Line Shapes in Electron Spin Resonance - Weizmann ... · Charles P. Poole and Horacio A. Farach ......

Line Shapes in Electron Spin Resonance

Charles P. Poole and Horacio A. FarachDepartment of Physics and Astronomy

University of South CarolinaColumbia, SC 29208, USA

1.II.III.IV.V.VI.VII.VIII.IX.X.XI.XII.XIII.XIV.XV.XVI.XVII.XVIII.XIX.XX.

INTRODUCTIONSHAPES AND MOMENTSLlNEWIDTH ANISOTROPY.CONVOLUTIONSEXCHANGEINHOMOGENEOUS BROADENINGPOWDER PATTERNSAXIAL POWDER PATTERNSCOMPLETELY ANISOTROPIC POWDER PATTERNS ;

SEMIRANDOM DISTRIBUTIONSAMORPHOUS OR GLASSY DISTRIBUTIONSINTERMEDIATE MOTIONS IN SOLUTIONSHINDERED ROTATIONS AND TUNNELINGCONCENTRATION AND TEMPERATURE DEPENDENCEHYPERFINE COMPONENT DEPENDENCEALTERNATING LINEWIDTHSFINE STRUCTURE EFFECTSSPIN LABELS . . .AQUEOUS ENVIRONMENTSTOPICS OMITTEDReferences

.. .162163

.164164166

.168169171173174175

.176178179184185186187

.188189190

I . INTRODUCTION

Most treatments of "static" electron spin resonance(ESR) emphasize the positions and spacings of spectraliines and the interpretation of these quantities in termsof Hamiitonian parameters. "Dynamic" ESR studies areconcerned with the measurement of relaxation timesand the explanation of relaxation processes in terms ofmicroscopic mechanisms. For many systems, theshapes and widths of ESR spectral lines contain a con-siderable amount of information on both static anddynamic aspects of the spin system.

The purpose of the present review is to survey the

types of lineshapes encountered in practice and to showhow to extract from them information on the Hamiitonianparameters and environmental interactions of the par-amagnetic species.

Typical ESR spectra consist of multiplets arising fromthe superposition of individual lines which are Gaussianor Lorentziah in shape. High viscosity liquids tend toproduce Lorentzian lines, dipolar broadening in solids ischaracterized by Gaussian shapes, and exchange nar-rowing causes lines to be Lorentzian in the center andGaussian in the wings. These shapes will be definedanalytically in the next two sections.

If the interactions which produce the ESR spectra aredirectionally dependent, then the appearance of thespectrum will vary with the orientation of the magneticfield relative to the crystallographic axes. If the sample isa powder or a glass, then the spectra arising from var-ious orientations will be superimposed and produce apowder pattern without angular dependence. In low vis-cosity fluids, the anisotropies will largely average out toproduce a spectrum arising from the isotropic parts ofthe interactions. At high viscosities, spectra are complexand only partially averaged.

This review will discuss the lineshapes arising fromthese and other cases. The emphasis will be on survey-ing the different lineshapes that arise in solid and liquidphases, rather than on providing exhaustive coverage ofthe literature. The lineshapes characteristic ofspecialized systems such as triplet states, conductionelectrons, magnetically and electronically orderedsystems, and one and two dimensional cases will betreated in a later review.

Duplication of Bulletin of Magnetic Resonance, in whole or in part by any means for any purpose is illegal.

162 Bulletin of Magnetic Resonance

II. SHAPES AND MOMENTS

An absorption spectral line Y(H) can be characterizedby the values of its various moments

J/ - Ho)" Y(H)dH (1)

where A is the area under the absorption curve

f (2)and Ho is defined as the magnetic field value whichmakes the first moment vanish

~ Ho) = 0

For first derivative lines

(3)

(4)

we have the corresponding expressions

(5)

A = - JZJH - Ho) Y'(H)

Y'(H) dH=0

The second of these expressions is the condition for theproper choice of the baseline, and the fourth is the con-dition for the proper choice of Ho. Analogous expres-sions can be written for second derivative presentation(D-

In addition to its moments, an absorption line has afull width AH,A between the half-amplitude points, and afirst derivative absorption line has a full peak-to-peakwidth AWpp between maximum positive and negative de-flections.as indicated respectively on Figures 1 and 2. It

I — H

GAUSSIANl.ORENTZIAN

Ym

HoMAGNETIC FIELD H

Figure 1. Comparison of Gaussian and Lorentzian absorptioniineshapes.

GAUSSIANLORENTZIAN

Figure 2. Comparison of Gaussian and Lorentzian first-derivative absorption lineshapesr-

is convenient to employ normalized moments,<Hn>/(AA/,/z)

n for the absorption line case, and< Hn>/(AHpp)n for the first derivative line case.

For symmetrical lines, all odd moments vanish and alleven moments exist. For a Gaussian shape

or its derivative

the moments are given by

and

nodd

(6)

(7)

(8)

< Hn> = (n - 1)!! (1/2AAVpp)n neven (9)

where the symbol ( n - 1 ) ! ! = ( n - 1 ) ( n - 3 ) . . . 5 - 3 • 1.A Gaussian shape has the particular parameters

AH% = (2 In 2)'A AAYpp = 1.1776AHpp

A=\ (x/ln 2)» AHViym = 1.0643 HV2 ym

m = 1.0332(AHpp)2 y'm

^= 0.8244 AHppy'm

)2 = 0.2500(AWpp)2


Vol. 1,No. 4 163

For a Lorentzian shape

+ (H -(12)

and its derivative

pp(13)

we have the expressions

i,A= 1.7321 AAV,

(14)

pp

* = | AH* y m = 1.5708 AH*

= 2w

V3

Ym=

Since the shape is symmetrical, all odd moments vanish.The wings of a Lorentzian line converge so slowly that alleven moments are infinite.

Sometimes it is useful to work with a lineshape that isalmost Lorentzian in the center and has a finite secondmoment. One such shape is the cut-off Lorentzian whichis Lorentzian in the range

about the center field Ho and is zero outside this range.All even moments are finite and have closed-form ex-pressions (1,2). Another such shape is the squareLorentzian

Y(H-HO) = , ym

+(w-H0)2r2]2 (16)

which has a finite second moment and infinite fourth andhigher-order even moments (3).

The degree of asymmetry of a resonance line may bemeasured by the magnitudes of its odd moments, and asimple parameter to characterize such asymmetry is theratio of the cube root of the third moment to the squareroot of the second moment, (H3)*/ (H2)'A.Svare(4) cal-culated this ratio for dipolar coupled spins at lowtemperature and obtained a finite value, which indicatesthat the lineshape is not symmetrical.

The shapes of experimental spectra can be conven-iently checked to ascertain if they are Lorentzian orGaussian (1). It has been customary to calculatemoments from experimental spectra by numerical or

graphical methods. Rakos et al (5) have described ananalogue electronic device for the automatic computa-tion of second moments.

Work up to 1967 on the use of moments in analyses ofESR line shapes was summarized by Poole (1). Buluggiuet al (6) discussed the effect of covalence on moments.Wenzel (7) obtained an expression for the secondmoment of ESR lines broadened by strong hyperfine orsuperhyperfine interactions. O'Reilly and Tsang (8) em-ployed lattice harmonics to analyze line shapes. Lin andKevan (9) determined the differences between thesecond moments of trapped electrons in sodium icecondensates with and without deuteration at 9 and 35GHz, and thereby obtained the isotropic and anisotropichyperfine coupling constants of the nearest neighborprotons.

III. LINEWIDTH ANISOTROPY

Van Vleck (10) showed that the width of an ESR ab-sorption line depends on orientation and that its varia-tion with direction depends on the crystal structure. Inmany cases, this width variation is much less than thewidth itself, and hence is not noticeable experimentally.A number of authors (11-48) have studied the angulardependence of the width, and typical results are listed inTable 1.

Sastry and Sastry (39) explained the linewidth aniso-tropy of Cu2+ in a tetramine thiocyanate complex interms of contributions from dipolar {AHM), hyperfine(AHhf), and exchange (Hexch) interactions. Their expres-sion for the linewidth AH is

AHVz = 2 ( 1 - | A H J + AHh2)/ Hexch (17)

For the case of tetragonal symmetry, Sastry and Sastryused the following expression from Bleaney et al (49)

+3cos24>) (18)

where <t> is the angle between the applied magnetic fieldand the tetragonal axis. The dipolar contribution AHdd isalso orientation dependent.

IV. CONVOLUTIONS

When a spectral line is broadened independently byboth Gaussian and Lorentzian effects, then the line-shape is a convolution obtained by integrating theproduct of the lineshapes, equations (6) and (12). If thefull half-amplitude Gaussian and Lorentzian widths, re-spectively, are denoted by GAHvi and LAH,A then the line-shape is g iven by (1)



Table 1. Selected Reports on Lineshape and Linewidth Anisotropies

ParamagneticSpecies

radicalradicalradicalradical

co2-CF2CONH2

BrCV"centersF centerVK centerVO2

2+

vo6Cr3+

Cr3+

Cr3+Cr3+

Cr3+

Fe3+

Mn 2 +

Mn2 +

MnMn2 +

Cu2 +

CuCu2 +

Cu2 +

ErGd

GdGdYb3+

Y

Pu3+,Am4+

Host

irradiated boneTMPDchloranilirradiated Na acetatenaphthaleneCaCO3

trifluoroacetamideKBrO3

ZrO2, HfO2

KCIKCI:LiVO2(NO3)2 • 6H2OV2MoO8

MgOAI2O3(ruby)

AI2O3(ruby)ZnWO4

alumZnWO4

(NH4)2Ni(SeO4)2-6H2OMg(HCOO)2-2H2OSi surface(NH4)2Ni2(SO4)3

Cu(trien)(SCN)2

Si surfaceCu[OC(NH2)NHNH2]CI2

Cu(NH3)2(CH3COO)2

Ag film on NaCILaSb

Y-AI-garnetY-Gd alloyZnS (sphalerite)

Y(C2H5SO4)3 • 9H2O

ThO2, SrCI2

Graph ofAWvs Angle

noyesnoyesyes

yesnonono

yesyesno

noyes

noyesnoyes

no

noyes

yesyesyes

noyesyes

yes

no

Comments

quasirandom distributiondipolar anisotropyanisotropy due to shiftstriplet pairstunnelingAW depends mainly on /W,2

uses jump model77 K, irradiatedexperiment and theoryAH for many samples300 K rapid reorientatione-hopping processstrain-modulated ESRAH± homogenous, AW|

inhomogeneousshape, width, equalitystudies Li+, Na+ charge

compensationlineshape angle-dependentconcentration dependenceAW field-dependent2-dimensional systemMn adsorbedMn2+-Ni2+ cross relaxationwidth due to exchange,

dipolar, and hyperfinecontributions

Cu adsorbedstudies exchange;

determines TC

different sites, weak exchangespace-dependent strainAH=A + H(4-20x2 + 15x4)2

where x = sin 8

explains anisotropyasymmetry due to

stacking faultsAW changes by factor

of approximately 10AWanisotropic

Reference

1112131415,161718192021,2223242526-28

2930,31

3233,343536373839

3740

414243

444546

47

48

Vol. 1, No. 4


165

(In 2)'A

GAHV,) times the integral (19)

e-*zdx

)2 In 2 + (2 (In 2)v* H H° - x]*

where

2 = (20)

authors present tables of phase changes and efficien-cies of collisions for spin lattice relaxation times that arelong and short compared with rc.

Heinzer (68) employed the density matrix theory in theLiouville approximation to derive the following generalfirst derivative lineshape formula for the case of in-tramolecular exchange between many sites

(23)

The integral (19) cannot be integrated in closed form, butPosener (50) has published tabulations of it. Castner (51)evaluated Y(H) in terms of the error-function integraland described how to determine the linewidth ratioLAW* / GAHV Farach and Teitelbaum (52) also showedhow the Gaussian and Lorentzian widths may bedetermined.

The convolution lineshape has been employed to treatthe case of inhomogeneous broadening with narrow Lor-entzian spin packets forming a broader Gaussian-shaped envelope (see Section VI).

Gaussian-Lorentzian convolution lineshapes werealso discussed by Strandberg (53), who gave new math-ematical properties of the integral, and by Stoneham(54), who derived an expression for the relationship ofthe observed peak-to-peak linewidth AHPP to the Gaus-sian and Lorentzian widths. Al'tshuler et al (55) used aconvolution for the profile of the ESR line from localizedspins in metals, and Kemple and Stapleton (56) em-ployed it to explain the shape of Ho3+ in yttrium ethyl sul-fate(YES).

V. EXCHANGE

A number of workers (57-60) have discussed theo-retical aspects of how the exchange interaction in-fluences ESR line shapes. Some of the recent exper-imental data obtained with exchange-dominatedsystems are summarized in Table 2.

Salikhov et al (62) presented a theory and experimen-tal verification involving exchange broadening in dilutesolutions containing free radicals with spin S t = V2 andparamagnetic complexes of transition ions with spin S2

> V2. They assume exchange J(t) to be operative onlyfor the duration TC of a collision

J(t)=J to<t<

J(t) = O t<t0, t

(21)

(22)

where to is the time for the start of the collision. The

where the imaginary part of AJM provides the line position,the real part of 5JM is a measure of the intensity and thereal part of 2AJM/y/3 is the width of the line j^i. Theimaginary part of A^ produces a deviation from a Lorent-zian shape. A computer program provides synthesizedspectra from equation (23). More recently, Heinzer (69)applied a least-squares procedure to the computeranalysis of isotropic-exchange-broadened ESRlineshapes.

Fedders (84) calculated spin-pair correlation func-tions for exchange narrowing using a microscopic der-ivation and a sum-rule moment expansion. The sum-rulemethod gave a frequency dependent linewidth. Theorywas compared with experimental results on Rb MnF3.

Boucher (102) showed that exchange-narrowed linesare essentially composed of the Fourier transforms ofcross-correlation functions, while motionally-narrowedlines contain only the Fourier transform of the self-correlation function.

Myles (103) treated the case of anisotropic exchangeenergy J together with uniaxial anisotropy energy D.They examined the dependence of the dipolar spectralfunction, quadrupolar spectral functions, exchange-narrowed dipolar linewidths and spin-diffusion coef-ficients as a functionof the spin S and the ratio3D2 /16S(S+1)J2 .

Cusumano et al (71,104) studied solid solutions of di-phenyl picrylhydrazyl (DPPH) in polystyrene and foundthat the exchange-narrowed shapes are Lorentzian inthe center and exponential in the wings. The experimen-tal behavior resulted from the spin-lattice relaxation inthis Heisenberg ferromagnet. Lazuta and Maleev (105)studied the dynamics of an impurity spin in a dielectricand also found an exponential lineshape. Misra et al(106-108) found that the linewidth in DPPH dependsupon the recrystallization solution.

Most calculations of linewidths by the Van Vleck (10)moment method make use of a truncated Hamiltonianwhich excludes weak, subsidiary spectral lines at the po-sitions 0, 2gf1H0 and 3gPH0. These lines are not ob-served at high fields, but do contribute to the evenmoments and, in particular, increase the second



Table 2. Exchange Effects on ESR Linewidths and Lineshapes

System State Comments Reference

S = 1/2, radicalS > 1/2, paramagnetic

metal ioncarbazylphthalocyanine dyeK tetracyanoethylenepiperidine radicalradical ion(CH3)3COPH3

DPPH in polystyrene

14N in diamondnaphthyl-n-alkanes

nitroxide, verdazyl,and hydrazyl radicals

nitroxyl radicals

TMPDchloranil

Wurster's blue perchlorate

Cr3+/ZnGa204

Cr carboxylatetetramethyl Mn chloride

Mn(TCNQ)2,Mn(TCNQ-dx)2

RbMnF3

CuMn

CuMn.AgMnAg,Mn/Sb, AuNiSiF6 • 6H2O

Cu(HCOO)2 • 4H2O

Cu2+ complexesCu2 + chelateCu2+ oxalateK2CuCI4 • 2H2OCu(NH3)4PtCI4

(M = Co2+,Sr2+,Ba2+)EuS

Gd in Au

GdxY1x(P,As,Sb)and Gd^Y^Sb

Gd/(RE)F3

Dy, Er, Yb in Ag, Al

solution

solutionsolutionsolutionsolutionsolutionmatrixpowder

solidpolymer

solid

crystal

crystal

powder andsingle crystals

single crystal

solid

polycrystallinesolidalloy

alloyalloysingle crystal

solid

solidsolution, glasspowdersingle crystalsingle crystal

solid

film

single crystal

powders

single crystal

alloy

mutual, linear concentrationbroadening

applies Kivelson (1960) theoryintramolecular exchangeslow and fast exchangemixed with Cu(NO3)2

theory, calculated spectramobility, proton exchangeLorentzian shape at center,

exponential in wingsexchange broadeninginter- and intramolecular exchange

in biradicalexchange-energy varied exponentially

with spin separationwidth due to weak exchange with

non-nearest neighborsAH angular dependence follows 2nd

moment of e-dipolar interactiontemperature-dependent exchange

anisotropy; observed 10/3 effectpairs; isotropic, biquadratic,

anisotropic exchange3-ion clusterstreats exchange narrowing

mechanically withoutexperimental verification

AFM Mn2+ couplingcalculates exchange narrowingtheoretical explanation,

conduction ESRlocal momentsHFS, exhange narrowing, dilutetemperature dependence,uniaxial anisotropytemperature dependence,

antisymmetric exchangedipolar and exchange interactionsmonomer, dimeramine complexesJ strongly temperature dependentexchange-narrowed Lorentzian;

observed 10/3 effect; 1-dimensionalexchange

300 K, indirect exchange

spin-wave absorption,ferromagnetic resonance

local moments, high concentration,exchange narrowing

Lorentzian center,asymmetric wings

exchange causes AH todepend on M

AH calculation. anisotroDic

61,62

6364,65666768,697071

7273,74

75

76

77

78

79-8259

838485

868788,89

90

9192

939495

96

97

98

99

100

101exchange


Vol. 1, No. 4 167

moment by the factor of 10/3 when the measurementsare carried out at low frequencies. A number of authorshave observed this 10/3 effect (see Table 2).

VI. INHOMOGENEOUS BROADENING

Homogeneous broadening occurs for a transitionbetween two spin levels that are somewhat broadenedrather than sharply defined. This type of broadeningarises from the dipolar interaction between like spins,spin lattice relaxation, motional narrowing, and other ef-fects. An inhomogeneously broadened line consists of aspectral distribution of individual resonant lines or spinpackets merged into one overall line or envelope. In-homogeneous broadening may be due to magnetic fieldnonuniformities, unresolved fine or hyperfine structure,and dipolar interaction between unlike spins.

Inhomogeneous broadening was analyzed by Portis(109,110) and Castner (51); their results were discussedin earlier reviews (1,111) and will not be repeated here.Castner analyzed the case of a Gaussian envelope ofwidth GAHvi composed of Lorentzian-shaped spin pack-ets of width /. AHl/i, and he obtained the convolution line-shape of equation (19). He presented graphs which maybe employed to determine the ratio GAHv/LAHv,. In usingthis article, one should be careful to note the use of un-conventional linewidth definitions. Coffman (112)analyzed inhomogeneously broadened high-spin Fe3+

spectra from biological specimens by making use ofmultiple contour integrals.

Cullis (113) treated inhomogeneously broadened linesin terms of a spin temperature in the rotating coordinateframe associated with the spin packets. He obtained ex-pressions which describe the system for both slow andfast passage conditions, and his treatment reduced toCastner's results (51) in several cases.

The relaxation parameter of spin packets can be mea-sured by the continuous saturation and spin echo meth-ods. Semenov and Fogel'son (114) showed that an in-vestigation of the dependence of the response signal onthe intensity and duration of a saturating microwavepulse yields the saturation factor and width ( 1 / T2) of thespin packets that make up inhomogeneous lines.

Baumberg et al (115) found coherence effects in theinhomogeneous lines of nitrogen in SiC irradiated with amicrowave pulse of duration less than T2. This pulseburns a primary hole at the point of application, andsmaller holes appear equally spaced and symmetricallyarrayed about the primary hole. These arise from the ro-tation of spin-packet magnetization vectors by thecoherent microwave pulses.

Epifanov and Manenkov (116) examined relaxationprocesses in inhomogeneously broadened lines by aquantum statistical method. Bugai (117) studied pas-sage effects and Zhidkov et al (118) used continuoussaturation to determine the width of the distributionfunction.

At liquid helium temperature, the 1/2 • • —1/2 ab-sorption line of Cr3+ in dilute ruby is found to be in-homogeneously broadened when the external magneticfield Ho is directed along the c axis, and is homogen-eously broadened when the external field is perpen-dicular to this axis (26-28). The broadening arises fromsuperhyperfine structure from 27AI nuclei which becomemagnetically equivalent in groups when Ho is along c,and which generate spectral spin diffusion in the elec-tron Cr3+ spin system, and hence increase the width.

Vugmeister et al (119) asserted that inhomogeneousbroadening is characterized by interactions which lackdynamic terms, i.e., terms which cause energy exchangebetween different spins. The dynamic portion of thedipolar interaction produces homogeneous broadeningwhen the energy exchange between separate spins oc-curs more rapidly than spin-lattice relaxation. For thecase g»n > gx, with Ho along the symmetry axis, thedynamic part of the interaction may be negligible,producing inhomogeneous broadening. At sufficientlylow temperature, the homogeneity will be angularlydependent.

Mailer et al (120), analyzed the theoretical and exper-imental effects of radio-frequency (RF) and modulationfield nonuniformities and sample length on thecontinuous-wave (CW) saturation characteristics of in-homogeneously broadened lines. They presentedcurves of signal-amplitude dependence on various par-ameters, and illustrated their results with experimentaldata on biological samples.

Strandberg (53) has treated correlation-time narrow-ing and narrowing by spin-stirring experiments with in-homogeneously broadened lines. He also elucidatedrelaxation and adiabatic fast-passage behavior.

Bowman et al (121) analyzed the saturation behaviorof an inhomogeneously broadened line with secondharmonic detection under slow-passage conditions.They presented curves which allow one to extract thewidth and relaxation times T-, and T2 from saturationdata on inhomogeneous lines.

Inhomogeneously broadened lines are readily studiedby electron-electron double resonance (ELDOR) inwhich one part of the line can be saturated and anotherobserved. See Kevan and Kispart (122).



VII. POWDER PATTERNS

Until now the discussion has centered around spectraobtained from single crystals. We have seen how the po-sition and width of a resonant line can depend upon thedirection of the applied magnetic field relative to thecrystallographic axes. It is best to obtain angular rota-tion data with single crystals if they are available, sincesuch data will maximize the information attainable on theprincipal values and direction cosines of the Hamiltonianparameters.

However, sometimes single crystals are not availableand it is necessary to work with powders, which haverandomly oriented paramagnetic centers. In this case,the single crystal spectrum must be averaged over allangles to produce a powder-pattern spectrum with a

shape characteristic of the Hamiltonian parameters. Insection VIII, we will discuss in turn powder patterns aris-ing from an axially symmetric Hamiltonian, totallyunsymmetric cases, partially ordered or semi-randomsystems, and situations such as those that arise inglasses where the paramagnetic ions occupy sites with arange of Hamiltonian parameters. Several authors(1,42,123) have reviewed powder-pattern lineshapes.Table 3 summarizes some experimental results.

To illustrate the principles involved in the calculationof lineshapes from randomly oriented or powder sam-ples, we consider the case of a single line arising from anangularly dependent g-factor

E = hv = g{d,<l>)iipH' (24)

where 6 and <j> are the polar angles of the magnetic field

Table 3. Powder-pattern Reports in Chronological Order

Symmetries Lineshape Comments References

g-factor9

HyperfineA

Zero-fieldD

8—delta functionL—LorentzianG—Gaussian

axialasymmetricaxialisotropicasymmetricaxialaxial

asymmetricisotropicisotropicisotropicasymmetricaxialasymmetricaxial

isotropicaxialisotropicaxialaxial

axialisotropicaxialisotropic

asymmetric

—————

axial—

—

asymmetricasymmetricasymmetric

—axial——

anisotropicaxial

asymmetric——

isotropicaxial—

asymmetric

axial

588888

L,G

85,GG8LG—

L

8,G8,GG—

L

88,LL,GG

NMR

spin = 3/2

orientation-dependenttransition probability

good graphs

spin = V2, nuclear Zeeman / = V4

spin > Vz, orientation-dependenttransition probability



49,124,125126127128129130

131,132

133134135136137138135136a

137a138a139140141

142143144145

L,G 146Continued on next page

Duplication of Bulletin of Magnetic Resonance, in whole or in part by any means for any purpose is illegal

Vol. 1, No. 4 169

Table 3. Powder-pattern Reports in Chronological Order (Continued)

Symmetries Lineshape Comments References

g-factor9

HyperfineA

Zero-fieldD

8—delta functionL—LorentzianG—Gaussian

asymmetricaxialaxialaxialasymmetricasymmetricasymmetricasymmetricaxialaxial

—axial

—axial

—asymmetricasymmetric

—anisotropic

axial

isotropic axial

L,GG

arbitraryL,GL

L

L

8

axial

axial

asymmetric

axial

axialisotropic

isotropic

isotropicasymmetricisotropic

axialasymmetric

isotropic

isotropicasymmetric

asymmetricasymmetricasymmetric

axial

axialisotropic

asymmetric

—

axial

axialisotropic

axial

axialasymmetric

—

—

Eaxial

axial

asymmetric

—

axial

—

axial

—

asymmetric

axialasymmetric

asymmetricasymmetric

—

asymmetric

— L,G

—

L

L,G

G

8

8

L,GG

G

—

8GG

8

—

computer calculation

valid for small anisotropylarge 19F anisotropynuclear Zeeman / = Vfe

computer simulationorientation-dependent

transition probabilityspin = Vfe, nuclear Zeeman / = V2,


shape-transform graphdiscusses singularitiesspin = 3/2, no orientation-

dependent transition proba-bility, central powderpattern folded back

gz/gx vs gx/gy, orientation-dependent transitionprobability

different principal directions,orientation-dependenttransition probability

quadrupole, nuclear Zeemanspin = 5/2, doubling gives

signs14N HFS axial; rotating

methyl groupspin > V2AH angular dependenceH =&J • g • H + aZJiHjEk

V*j=tk) Stark termspin = 1/2, vanadylnuclear Zeeman + asymmetric

quadrupolespin = 5/2; includes B4

zero fieldspin = 1, Calibrationspin = 1, different principal

directionsspin > Vzspin = 5/2, g$H < Dspin = V2; I, = l2 = V2; torsional-

oscillation axis distribution

h = k = k = 1/2- extrasingularities

spin = 5/2, gvs3E/D plot

147148149150151152153154155156

157

158

159,160161,162

163

164

165

166

167,168

169,170

171,172

173

174

175

176

177,178

179

180

181

182

81,82,183

184

185



H' in the principal axis system of the g-tensor. Since theexperiment is carried out at a constant frequency v, weequate hv to the product g0Ho of a mean or average g-factor g0 and resonant field Ho

= gofi(jHo (25)

The resonant field H' (9,$) corresponding to the direction8,4> is obtained by combining equations (24) and (25) andrearranging terms

Hog0 (26)

The angularly dependent g-factor g{8,<p) may arise froman anisotropic Zeeman term, or, in a more general case,it may arise from an effective g-factor which includeszero-field interactions. If the spectrum consists of widelyseparated lines, then equation (26) applies to each, withg(d,4>) being different for each line. For simplicity, we nowconfine our attention to the spectrum of a single line andwill treat hyperfine structure later.

If a study is made of a single crystal with one angularlydependent spectral line arising from equation (26), thenthe shape of the spectral line recorded at the orientation0, <£willbe

Single-crystal = (

hneshape v v ' (27)

H- H'The lineshape function V( AH ) in particular casesmay indicate a Lorentzian or Gaussian absorption orderivative lineshape defined by equation (6), (7), (12), or(13), and the function P(B,4>) takes into account the an-gularly dependent probability that the microwave fieldwill induce an ESR transition. In the general case, thetransition probability P, the linewidth AH, the line shapefunction Y, and the resonant field position H are all an-gularly dependent. In most cases encountered in prac-tice, the shape and width may be treated as constantsindependent of the orientation.

For a powder sample in which the microcrystallites arerandomly oriented, the powder-pattern lineshape can beobtained by integrating equation (27) over a unit sphere

Powder-patternlineshape

J O J II*- H'(8t<t>)AH(8,4>)) s i n 9 d ^ (28)

In general, this is not a convenient expression to workwith because /-/' is a complex function of position and isnot easy to integrate. Some authors have simplified it forspecial cases, as discussed below. Taylor and Bray (186)

have discussed the use of perturbation techniques insimulating powder spectra.

VIII. AXIAL POWDER PATTERNS

In the case of axial symmetry, there is no </> depen-dence and the single-crystal and powder-pattern line-shapes reduce to

Single-crystal = y,H - H'(8),lineshape w V AH(6) '

Powder-pattern = j ( H - f/' (0)}

lineshape J m v AH(6) '

The g-factor has the explicit form

g = (gfcos2d + g\ si

which gives for H' of equations (29)

(29)

H' =T^rgoHo

(30)

(31)

The field H' varies with the angle 6 between the limitingvalues W|| and H± given by

= g0H0 (32)

where H' = H\\ for 6 = 0 and H' = H± for 6 = TT/2.To transform equations (29) to more convenient

forms, it is desired to convert them from functions of 8 tofunctions of H'. To accomplish this, we observe that theprobability of finding a microcrystallite randomly orient-ed at an angle 6 is proportional to the area sin 8 d0 of anannular ring about the pole of a unit sphere. Associatedwith the range of angles from 8 to 8 + d0 is a range ofresonant magnetic field strengths from H' to H' + dW.The probability G(H') of finding a microcrystallite whichis resonant within this range is obtained from theexpression

G{H)dH' = sin 8 dd = d(cos 8)

which may be written

1dA/7d(cos0)

(33)

Carrying out the differentiation of equation (31), weobtain

H'2gf-gj (34)


Vol. 1, No. 4 171

which includes a normalization factor to make G(H<) = 1for H = H\.

The transition probability for a linearly polarized oscil-latory microwave field normalized to P(H.( = 1 for 0 = 0corresponding to H' = W| is given by (125)

(35)

As Table 3 indicates, many authors neglect this fact-or, which is equivalent to setting P{H,} = 1.

The single-crystal and the powder-pattern lineshapesare given by

Single-crystal _lineshape

Powder-pattern _ fH

lineshape

,H-H\

(36)_r H ~ H \

With the aid of equations (34) and (35), the latter expres-sion may be put in the form

Powder-pattern _lineshape ~

J "i (g'tHf-(37)

where the limits of integration were selected for the caseW|l< H±. For the opposite case H± < Wj, the integrationlimits are interchanged.

The general features of the lineshapes' angulardependence can be readily grasped by considering thespecial case of intrinsically narrow single-crystal lines,which corresponds to writing Y{H^ as a delta function

- H')

Delta-functionlineshape II

~ P(H) G(H)

A (38)

This lineshape exists within the range H\\ < H < Wx (orWx < W < W||) and vanishes outside it. Thus, the inte-grand P(H.} G{H) is often referred to as the delta-functionlineshape. Many of the published lineshape articles havedealt only with delta-function shapes, as shown in Table3.

Figure 3 shows the powder lineshapes for the cases ofa delta-function width and a finite angularly independent

Figure 3. Powder-pattern absorption lineshapes for an axiallysymmetric g-factor. Component linewidth is zero for solid line,finite for dashed line.

width. A number of authors have treated this case of anaxial g-factor, as Table 3 indicates. We see from thetable that they generally set P(H., = 1, and they rarelytake into account an angularly dependent linewidth.Some of the articles in the table analyze cases of morecomplicated Hamiltonians which include hyperfinestructure or zero-field terms. These latter cases involvethe superposition of more than one line in the spectrum,and the resulting powder pattern can be written as thesum of the patterns of the individual component lines.

Powderpattern

(39)

In general, each component can have its own character-istic functions P,, Yh AHh and G,, but in practice, Y, andAW,, are generally the same for each component, andoften one sets each P, equal to unity.

The powder-pattern problem for an axially symmetricHamiltonian with hyperfine structure corresponds to thefollowing Hamiltonian (124,130,169)

hv = g± X + HySy)Syly)

In terms of the parameter K defined by (124)

j A\h 2vz(g\ cos20 + g\ sin20)

G(H) for the /nth hyperfine component has the form

(40)

(41)



G(H) - -

(42)

m(

If the individual component lines have finite widths, thenthe powder pattern is given by inserting P(W) = 1 andG(H) from equation (42) into equation (36). Figure 4shows the powder pattern arising from one hyperfinecomponent in this axial case.

Shteinshneider and Zhidomirov (164) discussed thelineshapes obtained from a generalization of theHamiltonian, equation (40), to the case where the g-factor and hyperfine tensors are axially symmetric withdifferent principal directions. Various aspects of axialpowder patterns in the presence of hyperfine structurehave been treated by others (cf. Table 3).

Other Hamiltonian terms can also give axially sym-metric powder patterns, such as a nuclear electricquadrupole moment. We have shown elsewhere that azero-field D term SD^- S with a magnitude small relativeto the Zeeman energy, produces the following delta-function lineshape when g is isotropic, S > Vz, and nohyperfine structure is present (169.)

GIH\ =2(H0-H)

nDwhere D is in gauss, we assume PW) = 1, and

Ho= H + ~ (3cos20 -

(43)

(44)

Here n takes on the following sequence of values:

Figure 4. Powder-pattern absorption lineshapes for onecomponent of a hyperfine multiplet when the g-factor andhyperfine tensors have the same axis of symmetry (from refer-ence 1). Component linewidth is zero for the solid line, finite forthe dashed line.

±(2S-1), ±(2S-3), . . . , ±1 or 0, starting with thehighest pair ±(2S-1) and decreasing in steps of 2 tothe lowest value of 1 or 0. Thus, for example, for S = 3/2,we have n = 0, ± 2, and for S = 2, we have n = ± 1, ±3.The magnetic field produces resonant absorptionbetween the limits (169)

-1 <2(H0 - H)

nD < 2

with G{H) = 1 at the right hand limit and °° at the left handlimit, as may be shown by direct substitution in equation(43). When there is a finite component width, then G(H)

integrates to a finite area in equation (43).Mialhe et al (166) analyzed the powder pattern of

Mn2+ in an axially symmetric environment in terms of aHamiltonian with an isotropic g-factor and hyperfineinteractions and an axially symmetric zero-field D term.The probability function G(H) for the lineshape has theusual infinity for 6 = w/2, where cos 6 = 0, but in this case,it has another infinity at 6 corresponding to the condition

8COS20(1 + ^" o

(45)

which was obtained from perturbation theory. The resultwas a doubling of the allowed transitions which permit-ted the determination of the relative signs of the A and Dtensors. Their spectra also exhibited forbidden linesbetween the main hyperfine components.

IX. COMPLETELY ANISOTROPIC POWDERPATTERNS

This section is concerned with Hamiltonians whichlack^symmetryso that tensor operators such as *g, A,andl? have three principal values. We will begin with asystem which has only a Zeeman interaction but alsocomplete asymmetry. Table 3 lists articles which treatpowder patterns formed from various combinations ofHamiltonian terms some or all of which are completelyanisotropic.

For the case of a completely anisotropic g-factor,equation (36) is still valid and now the angular depen-dence contains the two polar angles & and <j>

g = (g*sin20cos2</> + g\sinWsin2<A + g\cos20)'A (46)

This expression reduces to equation (30) for axialsymmetry where g, = gz. The resulting lineshape for theconvention

Qz < 92 < 9\ (47)


Vol. 1, No. 4 173

Component ( 0ine-width I 0.5

For carbazyl. K bandH2 = 8230.0 G

Figure 5. Calculated powder-pattern lineshape for carbazyl,assuming both zero and nonzero component linewidths (fromreferences 1,129).

which corresponds to

H1<H2< H3 (48)

is illustrated in Figure 5. We see from the figure thatthere is an infinity at the magnetic field position H2 =hv/g2fiff, and this infinity integrates to a finite area. Thedelta-function lineshape corresponds to the probabilityG{H) which was introduced in the previous section, andwhen a finite linewidth exists, the lineshape Y{H) is cal-culated from equation (36). The transition probabilityP{H) for this case has been reported (163,188,189).Hauser and coworkers (171,172) calculated the powderpattern arising from a completely anisotropic g-factorwhen the component linewidth is also angularly depen-dent (see Table 3).

Table 3 lists several articles which treat asymmetriccases of powder patterns with hyperfine and zero-fieldterms. For example, Mohrmann et al (161) discussed acase of Cr3+ in (NH4)2 [In (H2O) Cl5] which has an almostisotropic g-factor (g^ = 1.987, g± - 1.982) and a com-pletely unsymmetric zero-field splitting (D = 641G,3E/D = 0.25). Two of the fine structure (AMS = ±1) linesproduced powder patterns which resembled the onewith a completely anisotropic g-factor illustrated inFigure 4. The third or central transition produced asimilar pattern with parts of it folded back on itself. Thisfolding back arises from the double valuedness of themaxima in the angularly dependent magnetic fieldH{0,4>). Measurements at 9.5 GHz and 35.4 GHz and the

availability of Hamiltonian parameters from previoussingle-crystal work helped the authors in their interpre-tation of the data.

X. SEMIRANDOM DISTRIBUTIONS

The previous three sections treated powder patternsfrom microcrystallites with randomly oriented axes. Themathematical procedure was illustrated by calculation ofthe geometric probability factor Gm for the case of anaxially symmetric g-factor.

Sometimes the microcrystallites are not randomlyoriented, but have a preferred orientation or distributionp(8), where 8 is the angle between the microcrystalliteaxis and the preferred direction. The microcrystallitesare assumed to be randomly oriented about thispreferred axis. For this situation, ESR measurementscan be carried out with the magnetic field set at variousangles \p relative to this preferred direction. For eachchoice of \p, the spectrum will be an average over mi-crocrystallite distributions, and so will appear powder-like. An analysis of a series of such spectra can providethe distribution function ,c#). More complex distribu-tions in terms of polar angles p(d,<t>) or Euler anglesp(a,j8,7) can also be treated.

Many examples of the magnetic resonance study ofsemirandom distributions exist (11,187,190-194). Forexample, McDowell et al (194) reported the ESR spec-trum of CIO2 molecules preferentially oriented in a rare-gas matrix at 4.2% with the molecular plane parallel tothe deposition surface. Other paramagnetic species aresimilarly oriented (195-197). Panepucci and Farach (11)found apatite microcrystallites in irradiated bone to bepreferentially aligned along the bone axis. Blazha et al(187) were able to reconstruct the semirandom defectdistribution around a paramagnetic center from ananalysis of the lineshape. Hentschel et al (198) treated adistribution function p(a,(l,y) expressed in terms of Eulerangles a, j8, y and used an expansion of this function interms of Wigner matrices. They discussed the applica-tion of these results to the magnetic resonance of drawnpolymers, partially ordered solids, and spin-labeledcomplexes.

To describe the situation mathematically for a cylin-drically symmetric microcrystallite distribution, weselect a Cartesian coordinate system with the appliedmagnetic field in the z direction, the preferred axis p forthe distribution function in the xz plane with a polarangle \p, and the crystallite axis c in a general 6,<j> direc-tion. In this coordinate system, the crystallite distributionfunction p(6) depends only on the angle 5 between thecrystallite direction c and the preferred axis p. The angle



8 can be written as a function of 6, <j>, and ip through theexpression

cos 8 = cos 0 cos f + sin 0 sin \p cos 0 (49)

If equation (30) is expressed in the form

with the aid of equation (32), then cos 8 can be written

c o s g _

B(H') cos <t>

(51)

(52)

As a result, p(5) can be treated as piH'rf) since \p is a con-stant for a given spectrum.

To treat the case of a semirandom distribution of thistype, equation (36) can be generalized to the form

Semirandompowder-pattern =lineshape

jHA_(H'>+Hf)YmdH>

"'(gfH2 - glH'2)v'H12 (53)

We should note that the <j> dependence conveniently fac-tors from the integral. Panepucci and Farach (11) treatedthe case of an experimental distribution exp («cos 8)where a was evaluated from the fit of calculated toexperimental spectra.

XI. AMORPHOUS OR GLASSYDISTRIBUTIONS

In sections Vll-X, we discussed lineshapes from spinsystems with definite Hamiltonian parameters for whichthe samples consist of microcrystallites with a random orsemirandom orientation. These situations occur with anionic or molecular crystal in which the paramagneticspecies has a definite site symmetry. A glassy or amor-phous substance is a different kind of material in which agiven ion or radical can exhibit a range of Hamiltonianparameters. The powder spectrum of such a material isan average not only over angles but also over Hamilton-ian parameters. As a result, different techniques areoften applied to calculate the powder spectra of amor-phous solids.

A simple example can illustrate how glassy samplesproduce spectra which differ from those of powders.Consider the case of an axially symmetric g-factor inwhich the most prominent powder-pattern features ap-pear in resonance absorption at the field value H = Hx,as indicated in Figure 3. If the paramagnetic species is ata site of axial symmetry with a fixed value of g\\ and arange of values of g±, then the resonance absorption ofthe g± points will be spread over a range of fieldstrengths, whereas the absorptions from all of the spinsat the <7n point will superimpose and add. As a result, themost prominent feature on the spectrum will correspondto the g\\ point. Thus, the basic principle in the interpre-tation of spectra from glassy samples is the determina-tion of field strengths that correspond to regions of ab-sorption by spins with a range of site symmetries. Suchfield strengths where many different Hamiltonian valuesproduce a superimposed absorption are referred to asstationary points.

As an example of how the spectra from amorphoussamples can be interpreted, consider the case of a 6Stransition ion such as Fe3+, which has the Hamiltonian(199)

H = gpH- S + D[Sl - 1/3 S(S+ 1)] + E{S2 - S2y)

where g is isotropic, D ranges over values greater thanthe Zeeman term

g/3H< \D\ <Dmax

and the asymmetry parameter •» = 3E/D can rangebetween 0 and 1

0<7J< 1

Since S = 5/2 for a 6S state, there are, in general, sixenergy levels, and the observed spectrum arises fromvarious transitions between these levels. Energy absorp-tion will occur at those magnetic field strengths H wherethe distance between two energy levels equals the mi-crowave energy hv. Associated with such an absorptionof energy, we can define an effective g-factor g =hv/fiBH. If we define the magnetic field Ho by setting hvequal to g^BH0, where g0 is the free electron value, then

Ho = hv/goHQ

and we can write the following expression for the magne-tic field H where resonant absorption occurs

H = Ho (-£) (54)

Associated with each value of D and y and with eachorientation of the magnetic field at a fixed microwavefrequency, there is a series of transitions, each of whichis characterized by an effective g-value. After solving theHamiltonian problem by diagonalizing the matrix as-sociated with H of equation (54), a series of graphs can


Vol. 1, No. 4 175

be made in which gjg is plotted against D/hv for a fixedvalue of ?; and a particular orientation (199). Figure 6 il-lustrates such a graph for 77 = 0.6, with the individualcurves corresponding to the magnetic field alignedalong the x, y, and z principle directions. The most char-acteristic feature of this graph is the fact that most of thecurves are fairly horizontal for values of D/hv greaterthan unity. The g-factors corresponding to such flatregions are called stationary values. As a result of suchstationary regions, crystallites with a range of zero-fieldsplittings will absorb microwaves at close to the samemagnetic field position. Plots as shown in Figure 6 canbe constructed for various values of 77 and for a series oforientations, and the g-factors where the curves are flatcan be compared with experimentally observed g-factors to deduce the range of D and E values in thesample. Aasa (200) has graphs similar to Figure 6, butplotted g/g0 and D/hv on logarithmic scales. Taylor et al(123) have reviewed this field. Table 4 summarizes theresults presented in some representative articles.

- i 0

Figure 6. Graph of g^lg vs D/hv for some effective g values(given on right hand ordinate axis). Labels on curves indicatethe transition, magnetic field orientation (x, y, or z principaldirection), and the relative transition probability (in paren-theses) for the region 3 < D/hv > 4 (from reference 199).

XII. INTERMEDIATE MOTIONS INSOLUTIONS

We now turn our attention from rigid crystalline oramorphous lattices to fluids in which the rapid molecularmotion averages out many of the features that dominatethe spectra of solids. High viscosity liquids lie in a transi-tion region where spectral features are intermediatebetween those of a rigid lattice and those of a fluid. Thefeatures vary continuously between the two extremes asthe viscosity changes. In this section we will consider theregion of intermediate motion.

Theoretical studies involving intermediate motion areavailable (219-229). Here, we will attempt only to delimitthe region of intermediate motion in terms of character-istic times, discuss some of the significant underlyingmechanisms, and illustrate lineshape changes.

The Brownian motion of a liquid is characterized by acorrelation time r which is a measure of the rapidity ofthe motion. In classical dielectric relaxation exper-iments, the molecules are treated as spheres of radius aundergoing isotropic rotational reorientation in amedium of viscosity y, and the resulting Debye correla-fen time rD varies inversely with the temperature in ac-cordance with the Einstein-Stokes relationship

f J _ 4 7 n ? a 3 #«vT°~2D~ kT ( 5 S )

w|iere D is the rotational diffusion coefficient (111). Thec||responding correlation time TC for the motion of the

cjnetic moments in an electron spin resonance exper-snt is one-third the classical Debye value

= 3T C

corresponding to

6D4-irria3

'' ZkT

(56)

(57)

which relates explicitly to the isotropic rotational model.From a more general viewpoint, if we consider rc as

the correlation time for random perturbations involvedin ESR measurements, and rD as the correspondingdielectric relaxation counterpart, then

rD = frc (58)

where /varies between 1 and 3 and assumes the limitingvalue of 3 for the isotropic Brownian model mentionedabove. If the random motion is described by a jumpmodel wherein the molecules reorient via large angularshifts of random length, then TC = rD, i.e., f = 1.

To understand the significance of the correlation timerc, consider the situation wherein a random field Hw,which perturbs the motion of a spin at a time t, tends tochange appreciably in a time t of the order of rc. Thismeans that for times t < TC, the perturbing field will havealtered very little from its value at t = 0, and for times t



Table 4. Selected Reports of Lineshapes in Glasses

g-factorg

isotropicisotropicisotropicisotropic

isotropic

isotropic

isotropicaxialaxial

isotropic

axialasymmetricasymmetricaxial

Symmetries

HyperfineA

isotropicasymmetric

isotropic

isotropic

—

isotropicaxial

—

axial

axial

Zero-fieldD

averaged

gives valuesrange

gPH

several

—

range

—

Center system

Mnz + borate glassirradiated borate glassMn2+ Fe3+ complexesMn2 + borate glasses

Mn2+ As-Te-I glasses

Gd3 +

electronE' center in silicaFe3+ ,Ti3+ , various

a lassesMn2+,Fe3+, alumina

Cu2+ alkali silicateCd2 + Na glassesMn3+ As glassesCr2+,V4+ phosphate

Comments

3E/D = O; basic calculationcomplex calculationgives 3E/D values, good graphsgives 3E/D range; computer

data fit3E/D = 1, discusses g ~ 30/7

lineseveral ZEID values, gjg vs

Df hv araDhsgood graphscomputer simulationdiscusses integration

gives 3 E/D range, go/g vsD/hvarsobs

determines quadrupole momentreviews theorystructure changescomputer-simulated glass AH

References

201202,203

200204

205

206

121,207,208209,210211-213

199

214,215216

35,217218

» Tp, the perturbing field will be randomly related to itsinitial value. The time interval TC is the crossover pointfrom appreciable to negligible correlation between therandom field H(t) and its previous value.

The situation just described may be expressed analy-tically by an autocorrelation function Ha(, + T) A/a(0 whichis a time average of one of the components Hx, Hy, or Hz

of H(t). A commonly used autocorrelation function is

= HJ exp( - | r | (59)

In this simple case, the spin-lattice and spin-spin relaxa-tion times 7"1t and T2, respectively, are given by

1 , , 7 7 * . 775, T c _ ( 6 Q )

1

which gives for the linewidth

= yAH,A = —

(61)

(62)_ 2

The perturbing fields generally are isotropic, Hx =Hy = Hz , which gives

I (63)C00TC

the former being typical of low-viscosity liquids and thelatter of solids.

The discussion thus far has concerned a singlesymmetrical resonant line having a width that varies inaccordance with the factor worc. Of greater interest is the

effect on powder-pattern spectra arising from asym-metric Hamiltonian tensors. In this case, the critical par-ameter is the ratio 2TC/ T2 which equals rcAccy2 by substi-tution from equation (62).

In the slow motion regime corresponding to TCACOV2 »1, a rigid lattice powder pattern is obtained, while at therapid motion limit TCA«,/2 «: 1, the powder pattern col-lapses to a singlet characteristic of a liquid. At inter-mediate rates of reorientation, where TCAU>V2 = 1, thelineshape is a poorly resolved or partially collapsedpowder pattern.

Sillescu (230) examined these various cases in termsof a fluctuating time-dependent axially symmetricZeeman Hamiltonian

H(t) - MB[9 [ X(5XW X

If one adopts the notation_ _ 1 ,

SyHy) + g{lS2Hz] (64)

(65)

(66)

then in the limit of | Ag \ « g0 the Hamiltonian becomes

6(t) - (67)

where the time dependence arises from random localfluctuations in the angle 0(f) with the correlation time rc.If TCAU>% greatly exceeds unity, then very little averagingoccurs and the observed spectrum is a powder patternof the type illustrated in Figures 3 and 7. In the oppositelimit, when rcAw% is much less than unity, then the factor


Vol. 1, No. 4 177

3cos20(t) - 1 is almost averaged out to produce theslightly asymmetric singlet illustrated in Figure 8. The in-termediate case, where TCAU)VZ ~ 1, corresponds to apartially averaged but still recognizable powder patternas presented in Figure 9. The presentation of curves forboth the rotational Brownian motion and random-jumpmodels shows the effect of random fluctuation on line-shape. We see that for the same value of TCACO%, theBrownian motion model leaves the powder-pattern fea-tures somewhat more prominent than in the random-jump model.

In the slow motion region, i.e., rc between about 10"9

and 10~6s, partially averaged powder-pattern spectraare observed. For shorter times, rc becomes of the orderof the period of microwaves (WOTC =J 1), and for even

Figure 7. ESR spectrum in the slow-reorientation regionTAWI/2 = 200 arising from a slightly averaged, fluctuating,axially symmetric Zeeman Hamiltonian-Brownian-motionmodel—jump model (from reference 230).

shorter times, C<JOTC «: 1. For longer correlation times, TC

> 10~6s, the rigid lattice limit is approached. Variousauthors (228,231-234) have treated the completely ani-sotropic and axially symmetric g-factor lineshapes at in-termediate correlation times, and some (235-240) in-cluded hyperfine interactions (see Table 5).

XIII. HINDERED ROTATIONS ANDTUNNELING

Hindered rotation or tunneling motions can influencethe number, position, and widths of the lines in a hyper-fine multiplet. For example, a hindered methyl group un-dergoing tunneling rotation at liquid-helium temperatureproduces an eight-line spectrum, while the same methylgroup freely rotating at high temperatures produces a1:3:3:1 intensity ratio quartet (246-248). In an argon ma-trix at 4.2 K, the a and 0 proton hyperfine tensors of theethyl radical are axially symmetric due to rapid reorien-tation about the C-C bond (249), and n-propyl andn-butyl radicals exhibit motional or tunneling effects(250). Maruani et al (251) published axial hyperfine cou-plings for the CF3 radical, and Vugman et al (252) report-ed axial rotation of H2CN.

Freed (253) analyzed the effects of internal motions ofmethyl groups on hyperfine patterns, taking tunnelinginto account. His results agree with classical theorywhen the hyperfine frequency exceeds the tunnelingrate, while quantum effects become important under theopposite condition.

When two or more isotropic hyperfine coupling con-stants are modulated by fluctuating fields at rates thatare comparable to their difference expressed infrequency units, then the spectra exhibit selective

Figure 8. ESR spectrum in the rapid-reorientation regionTACOI/2 = 0.6 arising from an almost completely averaged,fluctuating, axially symmetric Zeeman Hamiltonian-Brownian-motion model—jump model (from reference 230).

Figure 9. ESR spectrum in the intermediate-reorientationregion, rAwy2 = 4 arising from a partially averaged, fluctuat-ing, axially symmetric Zeeman Hamiltonian-Brownian-motionmodel—jump model (from reference 230).

Duplication of Bulfetin of Magnetic Resonance, in whole or in part by any means for any purpose is illegal.


Table 5. Lineshapes of Intermediate and Slow Motion Systems(See Table 11 for spin-label data.)

Symmetries Model Comments References

g-factor9

axial

asymmetric

isotropicasymmetric,

axial

asymmetricasymmetric,

axialisotropic

axialasymmetric

axial

axialaxial

HyperfineA

axial

axialasymmetric,

axial

axialaxial

isotropic,axial

asymmetric

wide range0.3<rcA<o<102

0.07<TCACO<70

wide range

3 X 10-12to10-6

2 X 10- 1 0 to4 X 1CT8

wide range

4 < T C A < O < 1 0 4

wide rangewide range

isotropic rotationaldiffusion

rotational diffusion

Brownian motionBrownian, jump, isotropic,

and anisotropic diffusion

rotational diffusionisotropic rotational

diffusionBrownian, rotational,

jump, isotropic, andanisotropic

Brownian, jumpjump

axial diffusion

spin pairsMarkov process

imino acid radicals,H2O glycerol

rotating methylsperoxylamine disul-

fonate(PADS),rcestimated

—Monte Carlo

peroxylamine disul-fonate(PADS)inglycerol-H2O

sulfur in 60% oleumpartially ordered

powdercontinued fractions,

axially symmetriczero-field D

projection operatorsgraphical techniques

241228

237

167,168239

236240

242,243

233235,238

244,245

231232

broadening. Krusic et al (254) discussed the casewherein hindered rotation caused such modulation viatwofold potential barriers. They present data on theCH2OH radical in which the a protons are equivalent at215 K and nonequivalent at 148 K. At intermediatetemperatures such as 183 K, tunneling broadens thelines.

When frozen in place at low temperatures, the CO2~radical ion in calcite has three equilibrium configurationscorresponding to three ways in which the two oxygens ofCO2~ can occupy the three CO3

2~ oxygen sites, andeach gives a separate spectrum (15). At high tempera-tures, free rotation causes the three lines to coalesceinto one. At intermediate temperatures, where the tun-neling jump rate is comparable to the Zeeman frequencygiiBH/h, the spectrum is broad and poorly resolved. Thecalculated angular dependence of the linewidth in the in-termediate temperature region agreed qualitatively withexperimental results.

Krishnamurthy (23) found that the vanadyl (VO2+) ionin VO2(NO3)26H2O undergoes a rapid reorientation orhindered motion at room temperature which produces asolution-like spectrum. At low temperatures, the vanadylion freezes in place with a random orientation and dis-plays a powder-pattern spectru m.

Raoux (255) explained the ESR spectrum of an Hcenter in irradiated LiF in terms of the rotation of themolecule about the average (111) direction induced bylibrations and pyramidal jumping between the four (111)directions. The inhomogeneous Gaussian line recordedat low temperature broadens above 150 K due to ahomogeneous Lorentzian contribution from reorienta-tional effects.

Chase (256) examined the resonance from Eu2+ in al-kaline earth fluorides and found a tunneling energy levelabout 5 cm~1 above the ground level. He calculated line-shapes for the competition between strain and tunnelingeffects associated with a dynamical Jahn Tellerdistortion.

Table 6 summarizes representative data on hinderedrotation and tunneling effects.

XIV. CONCENTRATION ANDTEMPERATURE DEPENDENCE

Free radicals and transition ions in solution or in rigidsolids have linewidths which tend to increase linearlywith concentration from a residual value at zero concen-tration, in accordance with the expression

Duplication of Bulletin of Magnetic Resonance, In whole or in part by any means for any purpose is illegal.

Vol. 1,No. 4 179

Table 6. Effect of Hindered Rotation and Tunneling on Lineshapes

System

n-propyl, n-butylCH3OH radical

C2H4 + H2

theoretical

CH3CH2 radicalL-alanine

azobisisobutyronitrileCF2CONH2

CH3COHCOO-

RCCH3 radicalp-x-acetophenone

calcite CaCO3

[Ni(NH3)6](BF4)2,related compounds

dinitrophenyl,dinitronaphthalene

A s O / -

Na, SO2,02

(GeOx-)-

LiF

UO2(NO3)26H2OCs2HfCI6, Cs2ZrCI6Cu(en)32+

CaF2

Group

CH2

CH2

wrigV-'Hg

CH3

CH3

CH3

CH3

C-CCH3

CH3

acetyl

co2-NH3

NO2

H bondtunn.NaSO2, NaO2

xSiO2

H center >VO2

Mn2+

Cu(en)33+

Eu2+

Temperature, K

4.2120 to 330

4.2wide

4.2 to 804.2 to 80

77 to 2902 to 20

4.2241 to 322

400 to 500

80 to 330

193 to 293

wide4.2

15 to 300

77 to 220

120 to 30012 to 400

771.5 to 4

Comments

axial Hamiltoniantheoretical explanationaxial Hamiltoniantheory, spin-rotation

coupling and rotationalrelaxation

explanation

explains temperaturedependence

computer simulationtorsional motionweakly hinderedENDORhindered rotation

activation energy

theoretical explanation

ordering temperaturefound

seesaw motion

jump theoryAr matrix3 orientations,

jumpinglibration

3-site axis jump

pulse radiolysisJahn Teller

Reference

250254249253

247248

167,16817

246257258

15,259

260

261

235,238262263

25523264265256

AH = AH0(1 + aC) (68)

where C is the concentration and a is a constant. Thefactor AH0 arises primarily from dipolar interactions.

When the concentration becomes large enough, ex-change effects become appreciable, and at high concen-trations the exchange interaction causes the lines to nar-row. The quantities AH0 and AH, are strongly tempera-ture dependent in solutions due to the motional narrow-ing influence of Brownian motion, and are less tempera-ture dependent in solids which lack Brownian fluctua-tions. There are exceptions to this rule, attributable tosuch mechanisms in solids as internal molecular rota-tions and tunneling.

To illustrate the manner in which the factors in equa-tion (68) depend on temperature, we will summarize theresults of a study by Ayant and coworkers (266,267) ofthe deuterated tanone radical (tetramethyl 2,2,6,6,piperidone - 4, oxide) which gives a triplet hyperfine

pattern due to the nuclear spin / = 1 of the nitrogen.Both AH0 and AAV, of equation (68) depend on r\l T, theratio of viscosity to temperature, through the correlationtime rc for which the Debye approximation is given inequation (56). Ayant showed that the concentration-independent width AHO has the relationship

AHo =a — +b + {A + BM+ CM2) ^ (69)V I

The first term, aT/rj, comes from the spin-rotationalinteraction; b is due to unresolved hyperfine interactionsand instrumental effects; and the third set of terms isrelated through M to the hyperfine component line (to bediscussed in section XV). The concentration-dependentwidth AH-i of equation (68) has the relationship

7"1 T V

which has a dipolar part dr]/T and an exchange partwhich depends on temperature through the complexfunction <£(77J?). Calculations agreed satisfactorily with



Table 7. Concentration Dependence of Linewidths

System

radical S = V4,metal ion S > V4

2ON(SO3)2

several radicalspiperidine radicalstanonCr3+,Fe3+

tFe(H2O)nF6.n]3-n

Mn(C104)2

Mn(NO3)2

Mn(NO3)2

MnCI2-KCI

ion implantation

9,10 dimethyl-anthracene

butadiene radicalsirradiated NaOHN in diamonddiamondn-typelnSbCdSSi-Ge alloyV4+/VOPOCr3+

Cr3+/(Cr,AI)2O3

Cr3+ alumCr3+/AI2O3

Mn(CH3COO)24H2O

Mn2/Mg(HCOO)22H2OMnO-MgOand

NiO-MgOCu(en)3

2+Gd(CoxNi1.J2

Gd2+/LaandLaAI2

Dy/GdAI2

State

solution

aqueousTHF, DGDEH2O, glycerolmethanolH2O glycerol

aqueousmethanol, DMF, DMSO,

CH3CN, acetonitrilesolution, H2OH2Oliquid

layer

silica alumina surface

silica gel surfaceaqueous glasssingle crystalsolidsolidsingle crystalamorphous ,glasssingle crystal

powder—

single crystalsingle crystal

single crystalsolid

glasspolycrystallinesolidsolid

A H-vs-concentrationgraph

yesyesyesyesyes

—yes

yesnoyes

yes

no

noyesnoyesyesyesyesyesyes

yes

yesno

yesno

yesnoyesyes

Comment

mutual linear concentrationbroadening

measured relaxation250 to 310 Kbroadening by transitions183 to 363 KH2O/glycerol ratio varied; X

and Q bandscomplexes n=0 to 6Oh and tetrahedral complexes

290 to 370 K200 to 360 Ktotal concentration

range studiedeffective donor

concentration variedsurface site dependence

hfs concentration dependenceH2O/D2O ratio plotN-N interactiontemperature dependencetheoretical explanationdonor concentration studiedAH increases with Ge/Si ratioantiferromagnetic couplingCr3+-Li+ pairs give

angle-dependent AHtotal concentration range

optically detected 2E stateAH angle- and temperaturedependent2-dimensional magnetic latticeMnO Lorentzian,

NiO Gaussiane-tunnelingtemperature dependencebottleneck relaxationtemperature dependence

References

61,62

271272

67,268266,267

273

274275,276

277278279

280

281

282208283284285286

287,28828931

269,27032290291

36292

265293294295

experiment from 183 K to 363 K for concentrations from0.2 x 10-3 to 4 x 10-3 mol l~1. Anisimov et al (268) foundthat equation (70) was satisfied by two piperidine-typeradicals when lines were broadened by the para-magnetic nitrates of iron-group transition ions, with cad-mium nitrate used as a blank to eliminate nonparamag-netic effects.

Concentration-dependent effects are also importantin solids. For example, the narrow lines arising from Cr3+

ions in ruby (AI2O3) broaden with increasing chromium

content, and then begin to narrow again for large Cr/AIratios (269,270). Tables 7 and 8 list results of studies ofconcentration and temperature dependence in solutionsand solids.

Pressure dependence has also been studied, forexample, by Filippov and Donskaya (335,336) on VO2+,Cr3+, and Mn2+ in aqueous solutions up to 6000 atm.Grouchulski et al (324) studied temperature dependencein MnTe at 1,2, and 4 kbar.

Continued on p. 184


Vol. 1, No. 4 181

Table 8. Linewidth Temperature Dependence in Liquids and Solids

System

(p-x-acetophenone)+

x=NO2,CH3,H, Br,and OCH3

benzene anion,tropenyl

C102

dinitro radicalsDPPHamino acid radicals

semiquinones

tanone

several radicalsTi(H2O)63+

Cr, Mo, Wdibenzyl iodide

MnCI2-KCIMn(CIO4)2

Mn(NO3)2

Mn(CIO4)2

Mn=+, Fe3+

K3Mo(CN)8,K3W(CN)8

Eu2+/AgCI

B

As-doped Ge

blue d'outre-mer

BrO32"/KBrO3

butadieneradical

O2/N2

CO3-/CaCO3

CIF6,BrF6,IF6/SF6

Ge 'H center/Li

n-type InSb

State orsolvent

DMF

THF, DME

TBPDMFtolueneglycerol

ethanol,butanol

isopropanoland others

THF, DGDEaqueous acid

ethanol

selfethanol, DMF,

DMSO, CH3CNH2O,H2O-

glycerol,H2O-sugar

DMG,DEF,DMSO

DMF, TBP,methanol,ethanol

methanol,ethanol

single crystal

amorphous,single crystal

solid

solid

single crystalsilica gel

surfacesolid .single crystalsolid

amorphoussingle crystal

solid

Atf-vs-temperaturegraph

vsi/r

vs j ; / T

vs Tlt\nono

VSJJ /T

no

vsi/rvsij/T

no

VSTJ/T

yes

—

vsi/r

vsi/r

vsi/r

yes

yes

yes

yes

yesno

noyesno

yesyes

yes

Temperature, K

241-322

150-275

variable193-293232-314200-291

77-300

140-300

250-310230-290

133-293

775-1275250-375

—

295-430

180-370

180-360

93-300

1.5-1000

1.5-4.2

77-525

26-25077-195

1-2077-40027-200

4-3007-150

1.5-40

Comments

acetyl hindered rotation

CW saturation

35CIHFSout-of-phase correlationspectra vs temperaturerotational diffusion,

anisotropicinteraction

tumbling studied

transition from normalto supercooled liquid

relaxationplot linear through

glass formationspectra vs temperature

dipolar and exchangeOh and tetrahedral

complexes—

fine-structuredependence

fine-structuredependence

14NSHFS,X and Q bands

activation energydetermined, bleachingstudied

activation energy,hopping frequency

concentrationdependence studied

AH increases withtemperature

jump modelirradiation, 3 radical

types—

3-site hopping2 sites with different

AWO2, annealing studiedreorientation motion

broadening above150 K

experiment and theory

References

258

297

298261299237

300

301-303

272304

305

279276,306

278

162

307

308

309

310

288

311

18282

"177259312

313,314255

285

182

Duplication of Bulletin of Magnetic Resonance,//? whole or in part by any means for any purpose is illegal.

Bulletin of Magnetic Resonance

Table 8. Linewidth Temperature Dependence in Liquids and Solids (Continued)

System

donors/CdS

SfilmP-doped Si

vinylV2MoO3

Cr2O3

CrBr3

Cr3+/GASH(Cd, Zn, Hg)Cr2

Se,S)4

Mn2+/MgTeMn2+/TIN3

Mn(CH3COO)2-4H2OMn2+/Mg(HCOO)22H2OMnTeMnO, MnS, MnF2,

KMnF3, RbMnF3>CsMnF3

Fe+,Fe3+,Mn2+/MgO

Fe+/voltaite

Fe°, Fe+/Si

F+,Fe3+ ,Mn2+ /Mg0NiSiF6 • 6H2OCu(HCOO)2 • 4H2O

Zn/Cu acetate

Ce, Nd, Sm inLaMg nitrate

GdtCo^NI,.^

GdNi,GdNi3,GdNi5

GdFe3

GdinY(Gd, Dy)AI2

Ho3+

State orsolvent

single crystal

amorphoussingle crystal

silica gelsingle crystal

polycrystallinesingle crystal

single crystalsingle crystal,

polycrystallinesingle crystalsingle crystal

single crystalsingle crystalpolycrystallinesolid, liquid

single crystal

powder

single crystal

single crystalsolidsingle crystal

single crystal

single crystal

polycrystalline

solid

film

single crystalsolid

single crystal

AW-vs-temperaturegraph

yes

yesyes

noyes

yesyes

noyes

noyes

yesyesyesyes

yes

yes

yes

yesyesyes

yes

yes

yes

yes

yes

yesyes

yes

Temperature, K

1.4-4.2

300-130020-300

77-3004-300

290-52520-550

1.6-300100-300

4,77,300253-353

77-37377,178,298

300-360300-1500

4.2-300

1.6-293

90-170

4.2-300—

4-260

20-120

2-26

100-300

77-700

4-550

3-20150-205

1.2-25

Comments

hopping frequencycalculated

annealing studiedlocalized and delocalized

electronsreversible changesAH angle-dependent,

e-hoDDina1%and6%Gaferromagnetic,TN =

32.5 K

T > fN, relaxationdiscussed

AH decreases astemperatureincreases

AW angular dependence2-dimensional system1—4.3kbarmeasured through

melting point

AH dipolar at lowtemperature,relaxation-limited athigh temperature

measured above 0.7 Kferromagnetictransition

Fe° —• Fe+ transition athigh temperature

relaxation narrowinguniaxial anisotropyAH angular dependence,

e-hoDDina\J 1 1 V* mj B̂ I I I ^M

Lorentzian above 50 K,Gaussian wimLorentzian tailsbelow 40 K

7",/7"2 measured

concentrationdependence studied

AH = a + bT, localizedmoments

AH frequencydependent 8.4-35 GHz

AH angle dependentAH depends on

Gd/Dy ratioTf measured

References

286

313315

316317

24318,319

320321

322323

29136324325

326

327

328

32688,89

24,90,329

330

331

293

332

333

.45295

56

Vol. 1, No. 4


183

Table 9. Systems with Hyperfine Component Widths AH = A + BM + CM2 + DM3

XV. HYPERFINE COMPONENTDEPENDENCE

Hyperfine multiplets from radicals or transition ions insolution often exhibit a linewidth that varies with themagnetic nuclear spin quantum number M from onecomponent to the next across the spectrum. Rogers andPake (337) recorded the hyperfine structure of the van-adyl ion VO2

+ at 9.25 and 24.3 GHz and found that thecomponent dependence of the width was also a functionof frequency. Kivelson and co-workers (220-222,229)explained this in terms of incomplete averaging of g-factor and hyperfine coupling anisotropies and derivedthe formula

AH,A = AHO + A + BM + CM2 (71)

where the linewidth AH is related to the spin relaxationtime T2, through a rearrangement of equation (62), by

2

in strong magnetic fields (e.g. 103 or 104 gauss), theterms A, B, and C are given by (111)

(73)

(74)

where the factors containing Ag and Aa arise from thedeviations of the g-factor and hyperfine tensors fromtheir respective isotropic parts g0 and ao

(AaF = (a^ - ao)2 + (aw - a0)

2 + ( a s - ao)2 (75)

= (ax>. - ao)(axx - go) ++ (azz-a0)(gzz-g0)

- go)(76)

and AHO is the residual linewidth which often comes fromspin-rotational interactions. In the limit of rapid tumbling

Other interactions can also contribute to A, B, and C.The factor (nBHo/h) causes the widths to depend on

the frequency band. The term BM gives the spectrum anunsymmetric appearance. A more refined theory adds

Paramagnetic species

anthracene, perylene,tetracene

p-benzoquinone anionCF3CONH2

dichloro acetamidedi-t-butyl nitroxidenitroxidenitroxideperoxylamine disulfonateVO2+VO2+

VO22+ acetylacetonate

VO22+ diketonate

VO22+ octaethylporphyrin

Cu2+,c-asparaginCu (CIO4)2

Cu(NO3)2

Nb(C5H5)4

(dihydroxydurene)+

(dinitrodurene)-,dinitromesitylene

(m-dinitrobenzene)~hydropyrenepyrazine-alkali

Solvent or matrix

silica aluminasurface

H2O, ethanol, DMSOtrifluoracetamide SCirradiated single crystalsupercooled H2Osolutionsolutionglycerol-H2Oaqueous solutionalkali halides

various solutions

several inert solventsglass, 100 Kaqueousaqueous and methanolliquid ammoniatoluene

solutionsolution

solutionDMG, THF, MTHF, H2SO4

THF, MTHF'

Comments

antisotropic rotation

—irradiationbroader upfield linescomputer simulation180—380 K, 14N, 15Nnematic-isotropic phases17O doped, theoryX, K bandsAH decreases as

temperature increasessecular, nonsecular,

residual17/7" plotsAH inversionpowder, glass, solutionNMR relaxation195-300 Ksolution and frozen,

nl 7"DlotSiff 1 |JlwlU

alternating widthsalternating widths

alternating widthsalternating widthstriple ions

1References

281

34017

342343,344345,346

347242,243

337348

220,229,349

339350351352353354

355356

357358-360361,362



the terms EM3 and FMA to equation (71), but these areusually small. For rotational diffusion in the Debye ap-proximation, rc is given by equation (57).

Wilson and Kivelson (229) sorted out the secular andnonsecular contributions to A, B, C, and D and obtainedthe residual linewidth AH0 by subtracting calculatedvalues of A from measured values of A + AH0. For van-adyl acetylacetonate, AHO came from spin-rotationalrelaxation and was proportional to Tli\. Residuallinewidths in solutions commonly derive from the spin-rotation interaction (338). Eagles and McClung (339)studied vanadyl /3 diketonate complexes and founddeviations from the Kivelson linewidth theory at lowtemperatures and high viscosities.

Equations (71) through (76) were derived for the highfields typical of X-band (3400 gauss, 104 MHz) spectra.Barbarin and Germain (340) derived analogous expres-sions for low fields (30 gauss, 80 MHz) and carried outmeasurements at both high- and low-frequency bandsto evaluate the contribution to the linewidth from the g-factor and hyperfine anisotropies.

Bogomolova et al (218) analyzed the powder spectraof Cu and V glasses with the aid of the expressions

(77)

i4, + BtM+ CtM2

2± = A±+ B±M + CXM2

AW2 = AH2 cos20 + AH\ sin20

Freed et al (341) used a stochastic Liouville approachto the theory of the hyperfine component dependence ofthe linewidth.

Table 9 summarizes other recent work on thisdependence.

XVI. ALTERNATING LINEWIDTHS

Freed and Fraenkel (363) considered linewidths fromhyperfine patterns in which nuclei are symmetrically butnot completely equivalent. This means that the nuclei areequivalent with respect to the zero-order Hamiltonian sothat they have the same gyromagnetic ratios (Y, = jj),spins (/, = Ij), and hyperfine interaction constants (At =Aj), but are not equivalent with respect to a perturbingHamiltonian such as the dipolar interaction (Wdd, =tWdd/). Completely equivalent nuclei also have the sameperturbing Hamiltonian (HMi = Hadj). The particular casestudied concerned two nuclei with widths that vary asCM2 when the nuclei are completely equivalent, i.e.,

where

and

M = M, + M2

\M\ <2/

(79)

(80)

If the two nuclei are incompletely equivalent, then thewidth of a hyperfine line with M = M-t + M2 depends onthe magnetic quantum numbers as follows

= 7n j22 M2 + (y12 + y21)

Since ju = j22 and y12 = y21, this becomes

A H * = / „ A*2 + 2(/12 - / „ ) Mi M2

(81)

(82)

AW,, = CM2 (78)

Each hyperfine line has a degeneracy of 1 if M^ = M2 and2 if M, ¥= M2, since in the latter case (Mf, M2) and (M2,MJ give superimposed lines.

Table 10 presents the results for AHV2 obtained withthe nuclei / = 1/2, 1, and 3/2. The last three columns ofthe table give the values for the in-phase case (y12 = j u ) ,where the nuclei are completely equivalent and equation(82) reduces to equation (78), the out-of-phase case(y12 = -y'n), and the uncorrelated case (y12 = 0), whichhave the respective widths

n u (M, + /W2)2 y12 = y'n in phase

AWy2 = / y'n (W, - M2)2 y12 = -ju out of phase (83)

O n ( w i 2 + M22) y12 = 0 uncorrelated

The general case produces linewidths which vary in acomplex manner from one hyperfine component toanother. For some choices of the ratio / 1 2 / / u , certainhyperfine components have very narrow widths andothers are broad. The theory explains some observedspectra in which the widths of successive componentstend to alternate between wide and narrow, and otherobserved spectra in which the component amplitudesdeviate from the binomial case due to the broadening ofsome components beyond detection. For example, inthe out-of-phase case with A, = l2 = 1, if y^ is very large,the M = ± 1 lines will be broadened beyond detectionand the hyperfine pattern will appear as an equal intensi-ty triplet arising from the M = 0, ± 2 lines, rather than asa 1,2,3,2,1 quintet. This phenomenon has been ob-served experimentally (363).

Goldman et al (242) fitted the linewidth data of170-labeled peroxylamine disulfonate in glycerol watermixtures to the expression


Vol. 1,No. 4 185

Table 10. Secular Linewidths, Isotropic Modulation of Two Nuclei (from reference 363)

/

Vz

1

V,

M

± 10

± 2± 1

0

± 3± 2± 1

0

(±y2, ±y2)[±y2, +y2]*(±1, ±D[±1.0]

(±1 , +1)(0,0)

±V2, ±V2

[±3A, ±y2][±3/2, +1/2]±y2, ±y2

[±y2, +3A][±y2, +y2]

Dk

12

1221

122122

General

1 /2(/ i1+/i2)

V2(7i 1-/12)

2(/n + /i2)/11

2(y" 11 /12)0

'zUH "•" /i2/

V {̂5y-i -j — 3yi2/

VH/11+/12)VaCfn- /^)1/2(/11 — /12)

In-phasecorrelated

7i2 = 7n

7n0

4 / n7 i i

00

9/ii4 / n/11

/11

00

Out-of-phasecorrelated/12 = -/11

0

711

0

7n4/n

0

0

/'114/n

0

9/n711

Uncorrelated/12 = 0

V2/1,

2/n/'112/n

0

V2/11

V2/11

Vzjn9 / 2/n

1 / 2 / 1 1

'Square brackets indicate inclusion of both (m,, m2) and (m2,mi) states.

BOMOMO

+ DMNM0 (84)

where the subscript N refers to the nitrogen nuclei (/N =1) and O denotes 17O (/0 = 5/2). The constant 71N is thewidth of the central line (MN = 0) in the undoped radical.The various factors were determined at 5,13, and 21 °C.High power studies indicate that the three hyperfine linesof the undoped radical saturate at different power levels,the central component saturating first. Table 9 lists arti-cles which discuss alternating linewidths.

XVII. FINE STRUCTURE EFFECTS

The linewidth within a fine structure multiplet can alsovary with the particular fine structure transition| M S > * * | /Ws±1> due to modulation of the zero-fieldHamiltonian term SDS. Using the notation

1 _ . _dxx= - -

(85)

we can write for the inner product of the traceless Dtensor

2 2 2

= | 7 o 2 + 3E2). " (86)The widths and intensities of the five fine structure linesfor the case of spin S = 5/2

± 3/2>

± (87)

depend upon the inner product (Ad)2 and the correlationtime TC.

Burlamacchi (307) calculated AA//(Ad)2 and the inten-sity as a function of WOTC for these transitions, as shownby the graphs in Figure 10. We see from these plots thatfor WTC « 1, the total intensity converges to the central| Vz>++1 - 1 /2> transition. For «OTC » 1, the other transi-tions broaden beyond detectability and only the|1 /2>**| —1/2> transition is observed. Nearworc ^ 1, thecentral transition reaches a maximum width, and nearO)OTC =s 1.5 to 2, the | + 3 / 2 > * * | ±1/z> transition partial-ly contributes to the observed spectra. Burlamacchi andcoworkers (276,306,307,364) confirmed this theoryexperimentally with Fe3+ and Mn2+ in various solventsover a wide range of temperatures. Also see(107,162,234,365).

Reed et al (366) calculated linewidths and transitionprobabilities for the equation (87) transitions as eigen-values and eigenvectors, respectively, of a 7"2 relaxationmatrix for Mn2+

(A D E 0 0s

D B 0 F 0

E 0 C 0 £ I = 0 (88)

0 F 0 B D

0 E D AJ

where the rows and columns are labeled by \MS> =|2>, |1>, |0>, | - 1 > , and | - 2 > , and the matrix



- 6

(a)

-2(6JO+

- ( 3 J 0 + 18 J, + 2ZJ2)

+ 7J2)

Figure 10. Theoretical curves for a sextet electron spin stateas a function of O>TO: (a) linewidth A H normaliied relative to thezero-field term D, (b) transition intensity (from reference 307).Transition is indicated on each curve.

elements are

A =

B=

C = ( ^ 2)

^ (89)

F=

Graphs of calculated widths and intensities versus TC re-sembled those in Figure 10.

Reed et al also found that an individual Mn2+ solutionspectrum was compatible with a range of TC and(£>2 + 3 f2). values, but that the simultaneous fit ofX-band (9.1 GHz) and Q-band (35 GHz) spectra from thesame solution corresponded to unique rc and (D2 + 3E2)values. Table 9 summarizes their results for variousMn2+ complexes. Rubinstein et al (367) obtained similarresults with aquo Mn2+ using 1H NMR.

Levanon et al (368) performed X-band and Q-bandstudies of a series of ferric aqueous complexes[Fe(H2O)nF6_,,]3-« that had linewidths which varied withvalues of n. They determined equilibrium constants forthe various complexes.

Maniv et al (264) studied Mn2+ in single crystals ofCs2HfCI6 and Cs2ZrCI6 and found fine structure effectsanalogous to the solution cases described above. Thesearise from jumps of the D-tensor axis of the (MnCI6)

3-group between three equivalent crystallographic cubicaxes defined by the missing Cl~ ions in the XCI6 octahe-dron that would be present in the absence of themanganese. Sharma (100) found that the linewidth ofGd3 + in rare earth trifluorides varies with the individualfine structure transition.

XVIII. SPIN LABELS

Nitroxides and other stable free radicals called spinlabels have been used as probes to monitor molecularmotion in liquids and polymers and especially inbiological systems. Spin labels are attached to or oc-cupy positions near active sites in proteins or othermolecules. They produce spectra which are sensitive tothe anisotropies of the local environment and to the ex-tent that molecular motion averages these anisotropies.Variations in linewidths and shapes can serve as indica-tors of changes in the environment. In addition, thehyperfine coupling constant A and the g-factor are sen-sitive to the extent to which the environment is hydro-phobic or hydrophilic.

The most commonly used spin labels are nitroxideswhich contain an «NO group with an unpaired electron onthe nitrogen atom. Examples of the molecular configura-tion of spin labels are peroxylamine disulfonate (I), andpiperidone-1-oxyl (II) which has been used, for example,in its 2,2,6,6, tetramethyl form. They exhibit a three-linehyperfine pattern from the / = 1 nucleus of 14N, and both

t

so3" so3-

fNo0

II


Vol. 1, No. 4 187

A and g are generally anisotropic with principal direc-tions oriented so that x is along the N-0 band direction,z is perpendicular to the «N-0 plane, and y is in this planeand perpendicular to x. The principal values of the A andg tensors are obtained by computer simulation for highviscosity solutions. Typical values are

membrane plane. They applied the distribution function

Pm = sintf exp[(0 - 6o)2/2<rz] (92)

gx = 2.008

gy = 2.006

gz = 2.003

Ax = 5.5G

Ay = 4.0G

Az = 30.0G

(90)

Freed (369) analyzed motional effects on nitroxidespectra in the slow-tumbling region where the correla-tion time rc is between 10"9 and 10~6 s (see section XII).When the rotational diffusion constant D is isotropic, TC

is approximated by the Einstein-Stokes relationship,equation (55). For peroxylamine disulfonate, D is axiallysymmetric, with D\\ along the y direction and D± in thexz plane of the coordinate system defined above. Therotational correlation time rR, which is analogous to TC inequation (57) is given by (369)

1 (91)

The value of rR and the ratio D\/Dj_ for a particulartemperature may be obtained by evaluating the factorsA, B, and C of equation (71) as explained by Freed (253).

Mason et al (370) treated the case of a riitroxideradical undergoing rapid rotation about a single bondwith the correlation time rR\ while it is attached to amacromolecule which is slowly reorienting with the cor-relation time rRX. They simulated spectra for variousrates of reorientation. Benton and Lynden-Bell (371)studied a similar case.

Livshits (372) calculated lineshapes for the slow rota-tional region, T between 10~7 and 10-*s, and obtainedexpressions for the widths of each hyperfine component.He plotted shifts of the m - ± 1 lines from their rigid lat-tice positions as a function of the correlation time. Sam-ples in this slow rotational region are convenientlystudied by an adiabatic fast-passage method and withthe aid of an ELDOR spectrometer:

Van et al (373) presented a general treatment of mo-tional averaging at the rapid motion limit T C « 10~8 S.After developing the general formalism, they consideredspecific examples of preferential rotation about a givenaxis, the nitroxide principal axes in particular; wobble,regarded as a random walk over a spherical region nearthe pole characterized by 0 < 0O; and motion restrictedto oscillations in a plane.

Libertini et al (191,192) treated spin labels diffusedinto fluid bilayer regions of biological membranes con-taining lipids which tend to be oriented normal to the

For small a, this is a Gaussian distribution, but for large ait becomes sinusoidal. Gaffney and McConnell (374)used both equation (92) and the more complex distribu-tion function

P'M = sin 0exp [(0-0o)2/2<r2] exp [^/2^] (93)

in their study of phospholipid membranes.Tenny et al (375) employed a stochastic method to

analyze the spin label results from rat liver and hepa-toma mitochondrial lipids. They found that the logarithmof the correlation time increased linearly with the re-ciprocal temperature.

Shiotani and Sohma (376,377) studied molecular mo-tions in polymethylmethacrylate in the temperaturerange 77 to 500 K with the use of spin labels. Thepowder-pattern lineshape was partially averagedbetween 360 and 410 K and was isotropic above 470 K.They analyzed the temperature dependence of line-widths and shapes to obtain correlation times and ac-tivation energies.

Nagamura and Woodward (378) used a nitroxideradical probe embedded in a polymer matrix to monitormolecular motion. In contrast to its use in biological ap-plications, the probe was only weakly held by, ratherthan bound to, the polymer chains. Poly (trans 1,4 bu-tadiene) produced a broad unsymmetric spectrum at lowtemperatures which froze polymer motion, and a sharptriplet at high temperatures which allowed the probemolecules to rotate freely. Bullock et al (379) alsostudied spin-labeled polymers.

The spin probes discussed so far have been ordinaryradical doublet states (S = 1/2). Biradicals (347) and tri-radicals (380-383) have also been used as spin probes.These multiple radicals have, respectively, triplet-singletand quartet-doublet zero-field splittings which are sensi-tive to the anisotropies of the environment. As a result,linewidth and shape variations occur which serve tomonitor such anisotropies and any motions that causeaveraging.

Table 11 summarizes some of the recent work on spinlabels.

XIX. AQUEOUS ENVIRONMENTS

The dielectric constant e of water has large real andimaginary parts

e = e' - )e" (94)

As a result, dielectric losses modulate the magnetic



Table 11. Representative Spin-label Reports with Linewidth and Lineshape Data

Spin label

di-t-butyl nitroxide5-doxylstearic acid

di-t-butyl nitroxide

imidoxyl

maleimide, tanolnitroxide, doxylsteroidcholestanol

tetramethyl-4-hydroxyIpiperidine-1-oxyl

nitroxide

nitroxidetanone, imidoxylnitroxidenitroxidenitroxide

12-nitroxide-stearicacid

nitroxyl

tanol, HDAtetramethyl-4-

piperidinol-1-oxyl

Substrate

egg lecithin

—

.—

hemoglobinpolymermembranesmembranes

polymethyl methacrylate

rod-shapedplastic

—rod-shapedpolystyrene

hepatoma lipids

—butadiene

Solvent

adamantane—

solution

glycerol,polyethylene

glycerolDMF

—

—

glycerol,polyethylene

supercooled H2O—

supercooled H2O—

toluene, cyclobenzene,chloronaphthalene

—

decalin,ethyl benzene

sec-butyl benzeneheptane, toluene

Comments

180-250 Korientational

distribution modelactivation energy,

correlation'timeanisotropy studied

—anisotropic rotation theoryrapid anisotropic motionRedfield relaxation

theorysee text

14N,15N HFS

240 - 288 Kinhomogeneous, ELDOR240 - 288 Ksimulated spectrarotating chain ends

stochastic-

anisotropy,additional broadening

ELDORactivation energy

References

384191,192

344

345

385370373374

376,377

345,346

343386343371379

375

387

385,388,389378

losses of paramagnetic spins in aqueous solutions. Theeffect is analogous to the modulation of magnetic lossesby conduction losses in the ESR of metals (126,390).

For aqueous samples that are thin compared with thewavelength, for example, 0.2 mm for the X-band, thedielectric loss effect is negligible and the lineshape is ab-sorptive. For thick samples, such as 1 cm for the X-band,the shape is a mixture of absorption and dispersion. Thiswas demonstrated (391) by measurements at 8.84 GHzwhere the dielectric constant is

e = €O(63 - J31.8) (95)

Most solution studies employ thin aqueous solution cellsfor which this effect is not very important.

To explain the results, a calculation was made (391) ofthe lineshape for a thin (0.1-mm) layer of spins bathed oneach side by a 4.65-mm-thick planar layer of dielectricconstant eo(63 - j50). It was found that the shapechanged from absorptive through dispersive to absorp-tive of the opposite phase as the dielectric constant ofthe spin-containing region varied from eo(23 — j50) toeo(43 - j50) to «O(63 — j50). The sample dielectric con-

stant that gave a purely dispersive signal varied with theimaginary part of the dielectric constant of the bathingsolution. The authors applied their technique to thestudy of biological membranes.

XX. TOPICS OMITTED

This review by no means exhausts all aspects of line-shapes and widths in electron spin resonance. Sig-nificant material could have been added to all the topicscovered, and in some cases theory was hardly touchedupon. A number of important topics were not covered,such as the ESR of ordered systems and phase transi-tions, acoustic and conduction ESR, Jahn Teller effects,liquid crystals, triplet states, and optical excitation.

ACKNOWLEDGEMENT

The authors wish to thank Mr. Willie Zimmerman forhis help in organizing the tables and references in thisarticle.


Vol. 1,No. 4 189

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