Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

Подождите немного. Документ загружается.

See also: Amorphous and Nanocrystalline Materials;

Amorphous Materials: Electron Spin Resonance;

Amorphous Materials: Nuclear Magnetic Resonance,

New Techniques; Amorphous Materials: Nuclear

Spin Relaxation; Amorphous Superconductors;

Magnetic Systems: Disordered

Bibliography

Chudnovsky E M 1995 Random anisotropy in amorphous al-

loys. In: Fernandez-Baca J A, Ching W-Y (eds.) The Mag-

netism of Amorphous Metals and Alloys. World Scientific,

Singapore, Chap. 3, pp. 143–74

Cote P J, Meisel L V 1981 Electrical transport in glassy metals.

In: Gu

ntherodt H-J, Beck H (eds.) Glassy Metals I. Springer,

Berlin, Chap. 7, pp. 141–66

Cusak N E 1987 The physics of structurally disordered matter.

In: Brew D F (ed.) Graduate Student Series in Physics. Hilger,

Bristol, UK

Dugdale J S 1987 Electron transport in metallic glasses. Con-

temp. Phys. 28, 547–72

Elliot S R 1983 Physics of Amorphous Materials. Longman,

London

Hamed F, Razavi F S, Bose S K, Startseva T 1995 Spin ﬂuc-

tuations in metallic glasses Zr

(Ni

1x

)

at ambient and

high pressures. Phys. Rev. B 52, 9674–8

Harris R, Strom-Olsen J O 1983 Low-temperature electron

transport in metallic glasses. In: Beck H, Gu

ntherodt H-J

(eds.) Glassy Metals II. Springer, Berlin, Chap. 10, pp. 325–42

Howson M A, Gallagher B L 1988 Electron transport prop-

erties of metallic glasses. Phys. Rep. 170, 267–324

Markowitz D 1977 Calculation of electrical resistivity of highly

resistive metallic alloys. Phys. Rev. B 15, 3617–9

Mooij J H 1973 Electrical conduction in concentrated disordered

transition metal alloys. Phys. Stat. Solidi (a) 17, 521–30

Mott N F, Davis E A 1979 Electronic Processes in Noncrystal-

line Materials. Clarendon, Oxford

Nath T K, Majumdar A K 1997 Quantum interference effects

in (Ni

0.5

)

1x

metallic glasses. Phys. Rev. B 55, 5554–7

Poon S J 1983 Superconducting properties of amorphous me-

tallic alloys. In: Luborsky F E (ed.) Amorphous Metallic Al-

loys. Butterworths, London, Chap. 22, pp. 432–50

Rao K V 1983 Electrical transport properties. In: Luborsky F E

(ed.) Amorphous Metallic Alloys. Butterworths, London,

Chap. 21, pp. 401–31

Ziman J M 1972 Principles of the Theory of Solids. Cambridge

University Press, Cambridge

R. Cywinski and S. H. Kilcoyne

University of Leeds, Leeds, UK

Amorphous Materials: Electron Spin

Resonance

Electron spin resonance (ESR) is a spectroscopic

technique that uses the spin magnetic moment of the

electron as a probe of its local environment. Only

unpaired electrons can participate in ESR. In most

insulating materials virtually all electrons are paired

and their magnetic moments cancel. However, a few

unpaired electrons may be present in the form of

electrons or holes trapped at point defects or located

in partially ﬁlled d or f shells of transition group ions.

ESR is sufﬁciently sensitive to detect unpaired elec-

trons in concentrations sometimes as low as a few

parts per billion.

1. ESR Theory and Practice

The spin S and spin magnetic moment m of the elec-

tron are related quantum mechanical vectors:

l ¼gbS ð1Þ

where b ¼7e7h/4pmc is the Bohr magneton, e is the

charge of the electron, m is its mass, h is Planck’s

constant, c is the speed of light, and g is a dimen-

sionless number. For the free electron g ¼g

2.00232, and l ¼l

g

bS. When any magnetic di-

pole l is placed in an external magnetic ﬁeld H, its

Zeeman interaction energy is given by the scalar

product

Zeeman

¼l  H ð2Þ

A quantum-mechanical property of spin angular

momentum is that its projections on the direction of

H are limited to the values M

¼S, S–1,y,–S, where

the scalar quantity S is termed the ‘‘spin.’’ For the

free electron, S ¼

. In general, atoms or assemblies of

atoms binding N electrons may assume net spins as

large as N/2, but in practice S ¼

for most point de-

fects, and S for transition group ions can be no larger

than

multiplied by the number of electrons in d and/

or f orbitals.

The most common ESR spectrometer exposes sam-

ples to microwaves of a ﬁxed frequency u and an

applied magnetic ﬁeld H. The allowed transitions be-

tween the Zeeman levels of Eqn. (2) are those for

which DM

¼71. Near thermodynamic equilibrium

there are more electrons in the lower energy Zeeman

states than the higher ones, resulting in a net ab-

sorption of microwave power when 7H7 ¼H

res

, given

by the ‘‘resonance condition’’

res

ð3Þ

There is also a magnetic moment associated with

the orbital angular momenta of electrons circling

about atoms. However, when atoms are embedded in

solids, local electric ﬁelds (‘‘crystal ﬁelds’’) tend to

‘‘quench’’ the orbital angular momentum of their

ground states leaving a magnetic moment due mostly

to the spin, i.e., lEl

. Nevertheless, the spin-orbit

and orbit-Zeeman interactions admix small amounts

of orbital angular momentum back into the ground

state. These residual orbital effects require the

Amorphous Materials: Electron Spin Resonance

replacement of gS in Eqn. (1) with g S. The so-called

‘‘g matrix’’ is expressed by (e.g., Weil et al. 1996)

g ¼ g

1 þ 2lK ð4Þ

where 1 is the unit matrix and l is the spin-orbit

coupling constant. The elements of the matrix K take

the form (correct to second order in perturbation

theory):

/07L

7nS/n7L

70S

ðE

 E

ð5Þ

where L

and L

are orbital-angular-momentum

operators and 70S and 7nS are the ground- and

nth-excited-state wavefunctions, respectively, of the

paramagnetic species and E

and E

are their energies.

The energetics of unpaired electrons subjected to

electric and/or magnetic ﬁelds are contained in the

‘‘spin Hamiltonian’’ H operator, (e.g., Weil et al .

1996). For the case of the Zeeman interaction in the

presence of crystalline electric ﬁelds

H ¼ bH  g  S þ S  D  S ð6Þ

where g is given by Eqns. (4) and (5) and the second

term on the right (sometimes called the ﬁne-structure

interaction) is nonzero only for S4

. When 7H7 ¼0,

S is quantized along the set of axes which diagonalize

D ¼l

K, where K is given by Eqn. (5). In its diag-

onalized form, the diagonal elements of D are com-

monly expressed as

DE,

D þE, and 

D, where D

and E (7E7p7

D7) are the axial and orthorhombic

ﬁne-structure splittings, respectively.

In general, solutions of Schro

dinger’s equation us-

ing Eqn. (6) as the energy operator yield resonance

conditions which depend on the angles between the

direction of the applied ﬁeld H and the principal axes

of g and D (which in turn relate to axes of crystalline

symmetry). For example, in the case of a paramag-

netic center with S ¼

located on a high-symmetry

site in a crystal with a three-fold axis of symmetry or

higher (i.e., a c axis), the value of g to be used in the

resonance condition of Eqn. (3) is found to be

g ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

cos

y þ g

sin

ð7Þ

where y is the angle between H and the c axis.

Many atomic nuclei have nonzero quantum me-

chanical spin vectors I and, by analogy with Eqn. (1),

nuclear magnetic moments l

g

I, where g

the nuclear g factor and b

¼7e7h/4pMc is the nu-

clear magneton (M is the proton mass). The projec-

tions of I on the direction of an imposed magnetic

ﬁeld can take only integral or half-integral values

given by M

¼I, I1, ...,I, where the scalar quantity

I is called the ‘‘nuclear spin.’’ A single unpaired elec-

tron whose ground-state orbital bears a ﬁxed geo-

metric relationship to a companion ‘‘magnetic’’

nucleus (Ia0) is therefore subjected to the vector

sum of the applied magnetic ﬁeld H and one of 2I þ1

possible magnetic ﬁeld values corresponding to the

2I þ1 possible orientations of the nuclear magnetic

moment vector l

. In these cases, a hyperﬁne term

¼S A I, must be added to the spin Hamilton-

ian of Eqn. (6). In an actual experiment (usually

involving 410

unpaired electrons) a paramagnetic

species with S ¼

which is associated with a single

nuclide of spin I gives rise to a manifold of 2I þ1

equally intense hyperﬁne lines with spacings which

depend on the angles between the ﬁeld H and the

principal axes of the hyperﬁne matrix

A ¼ A

iso

1 þ T ð8Þ

where A

iso

is the isotropic hyperﬁne coupling con-

stant and T is a traceless matrix accounting for the

angular dependence. A second-order-perturba-

tion-theory resonance condition for H ¼bH g S þ

S A I assuming axial symmetry and collinear g and

A matrices can be found, e.g., in Griscom (1990) or

Weil et al. (1996). For the special case of a p orbital

centered on the magnetic nucleus, T is diagonal with

elements B, B, and 2B.

Many point defects in insulators are characterized

by sp hybrid-molecular-orbital ground-state wave-

functions of the form

¼ c

7nsS þ c

7npS þ

S ð9Þ

where 7nsS and 7npS are, respectively, s- and p-state

valence orbitals of an atom at the apex of a triangular

or trigonal-pyramidal cluster of atoms and 7k

S are

the remaining valence orbitals of the cluster. Where

the apex atom incorporates a magnetic isotope, the

coefﬁcients c

and c

can be determined from the

experimental hyperﬁne coupling constants according

to c

¼A

iso

and c

¼B/B

, where A

and B

are

s-state and p-state coupling constants of the free atom,

respectively, available from tables (e.g., Weil et al.

1996). In general, there is a direct relationship between

the hybridization ratio c

and the apex bond angles

(see, for example, references in Griscom 1990).

2. ESR in Crystals and Powders

Figure 1(a) exhibits a plot of the expected positions of

res

(y) speciﬁed by Eqns. (3) and (7) for a particular

set of parameters (g

¼2.185, g

¼1.999, and

n ¼9.23GHz). If this crystal were to be crushed into

a ﬁne powder, the resulting ESR signal would become

‘‘smeared’’ between the ﬁeld values H

min

¼hn/g

and H

max

¼hn/g

b. However, this smearing is not

random. In practice, all solid-angle orientations O of

the particle’s c axes with respect to the direction of H

are represented with equal probability. This uniform

orientational distribution, P

(O) ¼constant, trans-

forms into a function P

res

) ¼P

(O) 7J

(O)7,

Amorphous Materials: Electron Spin Resonance

where J

(O) is the Jacobian of the transformation

O-H.P

res

) is called a powder pattern and ex-

presses the probability of resonance absorption per

unit interval of H.

Figure 1(b) is the powder pattern for the case in

point, where J

(O) ¼[@H

res

(cos y)/@ cos y]

1

and H

res

is given by Eqns. (3) and (7) (the vertical arrow marks

a mathematical singularity which would occur if the

single-crystal linewidth were inﬁnitesimally narrow).

The ﬁrst derivative of this envelope is shown as the

bold lines in Fig. 1(c). The thin curve in Fig. 1(c) is an

experimental ﬁrst-derivative spectrum (solid curve)

arising from superoxide ions (O



) in polycrystalline

. The dotted curve is a computer simulation

based on the (approximately valid) assumption of

an axially symmetric g tensor. This simulation is the

result of convoluting the powder pattern with a

Lorentzian single-crystal line shape of an appropriate

ﬁnite width (Griscom 1990).

3. ESR in Amorphous Materials

Glasses and other amorphous materials differ from

powdered crystals in a fundamental way. Whereas

each particle of a crystalline powder has a structure

and density identical to all others, the structure of

each nanoregion of a glass differs randomly from

other nanoregions of equivalent size. In particular,

glasses display spatial density ﬂuctuations. Thus, a

simplistic but instructive way to visualize a glass

would be as a nanocrystalline solid wherein the en-

semble of randomly oriented nanocrystallites is char-

acterized by a statistical distribution of densities.

3.1 How ESR Spectra of Amorphous Materials

Differ from Those of Powders

It can be shown theoretically that an O



ion in free

space (or in a material of zero density) is character-

ized by g

¼4, g

¼0, whereas in a hypothetical ma-

terial of inﬁnite density g

¼g

. Figure 2(a)

illustrates the predicted angular dependencies of the

ESR spectra of O



ions in a material of ﬁnite density

subjected to ten incrementally different isostatic pres-

sures. The decreases of g

and correlated increases of

with increasing pressure were calculated from the

theory of Ka

nzig and Cohen (1959) by approximat-

ing their formulae to axial symmetry. Figure 2(b)

shows the result of adding together the powder pat-

terns corresponding to each of these 10 cases accord-

ing to a bell-shaped weighting function. The subtly

rounded shape of the ‘‘glass pattern’’ which would be

obtained by replacing the 10-element summation of

Fig. 2(b) with a continuous integral over the same

weighting function contrasts with the sharply angular

powder pattern of Fig. 1(b).

Figure 2(c) (unbroken curve) illustrates the exper-

imental ﬁrst-derivative spectrum of O



ions in an

amorphous peroxyborate material (Griscom 1990).

The dotted curve in Fig. 2(c) is a computer simulation

based on the correlated skew-symmetric g-value dis-

tributions of Fig. 2(d), which in turn were derived by

assuming a Gaussian distribution P

(D) in the crys-

tal-ﬁeld-imposed splittings D of the 2pp

energy levels

of the O



ion in a solid (Ka

nzig and Cohen 1959).

Figure 1

ESR of crystals and powders using the example of a

resonance condition speciﬁed by Eqns. (3) and (7): (a)

simulated angular dependence data for a hypothetical

single crystal; (b) ‘‘powder pattern’’ absorption in the

case of an inﬁnitesimally narrow single-crystal linewidth;

with superposed experimental (thin curve) and simulated

(dots) spectra due to O



ions in polycrystalline Na

The simulation assumed an axially symmetric g matrix

(which is only approximately true of the experimental

spectrum) and a Lorentzian single-crystal line-

broadening function whose (ﬁnite) width was used as a

ﬁtting parameter.

Amorphous Materials: Electron Spin Resonance

The O



ion is a special case used above for illus-

trative purposes because g

and g

depend in differ-

ent ways on a single energy difference D. In more

common cases where the ground-state wavefunction

is given by Eqn. (9), g

while g

is calculated

from Eqns. (4) and (5). Redeﬁning D as E

– E

Eqn. (5), skew-symmetric distributions in g

can re-

sult according to P

(g) ¼P

(D) 7J

(D)7, where

(D)(@g/@D)

1

is the Jacobian of the transforma-

tion D-g (Peterson et al. 1974) and again P

(D)is

taken to be a Gaussian. Skew-symmetric g-value dis-

tributions arising in this way are endemic to glasses.

3.2 Use of Hyperﬁne Interactions to Determine the

Ground State of a Point Defect in an Amorphous

Material

One of the common radiation-induced point defects

in amorphous SiO

is the so-called ‘‘peroxy radical’’

(POR). As discovered from ESR (Friebele et al. 1979,

Stapelbroek et al. 1979, Griscom 1990), the POR

comprises an O



molecular ion covalently bonded to

a silicon in the glass network. The most important

knowledge concerning the POR derives from meas-

urements of its hyperﬁne interactions with the mag-

netic nuclides

Si (I ¼

, natural abundance 4.7%)

and

O(I ¼

, natural abundance 0.038%). Figure

3(a) shows the experimental spectrum of a neutron-

irradiated SiO

sample enriched to 36 at.%

O. The

inset to Fig. 3 shows the structural model of the POR;

this model implies that oxygens O(1) and O(2) are

chemically inequivalent. For ﬁnite amounts of both

isotopes, there are four distinguishable cases which

must simultaneously contribute to the observed spec-

trum:

O(1)–

O(2),

O(1)–

O(2),

O(1)–

O(2),

and

O(1)–

O(2). For 36%

O, the relative prob-

abilities of these four cases are calculated to be

0.41:0.23:0.23:0.13, respectively. The ﬁrst-mentioned

case effectively corresponds to zero isotopic enrich-

ment; the spectrum for this case (not shown) was ﬁrst

computer-simulated to determine the g-value distri-

butions (Stapelbroek et al. 1979).

Using these g-value distributions as given, the

synthetic spectra of Figs. 3(c) and 3(d) were devised

(Friebele et al. 1979) for the cases

O(1)–

O(2)

and

O(1)–

O(2), respectively. Finally, these two

synthetic spectra were added together in the required

1:1 ratio to achieve the simulation of Fig. 3(b), which

successfully replicates all of the main

O hyperﬁne

lines. (The central part of the spectrum due to the

case

O(1)–

O(2) is not simulated here.) The diag-

onal elements of the hyperﬁne matrices,

¼A

iso

þ2B and A

¼A

iso

B, were determined

from this simulation for both cases. As a check, the

same coupling constants were used successfully to

predict the positions of the more numerous but

weaker lines of the

O(1)–

O(2) case (vertical bars

in Fig. 3(a)). The resulting values of A

iso

and B were

Figure 2

Conceptualization of the ESR spectrum of a glass: (a)

simulated angular dependencies of the spectra of O



ions in a hypothetical single crystal subjected to a range

of isostatic compressions; (b) powder patterns of the

foregoing ten cases summed according to a bell-shaped

weighting function; (c) experimental spectrum of O



ions in an amorphous peroxyborate preparation (solid

curve) and its computer simulation (dots) based on a

statistically weighted set of powder patterns deriving

from (d) correlated distributions of g

and g

values

semiempirically determined under the constraints of the

g-value theory of Ka

nzig and Cohen (1959).

Amorphous Materials: Electron Spin Resonance

then used to determine the coefﬁcients in Eqn. (9),

namely, c

(1)Ec

(2)E0, c

(1) ¼0.26 and c

(2) ¼0.74.

Thus, the ground-state wavefunction of the POR in

amorphous SiO

was deduced to comprise a 2pp an-

tibonding orbital asymmetrically partitioned over

two chemically inequivalent oxygen atoms. As in-

ferred from ESR measurements of samples enriched

Si, the reason for this chemical inequivalence is

that one of the oxygens is bonded to a single silicon

atom in the glass network (c.f., Griscom 1990).

3.3 Transition Group Ions in Glasses

For a single electron (S ¼

)inad or f orbital of a

transition group ion involving a nonmagnetic iso-

tope, the resonance condition is given by Eqns. (3),

(4), (5), and (7). In this event, the skew-symmetric

g-value distributions which occur in glasses are sim-

ply calculated by assuming Gaussian distributions in

the energy denominators of Eqn. (5) and effecting the

transformation D-g as described above (Peterson

et al. 1974). However, in many cases where more than

one electron is found in a d or f shell, details of the

coordination sphere and crystal-ﬁeld strengths deter-

mine not only the matrix elements of D but also the

value of S (e.g., Griscom 1990, Weil et al. 1996). For

the spin Hamiltonian of Eqn. (6) applies. Per-

turbation-theory-based resonance conditions have

been derived for the cases hnb7D7 and 7D7bhn.

Notable is the case of Fe

3 þ

ions (S ¼

) in glasses,

where the conditions 7D7bhn and 7E7E

7D7 com-

monly prevail, leading to an isotropic absorption at

an effective g value of B4.3 (reviewed, e.g., in

Griscom 1990). However, situations where hn and D

have comparable magnitudes necessitate exact matrix

diagonalization of Eqn. (6).

Figure 4(b) shows the allowed transitions of an

S ¼

state determined by matrix diagonalization of

Eqn. (6) for H parallel to each of the principal axes of

D; these transitions are plotted here as normalized

resonance ﬁelds, g

res

/hn, versus the normalized

ﬁne-structure coupling constant, 7D7/hn, for the spe-

cial case 7E7 ¼7D7/3 (Griscom and Griscom 1967).

This diagram pertains to Fe

3 þ

and (neglecting

hyperﬁne interactions) also to Mn

2 þ

. ESR spectra

of Mn

2 þ

in an alkali borate glass recorded at two

separate microwave frequencies are displayed in

Fig. 4(a). Note that the normalized magnetic ﬁeld

scale for the Ka-band spectrum (n ¼35 GHz) is com-

pressed by a factor of B4 relative to the X-band

spectrum (n ¼9 GHz). Figure 4(c) represents a single

Gaussian distribution in 7D7 values plotted as

7D7

(7D7/hn) for two separate values of n (9 and

35GHz). On the basis of the correspondences indi-

cated by the dotted lines, it has been proposed that

a single bell-shaped distribution P

7D7

(7D7) might

account for all details of the experimental spectra at

both microwave frequencies (Griscom 1990).

In any event, the diagram of Fig. 4(b) appeared to

account for the Mn

2 þ

spectrum in the Li

O 4B

polycrystalline compound by assuming a single dis-

crete value of 7D7 corresponding to 7D7/hn ¼0.4 at X

band (Griscom and Griscom 1967). Actual computer

Figure 3

ESR spectra of peroxy radicals (PORs) in 36%

O-

enriched silica glass: (a) experimental spectrum and (b)

simulation of the highly structured parts of (a) assuming

a POR model with one

O(I ¼

) and one

O(I ¼0) per

defect. Under the assumption that the two oxygen sites

are chemically inequivalent (see inset), the simulation of

(b) perforce comprises an equally weighted sum of the

simulated component spectra (c) and (d).

Amorphous Materials: Electron Spin Resonance

line-shape simulations of glassy-state ESR spectra

based on matrix diagonalization in this parameter

regime have been carried out by only a few workers

to date (e.g., Kliava and Pura

ns 1980, Cugunov et al.

1994, Legein et al. 1993).

3.4 The Bigger Picture

For convenience, the examples given above have per-

tained to a single paramagnetic species, O



, and its

occurrences as point defects in the amorphous per-

oxyborates (Sect. 3.1) and irradiated silica glasses

(Sect. 3.2) and to a single transition group ion, Mn

2 þ

in an oxide glass system (Sect. 3.3). Critical reviews of

the application of continuous-wave (c.w.) ESR to

much broader ranges of point defects and transition

group ions in oxide glasses have been given by

Griscom (1980, 1990). The importance of carrying

out comparative studies of both glassy and polycrys-

talline forms of the same material system when at-

tempting to model paramagnetic centers in the former

has been stressed in a review paper featuring point

defects in the alkali borate system (Griscom 1993a).

Continuous-wave ESR of point defects in chalco-

genide and multicomponent ﬂuoride glasses have

been reviewed by Griscom (1980) and (1993b), res-

pectively. Examples of ESR investigations of

Figure 4

ESR of Mn

2 þ

in an alkali borate glass: (a) absorption spectra recorded at n ¼9GHz (solid curve) and n ¼35GHz

(dashed curve); (b) calculated resonance conditions (neglecting

Mn hyperﬁne splittings) g

res

/hn for H parallel to

each of the principal axes of the tensor D (for the special condition 7E7 ¼

7D7) shown as functions of 7D7/hn; (c) a

Gaussian distribution of 7D7 values plotted as P

(7D7/hn) for the two values of n pertaining to the spectra of (a).

Amorphous Materials: Electron Spin Resonance

transition metal ions doped into ﬂuoride and

chalcogenide glasses can be found in Legein et al.

(1993) and El Mkami et al. (1996), respectively. Street

(1991) has reviewed applications of ESR to hydro-

genated amorphous silicon (a-Si:H).

Continuous-wave ESR has also been employed to

good advantage in the characterization of ferro- and

ferrimagnetic particles precipitated in both synthetic

and natural glasses (Griscom 1980, 1984). ESR stud-

ies of ferrimagnetic particles in 65-million-year-old

glasses have a bearing upon the theory that an aster-

oidal impact triggered the extinction of the dinosaurs

(Griscom et al. 1999).

Modern pulsed-ESR spectrometers afford several

new capabilities over and above those inherent in c.w.

methods. These include the possibility of separating

overlapping spectra of paramagnetic centers charac-

terized by differing spin-lattice relaxation times and

the ability to resolve weak hyperﬁne interactions with

ligand magnetic nuclei by the study of electron spin

echo envelope modulation (ESEEM). ESEEM stud-

ies of defect centers in alkali borate glasses (Kordas

1997, Shkrob et al. 1999) appear to compel revision

of some of the defect models proposed earlier on the

basis of c.w. ESR results. Isoya et al. (1993) have used

ESEEM to probe the spatial distribution of deuteri-

um near dangling-bond defects in deuterated amor-

phous silicon.

Morigaki (1993) reviews the results of electron-nu-

clear double resonance (ENDOR) and optically de-

tected magnetic resonance (ODMR) studies of

dangling bonds in a-Si:H. Electron-hole recombina-

tion processes in a-Si:H have been investigated by a

time-resolved ODMR technique (Mao and Taylor

1995). Shkrob and Trifunac (1997) comprehensively

review time-resolved pulsed ESR and ODMR in ra-

diation chemistry, including applications to amor-

phous silicon dioxide and a-Si:H.

Bibliography

Cugunov L, Mednis A, Kliava J 1994 Computer simulations of

the EPR spectra for ions with S-greater-than-

by eigenﬁeld

and related methods. J. Magn. Reson. Ser. A 106, 153–8

El Mkami H, Deroide B, Zanchetta J V, Rumori P, Abidi N

1996 Electron paramagnetic resonance study of Mn

2 þ

and

2 þ

spin probes in (Ag

(GeS

)

1x

glasses. J. Non-Cryst.

Solids 208, 21–8

Friebele E J, Griscom D L, Stapelbroek M, Weeks R A 1979

Fundamental defects in glass: the peroxy radical in irradiat-

ed, high-purity, fused silica. Phys. Rev. Lett. 42, 1346–9

Griscom D L 1980 Electron spin resonance in glasses. J. Non-

Cryst. Solids 40, 211–72

Griscom D L 1984 Ferromagnetic resonance of precipitated

phases in natural glasses. J. Non-Cryst. Solids 67, 81–118

Griscom D L 1990 Electron spin resonance. In: Uhlmann D R,

Kreidl N J (eds.) Glass Science and Technology: Advances in

Structural Analysis. Academic Press, New York, Vol. 4B,

Chap. 3, pp. 151–251

Griscom D L 1993a Electron spin resonance spectroscopy

of radiation-induced defect centers in insulating glasses. In:

Simmons C J, El-Bayoumi O H (eds.) Experimental Tech-

niques of Glass Science. The American Ceramic Society,

Westerville, OH, pp. 161–84

Griscom D L 1993b Defect centers in heavy metal ﬂuoride

glasses: a review. J. Non-Cryst. Solids 161, 45–51

Griscom D L, Beltra

n-Lo

pez V, Merzbacher C I, Bolden E

1999 Electron spin resonance of 65-million-year-old glasses

and rocks from the Cretaceous–Tertiary boundary. J. Non-

Cryst. Solids 253, 1–22

Griscom D L, Griscom R E 1967 Paramagnetic resonance of

2 þ

in glasses and compounds of the lithium borate sys-

tem. J. Chem. Phys. 47, 2711–2

Isoya J, Yamasaki S, Okushi H, Matsuda A, Tanaka K

1993 Electron-spin-echo envelope-modulation study of the

distance between dangling bonds and hydrogen atoms in

hydrogenated amorphous silicon. Phys. Rev. 47, 7013–24

nzig W, Cohen M H 1959 Paramagnetic resonance of

oxygen in alkali halides. Phys. Rev. Lett. 3, 509–10

Kliava J, Purans J 1980 Simulation of EPR spectra of Mn

2 þ

glasses. J. Magn. Reson. 40, 33–45

Kordas G 1997 N( ¼1,2,3)-dimensional Fourier-transform–

electron paramagnetic resonance (FT-EPR) spectro-

scopy measurements in the B

–Li

O System. Phys. Chem.

Glasses 38, 21–4

Legein C, Buzare

J Y, Jacoboni C 1993 EPR structural inves-

tigations of transition metal ﬂuoride glasses (TMFG). J.

Non-Cryst. Solids 161, 112–7

Mao D, Taylor P C 1995 Optically detected ESR studies of re-

combination processes in a-Si:H. J. Non-Cryst. Solids 190, 48–57

Morigaki K 1993 ENDOR and ODMR characterization. In:

Sakurai Y, Hamakawa Y, Matsumoto T, Shirae K, Suzuki K

(eds.) Current Topics in Amorphous Materials: Physics and

Technology. Elsevier, Amsterdam, pp. 323–8

Peterson G E, Kurkjian C R, Carnavale A 1974 Random

structure models and spin resonance in glass. Phys. Chem.

Glasses 15, 52–8

Shkrob I A, Trifunac A D 1997 Spin and time-resolved mag-

netic resonance in radiation chemistry. Recent developments

and perspectives. Radiat. Phys. Chem. 50, 227–43

Shkrob I A, Tadjikov B M, Trifunac A D 1999 Magnetic

resonance studies on radiation-induced point defects in

mixed oxide glasses. I. Spin centers in B

and alkali borate

glasses. J. Non-Cryst. Solids 262, 6–34

Stapelbroek M, Griscom D L, Friebele E J, Weeks R A 1979

Oxygen-associated trapped hole centers in high-purity fused

silicas. J. Non-Cryst. Solids 32, 313–26

Street R A 1991 Hydrogenated Amorphous Silicon. Cambridge

University Press, Cambridge 104–14

Weil J A, Bolton J R, Wertz J E 1996 Electron Paramagnetic

Resonance: Elemental Theory And Practical Applications.

Wiley, New York

D. L. Griscom

Naval Research Laboratory, Washington, DC, USA

Amorphous Materials: Nuclear Magnetic

Resonance, New Techniques

The elucidation of the structure and dynamics of sol-

id or highly viscous amorphous materials represents a

major challenge for techniques of characterization.

Amorphous Materials: Nuclear Magnetic Resonance, New Techniques

This holds, in particular, if atomistic resolution is

desired. Here solid-state NMR offers unique possi-

bilities, owing to its selectivity and versatility, in par-

ticular if multidimensional techniques are employed.

This article describes advances in the ﬁeld, empha-

sizing multidimensional exchange and double-quan-

tum NMR. These techniques allow one, for example,

to probe the conformation of macromolecules and

their packing in both the solid state and the melt, as

well as the connectivity of building blocks in inorganic

glasses, and both the geometry and memory effects of

complex dynamics.

1. Multidimensional Exchange NMR

In two-dimensional (2D) exchange NMR, informa-

tion on the structure of the material is obtained by

monitoring the transfer of nuclear magnetization be-

tween different building blocks, which can be distin-

guished in the NMR spectrum if differences in the

chemical structure, conformation, or orientation of

the distinct residue are reﬂected in different resonance

frequencies. For optimum spectral resolution, magic-

angle spinning (MAS) is employed, while nonrotating

samples often yield better geometric information.

Magnetization exchange can result from the dipole–

dipole interaction between the nuclear spins; this

leads to through-space spin diffusion, from which the

spatial proximity and mutual orientation of molecu-

lar moieties, i.e., valuable structural information, can

be determined.

Alternatively, magnetization exchange can result

from a spatial movement of the moiety under study,

thereby changing its NMR frequency, and thus pro-

viding information about dynamics. From the time

needed for the exchange to occur, the time scale of the

dynamic process can be directly obtained in real time.

Moreover, the mutual orientation of molecular moi-

eties before and after the move can be read from the

off-diagonal exchange patterns (geometry of dynam-

ics). This experiment provides information on slow

dynamic processes on time scales between 1 ms and

100 s, which is equivalent to that available via inco-

herent neutron scattering on fast dynamics. By ex-

tending the NMR technique to three and four

dimensions, memory effects concerning the orienta-

tion as well as the rate of molecular dynamics become

experimentally accessible. For organic materials,

and

H are widely used, while in inorganic sys-

tems, spin 1/2 nuclei such as

P are most popular

(Schmidt-Rohr and Spiess 1994).

1.1 Structure

Multidimensional exchange and other solid-state

NMR techniques are used to determine the packing

of amorphous polymers in the glassy state (Klug et al.

1997, Tomaselli et al. 1997) and the connectivity of

building blocks in inorganic glasses (Eckert 1994).

Advanced approaches then combine the information

provided by spectroscopy with computer simulations

of the packing. Similarly, information about the con-

formation of macromolecules is readily available

from an analysis of the observed

C chemical shift,

making use of ab initio calculations of this spec-

troscopic parameter (Born and Spiess 1997).

1.2 Dynamics

The molecular dynamics is probably most complex in

amorphous melts close to the glass transition. Nev-

ertheless, applying various 2D exchange NMR tech-

niques it has been possible to show that molecular

dynamics, in both low molar mass glass-formers and

synthetic polymers, consists of small-angle ﬂuctua-

tions causing total displacements of 2–41 and occa-

sional jumps through 10–251 (Spiess 1997). By

extension to 3D or even 4D NMR, the orientation

of individual moieties can be probed at three, or even

four, subsequent times, and memory effects concern-

ing the geometry as well as the rate of the molecular

dynamics in vitrifying polymer melts can be eluci-

dated. Studies along these lines lead to the notion of

dynamic heterogeneities, as shown schematically in

Fig. 1. By combining 4D exchange NMR with spin

diffusion, the length scale of these heterogeneities has

been determined (Tracht et al. 1998).

2. Double-quantum Solid-state NMR

In 2D exchange NMR, direct magnetization transfer

cannot readily be distinguished from transfer via a

third spin or orientation. Likewise, it is difﬁcult to

distinguish transfer back to the original moiety or

orientation from no transfer at all. Such problems do

not occur in double-quantum (DQ) NMR, where a

DQ coherence is generated within selected spin pairs.

These can either be introduced by isotope labeling, in

particular

C–

C, or separated from others by fast

MAS. The latter approach even allows one to use

solid-state

H NMR, which makes the technique

widely applicable. Although mainly aimed at deter-

mining structure, the connectivity of functional

groups provided by this technique also yields infor-

mation about dynamic processes on mesoscopic

length scales (Spiess 1997).

2.1 Structure

The advantages of DQ solid-state NMR have already

been exploited for both inorganic and organic amor-

phous materials. In inorganic phosphate glasses, the

connectivity between coordination polyhedrons has

been elucidated with previously inaccessible detail by

fast MAS DQ NMR, with motifs involving sequences

of at least three polyhedrons of the same kind being

Amorphous Materials: Nuclear Magnetic Resonance, New Techniques

identiﬁed in such glasses for the ﬁrst time (Feike et al.

1998). The conformation of organic polymers can be

probed by the determination of torsional angles

between nearby groups via DQ NMR on systems

containing

C–

C labels, and clear-cut differences

between crystalline and amorphous regions have been

elucidated (Schmidt-Rohr 1996). Moreover, using

fast MAS DQ

H NMR, the structure of hydrogen

bonds in disordered organic materials is now readily

accessible. Owing to the high sensitivity of

H NMR,

such experiments can be performed with samples of

about 10 mg and measuring times as short as 30

minutes, thus seriously challenging the use of neutron

scattering for this purpose (Schnell et al. 1998).

2.2 Dynamics

The application of DQ NMR with MAS to viscous

melts even allows one to probe both the conforma-

tion of chains on mesoscopic scales as well as trans-

lational dynamics. This is due to the fact that the

chain motion in entangled polymer melts is not com-

pletely isotropic, resulting in residual dipole–dipole

couplings. From these residual couplings, dynamic

order parameters can be determined, which are de-

ﬁned in the same way as the conventional order pa-

rameters in liquid crystals. As shown in Fig. 2, the

spectral resolution of DQ MAS

H NMR of polymer

melts is high enough to probe the order parameters

for the chain direction. The values found are about an

order of magnitude higher than anticipated from

standard theory of polymer melts (Doi and Edwards

1986). From the time, molecular weight, and tem-

perature dependence of the dynamic order parameter,

information about translational chain motion as well

as the length scale of the regions with partially or-

dered chains could be deduced (Graf et al. 1998).

3. Conclusions

The specific examples of advanced solid-state NMR

discussed here indicate the potential of this technique.

Moreover, it is possible to cover all length scales from

molecular via mesoscopic to macroscopic levels

(Spiess 1997). For example, the combination of spin-

diffusion experiments with ﬁlters, which can discrim-

inate between different dynamic behavior, allows one

to probe the phase behavior and the interphase struc-

ture of heterogeneous polymers such as block copol-

ymers, core-shell particles and organic–inorganic

Figure 1

Schematic of (a) dynamic heterogeneities and (b) how

their time and length scales can be measured by

multidimensional NMR (after Tracht et al. 1998).

Figure 2

Schematic of (a) experimental scheme for measuring

residual dipole–dipole couplings for selected proton–

proton pairs in (b) trans-1,4-polybutadiene from (c) a

DQ MAS NMR spectrum (after Graf et al. 1998).

Amorphous Materials: Nuclear Magnetic Resonance, New Techniques

hybrid materials, while NMR imaging can detect

subtle differences in molecular dynamics resulting

from nonlinear plastic deformation of solid polymers.

Close analogies exist between solid-state exchange

NMR and incoherent neutron scattering. Moreover,

DQ solid-state NMR provides information even

equivalent to coherent neutron scattering, albeit for

much slower processes in the region of milliseconds.

In short, advanced solid-state NMR is already an in-

dispensable tool for the characterization of amor-

phous materials, and its importance is expected to

increase further in the future.

See also: Amorphous Materials: Nuclear Spin Re-

laxation; Nuclear Magnetic Resonance Spectrometry

Bibliography

Born R, Spiess H W 1997 Ab initio calculations of conformat-

ional effects on

C NMR spectra of amorphous polymers.

In: NMR—Basic Principles and Progress. Springer, Heidelb-

erg, Germany, Vol. 35

Doi M, Edwards S F 1986 The Theory of Polymer Dynamics.

Clarendon Press, Oxford, UK

Eckert H 1994 Structural studies of non-crystalline solids using

NMR. New experimental approaches and results. In:

NMR—Basic Principles and Progress. Springer, Heidelberg,

Germany, Vol. 33 pp. 125–98

Feike M, Ja

ger C, Spiess H W 1998 Connectivities of coordi-

nation polyhedra in phosphate glasses from

P double-quan-

tum NMR spectroscopy. J. Non-Crystalline Solids 223, 200–6

Graf R, Heuer A, Spiess H W 1998 Chain-order effects in pol-

ymer melts probed by

H double-quantum NMR spectros-

copy. Phys. Rev. Lett. 80, 5738–41

Klug C A, Zhu W L, Tasaki K, Schaefer J 1997 Orientational

order of locally parallel chain segments in glassy polycar-

bonate from

C–

C dipolar couplings. Macromolecules 30,

1734–40

Schmidt-Rohr K 1996 A double-quantum solid-state NMR

technique for determining torsion angles in polymers. Mac-

romolecules. 29, 3975–81

Schmidt-Rohr K, Spiess H W 1994 Multidimensional Solid-

State NMR and Polymers. Academic Press, London

Schnell I, Brown S P, Low H Y, Ishida H, Spiess H W 1998 An

investigation of hydrogen bonding in benzoxazine dimers by

fast magic-angle spinning and double-quantum

H NMR

spectroscopy. J. Am. Chem. Soc. 120, 11784–95

Spiess H W 1997 Multidimensional solid state NMR: a unique

tool for the characterization of complex materials. Ber.

Bunsenges. Phys. Chem. 101, 153–68

Tomaselli M, Zehnder M M, Robyr P, Grob Pisano C, Ernst R

R, Suter U W 1997 Local conformations in the glassy po-

lycarbonate of 2,2-bis(4-hydroxyphenyl)propane (bisphenol-

A). Macromolecules 30, 3579–83

Tracht U, Wilhelm M, Heuer A, Feng H, Schmidt-Rohr K,

Spiess H W 1998 Length scale of dynamic heterogeneities at

the glass transition determined by multidimensional nuclear

magnetic resonance. Phys. Rev. Lett. 81, 2727–30

H. W. Spiess

Max-Planck-Institut fu

r Polymerforschung, Mainz

Germany

Amorphous Materials: Nuclear Spin

Relaxation

Nuclear spin relaxation (NSR) techniques are widely

used to investigate the dynamic properties of amor-

phous materials over a very wide time scale ranging

from seconds to nanoseconds. Application of strong

magnetic ﬁeld gradients allows the measurement of

tracer diffusion coefﬁcients down to about

15

1

on mesoscopic-length scales (Geil 1998).

The techniques are extensively employed to amor-

phous polymers to study rotational and translational

motions of molecular groups as well as glass-transi-

tion dynamics. In particular, incorporated site-spe-

cific deuterium probes yield very detailed information

about molecular dynamics in polymers (Schmidt-

Rohr and Spiess 1994). Until now, however, only a

few NSR studies have been performed on amorphous

metals. For instance, (Tang et al. 1998) have observed

the diffusion of beryllium in a beryllium-containing

metallic glass by means of

Be NSR.

In inorganic glasses containing alkali ions, three

processes are important which occur in different tem-

perature regions. At low temperatures low-frequency

transitions take place between energetically nearly

equivalent positions of atomic conﬁgurations which

are intrinsic to the glassy state (Dieckho

fer et al.

1997). At elevated temperatures, ionic diffusion be-

comes dominant, whereas close to the glass-transition

temperature the shear viscosity decreases rapidly cor-

responding to fast-growing density ﬂuctuations. The

present contribution focuses on the study of ionic

diffusion by NSR.

1. Diffusion-induced Nuclear Spin Relaxation

Atomic diffusion gives rise to a time evolution of the

nuclear magnetization in a sample towards the ther-

mal equilibrium. The time constant (NSR time) T

the process is given by the relation

1=T

¼ Jðo

Þþ4Jð2o

Þð1Þ

where

Jðo

Þ¼Re

GðtÞe

jo

dt ð1aÞ

denotes the spectral density of the diffusional pair

correlation function

GðtÞ¼

ioj

ð0Þo

ðtÞS

therm:av

ð1bÞ

at the applied Larmor frequency o

of the nuclear

probes. o

(t) represents the time ﬂuctuation of the

coupling frequency between a resonant probe nu-

cleus i and a neighboring species j (resonant or

Amorphous Materials: Nuclear Spin Relaxation