rotor) and drops into a torsional state In

another but equivalent weil. These reorientational jumps over potential barriers

occur due to thermal energy at elevated temperatures. The third possibility is a

quantum mechanical tunneling. This becomes important at very low temperatures for

symmetric and light (low moment of inertia) molecular groups which are unable to

overcome the small potential barrier by a much smaller available thermal energy.

1.2.4. Theoretical Evaluation of the Sccond Moment

The line shape of an NMR spectrum of a dipolar solid is in general difficult to

calculate theoretically, therefore the Van Vleck formula for the second moment Is

usually used. The direct application of this formula has however, several

disadvantages. The tedious calculations must be carried out for each , of the

orientation of a single crystal and for a number of evenly distributed orientations

sufficiently large t o enable an estimate t o be made of the average over a powder.

Furthermore, It is not clear from the formula how the situation is affected by the

symmetry, how many lndependent parameters one may determine from a set of experimental data, o r how averaging over lattice vibrations may be carried out.

Evaluation of the second moment will become tedious when the cation + under

consideration Is executing different types of motions. Gutowsky,et a16 were the first

t o evaluate M2 theoretically using Van Vleck's second moment formula. They

consldered NH4 halides in general which have a cubic structure. The X-ray data were

used i n their calculation and rigid lattice second moment was in good agreement with

the experimentally obtained values. They divided the contribution t o the second

moment into three parts :- (1) intramolecular (2) intermolecular and (3) Contribution

due to unlike spins, The lattlce sum in this case was done by replacing all the

H-atoms at the nitrogen site in a CsCl structure. In 1959 McCau and ~ a m m l n ~ " * ' * proposed a method which allowed the separate evaluation of the orientational and

structural dependencies of the second moment in rigid solids and found that In

general fifteen quantities are sufficient to describe the structural dependence of the

second moment. O'reilly and ~ s a n ~ ' ~ published a more comprehensive study of the second moment using t l ~ o ovon coniplux sptiericel tiarrnonics upto and including

fourth order. Making extensive use of group theory, they derlved somewhat cumbersome expressions for the symmetry adapted functlon and tabulated the

number of Independent structural parameters for the different crystal symmetries.

~ e r e p p e ' ~ recently extended the procedure to include molecular motion, and

also derlved a convenient formula for the average over a powder. In 1969,

Falaleev et introduced a tensor approach which was later supplemented by

symmetry considerations. Their method is more convenient than the others referred

to but does not Include cases involving molecular reorientations. ~ j o b l o m " has

discussed a general procedure for the calcuiation of the theoretical NMR second

moment In dipolar solids. He has shown that the second moment of a dipolar solid

can be written as a sum of products of real second order spherical harmonic formula.

This sum can be contracted to a quadratic form q 5 q where q depends on the

direction of the magnetic field and s, the second moment tensor, only on the crystal

structure and assumptions made concerning the molecular motions. The effect of

symmetry, rlgld body reorientations and vibrational motion on s are investigated

quantitatively. A formula for the second moment of a powder is also given, This

has been applied to oxallc acid dihydrate. A computer program has been written by Sjoblom and ~ e g e n f e l d t ~ ~ ~ " based on this procedure. This has been successfully

applied by them20q21 In the case of dimethyl ammonium chloride, trimethyl ammonium

chloride and a good agreement between theory and experiment has been found. They

have considered three main models in their experiment : - (a) rigid structure, (b)

rotation of methyl group around the N-C axis, and (c) reorientation of the methyl

groups combined with a reorientation of the whole Ion among the possible eight

orientations.

Andrew et alS2 also have done such calculation in mono, di, t r i and tetra methyl ammonlum chlorides. Their values also agree very well with the experlmental results

indicating that the model proposed (for such solids) is satisfactory. Tsang and

Utton 23 have done similar calculations in the case of T M A C ~ C I ~ . The effect of lattice

vibration on the second moment has been calculated by ~olak* ' . In this case the

vibrational frequencies were evaluated from the second moment data and were

compared with the IR frequencies.

Reynhardt et a12= have theoretically evaluated the Mp values for both proton and

fluorine in NH4HF2 with all possible types of motions and their effect on second

moment. Their model was able to explain the M2 transitions for both 'H and ' g ~

absorptions. Jagannathan et a12= have evaluated M2 values for 'H in ammonlum

acetate. They have used the same method as that of Sjoblom ot a ~ ' ~ ' ' ~ and found a

good agreement between theory and experiment.

1.3. SI'I N LAI'I'ICE HELAXATION

13.1. Introduction to I'ulscd NMR

NMR in solids by using pulsed methods was introduced by or re^^' and ~ahn''.

The Important difference between the pulsed and CW technlquea is that In the latter,

the signal is observed in the presence of the rf field whereas. in the former the

observation Is done in the absence of the rf field which has a perturbing influence on

the spins. The spin system is subjected to a powerful rf pulse, then it is turned of f

and the response of the spin system induces an emf in tho receiver coil by tho freely

precessing resonant spins known as the free il;duction decay (FID). Thus pulsed NMR

is observed in the time domain in contrast to CW NMR'which is observed in the

frequency domain. The relaxation times of the spins can be determined from the

response of the spin system to a chosen sequence of rf pulses. A brief discusslon

regarding the origin of the FID and its relation to relaxation times Is given below.

Let us conslder a spin 112 system in a static magnetic field (Ho) applied along the

2-direction. Due to the torque (mxHo) exerted by the field, the spins precess around

HO with the Larmour frequency w o = y HO (Fig.l.3.). At equilibrium, Mr the net

macroscopic magnetization along the 2-direction is denoted by Mo. This picture can be simplified by transforming into a rotating coordinate system ( X ' , Y 9 , 2') rotating with

a frequency w = y HO around Z'axis which coincide with the Z axis of the laboratory

frame. I n this frame of reference, the magnetic moment vector looks static (Flg. 1.4.)

and HO vanishes. If a rf field HI is applied along the X' direction the effective field In

the rotating frame Is given by

(a>

Fig. 1.3a Precession of around

i,, due to torque ( x

(b) Fig.1.3b Ptocession of-an ensemble

of magnetic moments of nuclei ( 1 = 1 /2 1 gene mtes a net magnetization Mo in equilibrium.

At resonance, w l y nullifies the effect of t i o and the effective field is HI only. As

a result, in the cotating frame at resonance, the magnetization precesses around HI,

in the Y'Z' plane, with an angular velocity U I = y HI. If HI is applied for a short

duration, tp , then the magnetization would precess through an angle 0 given by

0 = UJ tp = yHttp. An rf pulse whoso duratioti i s such that the magnotization tips

through an angle of 90' is called a n 12 pulse. After the application of such a pulse,

the magnetlzation Is turned through 90' and is along the Y-direction. The pulse for

which the magnetization is turned away from tho Z direction through 180' and allgnod

along the -Z dlrectlon is called a n pulso. This would invert the magnetization.

If H i is turned off at the end of a n 12 pulse, the magnetization which Is initlally

tipped from Z to Y ' direction by this pulse begins to precess around Ho. Due to

spin-spin Interactlons, dophasing of spin vcctors occurs and the magnetization in the

XY'-plane decays. This decaying magnotizatlon will bo precgssing around HO (Flg.1,5.) with an angular velocity w o .: y HO and will induce an emf in a coil placed

in the transverse plane (the XY plane) and is called the free induction decay (FID).

The maximum amplitude of the FID is thus proportional to the magnotization in tho

direction of Ho, prior to the application of the n 12 pulso.

The time constant of decay of the FiD (lransverse magnotization) is the spin-spin

relaxation time (T2). But generally tlio FID duration Is also lnfluonced by (static)

magnetic field Inhomogeneties over tile sample volume which provides additional

spread In the larmour frequencies and cause faster decay of the FID. In such a caso

the duration of the FID is shorter and is ctioractcrizod by T$ Normally T ~ I S less than

T2 and in many practical situations tlio obscrvod FID Indicates bari id not Tz,

The classical picture of precessing m~gnet izat ion gives useful insight into the

spin-lattice relaxation processes also. At the end of a n I 2 pulso, there is no net

magnotization along the 2 ' - axis. Tho spin system exchanges enorgy with the lattice

and starts relaxing towards the equilibrium state, during which process M Z component

develops continuously till i t attains tho thermal equilibrium value M Z = MO (Fig. 1.5.).

This growth of Mz along the HO direction, in general is exponential with a characteristic

time T i , the spin lattice relaxation time.

The phenomenological equations govorning tho growth of magnetization Including

the effects of relaxation have been postulated by loch^' and are given below.

Here Mx, My and M, represent tho instantaneous magnetization alorrg the X, Y and

Z directions respectively, and other symbols have their usual significance.

1.3.2. Effect of Molcculnr hlotion 011 Spin L,ntticc Rc1:~xntion.

Spin-lattice relaxation arises becauso oach spiri experioncos a fluclualirig field at

its site because of the random naluro of tho internal motions. Tho first successful

relaxation theory was proposed by Bloornbergcn, Pound and Purceii (BPP)~ ' for

liquids and can be applied for solids also. If the fourier spectrum of 'this fluctuating

field contains a frequency componcrit at 0 , tho transilions are iriduced botween the

spins and the lattica. Tho interactions alnorig Ihe spins are weak compared to Zeeman

interaction, and hence can bc taken as a porturbalion to the Zeeman interaction. The

hamiltonian for this time-dependent perturbation can be written as a product of lattice

and spin variableslO,

where ~ ~ ( t ) is the lattice part which conlains informalion about the molecular

motion and is the spin part. The iattico variables ~ ~ ( 1 ) is a random function of

time because of molecular motions and is characterized by an auto correlation

function

For a stationary random porturbalion,

where the bar denotes the ensemble average. The strength of the Fourier com.

ponent in the fluctuation, at w o is indicated by the spectral density J (w ) which is

given by

The relaxation rate is proportional to the spectral density at wo, i.e.,

The precise form of the spectral density depends upon the nature of the molecular

motion. Usually an exponential correlation function with a characteristic correlation

time is used to describe these thermally activated molecular motions. Thus,

Fig. 1.6. shows the plot of J(w ) vs w . If r c is long, the motions are slow and

the spectral density is large at low frequencies. If T C is short tho molecular motions

are fast and the spectral density is constant over a wide spectrum of frequencies. The

T I not only depends on the spectral density at oo but also depends on the strength

of the perturbing Interaction. Out of several posslblo Interactions which can couple

spins to the lattice the important one is the magnetic dipole-dipole interaction.

The dipolar interaction hamiltonian discussed in section (1.2.1.) shows that A and

B (for homo nuclear case) are secular and will not cause spin lattice relaxation. The

first term A gives the spread in the larrnour frequency and hence the broadening o f

the resonance lines. The second term B is the spin-flip term giving rise to life-time

broadening. The energy is conserved in this case and this term will not give rise to

relaxation. Only C to F terms will contribute to relaxation of spins. The terms C and

D contribute to the spectral density at oo (A m = 2 I ) , the terms E and F contribute

t o spectral denslty at 200 and cause the simultaneous flipping ol two parallel spins I i i a n d j ( A m - $ 2 )

The final expression for Tr can be written as

The spectral densities at w o and 2tu o are calculated from C and D and E and F

terms respectively and the expression for TI becomes

where r c is the correlation time for the molecular motion under conslderation,wis

the Larmour frequency and C indicates tlie strength of tho relaxation Intoraction.

The above equation can be discusscd under three cases.

Case 1: w7,= < < 1 1 - = 5C 7; ' T1

(1.44)

This region is called the w t~ i l e spectral region. As the temperature is increased,

T C will decrease and hence T i will increase. T i is independent of the oporaling

frequency in this region.

In this region T i is proportional to tho square of the operating frequency. When

the temperature is decreased, r c will increase hence T i will increase.

, w7 ,t 1 Case 111 . This is in between the cases I and II. I t shows a minimum in TI and tho value of

TI depends on the operating frequency. .From the abovo discussion it is seen that TI vs temperature shows a minimum in TI a1 wrc = 0.62.

The correlation times vary with temperature according to Arrhenius law, since

these motions are thermally activated

? C - 7 exp ( E,/RT)

where E l is the activation energy required to overcome the potential barrier and

is equal to the energy separation of tho lowest ground stato and the free rotor stato.

T C O , the pre exponential factor Is largely determined by the life time of the free rotor

statess

The correlation times are determined from cxpcr imental values of TI uslng BPP

equation (1.43). From the Arrhenius law (1.47) tho plot of I n q vs l O O O / T Is obtained which Is Ilnear. The slope of the straigtil line ~ i v e s the magnitude of E., Thus the

temperature dependence of TI gives quantitativa information about rates of motlon over a wide range of froqucncios (1 o6 - 1013 Hz).

When more than one rotational mechanism is present then the correlation tlmes

will be different for different molions which have a proriounced effects on tho observed

relaxation.

1.4. SIBIN RO'I'A'I'ION INI'ERAC'I'ION

This interaction has been found to bo ono of tho dominant relaxation mechanlsms

in gases and liquids for a long ~ i m e ~ " ~ ~ . More recently, it is found to be present in

many s o ~ i d s ~ ~ . ~ ~ also. This arises due l o the interaction batween the nuclear magnotic moment and the molecular magnetic rnomcnl produced by tho oleclronlc

charge distribution in the molecule. This interaction whbn modulated by internal

motions can cause relaxation. The hamiltonian for spin-rotation interaction can bo

described as,

where I and J are tho nuclear spin and angular momentum operator and C is the

spin-rotation tensor, The molecular motion changes tho value o f angular momentum

and modulates the fleld at the nuclear site in a given molecule. This interaction was

first observed In liquids 37. Brown et a13* have advanced a 'transient rotation

model'in which the molecule jumps frorn one orientation to another at random arid

this interaction operates only during the sliort jump time. ~ u b b a r d ~ ~ has proposed an alternate mechanism based on 'rotational diffusion model' which assumes that only during collision with other molecules angirlar momentum changes and remailis

constant in between collisions. The above two models apply for fluids. Till now no

rigorous theoretical formalism exists for spin-rotational relaxation in soiids. How-

ever, with some reservations, the above two models can be applied to solids also.

A comparison of dipole-dipole and spin-rotation interaction reveals some interesting features. For the spin-rotation interaction, it is the fluctuation in the

angular velocity which is important in contrast to the dipole-dipole interaction where

th.e fluctuations in co-ordinates are instrumental in causing relaxation. Both these

processes have got opposite temperature effects. At low temperatures, the

relaxation Is dominated by the dipolar interaction. With increase in temperature, the efficiency of relaxation due to dipolar interaction goes down and TI becomes longer,

whereas at high temperatures spin-rotation interaction becomes a dominant

relaxation mechanism. Thus T i shows a maximum due to these competing mechanisms.

1.5. TUNNELING EFFECTS AT LOW TEMPERATURICS.

The semiclassical theory predicts that at sufficiently low temperatures, the motion

of the molecular groups will be frozen with respect to the NMR time scale and hence

NMR lines should broaden. But there are many experimental observations where

lines remain narrow even down to liquid helium temperatures Instead of attainlng the

rigid lattice v a i ~ e ~ ~ ' ~ ~ . Also spin- lattice relaxation time shows multlple minima in 46,47 this temperature range which could not be accounted for by the classical theory

According to semiclassical theory of relaxation (BPP theory) the changes in TI and

line width are determined by variations in the rate of random thermal reorientations.

As a consequence, there should be an agreement between the various nuclear

resonance observations with regard to their predicted random molecular jump rates,

which indeed is so in the intermediate and high temperature regions but not in the

very low temperature regions. To explain this, quantum mechanical tunneling, has been invoked.

When the activation energy is sufficiently small (weakly hindered solids) the wave

functions of the spins in the different potential wells overlap appreciably. This causes

the tunneling splitting of the torsional oscillator ground states. Then the tunneling

of the protons from one well to another becomes possible and thls leads to narrowing

of the resonance lines. The additional minima in TI, occur when the tunnel splitting

are equal to integral multiples of Zeeman ~ ~ l i t t i n ~ s ~ ~ ' ~ ~ , There are excellent

r e v ~ e w s ~ ~ ' ~ ~ which discuss in detail the effects of tunneling on line shapes and

relaxation.

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