Solidons and rotons in liquid helium

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Solidons and rotons in liquid helium

M. Héritier, G. Montambaux, P. Lederer

To cite this version:

M. Héritier, G. Montambaux, P. Lederer. Solidons and rotons in liquid helium. Journal de Physique

Lettres, Edp sciences, 1979, 40 (18), pp.493-498.

�10.1051/jphyslet:019790040018049300�.

�jpa-00231673�

Solidons

and

rotons

in

_liquid

helium

M.

Héritier,

G. Montambaux and P. Lederer

Laboratoire de _Physique des Solides _(*), Université de Paris-Sud, Centre d’Orsay, 91405 _Orsay, France

(Re~u le _21juin 1979, revise le _{16 juillet} 1979, accepte le _{Z4 juillet} 1979)

Résumé. 2014 La courbe de

dispersion

de _phonon d’un

_liquide

_{presque solide doit}

_présenter

un minimum de _type

roton, correspondant à un mouvement de translation d’atomes dans une

_petite

_région solide du

_liquide.

Ce _solidon,

excitation élémentaire de l’4He _super fluide ou fluctuation dans

3He,

fournit une

_description

satisfaisante du roton.

Abstract. 2014

The _phonon

_dispersion

curve of a _nearly solid

_liquid

is shown to _present a roton-like minimum,

which _corresponds to a translational motion of atoms in a small solid

_region

of the

_liquid.

This solidon, which

should exist as an

_elementary

excitation in

_superfluid

4He and as a fluctuation in

3He, gives

a _satisfactory

des-cription

of a roton.

Classification

Physics Abstracts

67.40 - 67.50 - 67.70

1. Introduction. -

Describing

the atomic motion of

_liquid

Helium II in terms of

_elementary

excitations has

_proved

to be a _very

_powerful

method to

_interpret

its

_{superfluids properties}

_[1].

_However,

Landau’s

assumptions

about the existence of an _{energy gap} for

the individual excitations and about the

phonon-roton

_dispersion

curve were introduced

empirically

rather than on the basis of

_rigorous

_quantum

hydro-dynamics.

In

_particular,

his

_{interpretation}

of a roton

as a _quantum of rotational motion was

_proved

to be

erroneous

_[2].

Still now, it is difficult to

_give

a

_simple

description

of the atomic motions involved in a roton.

Some authors have

_presented

it as

_essentially

one

4He

atom

_{travelling rapidly}

_through

the other atoms which

move out of its

_path

_[3].

However,

the best

_description

of a roton has been

_given

_by

_{Feynman [4]}

and

_Feynman

and Cohen

_{[5]. They}

_proposed

a variational wave

function of the excited state which accounts

qualita-tively

for the observed roton

_spectrum.

In their

_model,

a roton can be

_{pictured roughly}

as a vortex

_ring

of such small radius that

only

one atom can _pass

_through

the centre.

_Usually,

the _concept of roton is restricted

to the

_superfluid

state of

_liquid

_{4He. However,}

we show in this letter that a roton-like minimum in the collective excitation

_dispersion

curve can be

encoun-tered in other _quantum

_{liquids provided}

_they

are

nearly

solid

_(in

a sense which will be

_{precised later).}

The

_{corresponding}

excitation is

_interpreted

as a

trans-lational motion of atoms in a small solid

region

in

(*) Associe au C.N.R.S.

the

_liquid.

We call it a solidon. In

_principle,

a solidon could exist in a non

_superfluid

_quantum

_liquid,

even in a normal Fermi

_liquid.

However,

in most cases,

it would be

_strongly

scattered

_by

individual

excitations,

and then would constitue a solid fluctuation or a

parasolidon

rather than a well-defined

_elementary

excitation.

_According

to us,

liquid

4He and

_liquid

3He

are the best

_examples

of what we call a

_nearly

solid

liquid.

Therefore,

we _suggest that the solidon is the

right description

of a roton. In

fact,

we believe that the

_Feynman

and Cohen wave function is

_merely

an

approximate

solidon wave

function,

essentially

valid

at

_large

distances. This

picture

provides

a _very

~imple

description

of the roton and a

_transparent

_explanation

of the minimum in the

_dispersion

curve, as will be

discussed below. We also

_predict

the existence of

parasolidons

or _pararotons in

_{liquid 3He,}

which

might

have been observed

_recently

[6].

2.

_Description

of the solidon. - What do we call a

_nearly

solid

_{liquid ?}

_Usually,

in a classical

_liquid,

a

typical longitudinal

phonon

energy

kB eD

(for

wave

vectors about

_nla)

is not

_{large compared}

to the atomic kinetic _energy which is of the order of the _temperature

of the

_liquid

where

_0D

is the

_Debye

_temperature

of the solid.

On the _contrary, in _quantum

_liquids,

at lower

tem-peratures,

the

_zero-point

atomic kinetic energy is of

order t =

1i2/2

ma2 where

_in is the atomic mass and

L-494 JOURNAL DE PHYSIQUE - LETTRES

a is the mean interatomic distance. As

recently

out-lined

_by

_Castaing

and Nozieres

_[7],

in

_liquid

3He

near its

melting

curve

This

_inequality

reflects the

_nearly

solid character of

liquid

3He.

The

_liquid

is

_locally

_very

_rigid

because of

the

_high

vibration _energy

_{kB eD}

of the atoms in their own _cages.

_But,

it still _possess the

_plasticity

of a

liquid

because of the small _{energy t} to drift the _cages around each other

_[7].

Morever,

near

_enough

to the

_melting

curve, in any

liquid,

where

_Gs

and

_G1

are the free

enthalpies

_per atom of

respectively

the solid

_phase

and the

_liquid

_phase

at

a

_given

_{pressure p} and a

given

_temperature

T.

_Eq.

(2)

defines a

_region

of the

_{phase diagram}

in the

_(p,

_T)

plane,

more or less extended

_according

to the value of t. _{The purpose of this letter is} to

_study

the

conse-quences of these

properties

on the excitation _spectrum of the

_liquid.

We deal with

_nearly

solid

_quantum

liquids, namely liquids

where

_inequalities

₍₁₎

and

₍₂₎

are satisfied. Of course, this is the case of

_{liquid 3He,}

but also of

_{liquid 4He,}

where the orders of

_magnitude

are the same, near

_enough

to the

_melting

curve, and

of their mixtures. We can consider 2 D

_{liquid layers}

as well as 3 D bulk

_liquid.

In an

_{ordinary liquid,}

i.e.

_{where 9D T,}

the

pho-non

_dispersion

curve increases

monotonously.

The

larger

the wave vector _q, the

_higher

the

_phonon

_energy

m(q).

On the _contrary, in a

_solid,

the

_phonon

_spectrum

exhibits a

_{periodicity :}

where K is a

_reciprocal

lattice vector. This is a

conse-quence of the translational invariance of the

crystal.

Let us consider a vibrational mode with wave vector

q +

K,

in a

perfect crystal,

with an infinite number

of atoms. It can

_always

be considered as _the

super-position

of a translational motion with momentum _p

and of a

phonon

with wave vector _q and _energy

_~(q)

but zero momentum. The atomic wave function in

the

_crystal

can be described

_by

a

superposition

of

plane

waves, the wave vectors of which

_belong

to

the

_reciprocal

lattice.

_Hence,

the translation has a

momentum hK and an

energy 112 K2/2

Nm in a finite

crystal

of N atoms. The latter _energy, i.e. the kinetic

energy transferred to the

lattice,

vanishes in an infinite

crystal.

Therefore,

the

_energies

of acoustical

_phonons

with wave vectors

_belonging

to the

_reciprocal

lattice vanish.

Some _memory of this

_property

should remain in the

_longitudinal

_phonon

_spectrum

of a

_nearly

solid

liquid.

At any

point (p,

T)

of the

_{phase diagram,}

we can define a metastable

_crystal

structure which minimizes

_6’g(/?,

_T),

the free

_enthalpy

_per atom in the solid state. Let us call K the

_reciprocal

lattice vectors

of this metastable

_crystal

and

_Ko,

the shortest one.

At least at

_Ko,

the

_{phonon dispersion}

curve should

exhibit a

_pronounced,

_although

non-zero, minimum

which should vanish at the

_liquid-solid

transition,

discontinuously

in a first order

transition,

but

conti-nuously

in a second order one.

_Obviously,

the

_phonon

energy at wave vector

_Ko

can be lowered

_by

_forming

a well ordered lattice. The

_larger

_~~

the more

effective the

_lowering.

This is obtained at _{the expense}

of a free

enthalpy G,,(p,

T ) -

GI(p,

T )

per atom, which limits the number of atoms N in this lattice to

a finite value. Of course, the

phonon

energy of wave

vector

_Ko

does not vanish in this lattice of finite extension and increases when N decreases. The balance between these two

_opposite

effects determines the value of N which minimizes the

_phonon

energy. As stated

above,

the roton-like minimum occurs because

the

_liquid

is

_nearly

solid

_{(inequalities}

₍₁₎

and

_(2)).

It

is due to the formation of a local

_region

in the

_liquid

where the atoms are ordered as in a

solid,

in which

the atomic motions involved in a

phonon require

much less _energy than in the

_liquid.

For that reason,

we call it a

solidon,

inasmuch as it is a well-defined excitation with a

_{long lifetime,}

and a

_parasolidon

when its width becomes

_comparable

to its _energy.

Let us _try to describe the structure of this solidon. The solidon _energy can be considered as the sum of

two terms : the formation energy of this N-site lattice

Elat(N,

p,

T)

and the kinetic energy of the atomic

motion

_Ekin(N,

_p,

_T).

We

_discuss,

now, these two terms. When a solidon is

formed,

three

_regions

of

space should

be,

in

principle, distinguished.

In the

first

_region,

of radius

R,

the lattice is

_perfect

or

_nearly

perfect.

In the second

_region,

the order

_parameter

decreases more or less

smoothly

over a

length

I/K.

The third

_region

is

_completely

_liquid.

The first

_region

always

exists,

because

_{kB 8D}

is much

_greater

than _any other _energy in the

problem. Any

amount of disorder would increase

_appreciably

the

_phonon

_energy. The

physical

value of KR is not obvious. It may be

small,

near to the

_melting

curve, if the

solid-liquid

surface tension is

_{quite large,}

close to a first order

transition,

even

_though

one could

_expect

that a _very

gradual

decay

of the order is unfavourable.

However,

we

approximate Elat by

the sum of two terms :

In

_{(3), 5G(~ T)=G*(p, T)-G1(p,}

_T),

where

_G*(p,

_T)

is the free

_enthalpy

_per atom in the

lattice,

when

boundary

effects are

_{ignored. G*(p,}

_T)

is lowered below

_6s(~,

_{T) by}

relaxation effects. The second term N2~3 a

_represents

the

solid-liquid

interface _energy. In

fact,

this

_expression

is

_essentially

valid when KR is

large.

When this is not _true, one should add other

terms with smaller _powers of N. Such terms are not

expected

to be

_important

when

_inequalities

₍₁₎

and

₍₂₎

are satisfied and we shall

_disregard

them here. Of

phenomenological.

In

_particular,

a should not be confused with the

_macroscopic

_solid-liquid

surface tension coefficient.

In this lattice of limited extension a low _energy excitation with wave vector

_Ko

can be formed

_by

transferring

to the lattice a momentum

_nKo.

The translation of the center of mass costs a kinetic _energy

This term refers to the kinetic _energy of the atoms

inside the solid

_region.

_However,

the

_particle

current

density

does not

_stop

_suddenly

at the

_solid-liquid

interface. The current flow would not be conserved.

We must consider a backflow in the disordered

_region.

The current lines form

_loops

schematized in

figure

1.

If we assume that the

_only

_elementary

excitations in the

_liquid

are

_phonons

with a

_dispersion

relation

Fig. 1. - Schematic

representation of a solidon. The atomic flow is shown by solid lines, the _entropy flow _by dotted line. The black dots are atoms.

... ~

an exact

_analogy

can be made between the electric field in the vacuum and the

velocity

in the

liquid.

At

_large

distances

_compared

to

R,

the solidon can be considered as a source and a sink of atoms close

together.

The electrical

_analogue

is a

_dipole.

The blackflow kinetic _energy in the

_liquid

is

_equivalent

to the electrostatic _energy of a

_{dipole :}

where a is a

_geometrical

_factor,

of order

_unity,

which could be

determined,

in

_principle,

_by

a variational

procedure.

The resultant of the backflow momentum

outside of a

_sphere

is zero. _However, because of this

backflow,

the solidon can have a non-zero momentum

1iKo,

but a _group

_{velocity equal}

to zero when _p =

1iKo

[2].

We have

_ignored

any

possible

effect of other

soli-dons on the

_velocity

field in the

_liquid

and, therefore,

of the solidon-solidon interaction.

We obtain an

_{approximate expression}

for the soli-don free

_enthalpy

as a function of N :

which must be minimized with _respect to N. Two

_regimes

are considered. In the first one, the

last term in _eq.

_(6),

i.e. the surface term, is

negligible

compared

to the volume term.

_Then,

the solidon free

enthalpy

is

corresponding

to a number of atoms in the solid

region

As the

_{liquid-solid equilibrium}

is

_approached,

the number of atoms increases and the solidon free

enthalpy

decreases. However

_No

does not

_diverge

and

TM

does not _go to zero at the transition because of the surface term.

In the second

_regime,

the surface term is dominant

in eq.

(6),

and we

_neglect

the volume term.

Then,

the solidon free

_enthalpy

is

corresponding

to a number of atoms in the solid

region :

This second

_regime

is obtained in the

_vicinity

of the

melting

curve. The cross-over between the two

_regimes

occurs when the surface term and the volume term

are

_equal.

This defines a cross-over line in the

_(p,

_T)

L-496 JOURNAL DE _{PHYSIQUE -} LETTRES

It may

happen

that two

_crystal

structures are

nearly

degenerate,

but lead to

_quite

different values of a.

In the first

_regime,

the stable solidon structure

_will

always

be the lower _energy one. In the second

_regime,

i.e. near the

_melting

curve, it will be

governed by

the

lower value of a, which

_might

not

_correspond

to the

lower value of 56’.

In what

_precedes,

we have

_implicitly

assumed that the

_entropy

_per atom was the same in the

liquid

and in the solid. This is

_rigorously

exact

_only

at zero

temperature (however

this is a

_good

approximation

up to 1.2 K in

_4He).

_Therefore,

at finite _temperature,

the atomic flow

_through

the solid

_region

does not preserve the lower

entropy

which is _necessary to

obtain the solid

_ordering.

The atomic flow can achieve the

_required

loss of

_entropy

_by

_emitting

_phonons

before

_entering

the solid

_region.

The _entropy

conser-vation will be ensured

by reabsorption

of the

_phonons

by

the

_outcoming

flow of atoms. There is a flow of mass

_coming

_through

the solid

region,

and a distinct flow of _entropy,

which, instead,

avoids it

_(Fig.

_1).

Now, the solidon momentum

_is,

_{approximately :}

in which v is the translation

_velocity

inside the solid

region,

AS the excess _{entropy per} atom of the

_liquid

and c is the sound

_velocity.

The second term in

₍₁₀₎

represents

the

_phonon

momentum of the _entropy flow.

It contributes to

decrease v, and, thus,

to lower the

kinetic _energy of atoms inside the solid

_region.

_{Eq. (4)}

should be

_replaced

_by

The last term is

_only

a small volume term which

slightly

renormalizes bG in eq.

(6)

and can be

_neglected.

Therefore,

we see that the main effect of the

_phonon

flow is a

lowering

of the solidon _{energy :}

where

_r~,

_{T )}

is

_{given by eq.}

₍₇₎

or _eq.

(8),

_according

as we are below or above the cross-over line defined

by

eq.

(9).

Up

to now, we have restricted this

_study

to the solidon of smallest

_reciprocal

lattice wave vector. Of

course, solidons

_{corresponding}

to

_{larger K}

could exist. Not too close to the

_melting

curve, the solidon

energy increase

proportionally

to

K (eq.

(7)).

However,

since at

_large

momenta, the

phonon dispersion

of the

liquid

increases less

_rapidly,

the minimum

_might

be

very shallow and even not exist. This

_might

be true

also near the

_melting

curve, where the solidon energy

increases as K415.

In the first

_{regime (f~J~), ~}

solidons with wave

vectors

_Ko

have

_exactly

the same _energy as one

soli-don with wave vector

_nKo.

Therefore,

there is no

tendency

to solidon condensation since the condensed

state has lower _entropy. In the second

_regime

(T ~ K4~5),

the condensed state as a lower _energy.

Bound states of n solidons may exist for not too

_large

n.

Indeed,

for

_larger

and

_larger

n, the volume term in

eq.

(6)

becomes more and more

important,

so

_that,

again,

F is

_proportional

to n and the

_binding

_energy goes to zero.

3.

_Application

to

_liquid

Helium. -

Although

we

have

_ignored

this

_problem

in the

_preceding

section,

the solidon is

_{obviously strongly}

scattered

_by

the

individual excitations.

Generally,

it cannot exist as a well-defined

_elementary

excitation with a width small

compared

to _{its energy, but} rather like a fluctuation.

However,

we can consider two cases in which this

scattering

is much weaker. In the first case, the atomic

masses are so

_heavy

and the atomic velocities so small that the solidon

_dispersion

curve is well

sepa-rated in _energy from the individual excitations conti-nuum.

Then,

the solidon lifetime can be

long.

We are

not aware of an

_{experimental example}

in which this situation is realized. The second case is

_provided

_by

the

_{superfluid liquids}

in which _{the gap} for individual excitations _suppresses their

_scattering.

Of course,

superfluid

4He

is an ideal

_candidate,

since it is

_nearly

solid,

as discussed above.

Therefore,

we are led to

the conclusion that the roton minimum observed in

the collective excitation

_dispersion

curve is in fact due

to solidons. One should remark

that,

far from its core,

the solidon consists

_essentially

in a

dipolar velocity

field,

in close

_analogy

with the

_Feynman

and Cohen

description.

Then,

one should not be

_surprised

that

the two models lead to the same

_qualitative

predic-tions. A

_quantitative

check of our

_theory

is out of the

scope of this

letter,

first because our calculations are

very

rough,

and also because we lack

precise

data on

~6’(/?,

T),

except

near the

_melting

curve, and (x.

The value

_{of Ko in}

4He

_corresponds

to a b.c.c. struc-ture. This is _easy to understand

since,

at the lowest

densities,

this structure has the lowest _energy

_[8].

However,

at _very low

_temperature

and near the

_melting

curve, bG is

certainly

smaller in the

h.c.p.

structure.

But,

as shown

_by

the recent measurements of Balibar

et

_al.,

the

_macroscopic

surface tension is

_greater

_by

one order of

_magnitude

for the

h.c.p.

structure. It is

plausible

that it is also true for a. _Then, the b.c.c. solidon could be

stabilized,

at low pressure

by 6G,

at

_high

_pressure

_by

oc

_(above

the cross-over

_line).

An absolute value of the solidon _gap cannot be

given

far from the

_melting

curve, where bG is

badly

known. Near the

_melting

_pressure, an estimate of a can be inferred from the

positive-ion mobility

measu-rements

_[10],

which were

interpreted

by

a

solid-liquid

surface tension coefficient als = 0.04

erg/cm2.

This

provides

an order of

_magnitude

for the solidon gap

of 7.5

K,

in _very

_good

_agreement

with the observed value of 7.3 K at 25.3

_atmospheres.

The

correspond-ing

number of atoms in the solid

_region

is about 50. The increase of the solidon _{gap when the pressure} is lowered is in

_qualitative

_agreement

with our model.

In the

_{dipolar velocity}

field

_model,

the roton _gap

depends

on

_temperature

because of the roton-roton interaction.

_{Recently, Castaing}

and Libchaber

_[12]

have calculated this effect

_{by taking}

into account virtual

_exchange

of

_phonons

and of rotons between

rotons

_(Fig.

_2).

In

fact,

such an interaction is

compa-.. Fig. 2. -

Temperature dependence of the roton gap : - as

given by the roton-roton interaction _{(ref. [12]) :} dotted line ; - when

the _entropy flow also is taken into account : solid line ; -

experi-mental points : dots.

tible with our solidon model. It would lead to

tempe-rature variation of the solidon gap.

But,

this gap

already

depends

on the _temperature in the

indepen-dant solidon

_{approximation,}

because of the

_entropy

flow effect

_{(eq. (12)).}

This can be estimated

_along

the

melting

curve, where the excess _entropy of the

_liquid

is known. When the two effects are taken into account,

we obtain a

surprisingly good

_agreement

with

expe-riment

_(Fig.

_2),

if one considers the crude

approxi-mations involved.

In

_3He,

besides the zero sound branch observed at

low

_temperature,

the neutron

_scattering

_spectrum

shows a broad individual excitation continuum. In

this

continuum,

the

_peak

of maximum

_intensity

appears like a roton branch in the

_((D,

_{q) plane}

_[6].

The roton minimum occurs at the

_expected

wave

vector in a b.c.c. structure. The roton gap decreases

from 10 K at the saturated _{vapor pressure} to less

than 4 K at 20 bars

_[6].

This much

_greater

_change

than in

4He

would indicate a smaller value of a and a cross-over line nearer to the

_melting

curve. Of course, at

_{present time,}

the situation is still

unclear,

and we would need much more

_precise

data to draw _any definite conclusion about the solidon

origin

of this

peak.

Solidons could also exist in two dimensional

_nearly

solid

_liquids.

Adsorbed Helium films offer an

experi-mental realization of these _systems.

_Solid-liquid

tran-sitions have been seen in these films

_{[14]. They}

seem

to be second-order. The case of a continuous

tran-sition is

_particular

because both

_~6’(j~

T)

and a should vanish at the

_melting

curve. Therefore we

expect

the solidon mode to soften at the transition. Three

_regimes

can be

_{distinguished.}

In the first

regime,

the solidon radius is determined

by

6G.

Eqs.

(7)

and

_(7’)

are still valid in two dimensions.

Near the

transition,

at constant _pressure the _gap should _vary as :

T -

_r

~

where t

= 201320132013~,

_7~

is the critical

_temperature

and ð is the critical

_exponent

for

_specific

heat at constant

pressure.

In the second

_regime,

the solidon radius is deter-mined

_by

a : :

where d = 2 is the

space

dimensionality

and 6 is the critical

_exponent

for (x.

When the

_melting

curve is

_approached,

one

obtains,

first,

the one of the two

_{regimes corresponding}

to

the

_largest

critical

_{exponent, and, then,}

a cross-over

to the other one.

_Finally,

the third

_regime

_appears in which the solidon radius is determined

_by

the correla-tion

_{length ~.}

In

fact,

the

_{phonon frequency}

in the

solid

_phase

_cOg(j~o)

does not vanish because of the

absence of true

_long

_range order. What should vanish

at the transition is :

/ , .

since v,

the critical

_exponent

_{of ~}

satisfies the

_scaling

law d~ = 2 2013 ~.

Therefore,

we _expect that the critical behaviour of

the solidon mode at the

_liquid

solid transition pre-sents, a

priori,

two cross-overs.

_{Unfortunately,}

at

present

time,

reliable data on the

_elementary

excita-tions in

_liquid

Helium

_layers

are not available near

the

_{solidification,}

so that a

_comparison

with the

theoretical

_predictions

is

_impossible.

When

_completing

the

_writing

of this

letter,

we have

been aware of an abstract

_by

_O’Reilly

and

_Tsang

_[13]

about a New

_theory

_of

_{superfluid 4He, according}

to

which «

_regions

_{of low and}

_high

mass

_density

exist in

_liquid

helium and moreover are

_spatially

well-ordered in the

_superfluid

state with the result that

the

_ground-state

wave function is

_{periodic ».}

We

cannot discuss without further information the link of this new

_theory

of

_{superfluidity}

with our model

of

_nearly

solid

_liquid.

We

_{acknowledge helpful}

discussions with W. Kohn,

L-498 JOURNAL DE PHYSIQUE - LETTRES

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LANDAU, L. D., J. Phys. Moscow 11 (1947) 91.

[2] FEYNMAN, R. P., Statistical Mechanics, edited _by J. Shaham (W. A. _Benjamin) 1972.

[3] DE BOER, J., Liquid Helium, editor Carreri _{(Academic Press)} 1963.

CHESTER, G. V., ibid.

[4] FEYNMAN, R. P., Phys. Rev. 94 (1954) 262.

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[7] CASTAING, B. and NOZIÈRES, P., J. _Physique 40 ₍₁₉₇₉₎ 257.

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unpu-blished.

[13] O’REILLY, D. E. and TSANG, T., to be published.