Two-Level States in Glasses

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Rep. Prog. Phys. 50 (1987) 1657-1708. Printed in the U K
Two-level states in glasses
W A Phillips
Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, U K
Abstract
This review covers a wide range of experimental and theoretical studies of two-level
or tunnelling states in glasses. Emphasis is on fundamental physics rather than a
detailed comparison of experiment and theory. Sections cover the static and dynamic
properties of tunnelling states, their contribution to thermal properties and their
response to weak a n d strong electric and acoustic fields, both steady state and pulsed.
A section on metallic glasses focuses on the importance of electron tunnelling-state
interactions, and a final section illustrates approaches to a microscopic description by
means of selected examples.
This review was received in February 1987.
0034-4885/87/121657 + 52$09.00 @ 1987 I O P Publishing Ltd 1657

1658 W A Phillips
Contents
Page
1, Introduction 1659
2. Tunnelling states 1663
2.1. Static description 1663
2.2. Dynamics 1666
3. Thermal properties 1669
3.1. Thermal conductivity 1669
3.2. Heat capacity 1670
3.3. Thermal expansion 1673
3.4. Energy relaxation and irreversibility 1675
4. Response to external fields 1677
4.1. Weak fields 1677
4.2. Strong fields 1679
4.3. Relaxation 1682
5. Pulse echo experiments 1686
5.1. General theory 1686
5.2. Echo experiments 1687
5.3. Spectral diffusion 1691
6. Metallic glasses 1696
7. Microscopic descriptions of tunnelling states 1700
7.1. Relation to specific defects 1700
7.2. Microscopic modelling 1702
7.3. General theories 1704
7.4. suffix 1705
Acknowledgments 1706
References 1706
Two-level states in glasses 1659
1. Introduction
One of the more unexpected results in solid state physics was provided by the first
reliable measurements below 1 K of heat capacity C a n d thermal conductivity K in a
number of glasses (Zeller a n d Pohl 1971). Beforehand it had been argued that because
low-temperature thermal properties are dominated by phonons (quantised lattice
vibrations) of low frequency, and because in crystals these phonons can be described
as long-wavelength sound waves propagating through a n elastic continuum, there
should be little difference between glasses a n d crystals in this regime where the phonons
are insensitive to microscopic structure. In fact, as shown in figures 1 a n d 2 for vitreous
a n d crystalline silica, the heat capacity a n d thermal conductivity are radically different
in the two materials.
\ Vitreous silica
\
\ \
\
1 10 100
T IKI
Figure 1. The heat capacity C ( T ) of vitreous silica and crystalline quartz as a function of
temperature T (Jones 1982, after Zeller and Pohl 1971), plotted as C / T 3 against T
The results in a-quartz are typical of an insulating crystal. The heat capacity varies
as T 3 below 10 K, where T is the absolute temperature, as expected from the Debye
theory. This theory predicts that in the long-wavelength limit the density of phonon
states g ( w ) varies quadratically with the phonon (angular) frequency w if the velocity
of sound U , is constant so that w q = v,,,where q is the phonon wavevector. At higher
temperatures phonon dispersion initially gives a more rapid increase of g ( w ) with w ,
and so C increases more rapidly than T 3 ,but ultimately, above the Debye temperature
0, C approaches the classical limit.
The cubic temperature variation of K can be interpreted qualitatively by means of
the kinetic formula
K =fCV,l (1.1)
1660 W A Phillips
102
IO-‘
lo-’ 1 10
T iK1
Figure 2. The thermal conductivity K ( T )of vitreous silica and crystalline quartz (Jones
1982, after Zeller and Pohl 1971), plotted logarithmically.
where 1 is the phonon mean free path. At low temperatures phonons are scattered by
defects in the crystal or by the surfaces of the sample, so that 1 is independent of
temperature a n d K is therefore proportional to T 3 .Above 40 K the reduction of 1 by
phonon-phonon scattering leads to the peak and subsequent fall in K .
These ideas are well known and serve to emphasise the peculiarity of the results
in vitreous silica, where C varies approximately as T and at 0.1 K is about two orders
of magnitude greater in the glass than in the crystal. Below 1 K K varies as T19 but
varies only slightly with temperature between 4 a n d 20 K before increasing at high
temperatures to a value approaching that of crystalline quartz. Similar results are seen
in a wide range of other amorphous solids; oxide, chalcogenide, elemental, polymeric
and metallic glasses all show the same behaviour. A representative sample is shown
in figures 3 a n d 4.
The universality of the phenomena and the idealised temperature dependences of
C proportional to T a n d K proportional to T’ proved great attractions for theorists.
In retrospect this idealisation can be seen to have handicapped the search for a
theoretical explanation, suggesting as it did very general models which were not
supported by a detailed examination of the experimental results and which were
ultimately unable to give quantitative agreement with experiment. For example, the
first and perhaps most obvious explanation for C was in terms of electron states
(Redfield 1971). In the amorphous state the sharp distinction between energy gaps
1 I I I I
0 01 02 03 04
T2 iKZ)
Figure 3. C ( T ) / T plotted as a function of T 2 for a range of glasses (Stephens 1973).
and energy bands can be blurred by disorder, and it was suggested that the specific
heat was a result of a n almost constant density of states at the Fermi level, giving a
linear variation of C with T just as in a metal. However, it turns out that the density
of states estimated from optical experiments (less than J-' m-3 o r l o ' * eV-' cm-')
is much smaller than that deduced from the heat capacity ( J-' m-')).
Similar quantitative problems arise in a model for thermal conductivity based on
the scattering of sound waves by inhomogeneities in the glass structure, a form of
Rayleigh scattering by variations in the local velocity of sound. I n this case the
magnitude of density fluctuations estimated thermodynamically or from light scattering
experiments is too small to give significant scattering below 1 K, and although there
is no independent way of directly estimating the local variation in velocity, the required
magnitude is unreasonably large (Jones e? a1 1978).
Sound waves are known to exist in glasses. In addition to measurements of the
velocity of sound, limited to frequencies below about 1 GHz, evidence for the existence
of well defined transverse and longitudinal waves comes from Brillouin scattering
below 30 G H z (Vacher a n d Pelous 1976) and phonon interference experiments
(Rothenfusser et a1 1983) u p to 500GHz in vitreous silica. Consistent velocities are
1662 W A Phillips
Figure 4. K( T ) as a function of temperature for a range of glasses (Stephens 1973).
measured in all these experiments, and the phonon mean free path is much greater
than the wavelength. Further evidence is provided by measurements of thermal
conductivity in very thin glass rods which have roughened surfaces to ensure that
boundary scattering dominates (Pohl er a1 1974, Zaitlin and Anderson 1975). Experi-
mental values agree with those calculated assuming the existence of sound‘waves. It
is clear, therefore, that the unexpected thermal properties arise from additional excita-
tions which both scatter phonons and contribute to the heat capacity.
Of the various models originally proposed to account for the thermal data, the one
most widely and successfully used has been the tunnelling or two-level-system model
(Phillips 1972, Anderson et a1 1972). In this model, to be described in detail in the
next section, atoms occupying one of two adjacent minima are assumed to tunnel
quantum mechanically to the other, leading to a splitting of the ground state as in the
ammonia molecule. The inevitable variations in local environment present in the
amorphous solid give rise to a distribution of these splittings which is almost constant
in energy, a n d hence to a heat capacity which can be evaluated as
C ( T )=
I: n0(E2/4kBT2)s e c h 2 ( E / 2 k B T )d E
where no is the constant density of states a n d the other factor in the integrand is the
(1- 2 )
contribution of a single two-level state (Schottky anomaly). Evaluating the integral

gives
C(T)=(~’/6)nokg~T (1.3)
in broad agreement with experiment. These states scatter phonons, leading to a thermal
conductivity proportional to T2 and to an acoustic attenuation which can be saturated
at high acoustic intensities. It is this last property that provides the strongest evidence
for the model which will form the basis for the interpretation of the wide range of
experimental results discussed in this review. To a large extent the model can be used
in phenomenological form, and possible microscopic representations will be described
only in the last section.
This review is not intended as a detailed evaluation and comparison of experimental
results in specific glasses, but as a critical description of the way in which the tunnelling
or two-level-system model can be used to explain a wide range of data. By far the
largest number of experiments have been performed on vitreous silica, and this material
will be used as a ‘running example’, although reference will be made to other materials
when necessary.
2. Tunnelling states
2.1. Static description

In a perfect crystal each atom is constrained by symmetry to occupy a single potential
minimum. Many defects, however, can be represented microscopically as interstitial
or substitutional impurity atoms or molecules moving in a multi-minima potential
provided by the neighbours. Such states have been extensively studied in, for example,
alkali halides (Narayanamurti and Pohl 1970). At low temperatures a quantum
mechanical description is necessary, and tunnelling of the atom from one minimum
to another gives rise to the very small energy splittings (less than eV) needed if
the states are to be observed in thermal experiments at 1 K and below.
The tunnelling model proposes that similar states are intrinsic to glasses: an atom
or group of atoms can occupy one of two (or more) potential minima. This choice at
the microscopic level is consistent with the experimentally determined excess entropy
of the glass with respect to the crystal, although the number of atoms contributing at
low temperatures is only a small fraction of the total. The remainder are essentially
immobile below the glass-transition temperature Tg.
Each tunnelling state (TS) can be represented by a particle in a potential of the
form shown in figure 5(a), where the abscissa may represent the position of one atom
but more generally is a configurational coordinate describing a combination of the
coordinates of a number of atoms. The energy levels of the particle are conveniently
calculated using as a starting point the solutions of the single-well problem, a choice
of basis set 41 and d2 known as the well, non-diagonal or localised representation
(figure S(b)). Each state is the ground state of the appropriate potential Vl or V,
(assumed harmonic in figure 5 ) . The complete Hamiltonian H can be written as
H = HI + ( v - v,) = Hz + ( v - V2) (2.1)
where H I and H2 are the individual well Hamiltonians. In the localised representation
the Hamiltonian matrix becomes
If the extension of each localised wavefunction into the barrier is small, the terms
(4ilV - V,14i)can be neglected in comparison to Ei,and if the zero of energy is chosen
1664 W A Phillips
Figure 5. A double-well potential ( a ) ,together with the wavefunctions in the localised ( b )

and diagonal ( c ) representations.
as the mean of E , and E * , (2.2) becomes
(2.3)
where the tunnel splitting A, is given by

A,= 2{4llf442)* (2.4)
Notice that only if the wells are identical, apart from a relative shift in energy, is the
A of (2.3) identical to that of figure 5(a).
A, can be evaluated for specific potentials: two overlapping harmonic potentials
in two a n d three dimensions (Mertzbacker 1970, Phillips 1981b) and Mathieu’s equation
for a rigid rotator in two-fold and three-fold potentials (Isnard a n d Gilchrist 1981)
are typical examples. The results all imply a similar exponential dependence of A, on
the barrier height and well separation, although precise numerical values change from
one model to another. The use of a particular detailed form cannot be justified in
amorphous solids (as opposed to crystals) because the microscopic picture is uncertain,
a n d so a simplified expression is usually adequate:
A,= h R e-A = ha exp[-d(2mV,/h2)”*] (2.5)
where h R is approximately equal to ( E ,+ E 2 ) / 2 ,d is the separation and V, the barrier
between the two wells, and m is the particle mass.
Typical values of the tunnelling parameter A can be estimated from the requirement
that A, must be approximately equal to k B T if the tunnelling state is to contribute to
thermal properties at a temperature T. At 1 K this requires a tunnel splitting of eV,

which with h R equal to eV gives approximately 5 for A. This is equivalent to a
bare proton tunnelling across a barrier of 0.1 eV with d = 0.7 A.
The matrix (2.3) can be diagonalized to obtain the eigenstates, the true, diagonal
o r energy representation. These eigenfunctions, illustrated in figure 5 , have energies
* E l 2 where
E’=(A2+A~)1’2 (2.6)
a n d can be written in the analytic forms
9, = cos 0 + 4 2sin 0 (2.7)
~2=~1sinO-~,cos0 (2.8)
where tan 2 0 = A , / A . I n the symmetric case where A = O these equations give the
expected symmetric a n d antisymmetric solutions.
The dipole moment of the tunnelling state can be written in terms of G1 and I,!J2 as
PI = I 9Tqx*1 d x
=q [ (4Tx41 cos’ 0 + 4Tx4, sin2 O + 4 7 ~ sin

4 ~2 0 ) d x
if the term involving 4Tx42 can be neglected, and where p o is the dipole moment when
the particle is located in one well:
po = [ 4Tx4, dx = -
I 4 $ x 4 2 dx.
Obviously p1 is zero for the symmetric case.

For this, as for any problem involving two energy levels, there is a formal analogy
with the problem of a spin-; particle in a magnetic field. The Hamiltonian matrix (2.3)
can therefore be rewritten in terms of the Pauli spin matrices
(2.10)
( o r equivalently in terms of the spin operators S, =+ha,).After diagonalisation, the

Hamiltonian takes the obvious form
H =$EuZ.
This analogy has important advantages when discussing non-linear a n d coherent effects
in the interaction of tunnelling states with acoustic and electric fields ( § 5).
A basic feature of the tunnelling state model as applied to amorphous solids is the
existence of a wide range of values of both the tunnelling parameter A and the
asymmetry A . Calculated properties of the states will depend critically on the form
of the distribution function f ( A , A,) for A a n d Ao, the latter distribution derived from
that for A.
In the case of the asymmetry A it is argued that the distribution function must be
symmetric because both positive a n d negative values of A are equally likely. (No
singularities are expected for A = 0 because the eigenstates remain non-degenerate.)
The scale of energy variation is determined by the thermal energy available at the
1666 W A Phillips
glass-transition temperature where the fluctuating local potentials of the liquid are
frozen in the structure, Since Tg is between 200 and 1OOOK for most glasses, this
energy is of the order of 0.05 eV, much larger than the thermal energy available at 1 K.
The low temperature properties are therefore sensitive to the centre of a broad symmetric
distribution, so that f ( A , A,) can be taken as independent of A.
The variation with A, is likely to be sensitive to the particular microscopic motion
involved, although the general form can be deduced from the distribution of the
tunnelling parameter A. Because of the exponential dependence of A, on A, only a
relatively small range of A is sampled for a large range of A, and over this limited
range the distribution of A can be assumed constant. The resulting distribution function
can then be written
f ( A , A,) = P/AO. (2.11)
This general result is only slightly modified (by a logarithmic factor) if the distribution
of A varies slowly with the energy, but in general the precise dependence on A. will
vary from one material to another, and can be regarded as a parameter to be determined
by experiment (Frossatti et a1 1977, Phillips 1981b).
The density of states n ( E )can be calculated from (2.6) and the distribution function
f ( A , A,) only if a lower cut-off value for A, is introduced when integrating (2.11). Such
a cut-off could arise from a maximum tunnelling parameter (Lasjaunias et a1 1978) o r
from characteristic time scales introduced by experiment. More precisely, the two
parameters A and A. describing the states must be replaced by a second pair of
parameters, often chosen as the energy E and the relaxation time T. The resulting
distribution function for E and T will be derived in 0 3.
In many cases the results of experiment can be interpreted and related in terms of
a simpler model which ignores the detailed origin of the two levels. Instead, of two
parameters A and A,, this two-level-system model considers the total energy E as the
only variable. The distribution function is assumed to be constant. Although in many
cases it is clear that the two-level-system model is inadequate for detailed understanding,
in others the simplification introduced by its use is very helpful.
2.2. Dynamics
The behaviour of a tunnelling state is defined by specifying the complex amplitudes
a , ( t ) a n d a , ( t ) of the two lowest states with eigenfunctions $,(r) and $*(r). Neglect
of higher states is justified because these two states are much closer in energy to each
other than they are to other excited states, although the possibility that higher energy
states might be important must be considered in certain cases (such as transition rates
at temperatures above 1 K). The general time-dependent wavefunction takes the form
V ( r , t ) = a , ( t ) $ l ( r ) exp(-iE,t/A)+a,(t)$,(r) exp(-iE,t/h) (2.12)

where E, and E2 are the energies of states $, and $z and for normalisation a l a T +
a2aT = 1.
The complex nature of a , and a2 means that two quantities are necessary to define
the two-level system, normally taken as the difference in probabilities of finding the
system in the two states, a 2 a ~ - a , u and
~ , a quantity related to the phase difference
between a , and a 2 , a l a $ . Measurable properties of the system depend on these two
parameters, as can be seen for example by calculating the time-dependent dipole
moment of a one-dimensional tunnelling state. Using p, = -p2 = ( $ , I X ~ $ ~ ) and p12=

p2, = ($llx[I,b2),the average dipole moment ( p ( t ) ) can be written
where hwo = E2 - E , .
Variations of a,( t ) and a2(t ) are determined through the time-dependent
Schrodinger equation by interaction with external fields. Any two-level system in a
solid will be subject to randomly varying strain fields which can be treated as the
superposition of independent phonon modes. The strain field of each is weak, and
interaction with the two levels can be treated using perturbation theory, ignoring phase
coherence, equivalent to the interaction of an atom with electromagnetic radiation in
a black-body cavity (Phillips 1981b). Experimentally the properties of the states may
be probed by strong external applied electromagnetic or strain fields where coherence
of the two wavefunctions I,bl and I,b2 must be taken into account, although for weak
external fields perturbation theory can still be used.
Two related simplifications can be made. The frequency needed to induce resonant
transitions from one level to the other is less than 20GHz for states contributing to
thermal properties below 1 K. Corresponding wavelengths are 10 mm for photons and
100 nm for acoustic phonons, in both cases much larger than the spatial extent of the
tunnelling or two-level state. The dipole approximation, where the local electric or
strain field is taken as uniform, is therefore valid. The dominant effect of these uniform
fields is to affect the energy of the tunnelling state by changing the asymmetry energy,
and changes in the barrier height can usually be ignored (Phillips 1981b, Anderson
1986). Any external perturbation is therefore diagonal in the local representation
( 41,&) which when transformed into the diagonal representation ( $ 1 , $2) has the form
cos24 sin24
(2.14)
sin24 -cos24
In terms of the Pauli matrices the general interaction Hamiltonian can be written
l::l
H i n t =- u z + - u x l::l
p o * & + -uz+-ux ye (2.15)
in the presence of an electric field 5 and a strain field e. The two parameters po and
y, defined as f d A / d g and ;aA/ae respectively, are equivalent to the electric and elastic
dipole moments of the equivalent classical problem of a charged particle moving in
the double well potential. In (2.15) the vector character of is preserved, but the
tensorial nature of e has been ignored and ye written as an average over orientations.
For the two-level-system model (2.15) must be replaced by an equivalent form for
Hintin which the relationship between the diagonal ( u Z and ) off-diagonal ( u x )terms
is ignored. In this model (2.15) is usually written
Hint=( t ~ u , + ~ ’ u x ) 5 + ( t D u z + M u x ) e . (2.16)
In the remainder of this section the phase coherence of I,b, and I,b2 will be ignored
in a calculation valid for interaction with thermal phonons and weak external fields.
The only relevant parameter is therefore the occupation probability p, = a l a r , with
p1+p2 = 1. Using a rate-equation approach (Golding and Graebner 1981)
P1= -PIwl2+P2w2I (2.17)
li2 =PIw12-p2~12 (2.18)
1668 W A Phillips
where w I 2 is the transition probability from state 1 to state 2. In thermal equilibrium

at temperature T, p 1 and p 2 are time independent, with values p y and p : , so that
P h 2 =P b 2 1
where
p:lp?= exp(-ElkT) =wI2lw2, (2.19)
if E is the energy difference between the two levels. Replacing p 2 in (2.17) gives
$1 = -PI(Wlr+ w21) +U21 (2.20)
defining a relaxation time T, where
-1
T = ( w 1 2 + w 2 1 ) = w , 2 [ 1 + e x p( E l k T ) ] . (2.21)
This relaxation time refers to the occupation probabilities, and so governs the relaxation
of the term a l a ? - a2a; in the expression for ( p ) in (2.13). To distinguish it from the
relaxation time for terms of the form a l a : (see 0 4.2) it is often referred to as TI, or
the spin-lattice relaxation time.
The transition probability can be calculated for a weak strain field using
time-dependent perturbation theory to give
(2.22)
where n B , the Bose factor, is [ e x p ( E / k T ) - l]-', g ( E ) is the phonon density of states,

and the summation is over phonon polarisations a. For an acoustic wave 2e, cos of
the strain amplitude per phonon is given by equating the time average classical energy
per unit volume to the energy of a phonon:
2pu2w2= ho
where U is the displacement amplitude and p is the bulk density. The strain is given
by
(2.23)
where qa is the phonon wavevector, and the matrix element, using the term in ax in
Hint(a,gives nothing), becomes
1/2
(2.24)
The Debye density of phonon states, valid in this low-energy limit, is

g ( E ) = E2/2.rr2h3vi (2.25)
per unit volume, where U, is the sound velocity for polarisation a. Collecting together
(2.21), (2.22) and (2.24) gives
(2.26)
In the equivalent result for the two-level system A i is replaced by E', but it should
be noted that this removes the very wide range of relaxation times predicted by (2.26)
through the broad distribution of A,, and found experimentally ( 3 3 1. At higher
temperatures additional higher-order processes will of course contribute 1 o relaxation,
but these have not been clearly observed in glasses.
3. Thermal properties
3.1. Thermal conductivity

The interaction between TS and phonons reduces the phonon mean free path, and is
the dominant scattering mechanism in glasses below 1 K. An expression for the free
path, valid for weak fields (see § 4) can be obtained from the rate equations, (2.17)
and (2.18), using detailed balance between energy absorbed by the T S ~and energy lost
by the phonons. For a single TS of energy E
g(E)fiptI=P1.
Using (2.20) a n d (2.22) this can be written as
(3.2)
defining a phonon scattering rate
- 77y2,wA' tanh ( E / k , T ) (3.3)

pv: E2
for the scattering of phonons of polarisation a by a single state with energy E and
tunnel splitting A,. The total scattering at energy E is obtained by integrating over
A. using
f ( A , A,) = PIA, (2.11)
(noting that both + A and -A contribute at a given E ) which gives
for the free path of a phonon of angular frequency w and polarisation cy.
The thermal conductivity K ( T ) is evaluated on the assumption that heat is carried
by non-dispersive sound waves, consistent with the calculations leading to (2.26) a n d
(3.4), and that scattering from TS dominates to give
This gives general agreement with Lhe experimental results below 1 K both in respect
of temperature dependence and magnitude, using the constant density of states derived
from the heat capacity and a coupling constant y of order 1 eV. A more precise
quantitative analysis requires information from acoustic experiments to separate longi-
tudinal a n d transverse polarisation contributions in (3.6), and a more careful analysis
of the heat capacity. Notice, however, that the interaction with phonons is dominated
by TS with small asymmetry.
1670 W A Phillips
Experimentally the temperature dependence is found to range from to T’.9

indicating that the distribution function (2.1 1) is not strictly accurate. Eilack (1978)
has pointed out that a distribution function of the form P ( h ) = P e-uA leads to K ( T )
proportional to T Z P asuggesting
, a width to the A distribution in the range 5-10. This
is not unreasonable, but the relatively small change in temperature depenldence shows
the insensitivity of the calculation of K ( T ) to the detailed form of the distribution
functions.
At higher temperatures the thermal conductivity passes through a ‘plateau’ at
temperatures between 4 and 20 K which has been explained in terms of increased
scattering from a quadratically increasing density of TS (Zaitlin and Anderson 1975).
Although there is some evidence from high-frequency acoustic measurements that the
density of states may increase at higher energies (Pelous and Vacher 1976), it is
extremely difficult to identify this contribution in the presence of relaxation scattering
from the TS ( 3 4), higher-order phonon processes, structure scattering or even effects
of phonon localisation.
3.2. Heat capacity

The energy E and relaxation time T are more convenient parameters than A and A,
when discussing heat capacity and other physical consequences of the distribution of
tunnelling states. A distribution function ( E , T ) can be related to f(A, A,) by the
Jacobian transformation
This is a general result, but making use of the specific form f(A, A,) = P / A o ,
D
where P is the same constant that appears in 1;; (3.4) and T , ~ ~ ( Edefined
), by replacing
A, by E in (2.26), is the shortest relaxation time for states of energy E. The density
of states is then given by integrating over all relaxation times from T , ~up ~ to a value
equal to the time scale to of the experiment:
=$P ln(4t,/~,~~). (3.9)

The heat capacity can be calculated as in (1.1) to give
~ ( t , =) ; P 1n(4t,/~,,,) lo-$$
m
sech’ (“)2 k B T dE
= &,rr2PkiT ln(4t,/~,,,) (3.10)

where the small energy variation of the logarithmic term, in the narrow range of E
close to k B T which dominates the integral, has been neglected.
Equation (3.10) encapsulates the non-equilibrium (non-ergodic) nature of amor-
phous solids and not surprisingly has been the subject of numerous experimental
studies. Before examining the results on short time scales, it is helpful to assess the
accuracy of (3.10) in respect of 'normal' measurements, where to is typically 10 s.
Figure 6 shows C ( T ) in three samples of vitreous silica, Spectrosil B (about
1200ppm water), Suprasil (1000ppm water) and Suprasil W (almost no water). We
will concentrate here on Suprasil W,where C ( T ) varies as T' in the range 25 mK < T <
250 mK, and where P = 3 x J-' m-3, yI = 1.6 eV and 7, = 1 eV can be deduced from
K ( T )and phonon echo experiments ( § § 3.1 and 5). The contribution of these states
to C at 0.1 K can be calculated from (3.10) as 0.35 J kg-' K-' (taking t o = 10 s and
evaluating T,,, as 1 ps) in comparison to the experimental value of 0.7 J kg-' K-'. The
logarithmic factor varies by 1.6 between 25 and 250 mK, giving a temperature variation
approximately proportional to T' '. Bearing in mind a possible small energy depen-
dence of P, the predicted temperature variation of C is in good agreement with
experiment, although the magnitude is too small by a factor of two (Black 1978).
This discrepancy could be ascribed to experimental uncertainties in the determina-
tion of y I and y , were it not for experiments at short time scales, which provide
additional evidence that the reality is more complicated than that described by a single
set of TS distributed according to (3.9). Such experiments are, however, notoriously
100
10
--
-
Y
m
m
L
I
al
-
h
I
L.
I I 1 1 1 1 1 I I I I I 1 l l l l
0 02 005 01 02 05
T IKI
Figure 6. The heat capacity of 'water-free' and 'wet' vitreous silica down to 25 mK. Also
shown as a broken line is the Debye phonon contribution calculated from the velocities
of sound (Lasjaunias et a [ 1975). 0 , 0,Suprasil W (<1.5 p p m O H ) ; 0, Suprasil
(1200 ppm O H ) ; A, Spectrosil B (-1000 ppm O H ) (Zeller and Pohl 1971).
1672 W A Phillips
difficult (Goubau and Tait 1975, Kummer et a / 1978, Lewis et a1 1978, Loponen et a /
1980, 1982, Meissner and Spitzmann 1981). In general a short time scale is achieved
by using thin samples of glass; at 0.4 K a sample 0.1 mm thick has a diffusion time of
typically a few j ~ s .With a thin film heater on one side and a thermometer on the
other the propagation of heat pulses through the sample can be followed to give either
the diffusivity by fitting a diffusion equation to the measured temperature profile or
the heat capacity by measuring the maximum temperature rise. One problem with this
experimental arrangement is that even very thin metal films used as heaters or ther-
mometers can have heat capacities comparable with the glass sample so that modelling
thermal diffusion becomes difficult.
There is limited agreement between the various experiments. Although all claim
to see a time-dependent heat capacity, the size of the effect differs greatly from one
experiment to the next. The most consistent results are those of Meissner and Spitzmann
(1981) (figure 7) and Loponen et a1 (1982) on Suprasil W, where the results tend to
that predicted by (3.10) at time scales below 100 ps at temperatures below 1 K. For
longer time scales C( t ) changes more rapidly than expected, indicating another
contribution to the heat capacity with a relatively well defined minimum relaxation
time of about 100 ps at 0.3 K, and which when added to that derived from P gives
approximate agreement with long time scale experiments below 1 K.
10-6 10-2
i Is)
Figure 7. The measured heat capacity of vitreous silica as a function of the time scale of
measurement at different temperatures: B 0.4 K; C, 0.6 K; E, 0.89 K; F, 1.0 K. The open
circles represent the long-time data, and the broken lines the total heat capacity predicted
by the tunnelling model. (Adapted from Meissner and Spitzmann (1981 ).)
It is worth mentioning at this point that although an all embracing theory based
on a single type of tunnelling state is attractive, there is ample evidence to suggest that
many different types of impurity could act in this way. It is clear from figure 6, for
example, that the presence of water can increase the heat capacity of vitreous silica.
3.3. Thermol expansion

The Gruneisen parameter r conveniently relates the measured thermal expansion to
microscopic theory. Macroscopically
r = pv/XTcv (3.11)
where P is the volume expansivity, xT is the isothermal compressibility and Cv is the
heat capacity at constant volume of volume V of the solid. Microscopically the
significance of r can be understood by writing the total entropy of the solid in the
( T ) where the normal mode frequencies U , are
quasiharmonic form as X , S 8 ( w , V)/
explicit functions only of volume, and where it is understood that experimentally
measured frequencies should be used in the formula (Barron 1965, Hui and Allen
1975, Phillips 1981b). Then
($)T =p;(%)
1
(3.12)
7
and
where S: = d S , ( x ) / d x . Eliminating S: a n d using C, = T ( a S , / a T ) , gives
which can be summed over i to give
(3.13)
Now
so that with Cv = CC,, (3.13) becomes
(3.14)
Each excitation therefore contributes to the thermal expansion by a n amount propor-

tional to the microscopic Gruneisen parameter -a In w , / a In V, weighted by the contri-
bution of that excitation to the heat capacity.
In crystalline solids (3.14) predicts that p can be calcuhted from the pressure
dependence of the elastic moduli, which determine phonon frequencies at low tem-
peratures. This prediction is confirmed by experiment, but it is not surprising that in
glasses, where the heat capacity is dominated by TS below 1 K, anomalous results are
found.
1674 W A Phillips
Measurements have been made only on a limited number of materials, often with
conflicting results, but results for vitreous silica, showing (figure 8) that below 1 K
becomes very large and negative, have been confirmed both by measurements of p
using a sensitive dilatometer to detect changes of 1 in 10l2 (Ackerman et a1 1981) a n d
by direct measurements of using the thermoelastic effect (Wright and Phillips 1984).
More generally there seems to be no ‘universal’ behaviour (Anderson 1986) suggesting
that thermal expansion is much more sensitive to details of the microscopic model,
including the distribution functions, than are heat capacity and thermal conductivity.
For a tunnelling state with energy E = hw the microscopic Gruneisen parameter
can be written
(3.15)
Changes in both A a n d A, with volume can contribute significantly to r. The term in

A, which dominates the interaction Hamiltonian ( P 2.2) leads to r for a single tunnelling
state of order lo4/ T, the ratio of (aA/a V), typically 1 eV, to the energy E = k,T, equal
to at 1 K. However, there is no reason to presuppose a consistent trend among
all states for the asymmetry to decrease as the volume is increased, thereby giving a
large negative F. Formally, the asymmetry contribution to thermal expansion is
proportional to (AYh/E), where Yh is the coupling constant for hydrostatic strain (2.15),
in contrast to phonon scattering which is proportional to (A2y2/E2). Only if the
distribution function for A is asymmetric will this term give a non-zero contribution
to r. However, since the magnitude of r is less than 100, this asymmetry need be only
1’4, too small for independent measurement by other techniques.
It is known from studies of tunnelling defects in crystals that hocan be very sensitive
to volume changes. In part this results from the exponential dependence on the local
potential, but also because this local potential is itself very sensitive to volume. In a
crystal, for example, the existence of local multi-minima potentials depends on a
II
t
GI
I 1 I I I 1 1 1 1
10
I I l 1 1 l 1 1
10
T (K1
Figure 8. The low-temperature Gruneisen parameter of vitreous silica. The points are from
direct measurements in Spectrosil B (squares) and Vitreosil (circles) (Wright and Phillips
1984) and the lines a and b for Spectrosil B and WF respectively are derived from thermal
expansion (Ackerman et a / 1984).
mismatch between the size of the impurity a n d the space it occupies, so that a very
small decrease in volume of the crystal can force the impurity into a central single
minimum. Similar arguments may well apply in amorphous solids, so that for a single
tunnelling state the factor (aAo/d V) can be large. As before, the total contribution will
depend on the distribution functions, a n d is smaller for TS in glasses because for a
given E most of the TS have very small values of A,, (Phillips 1973). However, in spite
of this reduction the effect of volume changes o n A. appears sufficiently large to explain
the observed magnitude of r (Wright and Phillips 1984) and also shows, through the
variability of local microscopic potentials, why n o universal behaviour is expected.
This tunnel splitting contribution to r should show little temperature dependence, but
of course the measured magnitude of r increases with decreasing temperature as a
result of the increasing contribution of TS to the total heat capacity below 1 K.
Experimentally it is observed (figure 8) that r is proportional to in vitreous silica,
suggesting that both asymmetry and tunnel splitting are important. The subtle balance
between the various factors explains why r varies so much from material to material.
3.4, Energy relaxation and irreversibility

The long relaxation time part of the distribution function (3.8) has been probed in
experiments which monitor the flow of energy from a system perturbed from equilibrium
(Zimmermann and Weber 1981). If a solid at temperature TI is suddenly connected
to a reservoir at a slightly lower temperature To ( T , > To) by a thermal link of
conductance a,the temperature of the sample is normally assumed to relax exponen-
tially with time from TI to T o , ignoring the small change of thermal properties with
temperature. In a glass the broad distribution of relaxation times leads to very different
behaviour, as shown in figure 9 where for long times the temperature varies as 1/t,
much more slowly than expected.
These results can be understood in terms of the distribution of relaxation times
(3.8). The rate at which heat is evolved from the sample is given by a t and at long
times will be the result of a slow evolution of heat from TS with long relaxation times.
Energy associated with the phonons decays exponentially with the thermal time
constant, a n d is negligible at long times. The instantaneous rate of energy loss q from
\I 53K
t Is)
Figure 9. The rate of flow of energy 0 from a sample cooled from a 'high' temperature,
given against each curve, to 0.0200 K. 0 is measured by monitoring the rate of decrease
in temperature, which is non-exponential (Zimmermann and Weber 1981).
1676 W A Phillips
a single state of energy E is
(3.16)
7
where p z , the instantaneous value of the occupation probability of the upper state,
has time dependence
p 2 - p : = ( p : - p : ) e-''T (3.17)
and p i and p ! correspond to equilibrium values at temperatures T I and TO. Summing
over all states the total rate of energy loss Q per unit volume is given by
(3.18)
Integrating over T in the limit t >> T,~" (which is always true in these experiments) and
using the explicit form of g(E, r ) from (3.8) gives
1 (dE. (3.19)
=
: [om E 11 + eXp( EIkBTI) - 1-k eXp(E/ kBTo)
Finally
7TL
Q ( T , ,t ) = - P k ~ ( T : - T ~ ) t - ' . (3.20)
24
Equation (3.20) agrees well with the results both in magnitude and time dependence,
and the derivation shows how a broad distribution of relaxation times leads quite
generally to a t -' dependence. These experiments demonstrate clearly the existence
of long relaxation times in glasses, up to 5 x lo3s at 1 K.
Associated with Q is an increase in entropy, and one interesting aspect of a two-level
system is that entropy changes and irreversibility can be followed in detail (Wright
and Phillips 1984). This is possible because the thermodynamic properties are uniquely
specified by the instantaneous occupation probability p which plays the role of an
internal variable in general irreversible thermodynamics. For a subset of TS with energy
E and relaxation time T an effective temperature T' can be defined by
(3.21)
(3.22)
where S p = p - p o is a small departure from the equilibrium value p o . The energy flow
q = Ep results in a rate of entropy change
(3.23)
(3.24)
In an experiment such as that described at the beginning of this section, the entropy
changes can be followed from a knowledge of the variation of p - p o . For larger
departures from equilibrium (3.21) and (3.23) can be used without assuming Sp is
small. This analysis has been used to relate thermoelastic processes measured under
different experimental conditions (Wright 1986, Wright and Phillips 1984).
4. Response to external fields
The response of two-level systems to external fields is a problem that has been treated
extensively in connection with the interaction of light with atoms a n d in magnetic
resonance. Although the basic ideas are the same for TS in glasses, additional features
arise because of the strong coupling to phonons, and because of the broad distribution
of energies and relaxation times. The interaction with both acoustic a n d electric fields
has been intensively studied, although the electrical case has the important sim-
plification from both experimental and theoretical points of view that the fields are
uniform across the sample. I n most of § 4 we will present the argument for the electrical
case, extending where necessary to acoustic fields.
4. I. W e a k fields
In this limit the interaction can be considered as in § 2.2 by ignoring the phase
relationship between the wavefunctions describing the two energy levels. This is
possible when the perturbation produced by the external field is small, a condition
that is defined more precisely in § 4.2. In this limit the attenuation is given in the
acoustic case by (3.4)
I-’ = (x) 2
TY,W P tanh( h w / 2 k B T ) .
(3.4)
The equivalent result for electric fields is obtained on replacing ye by the dipole
moment p o (see (2.15)) a n d the amplitude of the strain field ( h ~ 0 / 2 p v ~ ) ”by
’ the
equivalent electric field amplitude ( ~ W / ~ E ~ E , ) ”where
* , E , is the relative dielectric
constant. The attenuation coefficient a is given by
Ly= (c:!:)
- P tanh( h w / 2 k B T ) (4.1)
where c is the velocity of light in the solid and the factor of three arises from averaging
( p o 5 ) over relative orientationis. Both (3.4) a n d (4.1) have been tested experimentally
(Golding et a1 1976b, Schickfus and Hunklinger 1977) taking care at low temperatures
to satisfy the weak field condition. Results for 1;’ in the acoustic case (figure 10) show
not only the predicted tanh dependence on h w / k , T but also give values of P y 2 which
300 -
7 200-
E
--
I
0 1 2 3 4 5
liT llO-* K - ’ l
Figure 10. The phonon free path I-’ for transverse and longitudinal waves at 0.6 GHz
plotted against 1/ T (Golding et a [ 1976b).
1678 W A Phillips
can be used to calculate a value for the thermal conductivity within a factor of two of
the measured value. The slight energy dependence of P, implicit in the departure from
a T 2 dependence of K ( T ) , is such as to improve the agreement between parameters
measured in acoustic measurements at 0.6 G H z a n d those contributing to K ( T ) at 1 K.
Associated with the absorption is a temperature-dependent variation of velocity.
The velocity, calculated from the real part of the response function, is given by the
Kramers-Kronig relation (Landau and Lifshitz 1984) as
dw’ (4.2)
where the principal part of the integral is to be taken. Using (3.4) for I-’ and measuring
U ( T ) with respect to T = 0, the velocity is given by (Hunklinger a n d Arnold 1976)
where W is the digamma function (Abramowitz and Stegun 1964). The first term is
important only for hw comparable to kBT, which for frequencies of 1 G H z means
working at temperatures below 5 0 m K . Such measurements (figure 11) confirm the
detailed form of (4.3) (Golding et a1 1976a) but in most experiments it is sufficient to
use the result for hw << kB T, referring the velocity to a temperature To to give
PY2
U( T )- U( To)= -In( T / To). (4.4)
PV
This equation has been used extensively to derive values of P y 2 even in materials such
as amorphous metals ( 8 6 ) where the contribution of rs to the thermal properties is
L 10-2 lo-’
L
10
T iK1
Figure 11. The variation of acoustic velocity with temperature between 25 m K and 1 K
(Golding et al 1976a).
difficult to identify. One feature that makes (4.4) particularly useful is that it is
insensitive to the strength of the acoustic intensity. The reason for this is that the
major contribution to the integral (4.2) comes from states for which h o / k B T is close
to unity, and which are therefore much more difficult to saturate than those for which
.hw << kBT. This feature also means that the TS sampled in a typical velocity measurement
at frequency w have a larger energy splitting than those probed in attenuation measure-
ments at the same frequency. Derived values of Py2 therefore need not be identical
in the two experiments if P is energy dependent.
The electrical equivalent of (4.4) can be expressed in terms of the relative dielectric
constant E,( T ) as
2
E,( T )- E,( To)= - p i p In ( T / To). (4.5)
EO
Experimental results shown in figure 12 confirm the form (4.5) and also demonstrate
the additional information obtainable by measuring both acoustic and electrical proper-
ties. Different types of silica give almost the same curve for the variation of acoustic
velocity, but figure 12 shows clearly that the contribution of TS to the dielectric constant
is proportional to the concentration of water, probably as OH groups. This has been
interpreted as additional TS produced by hindered OH rotation (Phillips 1981~).
I I
.,
05 1 2 5 10
T (KI
Figure 12. The temperature variation of the dielectric constant of vitreous silica containing
differing concentrations ofhydroxyl (A,<1.5 PPM: 0 , 1 9 0 PPM; 1000 PPM; 7 , 1 2 0 0 PPM)
(Schickfus and Hunklinger 1976).
4.2. Strong fields

The response to large applied fields is critically dependent on the coherence between
the two eigenfunctions and (L2 ( 8 2.2) and can be treated only by solving the coupled
equations for the time evolution of the probability amplitudes a , and a,. Formal
methods for solving the problem developed for optical saturation (Sargent et a1 1974)
and magnetic resonance (Slichter 1980) have been applied to TS in glasses (Golding
and Graebner 1981) but for completeness we will present here a relatively straightfor-
ward approach for a spatially homogeneous electric field. The response to the perturba-
tion Hi,,(t ) is determined by the parameters p , = - p 2 = ($llqx~$l)and p12= p 2 , =
(+,~qx~~)~),where q is the charge associated with the tunnelling particle.
1680 W A Phillips
Using the general state function (2.12) in the time-dependent Schrodinger equation
av/
+
( Ha Hint(t ) )v/ = i h -
at
(4.6)
gives equations for U , and U * :

1
b1 = ~[(+ll~int(~)l+l?~l +(+llHint(t)1+2?a>exp(-iwot)l (4.7)
1
U2 =i,[(+llHint(t)l+dal ex~(iwot)+ ($dHint(t)l+2?a21. (4.8)
Writing Hint( t ) = -[( t ) q x and considering for simplicity the symmetric double well
where p, = p 2 = 0 (for a more general discussion see Pippard 1983), the equations
become
-5(t)p12
a, = a2 exp( - h o t ) (4.9)
ih
(4.10)
The time-dependent dipole moment from (2.13) is given by U, where
U = alaf exp(iwot)+ a2aT exp(-iw,t). (4.11)

Using (4.9) and (4.10),
zi = iw,[a,af exp(iw,t) - a2aT exp(-iwot)] = wov (4.12)
where v = i[a,af exp(iwot)- a2aT exp(-iwot)]. In a similar way
(4.13)
(4.14)
Only the interaction with the driving field has been included in (4.12)-(4.14), but the
incoherent interaction with thermal phonons can be included by introducing
phenomenological relaxation times TI and T 2 . The population difference w relaxes
back to the equilibrium value w o= -tanh(E/2kBT), where E = hw,, with a relaxation
time TI given by (2.26)l. The off-diagonal coherence between $, and relaxes back
to zero with a time constant T,, known as the dephasing time. The resulting equations
are then
U
u=--+w OV (4.15)
T2
(4.16)
(4.17)
Equations (4.15)-(4.17) are the Bloch equations of magnetic resonance.

In the steady state, when transient effects governed by TI and T2 have decayed to
zero, we can look for steady state solutions. In the absence of a driving field only w
is non-zero, so that both U and U will vary at least linearly with ((t). The final term
in (4.17) is therefore quadratic in 6, a n d will give a steady average value E which
differs from wo. With w = E and constant in (4.16),
(4.18)
For a driving field (( 2 ) = (,[exp(iwt) + exp( -iwt)] the second exponential term governs
the behaviour of p = po exp( -iwt), giving
(2i 5oP 12 T2 I3 1 1
(4.19)
Po =
h +
[ 1 i( wo - w ) TJ ’
The imaginary part of po gives the amplitude of U which when introduced into (4.17)
gives a value for % of
(4.20)
The result for po shows, after substitution of (4.20) in (4.19), that the response at
resonance is reduced at large fields by a factor [ l + ( 2 p 1 2 ~ 0 / h ) 2 T , T and
2 ] that the
response curve is broadened, defining a n effective relaxation time Ti where
(4.21)
In a glass the total response is the sum of contributions from states which are within
a linewidth of the exciting frequency 0. The number of these states, of order P( h / T i )
where P is the constant density of states, is increased at large fields by a factor Ti/ T 2 .
However, because the contribution of each is reduced by a factor (TJ T;)’, the effect
of a large field is to reduce the attenuation by a factor
(4.22)
Equation (3.4) must therefore be modified to give
1-1 ( r y t w ) P tanh(hw/2kBT)
=~
a (4.23)
(Pd) (1 +”
where in the acoustic case the intensity ratio
I/I,=-- (2ye)* T2 (4.24)

h2
with I = 2peiv3 (in W m-2) for a strain field 2e,, cos w t a n d where
I,= h 2 p v 3 / 2 y 2 T 1 T 2 . (4.25)
In the electrical case
I,= 3 h 2 ~ , ~ , c / 2 p ~ T , T 2 (4.26)
after averaging over dipole orientations, and where p I 2has been replaced by po used
earlier.
1682 W A Phillips
Equation (4.24) can be given a simple physical interpretation. The cross-over

between weak and strong fields occurs when the acoustic-phonon energy per unit
frequency range is equal to that of thermal phonons. If the critical energy density is
E , then for hw << k B T
( E , / hw T i ’ ) = ( W * / T U ~ ) k( B T /h w )
if the energy width of the acoustic pulse is determined by T2 and not by the pulse
length. The critical intensity is
I,= v o r E c = ( w 2 k B T / ~ v , T 2 )
which on using (2.26) with E = A o = hw, gives a result of the same form as (4.25).
Saturation has been observed in both electrical (Schickfus and Hunklinger 1977)
and acoustic experiments (Hunklinger et a1 1972, Golding et a1 1973, 1976b, Graebner
et a1 1983) below 1 K (figure 13). In general the agreement with (4.23) is good, although
at very low temperatures the steady state solutions of the Bloch equations, assumed
,
in deriving (4.23), may not apply if the length of the acoustic pulse is less than T I .
, 19-8
Energy / pulse (erg cm-’)
I;-~ iy6 ~q-~ 1y4
Peakacoustic intensity ( W ~ m - ~ l
Figure 13. Acoustic attenuation as a function of acoustic intensity in vitreous silica (Golding
er a/ 1976b). T = 0.023 K. 0 , transverse, 0 / 2 =~ 0.534 GHz; 0, longitudinal w / 2 7 r =
0.592 GHz.
4.3. Relaxation
Two-level systems can give a large response in non-resonant electric and acoustic fields.
Formally this can be treated together with resonant processes by solving the Bloch
equations (Jackle et a1 1976) but in practice it is convenient to treat the two separately.
The distinction can most easily be seen by subjecting a thermal distribution of tunnelling
states of energy E, asymmetry A to a step function field. In the electrical case the
dipole moment in the absence of the field vanishes for a random angular distribution
of the dipole moments, where each dipole is poA/E (2.9). Considering only those
pointing along the field, in thermal equilibrium the average moment is
A
(4.27)
and the mean polarisability is given by
where the effect of the field is to change the asymmetry by 2p05. Using (4.27)gives
for the average contribution from all states
(4.28)
where the factor of three comes from averaging over orientation. The first term in
(4.28)reflects the change in the dipole moment of each state, and the second the effect
of redistribution between the energy levels when the system moves out of thermal
equilibrium as E is changed. The redistribution occurs relatively slowly on a time
scale determined by coupling to the surroundings, while the change in dipole moment
occurs on a time scale h / E . In fact (4.28)represents a time average on a scale longer
than h / E, and the time response is oscillatory, as shown in figure 14. For non-resonant
fields the dominant response is by relaxation, represented by the second term of (4.28).
t -
Figure 14. The response of a two-level system to a step-function applied electric field,
showing the rapid oscillatory resonant variation of the average dipole moment together
with the s:swer variation arising from relaxation (after Pippard 1983).
With this preamble, it is possible to write down an equation governing the population
difference w between the two levels. Relaxation proceeds according to
w-
-- w( t )
(4.29)
7
where Lt. is the instantaneous equilibrium value of the population difference, given in
the electrical case by
w = WO+- 8aWO
A
2p05(t ) (4.30)
1684 W A Phillips
and T is conventionally written for the relaxation time T I . Combining (4.29) and (4.30)
gives for 6w = w - w,,
For if( t ) = 5, exp( -iwt) the solution is of the form

aw, toexp(-iwt)
6w=-2p,
i3A l-iwt ‘
The corresponding change in average dipole moment 8 p ( w ) = p o ( A / E ) 6w defines

a complex susceptibility x ( w ) = S p ( w ) / ( ( w ) , where
(4.32)
with
~ ( 0= )--sech’
Pi A2
(4.33)
3kBT E 2
In the acoustic case a n equivalent calculation gives the same form for x ( w ) (equivalent
to the modulus) with pi/3 replaced by y 2 (where the orientational average is incorpor-
ated). Restricting attention to the acoustic case, where most experimental information
is available, a n d using the results
ff =-w x ” ( w 1
PO3
and
where ~ ’ ( wand
) ~ ” ( are
0) the real and imaginary parts of x ( w ) , gives, after introducing
the distribution function g ( E , T ) ,
(4.34)
where A2/E2is replaced by ( 1 - T,,,~,,/T). Equation (4.34) can be evaluated analytically
in the limits WT,~,,<< 1 and wmin>> 1 to give
(4.35)
and
(4.36)
using (2.26) for T,~,, with A, = 11 No saturation is usually expected for these relaxation
contributions to the attenuation because the driving field does not itself cause transitions
between the two energy levels.
The corresponding velocity changes can be calculated in the same limits. For
1 the effect of relaxation processes on the velocity is small in comparison to
WT,,,~,,>>
the resonant process, but for W T , , , ~<<~ 1
(4.37)
for WT,~,,<< 1 and hw < kBT, where To is a reference temperature. This result has the
same form as that for resonant scattering, and differs only by a factor of 5. The factor
3 is a direct consequence of the T 3 dependence of T,,,~", so that higher-order phonon
processes will give an even more rapid negative change of velocity with temperature.
At any temperature T,~,, can be estimated from (2.26) with E = A, = k BT as approxi-
mately 1 ns at 1 K, varying as T - 3 , so that WT,,,~,, for a typical ultrasonic frequency of
100 MHz is unity just above this temperature. The regime WT,,,~,, >> 1 therefore coincides
with the onset of higher-order phonon processes, which cannot generally be included
by a simple power law dependence of T,,,~,, on temperature. However, a maximum in
A V / U occurs when the effect of relaxation dominates the logarithmic resonant contribu-
tion to A v / v seen at lower temperatures. The temperature of this maximum gives
another way of estimating coupling constants, using (2.26) and the condition WT,~,, 1 -
(Hunklinger and Raychaudhuri 1986).
In order to examine in detail the accuracy of (4.34)-(4.36) measurements must be
made at low frequencies. This has been done at about 1 kHz (Raychaudhuri and
Hunklinger 1984) and close to 1 Hz (Wright and Phillips 1984), where W T , ~ , , is unity
at 20 mK and 2 mK respectively. Results at 1 kHz shown in figure 15 demonstrate the
two expected logarithmic regimes. The combination of opposing resonant and relaxa-
tion terms (4.4) and (4.37) observed above 100 mK gives as expected a slope half that
of the resonant term alone below 100 mK. At these low frequencies the attenuation
is more conveniently described by the quality factor Q where
ffu (Py')
Q-' =;=y 7r
0' (4.38)
The frequency independence of Q for W T , ~ , , >> 1 is confirmed by experiment and takes

a value of about 3.5 x below 4 K i n silica. Although the average y 2 entering (4.38)
0 01 01 1
1 , .., 1 10
T (K)
Figure 15. The temperature variation of the velocity of sound measured at low frequencies,
showing the two logarithmic regimes with slopes in the ratio -2 (Hunklinger a n d
Raychaudhuri 1986). Cover glass 1028 Hz.
1686 W A Phillips
differs for different experimental techniques, the values of P y 2 in S i 0 2 are in good

agreement with those derived from high-frequency velocity measurements. At lower
temperatures (Raychaudhuri and Hunklinger 1984) Q-' falls as T 3in accord with (4.36).
In contrast to these very-low-frequency examples, relaxation attenuation of thermal
phonons can be a significant contribution to the total scattering at a few degrees, and
can give a decrease with increasing temperature of the thermal conductivity K . Even
though extension of relaxation effects to frequencies above 10" Hz involves an extrapo-
lation of (2.26) and (4.36) into regimes where no independent test of their validity is
possible, fitting to thermal conductivity data gives reasonable agreement with predicted
values (Zaitlin a n d Anderson 1975).
5. Pulse echo experiments
5.1. General theory

In contrast to the steady state response to large applied fields discussed in 8 4.2, this
section concentrates on pulse phenomena where the pulse length T~ is less than either
TI, T2 o r both. The experiments can be understood in terms of an analysis based on
(4.9) and (4.10), which for a resonant applied field to[exp(iwot) +exp(-iw,t)] take
the form, with neglect of the non-resonant exponent,
. top12
a1 - a2
ih
. top12
a2 - ai. (5.2)
ih
Combining these equations gives
U, = - ( t 0 p 1 2 / h ) 2 a ,= - w : a , (5.3)
a n d similarly for a2 where w i = t0p,2/h. If the system is initially in state the
wavefunction at time t later is
P(t)= c o s ( w , t ) + , exp(iwot/2)+sin exp(-iwot/2). (5.4)
After a time .rr/2wI = 7rh/25,p1, the system is in the state (L2. After a .rr/2 pulse, of
length .rr/4wI, the system is in a mixed state (9, + (L2)/J2, assuming that relaxation
can be neglected during the pulse, i.e. T ~ < <T , and T 2 .
A typical experiment consists of a 7r/2 pulse followed a time to later by a 7r pulse.
At the end of the first pulse the wavefunction is in the state ($,+(L2)/J2 if W , T ~ < <1,
and after a time to has evolved into
Wtd = (1/J2)[$1 exp(iwoto/2) + ( L exp(-iwoto/2)I.
~
During the second pulse the probability amplitudes must satisfy (5.1) a n d (5.2), so that
Wt) = (1/J2){ILi cos[w,(t - to)] + ( L sin[w,(r
~ - to)]) exp(iwoto/2)
+(1/J2){+, sin[ai,(t-to)]++2 cos[wi(r -to)]} exp(-iw,to/2)
to < t < to+ 2 r p
and at the end of the pulse
Y ( t o + 2 ~ , )= ( 1 / J 2 ) [ 9 , exp(iwoto/2)+ +, exp(-iwoto/2). (5.5)
The subsequent time variation has the form for to >> T~

T(t ) = (1/d2){4b2 exp(iwot/2) exp[-iwo(t - to)/2]
+ GI exp(-iwot/2) exp[iwo(t - to)/2]}.
The dipole moment ( p ) can be calculated as in § 2.2, and will be a maximum when
a, = a, = 1 / J 2 , at time t = 2t0. The signal produced at t = 2t0 is known as a spontaneous
echo. For a single subset of TS with energy E = hw, this maximum would not be sharp,
but because it occurs at a time independent of w o it is very sharp in an amorphous
solid where TS with a range of energies are inevitably excited as a result of the finite
length of the pulse. One of the fascinating aspects of echo phenomena is the way in
which a broad distribution of parameters leads directly to the sharpness of the response.
This analysis has ignored the effect of relaxation, justifiable during short pulses if
rp<<TI and T,, but which must be included during the time interval 2t0. Both the
energy exchange with phonons, which contributes to TI and T2,and interaction between
TS which produces dephasing important in T,, reduce the echo amplitude. Indeed, as
will be described in the next section, by a suitable choice of pulse pattern either TI
or T2 can be measured.
5.2. Echo experiments

Photon and phonon echoes can be observed only at very low temperatures. Unlike
the equivalent magnetic experiment where the large steady field prepares the system
in a well defined initial state, in amorphous solids each TS can be prepared in state
t,bl only by cooling to a temperature T less than h w / k , , where w is the angular
frequency of the exciting pulses. For example, in a typical spontaneous electric echo
experiment, similar to that analysed in § 5.1, two pulses of length T,, and 2rp (typically
-
< 100 ns) at a frequency of 1 GHz ( h w / kB 40 mK) are applied across a capacitor
containing the sample at a temperature of 10-20mK in a dilution refrigerator. At
these temperatures T~ is less than TI or T2 so that relaxation can be ignored during
the pulse. The echo amplitude is observed as a function of to,the electric field
amplitude, or to, the time between pulses.
Echo experiments on TS have been designed to give different kinds of information.
The amplitude of the echo as a function of the strength of the applied resonant field
gives a measure of the induced electric dipole moment or the phonon coupling constant,
the echo amplitude as a function of pulse separation in a two- or three-pulse experiment
can be used to give T2 or TI respectively. Analysis of these experiments follows the
equivalent magnetic case, except that the TS problem is complicated by three factors.
First, the elastic or electric dipoles are not aligned with respect to the driving field and
a calculation of the echo amplitude involves an average over orientation. Secondly,
for a given resonant frequency w o there exists a distribution of induced moments
(elastic or electric) and relaxation times which should be included in the analysis.
Finally, in electric echo experiments the local field seen by the TS is not equal to the
applied field, and a local-field correction factor must be used when evaluating absolute
values of the dipole moment.
The various experimental and theoretical complications and the relative scarcity
and inconsistency of results make it inappropriate to analyse and compare in detail
the experimental results. In this and the next section the emphasis is on basic physical
ideas at the cost of idealising what is still a confusing experimental and theoretical
problem.
1688 W A Phillips
The echo amplitude measured for small pulse separations in an ideal double-pulse
experiment, ignoring relaxation, is a maximum (from the analysis given in 9 5.1) when
the amplitude of the driving field (25, COS(WT)
or 2e, COS(WT)) is given by
in the electric case, or

e,y.r,
-=-
(5.8)
h 4
for acoustic pulses. (Experimentally it is often convenient to use two identical pulses,
in which case the echo amplitude is a maximum when ~ , ~ ~ = 2 7 ~The / 3relationship
. )
between p l z , derived from (5.7), and p , , derived from an attenuation or velocity
measurement (4.1) or (4.5), is not straightforward because averages over the distribution
functions are weighted differently in the two cases. For example, analysis of an electric
echo experiment has shown that p I 2= 0.6 p o (Phillips 1981c), although in practice
experimental difficulties involved in an accurate determination of 6, or e, give the
greatest uncertainties in the derived values of po or y.
Typical results for the electric echo amplitude as a function of 5, are shown in
figure 16. It is clear that two distinct dipole species exist in those samples of silica
(Suprasil I and Herasil) that contain OH groups, with dipole moments differing by a
factor of six. The larger moment of approximately 5 x lop3' C m (taking into account
local field factors) has been associated with rotation of the proton in OH (Phillips
1981c), but the smaller dipole appears to be intrinsic to silica. By adjusting the
frequency and magnitude of acoustic or electric fields particular TS can be selected
from broad distributions by means of their energy and coupling strength. This selectivity
of echo experiments is in marked contrast to most experimental studies of the amor-
phous state.
The variation of the echo amplitude with separation of two equal pulses is shown
in figure 17 for the acoustic case (Graebner and Golding 1979) with the field adjusted
1201 1
100
al
U
80
-
c
60
0
r
w
U
40
20
0
10-2 1
Relative electric field amplitude
Figure 16. Amplitude of the spontaneous electric echo in different samples of vitreous
silica (Golding e t a [ 1979). The OH content ofthe samples is as follows: Suprasill 1200 P P M ,
Herasil 200 PPM, Suprasil W and Infrasil less than 10 PPM. T = 19 mK.
I I I I I I I I I
4 8 12 16
2T,* Ips)
Figure 17. Variation of the spontaneous acoustic echo as a function of pulse separation
at a number of temperatures below 100 m K (Golding and Graebner 1981). Suprasil W,
f = 0.692 GHz.
to maximise the contribution from the intrinsic dipoles. Decay times fitted to the
approximately exponential variation give values for the dephasing time T2 which
decrease with increasing temperature as T - 2 , from a value of 16 ps at the lowest
temperature of 18 mK. Similar results were obtained for electric echoes (Golding et
a1 1979) also probing ‘intrinsic’ TS in silica, but those shown in figure 18 illustrate the
I \- I I
50 100 150 200

2T ips1
Figure 18. Variation of the spontaneous electric echo amplitude as a function of pulse
separation (Bernard el al 1979). P,,, = -43/40 dBm.
1690 W A Phillips
sensitivity of these echo measurements to the precise experimental conditions. In this

set of experiments the strength of the electric field was lower, to probe the OH-related
dipoles (Bernard et a1 1979), but the time constants of the exponential decays vary as
T-', and at 20 m K take the value 40 ps. The microscopic origins of the dephasing
time T2 are discussed in 0 5.3.
Experiments to measure the energy relaxation time TI require three pulses but for
the usual stimulated echo (Black and Halperin 1977) the results give a complicated
combination of T, and T 2 . A more direct measure of T, is obtained by using an initial
T pulse, followed a time to later by closely spaced 7r/2 and n- pulses (as for spontaneous
echoes). It is important, for reasons that are explained in 0 5.3, that the initial pulse
is significantly shorter (and therefore of higher intensity) than the succeeding two. If
this condition is satisfied the second and third pulses probe TS well within the frequency
range excited by the first (remembering T ~ < T < , , T J . The initial n- pulse takes the TS
from the ground state +, to the state &. During the time to a fraction exp(-to/ Ti) of
the systems will relax back to the state $l. The double-pulse sequence gives a
spontaneous echo as before, but the contribution of those systems which decayed back
to I),will be opposed by those remaining in (L2. The echo amplitude, independent of
any dephasing processes, is equal to the sum of exp(-to/ T , ) from and -[I -
exp( - to/ T1)]from o r -[ 1 - 2 exp( -to/ T , ) ] .Measurements on silica (Golding et a1
1979, Bernard et a1 1979) show that this simple exponential relaxation is not obeyed
for larger values of t o , indicating as expected from 9 2.2 a distribution of relaxation
times (figure 19). The minimum value measured at short times is in good agreement
with (2.26) (Bernard et a1 1979) with values of y', a n d y$ close to but slightly larger
than those deduced from the variation of spontaneous echo amplitude with acoustic
intensity.
One further ingenious electric echo experiment deserves mention. Bernard et a1
(1978) applied a DC electric field pulse between the two oscillating field pulses in a
spontaneous echo experiment. The effect of this field is to change the energies of states
and $2 so that a n additional phase difference of 25pr0/h is introduced in ( 5 . 6 ) ,
rn
G
200 400 600
.,
T,2 ip s I
Figure 19. The amplitude of the stimulated electric echo in vitreous silica as a function of
the time between the first and second pulses (Golding et a [ 1979). T = 19 m K , f = 0.72 GHz.
A, Infrasil; 0, Herasil; Herasil -OH; 0 , Suprasil I -OH.
where T~ is the length of the DC pulse, .$ the electric field strength and p the component
of the static dipole moment along the field direction. For a given TS this gives a n
additional factor in the echo of cos ( 2 [ p o / h ) , because the phase factor is not com-
pensated in the time interval to< 1 <: 2t0, and so the total resultant is
(5.9)
where nZ(p)d p is the number of TS with component moment p, weighted according

to their contribution to the echo. Measurements of E ( [ , TJ as a function of T~ (or 6)
give the Fourier transform of nZ(p). Results indicate a n exponential dependence of
E ( [ , T ~ on
) T~ and hence a Lorentzian distribution function for nZ(p), but this experi-
ment has not been analysed in detail with inclusion of all weighting factors.
5.3. Spectral diffusion

The short dephasing time T2 is the result of interactions between tunnelling states. In
principle a n interaction between two TS with the same energy could take place by
mutual excitation a n d de-excitation with exchange of a resonant phonon (through the
term in cr, in the interaction Hamiltonian (2.15)). In a glass, however, the density of
mutually resonant TS is too small for this to be a n effective dephasing process. Much
more effective is the non-resonant process (involving U , ) in which a transition of one
TS, equivalent to the reorientation of a n elastic dipole, gives rise to a change in the
strain field experienced by a neighbour, and hence to a n energy change. Because all
TS with energy less than about 2 k B T undergo transitions at temperature T, a given
state interacts with many more neighbours than is the case for resonant interaction.
A detailed quantitative calculation of T2 is complicated (Klauder and Anderson
1962, Hu a n d Walker 1977, Laikhtman 1985), but the following semiquantitive explana-
tion contains the essential physical features. Each TS experiences a fluctuating local
strain field which in the steady state gives rise to energy fluctuations of magnitude
AEo. In a n echo experiment the initial field pulse, of duration much less than the
time scale of the fluctuations, selects from the distribution of TS a subset which instan-
taneously have energy Eo = hwo (figure 20( b)). Because the mean energies of TS within
this subset are different, the energies of the excited systems gradually spread out over
an energy range AEo,as shown in figure 20( b ) . At times long compared to the time scale
1 2 3
I
E=/w
I
Lu
a
la1
Figure 20. A schematic representation of the way in which spectral diffusion leads to an
increase in linewidth with time.
1692 W A Phillips
of the fluctuations, governed by the average energy relaxation time T I , the width of
the energy distribution of the subset reaches A E o , and is then independent of time.
However, the most interesting physical aspects of the interaction involve the time
variation of this width AE ( t ) .
The magnitude of AEo can be evaluated from the strength of the coupling between
TS and phonons. Ignoring the tensorial character of the strain field (Black and Halperin
1977), the interaction energy between a pair i, j of TS separated by distance r!, can be
written
(5.10)
where y , A , / E, and yl A,/ E, are the static elastic dipole moments and C is a constant
of order unity. Replacing l / r i by the concentration of thermally excitable TS, Pk,T,
and averaging over neighbours j gives
y 2Pk, T A
AEo= CT - (5.11)
PO E
for a state of asymmetry A. The factor A / E is included to show explicitly that the
interaction energy which determines T2 is correlated with the induced dipole moment
given by yAo/E, but will be ignored in the remainder of the analysis.
The effect of this ‘spectral diflusion’, in which the excitation is spread out over an
energy range AEo, is dependent on the time interval to between the two pulses in the
spontaneous echo experiment. In the long time limit to >> T,,, where T,,, is defined as
-
the shortest energy relaxation time for TS with E k B T and is given by (2.26) with
E = A,= k,T, T2 is the time for which the spread in phase AEotlh is of order ~ / 2 .
This condition gives
(5.12)
Because the phase difference increases with t the echo decay is essentially exponential
for to >> r,,, , with T2 inversely proportional to T.
At short times, to<<. T,,,, the width of the energy distribution AE(t) increases
approximately as AEo[ 1 - exp( - t / T,,”)], so that
AE ( t ) = AEot/r,,, . (5.13)
The dephasing time is given by the condition
(5.14)
leading to
T:= A.rrrmi,/AEo. (5.15)
The decay is non-exponential, varying with to as
exp(-AE,t~/h~,,,). (5.16)
The echo amplitude therefore decays to l / e of its initial amplitude in a tirne T2 which
varies as T-’ for to<< r,,,, using r,,, proportional to T-3 (0 4.3).
The extent to which spectral cliffusion is observable in a spontaneous echo experiment
depends on the relative magnitudes of to a n d T,,,. Using (2.26) and the measurements
of T , described in the last section, rmlncan be estimated as 100 ps at 20 mK in silica.
The condition to<< T,,, is therefore well satisfied for to = 10 ps at 20 mK, a n d the results
for the lowest temperature shown in figure 17 for 'intrinsic' phonon echoes should
demonstrate spectral diffusion. The dependence of echo amplitude on to is however
more nearly exponential than Gaussian, although T2 varies as T-' and the value of
l o p s calculated from (5.15) is in agreement with the measured value. Results for
electric echoes from intrinsic TS in silica also give the same value for T2 at 20 mK,
although the decay is again non-Gaussian but with T2 varying as T-I.
There seems no obvious explanation for this discrepancy, although it may be a
result of the extreme sensitivity of the results to experimental conditions consequential
on the broad distribution of TS parameters. In particular, the pulses probe states of
energy E = hw but with a range of A and hence a range of interaction energies AEo
(5.11). The resultant decay will therefore be a superposition of terms of the form given
by (5.16). Smaller values of AEo give a slower decrease of echo amplitude which at
long times can make the decay approximate to a n exponential instead of a Gaussian
form. However, the shape depends significantly on the precise tuning condition, and
a critical assessment of data is difficult on the basis of published information.
Spectral diffusion involving TS is also important in controlling the homogeneous
linewidth (equal to h / T2)of optically active impurities in solids. In these experiments
the energy hw, of the relaxing species is very much greater than k,T, in contrast to
TS echo experiments where the two are comparable. Examples include Nd3+ in S i 0 2
where T2 was measured by a photon echo experiment (Hagarty et a1 1982), a n d organic
glasses (Thijssen et a1 1982) where the linewidth was measured by photochemical hole
burning. Values of Tz fall in the range 10 ns < to < 1 ps at 1 K, but all decrease with
increasing temperature as T-' in the range 0.1 < T < 10 K. Three separate publications
(Lyo 1981, Hunklinger and Schmidt 1984, Huber et a1 1984) have contained explana-
tions based on spectral diffusions, all in essence giving the arguments presented above
albeit in slightly different forms. The only significant difference is the extension to
more general forms of interaction, replacing (5.10) by a potential varying as l / r m , a n d
by the inclusion of higher-order phonon processes in determining T,," above 1 K.
The experiments are carried out in the long-time regime, so that T2 is given by
(5.12) in contrast to the TS echo experiments. This difference may also justify the
inclusion of a n energy dependence of P in the optical experiments to give the observed
temperature dependence of T - ' 3, while ignoring it for TS echoes. In fact Hunklinger
and Schmidt (1984) derived the additional T o factor by the dependence of the effective
density of states on time, as in the discussion of § 3.1.
The analysis of saturation in pulse experiments (§ 4.2) is complicated by spectral
diffusion. In an experiment where the width of the 'hole' burnt into the TS distribution
by the saturating pulse is constant with time, a measure of th: attenuation of a second
weaker pulse as a function of the time separation between the pulses allows measure-
ment of 'I,,. The extent to which this ideal arrangement can be realised in practice
depends on the relative magnitudes of T~ the pulse length, TI a n d T2 for the TS of
energy h w o , and T,,,, , the characteristic minimum relaxation time for neighbouring
states which determines the time scale of spectral diffusion. If h a o < kBTthen T,,, < T ,
and vice versa. Different regimes may be classified as follows.
( i ) T,, < T I ,T 2 . The hole burnt into the TS distribution has a width h / i p , larger
than that produced by spectral diffusion, which will therefore not be important. TI is
measured.
(ii) T2< T,, < T , . Under these conditions spectral diffusion may be important
because the hole in the distribution can broaden and become shallower with time
1694 W A Phillips
following the initial pulse. Only if T ~ T,,,~,,

< is this a major factor, because otherwise
spectral diffusion during the pulse will allow saturation of a hole of width A / T 2 ,a n d
subsequent spectral diffusion will have little effect. Difficulties of interpretation may
occur if hao/ k BT is not much less than unity.
(iii) T,> T 2 ,T I . In this steady state regime spectral diffusion is only important if
hwo/ kBT>> 1, unlikely at high temperatures where relaxation times are short.
-
The most straightforward results to interpret are those for which 7, T2or less, as
was the case for those shown in figure 21 (Golding and Graebner 1981). Although
the attenuation recovery is not a n exponential function of pulse separation, the
asymptotic behaviour at small times defines a relaxation time which is in good agreement
with that calculated for a one-phonon process (2.26) with y = 1.5 eV. The slower
recovery at longer times may reflect the broad distribution of TS relaxation times.
0 011
0
\
1
200
I
400
I
600
I
T.2 [psi
Figure 21. The change in attenuation of a weak probing pulse applied a time T , after
~ a
saturating pulse (Golding and Graebner 1981). Suprasil W, f = 0.692 GHz, L = 0.635 cm.
One further experimental feature can complicate the interpretation of saturation

recovery experiments. It is convenient for experimental reasons to observe the recovery
using a second pulse which is not weak (Hunklinger and Arnold 1976). Consequently
the relationship between attenuation and occupation probabilities is non-linear.
Hunklinger and Arnold (1976) used the steady-state solutions of the Bloch equations
(4.15)-(4.17) to relate the two, but it is difficult to ensure that the necessary condition
T,>> T, is indeed valid.
Direct observation of spectral dijfusion is possible by using an intense saturating
pulse of variable frequency w followed by a weak pulse of fixed frequency w o . (It is
obviously more convenient to fix the frequency of the weak pulse, although in practice
even the probing pulses are sufficiently intense to be affected by saturation.) By
measuring the attenuation of the second pulse as a function of w at a fixed and very
short time interval between pulses, the linewidth can be measured directly. Experi-
mental results at 0.4 K in silica (Arnold et a1 1978) show not only a linewidth much
greater than A / T ~or A / T I ,but an increase in linewidth with increasing pulse length,
giving direct evidence for spectral diffusion (figure 22). The results are in good
agreement with calculations based on the physical ideas leading to (5.15) and (5.16)
(Black a n d Halperin 1977).
i
i
0-
c-
2 ' 5 0 2-
mm
LO
i-
o c
5: o i -
+ a
?k
L
3%
O -
0.6
~
-
Two-level states in glasses
1.2 ps
o 7 ps
I I
Measurements at fixed time interval in a borosilicate glass BK7 have been collected
together in figure 23 (Graebner and Golding 1981). Experimental values of the
linewidth A w are compared to the predictions of a spectral diffusion model, showing
that Au is proportional to T4 at low temperatures where the short time limit (5.16)
applies. (The calculated curve in figure 23 is for a pulse separation of 1 ps, a reasonable
approximation for simultaneous 1 ps pulses and for shorter pulses with 1 ps separ-
ation). At high temperatures Aw is given by A E o / h from (5.12) in the limit T,;"<< 1 ps.
The agreement between theory and experiment gives further strong support for spectral
diffusion a5 a dominant contribution to T 2 .
T (Ki
1695
Figure 23. Measurements of the linewidth measured at different frequencies and tem-
peratures in a borosilicate glass BK7 (Golding and Graebner 1981). The full curve is
calculated from a spectral diffusion model. A, 0.692 GHz; 3 , 0.738 GHz; 0, 8.9 GHz.
1696 W A Phillips
6. Metallic glasses
There have been a number of recent review articles on the low-temperature

properties of amorphous metals, (Black 1981, Lohneysen 1981, Hunklinger and
Raychaudhuri 1985) a n d so this section will be limited to a survey of the way in which
the interaction between TS and electrons modifies the results described in the previous
sections. There have been detailed theoretical examinations of the coupled electron
TS problem based on analogies with the Kondo problem, but in general the only well
established experimental consequence of the interaction is the greatly reduced relaxa-
tion time of the TS.
The interaction between electrons and TS is represented formally by a Hamiltonian
H 2 of the form (Golding et a1 1978, Black et a1 1979)
kk
in spin notation, where C:. and Ck are electron operators. An electron in state k is
scattered into state k' with loss of energy E transferred to the tunnelling state (or vice
versa). Normally k and k' define the momentum of the electrons represented as plane
waves, but are used here simply as labels in view of the fact that the random structure
of amorphous metals invalidates the concept of a well defined electron momentum.
The Hamiltonian assumes that the spatial extent of each TS is smaller than the inelastic
electron scattering length. K is the Ts-electron coupling constant.
Following Q 2.2 the TS relaxation time can be calculated from the transition prob-
abilities w I 2 and w z l , where w , ~ for
, example, is given by
The Fermi functionf, represents the probability of finding state k occupied. Replacing
the summation in (6.2) by an integral
where a factor of arises from electron-spin conservation if n ( ~ is~ the

) toial electron
density of states at energy E k . The major contribution comes from states d o s e to the
Fermi level, so that on integration
7T
w I 2= - [ ~ ( E , ) K ] * ( A , , / E ) ~ E [-
~ exp(-E/k,T)]-'.
h
Combining with a similar expression for wzl gives
T-' - ?( n( E ~ ) K ) ~ ( A ~ / Ecoth(
) * EEI2kBT). (6.3)
-h
The strength of the electron--rs interaction can be gauged from the upper limit on TI
imposed by saturation recovery experiments (§ 5). A large intensity saturating pulse
has no measurable effect on the propagation of a second weak pulse even for pulse
separations as short as 100 ns (Golding et a1 1978). An upper limit of 25 ns for TI at
10 m K should be compared with the values of approximately 200 ps for Tl and 50 ps
for T2 in silica at the same temperature (§ 5.2).
In metallic glasses the electron interaction determines both T I a n d T2 at low

temperatures with T,(min) = T,~,, varying as T-‘. As the temperature is raised, the
phonon-Ts interaction becomes relatively more important, dominating above a few
degrees K (in much the same way as the phonon heat capacity dominates the electronic
contribution) and giving T , proportional
~ ~ to T 3 .The lifetime broadening of the TS
levels is significantly increased, but because of the already broad distribution of energies
in general this is expected to have little effect (Thomas 1983).
Because of the reduced T,~” relaxation processes are observable at lower tem-
peratures in metallic glasses. Results equivalent to (4.34), (4.35) a n d (4.36) can be
derived by using (6.3). In the limit UT,^,, >> 1
/-’=--7 T 3 ( P Y 2 ) ( n ( + ) K ) 2 k,T
(6.4)
24 (pv’) hv
and the velocity change is negligible. For WT,~,<< 1,
as in an insulating glass because the result is independent of the form of the relaxation
time, and
where the factor of in the insulating glass is replaced by f. In a metallic glass the
condition WT,~,, = 1 is satisfied for frequencies of the order of 500 MHz at approximately
100 mK, so that the logarithmic slope measured in a velocity experiment below 1 K
contains both resonant and relaxation contributions of opposite sign giving a resultant
A V- 1 (Py’)
_ - +- -In ( T / To). (6.7)
U 2 (PV’)
The combination of the two contributions to the attenuation is additive ((3.4) a n d

(4.35)) so that values of y 2 deduced naively from A v / v using (4.4) give much smaller
values than those given by the thermal conductivity o r ultrasonic attenuation. Detailed
fitting of experiments is needed to give good agreement with theory, as shown in figure
24.
At higher temperatures where phonons take over from electrons as the dominant
cause of TS relaxation, the velocity variation given by adding (4.4) and (4.37) is negative.
In practice higher-order phonon processes modify the logarithmic dependence to give
a n almost linear decrease of velocity with temperature. However, it should be noted
that the maximum in A v / v seen typically at 2-3 K in metallic glasses differs in origin
from that seen in insulating glasses. The latter is produced when the resonant increase
in A v / v is balanced by the decrease caused by relaxation, and occurs when wr,,, - 1
at a temperature which scales as w 1 I 3 . In a metallic glass the maximum occurs at a
temperature T, which is almost frequency independent.
The phenomenon of saturation is also complicated by electron-Ts interaction. The
critical intensity I , needed to saturate resonance absorption (4.25) is greatly increased
in metals because the relaxation times are much smaller. Figure 25 shows attenuation
as a function of acoustic intensity in a n amorphous metal PdSiCu. Before comparing
with the corresponding curve for silica (figure 13) it is important to make sure that
resonant scattering dominates at 62 m K and 720 MHz. Because WT,~,> 1 for PdSiCu
1698 W A Phillips
10-2 lo-’ 1 10
T IK)
Figure 24. The attenuation and velocity of acoustic waves in the metallic glass PdSiCu as
a function of temperature, together with the behaviour expected from the tunnelling model
(full curve) (Golding et a1 1978). f = 0 . 9 6 G H z .
E
.-
.I-
3
W .
L
+
a
Acoustic f l u x ( Wcm-’)
Figure 25. Attenuation as a function of acoustic intensity in amorphous metallic PdSiCu.
T = 0.062 K, transverse f = 720 MHz.
under these conditions, the total attenuation is the sum of resonant and relaxation
contributions given by (3.4) and (6.4) respectively. Evaluation of the terms shows that
the resonant term is larger by a factor of about 10, and so comparison with the case
of silica is indeed valid. As expected I , is much larger in the metallic glass.
Saturation is also observed when relaxation dominates the attenuation, at low
frequencies and temperatures above 100 mK (Araki et al 1980, Hikata et a1 1981).
This does not happen in an insulating glass because the intense pulse saturates only
those TS within an energy range h / T2 about the exciting frequency, small in comparison
with the effective width k B Tof the distribution of states contributing to relaxation. In
contrast the very short relaxation times in the metallic glass allow saturation of a
significant fraction of the TS contributing to the relaxation. At 0.2 K, for example, T 2 ,
approximately equal to T, , is about 1 ns, leading to a broadening equivalent to 0.1 K.
In practice the effect is enhanced by the power dependence of the linewidth.
Formally this can be taken into account by including the steady state solutions of
the Bloch equations in the treatment of relaxation. Strictly speaking, the Bloch
equations are not valid in the limit wTz<<1 but have been generalised to describe
metallic glasses (Continento 1982, Arnold et al 1982). The latter paper gives a detailed
fit to experiment in PdSiCu showing clearly the relative contribution of resonant and
relaxation terms, and demonstrating that both can be saturated.
The fact that a range of TS energies comparable to k,T can be saturated by a single
pulse suggests that acoustic velocity is also a function of acoustic power in metallic
glasses. An increase of velocity with increasing power has indeed been observed in
PdSi (CordiC and Bellessa 1981) and explained on the same basis as the power
dependence of relaxation attentuation. The data allow a determination of T2, approxi-
mately 2 ns at 10 mK in agreement with experiment.
The effect of electrons in determining TI for TS in amorphous metals is dramatically
confirmed by acoustic measurements in amorphous superconductors. Below the transi-
tion temperature T, the number of effective electrons decreases with decreasing tem-
perature, leading to an increase in TI (Black and Fulde 1979). There is a close parallel
between the behaviour of T;' and ultrasonic attenuation in (crystalline) superconduc-
tors, both reduced from the normal state value by a factor f(A,/ kT), where f is the
Fermi factor and A, the superconducting energy gap, in the limit where the TS or
phonon energy is small compared to A,.
The behaviour of the acoustic attenuation in an amorphous superconductor depends
critically on the relationship between T, and T, the temperature at which resonant
and relaxation contributions are equal, and on the value of W T , ~ , at T,. For the results
shown in figure 26, T,> T, and WT,,,<~ 1 at T,. The attenuation is dominated by
relaxation, which in the limit UT,,,<< 1 is, from ( 6 . 5 ) , insensitive to the form of T,~"
and proportional to U , as observed (Arnold er a1 1981). Nothing happens at T, because
the Ts-phonon interaction dominates, but at lower temperatures the attenuation falls
more rapidly in the superconducting than in the normal state. In the normal state the
electron-Ts interaction becomes more important than the phonon-Ts contribution at
about 1 K, but because the condition UT,,,<< 1 is still satisfied, the attenuation does
not change. By contrast, in the superconductor where both the electron and phonon
interactions are decreasing rapidly with temperature, UT,,, increases and the attenu-
ation is now (from (4.36)) dependent on the form of T , . Below about T,/2 the
attenuation decreases as T', characteristic of a one-phonon process.
Although these ideas explain the main features of the results the picture is not yet
complete. Measurements below 1 kHz in Cu,,Zr,, and Pd,oZr,, show a large increase
1700 W A Phillips
/
01 02 05 1 2 5 10
T iK)
Figure 26. Attenuation as a function of temperature in the amorphous superconductor
Pd,,Zr,, (Arnold et al 1981). Longitudinal waves: -, theory; 0, 0 , x , +, experiment.
in attenuation well below T, (Esquinazi et a1 1986). The difference is consistent with

a n increase in Py2 in the superconducting state, and is presumably observed clearly
at low frequencies because the condition WT,~"<< 1 is valid to lower temperatures.
7. Microscopic descriptions of tunnelling states
In a limited general review of this kind a detailed survey of specific microscopic models
is impossible and it is more appropriate to illustrate approaches to a microscopic
description of TS by means of well chosen examples. Generally models for tunnelling
states have been considered in three distinct categories: by relating tunnelling states
to established defects in amorphous or crystalline materials, by computer modelling
and thirdly as a consequence of general (and often imprecise) theories of the glassy
state. Each of these will be considered in turn, but only theories relating specifically
to TS are included.
7.1. Relation to s p e c & ?defects

The earliest and still one of the more convincing descriptions of a tunnelling state is
based on measurements in a mixed phase metallic NbZr crystalline alloy (Lou 1976).
The material is disordered because it contains small regions of one phase ( w ) in a
matrix of another ( p ) . For example, in Nb,,Zrpo the isolated regions of w phase are
typically 0.5 nm in diameter, and are formed by suitable heat treatment at a concentra-
tion of about m-3. Measurements of the heat capacity and thermal conductivity
show 'glassy' behaviour, dependent on the concentration of w phase. Microscopically
the tunnelling states in this case can be identified with the atomic displacements
occurring in the p - w phase transition (figure 27) in which two atoms move simul-
taneously through a distance of about 0.05 nm. Lou (1976) argues that this configur-
ational change can be described by a double-well potential, and presents experimental
evidence to support the existence of a wide range of barriers. Such a microscopic state
is of exactly the form suggested by the tunnelling model and could undoubtedly exist
in true glassy metals.
la I
rioi!
H
ro = 3.0 7
ibi
\ 20.58. k
- 11111
0 1
fir,
Figure 27. Atomic displacements a n d the local potential barrier involved in the p - w phase
transition in NbZr (Lou 1976).
Another known impurity, water, has been identified as a tunnelling state in vitreous
silica. Existing in the form of hydroxyl ions, it couples strongly to electric fields a n d
can be readily studied through measurements of the dielectric constant (§ 4.1) or in
electric echo experiments (0 5.2). This particular impurity can be analysed in detail
because the geometry of the displacement is established, and the density of states
depends only on the distribution of barriers ( V , in (2.5)). Using a distribution of V,
derived from higher-temperature dielectric measurements, the effect of OH on the
thermal a n d dielectric properties below 1 K can be understood in quantitative detail
(Phillips 1 9 8 1 ~ ) .
An interesting connection has been made between tunnelling states and the states
important in the optical a n d electrical properties of chalcogenide (and oxide) glasses,
with particular reference to As&. These glasses show a n activated conductivity a n d
no electron spin resonance signal as normally prepared, implying an absence of
unpaired electrons in contrast, for example, to pure amorphous germanium.
These and other experimental results were brought together and explained by Street
and Mott (1976) using a model based on a spin pairing idea of Anderson (1975) which
demonstrated the consequences of a strong attractive interaction between an electron
and its surroundings. The energy of each electron can be reduced by local distortion,
even to the extent of allowing two electrons to occupy the same site if the decrease in
1702 W A Phillips
individual electron energy more than compensates for the repulsive Coulomb interac-
tion between the two electrons. This pairing of electrons into ‘bipolaron’ or D- states
(Street and Mott 1976) explains why no free spins are observed in chalcogenide glasses,
although it must be pointed out that no detailed microscopic justification has yet been
achieved. However, support for the model is provided by the fact that it can explain
the large difference in frequency between absorbed and emitted light in luminescence
experiments, and the photoexcitation of free spins. A spin resonance signal can be
induced by irradiation with photons of energy slightly less than the band gap, and
destroyed by annealing or by irradiation with infrared light of about half the inducing
energy.
These paired electron ‘defects’are mobile, but can move only between sites separated
by no more than one or two atoms (Phillips 1976, Elliott 1979, Licciardello et a1 1981,
Karpov 1985) so that there is significant overlap between local distortions on the two
sites. This mobility gives rise to AC conductivity (Elliott 1979, Long 1982) and also to
the possibility that these defects can contribute to the low-temperature heat capacity
(Anderson 1975) both in chalcogenide glasses and in SiOz (Russo and Ferrari 1984).
It is clear that for small site separation there is no clear theoretical distinction between
the motion of electrons accompanied by local displacements of ions and the atomic
motion proposed in the tunnelling model.
On the experimental side there is clear evidence of a close connection between
these defects and tunnelling states. Fox er a1 (1982) show that the low-temperature
photon echo signal in AsZS3is sensitive to bandgap light, decreasing on irradiation.
Conversely the signal increases on annealing and under illumination with intense
infrared light, and so behaving in a way expected for paired electrons. The implication
of these experiments is that paired-electron defects behave as TS at low temperatures.
It is not possible to state the converse, that TS are the same as paired electron
states, because careful analysis of the experiment suggests that the density of optically
sensitive TS is much less than the total density of TS (Phillips 1985a). The fact that
the dielectric constant (0 4.1) is insensitive to light indicates that the echo experiments
probe only a small proportion of TS with large dipole moments ( § 5.2). This is not
altogether surprising, because the density of states deduced from heat capacity data
in As2S3is known to be much greater than that obtained from AC conductivity, and
perhaps the most interesting feature of the experiments is confirmation that the
tunnelling picture is a very general one. At sufficiently low temperatures any almost
degenerate coupled localised states will, in the random environment provided by a
glass or otherwise disordered solid, give rise to properties describable by the tunnelling
model.
7.2. Microscopic modelling

Random-network models of amorphous structures have been extremely useful in
providing atomic coordinates for the calculation of a wide range of physical properties.
Most structural calculations use as a starting point physically constructed models which
are subject to straightforward rules imposed by atomic coordination and limited
variations in bond length and angle (Etherington et a1 1982). These constructions are
subsequently ‘relaxed’ by computer to minimise the energy, usually using a form of
interatomic potential described by Keating (1966). The success of this approach has
led to attempts to search structural models in order to discover double-well potentials
or locally metastable states, although in general the models have been constructed
Two-leoel states in glasses 1703
from the outset by computer simulation, in order to avoid the constraints inherent in
physical modelling.
Smith (1978) started with a random arrangement of 250 atoms (subject to periodic
boundary conditions to avoid surface effects) and, by moving each atom in turn,
minimised an interatomic potential energy which included spherically symmetric long-
range attractive and core repulsive terms, together with an additional term which was
a minimum for tetrahedral coordination. The resulting model gave reasonable agree-
ment with experimental radial distribution functions, although the variation of bond
angles was greater than that found when starting from physically constructed models.
The most interesting feature of the results appeared in a search of local potentials,
carried out by calculating the energy of each atom in turn as it was displaced by finite
amounts from its equilibrium position. Approximately four local minima per atom
were found, although for the vast majority the additional energy in the second minimum
was much larger than the bond energy. The distribution of barriers also extends out
to similarly large energies, but the calculation appears to show that there are sufficient
states of low asymmetry and sufficiently small barriers to be interpreted as TS.
Wooten and Weaire (1984) show that by interchanging sufficient bonds in crystalline
Ge the resulting disordered Ge does not revert to the crystal on subsequent energy
relaxation, but gives a structure similar to that expected for the amorphous form. This
model can be ‘searched’ for single-atom double-well potentials, and as before the
calculation indicates a finite density of the resulting TS, although the range of asymmetry
is much less.
Two basic problems complicate the significance of this work on Ge. The first is
that the experimental evidence for TS in idealised amorphous Ge (i.e. a continuous
random network) is not clear, the available measurements showing a decrease in the
TS density of states as the physical density increases towards the ideal value (Duquesne
and Bellessa 1983, Lohneysen and Schink 1982, Graebner and Allen 1984, Phillips
1985b). Secondly, and this problem is not restricted to Ge, a search for double minima
in which only one atom is moved at a time is not an adequate representation of the
real atomic motion, and will seriously overestimate barrier heights. This second point
is not easily overcome, and indeed is an example of a general problem of optimisation.
A related approach has been described by Brawer (1981), who used a molecular
dynamics calculation with completely spherically symmetric potentials as a representa-
tion of the relatively ionic BeF,. The main conclusion of this work is that defects in
the form of threehold coordinated Be and F ions can be quenched into the glass (on
a very rapid timescale), and that these defects result in local motion which is compatible
with that needed to form TS. It is doubtful, however, that spherically symmetric
potentials allow glass formation on a laboratory timescale.
Guttman and Rahman (1985) adopted a slightly different approach in which they
studied very-low-frequency modes in a relaxed structure of SiO, in order to give a
microscopic description of TS. Using the eigenvectors as a guide to the cooperative
motion involved at very low frequencies, the potential energy was calculated as a
function of displacement in configurational space, a combination of a large number
of individual atom displacements. A double-well potential was observed, correspond-
ing to the correlated displacement of about 10 (at least) SiO, tetrahedra, although the
barrier was very small. This approach is interesting in that it automatically involves
cooperative motion, and is consistent with low-frequency inelastic neutron scattering
experiments (Buchenau et a1 1984, 1986) which indicate similar local motion (figure
28). However, values of energy barriers and displacements are likely to depend critically
1704 W A Phillips
Figure 28. Coupled rotation of SiO, tetrahedra as deduced from low-frequency inelastic
neutron experiments (Buchenau et al 1984).
on the particular choice of force constants. Calculated eigenvectors are particularly

sensitive to this choice.
7.3. General theories

In contrast to the more specific approaches described in $57.1 and 7.2, there have
been a number of theories which introduce TS as a consequence of a general discription
of the glassy state or of the glass transition. As a result of their generality it is often
difficult to derive even semiquantitative parameters from these theories, although several
qualitative predictions agree with experiment. However, the universal occurrence of
TS in disordered solids suggests a basic connection with structure, although it must be
emphasised that TS are not restricted to glasses. It may well be that variations in local
environment resulting from any kind of disorder are sufficient to give the behaviour
described in this review.
The relationship between physical properties and the ‘rigidity’ of a n amorphous
network has been examined in some detail in germanium selenide glasses by considering
constraints acting on the system. Atoms in a network where bond lengths and bond
angles take (ideally) fixed values are subject to constraints; each constant bond length
and independent bond angle gives a relation between coordinates of neighbouring
atoms. An m-fold coordinated atom provides m / 2 constraints to the network from
constant bond lengths, a n d 2m-3 from constant bond angles. Applied to Ge,Se,-,
this gives a n average of 5x + 2 constraints per atom, and hence to a critical composition
of x = 0.2 where the total number of constraints per atom just balances the three degrees
to freedom.
This critical composition marks a boundary between underconstrained ‘floppy’
glasses for x < 0.2, and overconstrained glasses with unrelieved strain energy. The
strain energy will increase with x, so that it is not surprising to find that bulk glasses
cannot be prepared for x > 0.42. For 0.2 < x < 0.42 the model has been used as a basis
for claims that the random network model is inadequate (Phillips 1979), and that
glasses in this range of composition are built u p from tightly coordinated clusters held
together by much looser bonding.
Thorpe (1983) has linked the number of tunnelling states to the number f of
unconstrained degrees of freedom. In the simplest approximation f = 1 - 5x but in
practice the approach to zero is likely to be smoother than this. Qualitatively the
model predicts that the density of TS should decrease as x increases, becoming
vanishingly small for x > 0.3. A contrasting approach was taken by Phillips (1981),
who associated TS with chalcogenide atoms on the surfaces of the clusters for x > 0.2.
In this case the number of TS should increase with x.
Ultrasonic velocity measurements (0 4.1) in Ge,Se,-, show a logarithmic tem-
perature dependence which indicates that Pyz decreases as x increases (Duquesne a n d
Bellessa 1985) and, less directly, that y is almost independent of x. This behaviour
supports the idea that TS are related to the unconstrained degrees of freedomf; although
agreement is only qualitative.
Other attempts have been made to associate TS with particular aspects of the glassy
state. Mon and Ashcroft (1977) point out that diffraction studies in glasses commonly
show a split second peak in the radial distribution function, indicating two possible
next-nearest-neighbour configurations. From this they conclude that TS are a subgroup
of these configurations in which a number of atoms can move from one to the other,
although as mentioned earlier no realistic estimate of barrier potentials is possible.
Applied to SiO, tetrahedra in silica this idea resembles a n earlier one due to Vukcevich
(1972). For completeness a n abstract theory based on the lack of range order in certain
glasses should also be mentioned (Duffy a n d Rivier 1983), as should a generalised
theory of local anharmonic potentials (Karpov et a1 1983).
A second group of general theories is based on the observation that the density of
TS decreases as the glass-transition temperature Tg increases from one glass to another
(see, for example, Hunklinger and Raychaudhuri 1986). Although this result follows
generally from the arguments given in § 2.1, a recent formulation of the well established
free-volume theory of the glass transition by Cohen and Grest (1980, 1981) has given
a correct order-of-magnitude estimate of the density of TS. They point out that the
number of TS depends on the ratio of molecular volume to the free volume which is
frozen into the glassy state at T g . Moreover, the same parameter determines the
viscosity of the supercooled liquid at the glass transition, usually defined to occur
when the viscosity reaches a particular value. This means that the basic parameter of
the theory is determined by Tg, and so all materials shoiild have a similar total number
of TS, spread over an energy range of kBTg. The density of TS in energy should therefore
decrease as 1/ Tg in approximate agreement with experiment.
7.4. s u f l x
The ideas presented in $0 2-6 have been very successful in understanding and relating
a wide range of low-temperature physical phenomena in glasses on the basis of a
general tunnelling state model. The general features of this model follow from the
basic properties of the amorphous or disordered state, namely a wide range of local
environments in which atoms d o not have well defined local minima. However, two
basic questions remain: why is the number of tunnelling states comparable in all
glass-forming systems, a n d what (in the absence of impurities) is the microscopic
structure of a tunnelling state?
These two questions will not be answered in a single all-embracing theory. An
answer to the first may follow from a refinement of the general considerations described
in 0 7.3, but also, on the experimental side, from correlations between low-temperature
1706 W A Phillips
and glass-forming properties (Hunklinger and Raychaudhuri 1986). It is worth

emphasising that reasonably complete data is available only on a very limited number
of materials, notably vitreous silica and A&.
A convincing microscopic description of a tunnelling state in any amorphous
material remains a problem to tax the solid state physicist. The key word is ‘convincing’.
Any analysis requires careful justification of structure and interatomic force constants
(bearing in mind the difficulty of handling long-range forces in the amorphous state),
related through the relaxation of the structure. Furthermore, there is no well defined
procedure for calculating energy barriers between atomic configurations, although the
technique of searching for low-frequency local vibrational states may provide a way
forward (Guttman and Rahmann 1985).
Acknowledgments
I would like to thank Karin Arnold and Keith Papworth for invaluable help in the
preparation of the manuscript.
References
Abramowitz M and Stegun I A 1970 Handbook of Marhematical Functions (New York: Dover)
Ackerman D A, Moy D, Potter R C , Anderson A C and Lawless W N 1981 Phys. Rev. B 23 3886
Anderson A C 1986 Phys. Rev. B 34 1317
Anderson P W 1975 Pbys. Rev. Lett. 34 953
Anderson P W, Halperin B I and Varma C M 1972 Phil. Mag. 25 1
Araki H, Park G, Hikata A and Elbaum C 1980 Phys. Rev. B 21 4470
Arnold W, Doussineau P, Frinois C and Levelut A 1981 J. Physique Lett. 42 L289
Arnold W, Doussineau P and Levelut A 1982 J. Physique Left. 43 L695
Arnold W, Martinon C and Hunklinger S 1978 J. Physique Colloq. 39 C6 961
Barron T H K 1965 Lattice Dynamics ed R F Wallis (Oxford: Pergamon) p 247
Bernard L, Picht L, Schumacher G and Joffrin L J 1978 J. Physique Colloq. 39 C6 957
- 1979 J. LOW Temp. Phys. 35 411
Black J L 1978 Phys. Rev. B 17 2740
-1981 Glassy Metals I (Topics in Applied Physics 46) ed H J Giintherodt and H Beck (Berlin: Springer)
p 167
Black J L and Fulde P 1979 Phys. Rev. Lett. 43 453
Black J L, Gyorffy B L and Jackle J 1979 Phil. Mag. B40 331
Black J L and Halperin B I 1977 Phys. Rev. B 16 2879
Brawer S A 1981 Phys. Rev. Left. 46 778
Buchenau U, Nucker N and Dianoux A J 1984 Phys. Rev. Lett. 53 2316
Buchenau U, Prager M, Nucker N, Dianoux A J, Ahmad N and Phillips W A 1986 Pbys. Rev. B34 5665
Cohen M H and Grest G S 1980 Phys. Rev. Lett. 45 1271
-1981 Solid State Commun. 39 143
Continento M A 1982 Phys. Rev. B 25 7820
Cordit P and Bellessa G 1981 Phys. Rev. Lett. 47 106
Duffy D M and Rivier N 1983 Physica lOSB 1261
Duquesne J Y and Bellessa G 1983 J. Phys. C: Soltd Srafe Phys. 16 L65
-1985 Phil. Mag. B 52 821
Elliott S R 1979 Phil. Mag. B 40 507
Esquinazi P, Ritter H M, Neckel H, Weiss G and Hunklinger S 1986 Z. Pbys. B 64 81
Etherington G, Wright A C, Wenzel J T, Dore J C , Clarke J H and Sinclair R N 1982 J. Non-Cryst. Solids
48 265
Fox D L, Golding B and Haemmerle W H 1982 Phys. Rev. Lerr. 49 1356
Frossatti G, Gilchrist J le G, Lasjaunias J C and Meyer W 1977 J. Phys. C: Solid State Phys. 10 L515
Golding B and Graebner J E 1981 Amorphous Solids: Low Temperature Properties (Topics in Current Physics
24) ed W A Phillips (Berlin: Springer)
Golding B, Graebner J E, Halperin B and Schutz R J 1973 Phys. Rev. Lett. 30 223
Golding B, Graebner J E and Kane A B 1976a Phys. Rev. Lett. 37 1248
Golding B, Graebner J E, Kane A B and Black J L 1978 Phys. Rev. Lett. 41 1487
Golding B, Graebner J E and Schutz R J 1976b Phys. Rev. B 14 1660
Golding B, Schickfus M v, Hunklinger S and Dransfeld K 1979 Phys. Rev. Lett. 43 1817
Goubau W M and Tait R A 1975 Phys. Rev. Lett. 34 1220
Graebner J E and Allen L C 1984 Phys. Rev. B 29 5626
Graebner J E, Allen L C, Golding B and Kane A B 1983 Phys. Rev. B 27 3697
Graebner J E and Golding B 1979 Phys. Rev. B 19 964
Guttman L and Rahman S M 1985 Phys. Rev. B 3 3 1506
Hagarty J, Broer M M, Golding B, Simpson J R and MacChesney J B 1982 Phys. Rev. Lett. 51 2033
Hikata A, Cibuzar G and Elbaum C 1981 J. Low Temp. Phys. 49 341
Hu P and Walker L R 1977 Solid State Commun. 24 813
Huber D L, Broer M M and Golding B 1984 Phys. Rev. Lett. 52 2281
Hui J C K and Allen P B 1975 J. Phys. C: Solid State Phys. 8 2923
Hunklinger S and Arnold W 1976 Physical Acoustics vol XII, ed W P Mason and R N Thurston (New York:
Academic) p 153
Hunklinger S, Arnold W, Stein S, Nava R and Dransfeld K 1972 Phys. Lett. A 4 2 253
Hunklinger S and Raychaudhuri A K 1986 Prog. Low Temp. Phys. vol. IX, ed D F Brewer (Amsterdam:
Elsevier) p 265
Hunklinger S and Schmidt M 1984 2.Phys. B 54 93
Isnard R and Gilchrist J le G 1981 Chem. Phys. 52 1 1 1
Jackle J, PichC L, Arnold W and Hunklinger S 1976 J. Non-Cryst. Solids 20 365
Jones D P 1982 Thesis University of Cambridge
Jones D P, Thomas N and Phillips W A 1978 Phil. Mag. B 38 271
Karpov V G 1985 Sou. Phys.-Semicond. 19 74
Karpov V G , Klinger M I and Ignat'ev F N 1983 Sou. Phys.-JETP 57 439
Keating P N 1966 Phys. Rev. 145 637
Klauder J R and Anderson P W 1962 Phys. Rev. 125 912
Kummer R B, Dynes R C and Narayanamurti V 1978 Phys. Rev. Lett. 40 1187
Laikhtman B D 1985 Phys. Rev. B 31 490
Lasjaunias J C, Maynard R and Vandorpe M 1978 J. Physique Colloq. 39 C6 973
Lasjaunias J C, Ravex A, Vandorpe M and Hunklinger S 1975 Solid Stare Commun. 17 1045
Landau L D and Lifshitz E M 1984 Electrodynamics ofcontinuous Media (Oxford: Pergamon) 2nd edn
Lewis 15, Lasjaunias J C and Schumacher G 1978 J. Physique Colloq. 39 C6 967
Licciardello D C, Stein D L and Haldane F D M 1981 Phil. Mag. B 4 3 189
Lohneysen H v 1981 Phys. Rep. 79 161
Lohneysen H v and Schink H J 1952 Phys. Rev. Lett. 48 1121
Long A R 1982 Adv. Phys. 31 553
Loponen M T, Dynes R C, Narayanamurti V and Garno J P 1980 Phys. Rev. Lett. 45 457
-1982 Phys. Rev. B 25 4310
Lou L F 1976 Solid State Conimun. 19 335
Lyo S K 1982 Phys. Rev. Lett. 48 688
Meissner M and Spitzmann K 1981 Phys. Rev. Lett. 46 265
Mertzbacher E 1970 Quantum Mechanics (New York: Wiley) 2nd edn
Mon K K and Ashcroft N W 1977 Solid State Commun. 27 609
Narayanamurti V and Pohl R 0 1970 Rev. Mod. Phys. 42 201
Pelous J and Vacher R 1976 Solid State Commun. 19 627
Phillips J C 1979 J. Non-Cryst. Solids 34 153
-1981a Phys. Rev. B 24 1744
Phillips W A 1972 J. Low Temp. Phys. 7 351
-1973 J. Low Temp. Phys. 11 757
-1976 Phil. Mag. 34 983
-1981b Amorphous Solids: Low Temperature Properties (Topics in Current Physics 24) (Berlin: Springer)
-1981c Phil. Mag. B 43 747
-1985a Physics ofDisordered Materials ed D Adler, H Fritzche and S R Ovshinsky (New York: Plenum)
p 1329
-1985b J. Non-Cryst. Solids 77-78 1329
1708 W A Phillips
Pippard A B 1983 The Physics of Vibrations vol 2 (Cambridge: Cambridge University Press)
Pohl R 0, Love W F and Stephens R B 1974 Proc. 5th Int. Conf. on Amorphous and Liquid Semiconductors
ed J Stuke and W Brenig (London: Taylor and Francis) p 1121
Raychaudhuri A K and Hunklinger S 1984 2. Phys. B 57 113
Redfield D 1971 Phys. Rev. Letr. 27 730
Rothenfusser M, Dietsche W and Kinder H 1983 Phys. Rea. B 27 5196
Russo G and Ferrari L 1984 Phil. Mag. B 49 311
Sargent M , Scully M 0 and Lamb W E 1974 Laser Physics (Reading, MA: Addison-Wesley)
Schickfus M v and Hunklinger S 1976 J. Phys. C: Solid State Phys. 9 L439
-1977 Phys. Lett. 64A 144
Slichter C P 1980 Principles of Magnetic Resonance (Berlin: Springer)
Smith D A 1978 Phys. Rev. Left. 42 729
Stephens R B 1973 Phys. Rev. B 8 2896
Street R A and Mott N F 1976 Phys. Rev. Lerr. 35 1293
Thijssen H P H, Dicker A I M and Volker S 1982 Chem. Phys. Lett. 92 7
Thomas N 1983 Phil. Mag. B48 297
Thorpe M F 1983 J. Non-Crysr. Solids 57 355
Vacher R and Pelous J 1976 Phys. Rea. B 14 823
Vukcevich M R 1972 J. Non-Cryst. Solids 11 25
Wooten F and Weaire D 1984 J. Non-Crysr. Solids 64 325
Wright 0 B 1986 Phil. Mag. B 5 3 477
Wright 0 B and Phillips W A 1984 Phil. Mag. 50 63
Zaitlin M P and Anderson A C 1975 Phys. Reu. B 12 4475
Zeller R C and Pohl R 0 1971 Phys. Rev. B 4 2029
Zimmerman J and Weber G 1981 Phys. Reu. Lett. 46 661

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Two-level states in glasses

This review was received in February 1987.

0034-4885/87/121657 + 52$09.00 @ 1987 I O P Publishing Ltd 1657

Figure 4. K( T ) as a function of temperature for a range of glasses (Stephens 1973).

contribution of a single two-level state (Schottky anomaly). Evaluating the integral

2.1. Static description

Figure 5. A double-well potential ( a ) ,together with the wavefunctions in the localised ( b )

as the mean of E , and E * , (2.2) becomes

where the tunnel splitting A, is given by

thermal properties at a temperature T. At 1 K this requires a tunnel splitting of eV,

=q [ (4Tx41 cos’ 0 + 4Tx4, sin2 O + 4 7 ~ sin

Obviously p1 is zero for the symmetric case.

( o r equivalently in terms of the spin operators S, =+ha,).After diagonalisation, the

V ( r , t ) = a , ( t ) $ l ( r ) exp(-iE,t/A)+a,(t)$,(r) exp(-iE,t/h) (2.12)

moment of a one-dimensional tunnelling state. Using p, = -p2 = ( $ , I X ~ $ ~ ) and p12=

where w I 2 is the transition probability from state 1 to state 2. In thermal equilibrium

where n B , the Bose factor, is [ e x p ( E / k T ) - l]-', g ( E ) is the phonon density of states,

The Debye density of phonon states, valid in this low-energy limit, is

3.1. Thermal conductivity

defining a phonon scattering rate

- 77y2,wA' tanh ( E / k , T ) (3.3)

Experimentally the temperature dependence is found to range from to T’.9

3.2. Heat capacity

=$P ln(4t,/~,~~). (3.9)

= &,rr2PkiT ln(4t,/~,,,) (3.10)

3.3. Thermol expansion

where S: = d S , ( x ) / d x . Eliminating S: a n d using C, = T ( a S , / a T ) , gives

which can be summed over i to give

so that with Cv = CC,, (3.13) becomes

Each excitation therefore contributes to the thermal expansion by a n amount propor-

Changes in both A a n d A, with volume can contribute significantly to r. The term in

3.4, Energy relaxation and irreversibility

a single state of energy E is

4. Response to external fields

4.2. Strong fields

gives equations for U , and U * :

The time-dependent dipole moment from (2.13) is given by U, where

U = alaf exp(iwot)+ a2aT exp(-iw,t). (4.11)

Equations (4.15)-(4.17) are the Bloch equations of magnetic resonance.

Equation (3.4) must therefore be modified to give

I/I,=-- (2ye)* T2 (4.24)

Equation (4.24) can be given a simple physical interpretation. The cross-over

I;-~ iy6 ~q-~ 1y4

and the mean polarisability is given by

For if( t ) = 5, exp( -iwt) the solution is of the form

The corresponding change in average dipole moment 8 p ( w ) = p o ( A / E ) 6w defines

The frequency independence of Q for W T , ~ , , >> 1 is confirmed by experiment and takes

differs for different experimental techniques, the values of P y 2 in S i 0 2 are in good

5. Pulse echo experiments

5.1. General theory

The subsequent time variation has the form for to >> T~

5.2. Echo experiments

in the electric case, or