Mobility of ions in solid helium

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Submitted on 1 Jan 1973

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D. Marty, F.I.B. Williams

To cite this version:

D. Marty, F.I.B. Williams. Mobility of ions in solid helium. Journal de Physique, 1973, 34 (11-12),

pp.989-999. �10.1051/jphys:019730034011-12098900�. �jpa-00208121�

MOBILITY OF IONS IN SOLID HELIUM

D. MARTY and F. I. B. WILLIAMS

Service de

Physique

^{du Solide et} de Résonance

Magnétique,

Centre d’Etudes Nucléaires de

Saclay

BP

2,

⁹¹¹⁹⁰

Gif-sur-Yvette,

^France

(Reçu

^le

27 . février 1973 )

Résumé. ²⁰¹⁴ Nous avons étudié les mobilités des ions

positifs

^et

négatifs

dans l’hélium 3 et 4 solide,

par ^mesure directe du temps ^{de vol en}

présence

^d’un

champ électrique.

^Ces mobilités varient _expo- nentiellement avec l’inverse de la

température,

^ce

qui

^{conduit à une}

énergie

d’activation. De

plus,

nous avons détecté un

piégeage

pour les ions

négatifs. Enfin,

nous avons observé, ^{dans l’hélium 3}

hexagonal,

^une discontinuité dans la variation de la mobilité des ions

positifs

^en fonction de la

température.

Abstract. ²⁰¹⁴ We have studied the mobilities of

positive

^and

negative

ions in solid helium 3 and 4,

by measuring directly

the time of

flight

^{in an} electric field. These mobilities _vary

exponentially

with inverse temperature,

leading

^{to an} activation energy. In addition, ^we have detected

trapping

of

negative

^ions.

Finally,

ⁱⁿ

hexagonal

^helium 3, ^we have observed a

discontinuity

^{in the} temperature variation of the

mobility

^of

positive

^ions.

Classification Physics ^Abstracts

14.95

1. Introduction. ^- We

report

measurements of the mobilities of ions in solid helium 3 and

4,

ⁱⁿ

tempe-

rature and pressure ranges 1.35-3.80

K,

and 27- 150

atmospheres.

Consideration of the

mobility

^{- or,}

equivalently,

the diffusion ^- of a

probe

^{in a substance involves}

three

problems :

^{the structure of the}

probe,

^the pro-

perties

^{of the}

system

of thermal excitations of the substance which governs its diffusion and the interac- tion between these two.

Evidently

^{it is not}

necessarily

trivial to unravel

unambiguously

^each

aspect

^from the

mobility

measurements alone.

The case of the

mobility

of ions in

liquid

^{helium 4}

provides

^a

good

illustration as well as a useful intro- duction to the solid

(see

^{e. g.} the review article

by

Gamota

[1]).

-

The structure of ions of both

signs

^{in the}

liquid

is now

quite

^well established both

by theory

^and

by

a wide

variety of experiments (e.

g.

trapping

^{in vortexes}

and at

surfaces, photoconduction,

^effective

mass).

The

positive

ion surrounds itself with a ball of solid He of radius about 7

À

held

together by

^the

polari-

zation

force,

while the

negative

^{ion is an electron}

trapped

^{in a} bubble of radius about 12

A

due to the balance between its kinetic energy of localization

(it repels

^{a He atom and so} has its free electron

mass)

and the

binding

energy between ^atoms of the

liquid («

surface tension

»).

Above the

À-point

^{the ions}

interact with the

damped single particle

excitations

(Stokes

law viscous

regime)

^{while below it}

they

^see

the collective excitations ^- rotons and

phonons -

and

finally

the residual dilute _gas of

3 He impurity

atoms. The forms of these excitations ^{are now}

quite

well known and the interaction between them and the bubble is the

only missing link,

^{to be derived}

from the

mobility

values. It is the

temperature depen-

dence of the

mobility

^{which tells us} with which

system

of excitations the ion is

interacting

^{the most}

strongly (the

^most

striking example being

the activation energy behaviour

reflecting

the number of rotons

excited,

^so

providing

^a measurement of the roton

gap).

Much less is known about ions in the solid. Cohen and Jortner

[2]

^have

proposed

^{that the}

negative

^ion

should have a similar structure to that in the

liquid,

but be somewhat smaller due to the

p V compression (typically

⁹

Â).

^The

positive

^ion

presumably

^carries

with it a localised

deformation,

^{but the idea of a}

local

phase change

^no

longer

^{has much}

meaning.

To our

knowledge,

^{no very}

specific

^{models have been}

advanced.

Quite

^{a lot is}

already

known about the excitations in solid He and

they

appear ^to be reaso-

nably

^{accounted for in terms of}

phonons

and vacancies.

We

might

^thus

hope

^to

identify

the excitation

system

in

strongest

interaction and

perhaps thereby

^see

some of its

properties ;

information on the interaction

65

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019730034011-12098900

itself is

necessarily

^tied up with the ion structure,

so that before

being

^{able to} unvavel the two we

would need to know ^more about this.

We find

immediately

^an

important

difference bet-

ween ionic diffusion in the solid and in the

liquid

over our range of measurements. In the

liquid

^the

starting point

^{was a free}

particle

^{whose movement}

is hindered

by

the excitations of the host whereas

we find that in the solid the

starting point

^seems

to be a bound

particle

^{whose diffusion} is activated

by

^the excitations of the host. Further conclusions

are less _easy.

It turns out that even the identification of the dominant

system

of excitations is not evident. Our results do not seem to be commensurate with the vacancy model

proposed by

^Shikin

[3], [4],

but certain features would

suggest

interaction with the

phonons, although

^no

sufficiently

^{detailed model has been}

presented

^to

give quantitative

theoretical

support.

We have limited ourselves in this paper ^to

presenting

our

experimental

^results

together

^{with certain} argu- ments, based

essentially

^on

comparison

^{with NMR}

atomic diffusion measurements, which make the vacancy

hypothesis unlikely. Among

^the

important questions

^{we leave} open ^are the structure of the

positive

ion and the form of the interaction with the thermal excitations.

In addition we

present experimental

^{evidence for}

the

trapping

^of

negative

ions which we

suggest

^is due to the stress field of dislocations.

On the

experimental side,

^we

point

^{out that our}

work was

inspired by

^and

supplements

^the

pioneering

current measurements of

Shal’nikov, Mezhov-Deglin

et al.

[5], [6]

^{on diodes} of solid He. Since

then,

^Sai-

Halasz and Dahm

[7]

have made further current measurements and have inferred mobilities from both their own and the Moscow

group’s

^results.

The

originality

^{of this}

experiment

^{is in}

measuring directly

the transit time of the ions in the

crystàl (in

^the presence of ^an electric

field),

^{free from} any

grid,

^{between two} electrodes. The ions are

produced by photo-electrons ejected

^{from a}

gold plated

^elec-

trode

by X-ray pulses

rather than

by

^a radioactive

source. We mesure the

intensity

^{and duration of the}

current

produced by

^a

pulse

^of

X-rays

^{in a}

given

electric field.

2.

Experimental apparatus.

^{- 2.1 HIGH PRESSURE}

CELL AND CRYSTAL GROWTH. - The main

problems arising

^{from the use of}

X-rays

^{are :}

- The necessary

permeability

^{of the cell to}

X-rays ;

- The evacuation of the heat

produced by X-ray impact

^{on the}

gold-plated

electrode. About 98

%

^of

the

X-ray

beam power is lost in heat

(the

^maximum

X-ray

power used was about 20

mW).

The first

problem

is solved

using beryllium

^{for the}

cell.

For the heat

evacuation,

^the

gold-plated

^electrode

is in direct contact with the

liquid ^4He

bath. This

proved

^to be sufficient to

prevent

any

extra-heating

of the

sample :

^for

pulsed irradiation,

^{we did not}

detect _any

temperature

^increase

(with

^a

precision

of 5

mK) ;

under continuous

irradiation,

^the tempe-

rature increase never exceeded 10 mK.

As shown in

figure 1,

the cell is formed of two

cylindrical

^coaxial electrodes made of

beryllium,

embedded in

Stycast,

^a

polymerised

^resin

(Stycast

2850

GT,

available from Emerson and

Cuming Inc.).

A thin foil

(10 Il thick)

^of

gold

^is

electrolitically depo-

sited on one half of the inner surface of the outer

electrode,

which forms the wall of the

high

pressure cell. The

high voltage

^is

applied

^{to the outer}

electrode, producing

^an electric field between the two electrodes

(separated by

¹

mm). Referring

^to

figure 1,

^a

pulse

of

X-rays coming

^{from the left crosses the whole}

probe

^and

ejects energetic

^{electrons from the}

gold- plated électrode ;

^these

primary

^{electrons in turn}

ionize the solid helium within a

depth

of about 10 _{p ;} the

sign

^{of the}

high voltage

^{selects the}

sign

of the ions which drift to the inner

electrode,

where the ionic current is collected. With this

device,

^the

density

of ions inside the thin cloud of ions

drifting

^{in the}

bulk reaches

109-101° ions/cm3 (the

decrease in electric field due to this _space

charge

^{does not exceed}

1

V/cm).

^Another

advantage

of this method is that

FIG. 1. ^- Cross-sectional view of the cell. 1 : Helium fill capillary.

2 : Helium _{gas. 3 :} Beryllium electrodes. 4 : Stycast. 5, ^{6 : Ther-}

mometers. 7 : Solid helium. 8 : Copper ^cold finger. ^{9 : Gold} plating. ^{10 :} Liquid ^{4He bath.}

the

density

^of

charge

^{carriers can be varied}

by varying

the _power of the incident

X-ray

^{beam. The}

X-ray

flux at 10 cm from the anode could be varied from 5 x

103

to 4 x

104 R/min,

^{i. e. from 30 to 200}

mW/cm2, by

^{the tube current control.}

2.2 GROWING OF HELIUM SINGLE CRYSTALS. ^- As

was

pointed

^out

by Mezhov-Deglin et

^al.

[5],

^the

quality

^{of the}

crystal

^{has a drastic influence on} elec- trical

conductivity.

^{So we tried to} grow

single crystals,

at constant pressure,

using

^a

temperature gradient.

During

^the

solidification,

^{a cold}

point

^is

produced

at the bottom of the

sample, by

^a copper cold

finger refrigerated by

^a flow of helium gas

pumped through

a leak from the

’He

bath.

Monitoring

^the

pumping speed, temperature

differences

(between

^the

top

^and the bottom of the

cell)

^{from 0 to 0.2 K could be}

produced,

^{at 3 K. In order to}

prevent

solidification at constant

volume,

^{the fill}

capillary

^{is heated}

by

^a

wound resistance. In our

experiments,

^the pressure is held constant and the

crystal

grows in 5-10

minutes,

on

slowly cooling

the whole

’He

bath. After the

solidification,

^the

heating

^{current is cut}

^off,

^{so that}

solid helium is formed inside the

capillary

^{and the}

volume of solid helium inside the

probe

^remains

constant.

We did not check

directly

^the

quality

^{of the}

crystals :

we measured ionic currents of the same order of

magnitude

^{on all the}

samples,

^{even those}

supposed

to be of _poor

quality (grown

very

quickly).

The

temperature

is measured with two Allen-

Bradley

^{10 Q}

resistances,

^{and the}

high

pressure is measured outside the

cryostat

^{on the}

high

pressure oil pump. There ^was

generally

^a

good agreement

between solidification pressure and

temperature

^mea- surements,

though

^{a certain}

discrepancy reaching

0.1 K at 3 K was

observed,

^at

high

pressures

(140 atm).

This

discrepancy

^{is not}

explained.

The

temperature

^could be varied from 4.2 K to 1.35 K and the _pressure from 1 to 150 atm;

they

were measured with ^an _accuracy of 5 mK and 1 atm.

2.3 ELECTRONICS AND MOBILITY MEASUREMENTS. ^-

The non

uniformity

^of the electric field due to the

cylindrical geometry

is less than 10

%.

But the

spacing

of the

electrodes, d,

^{was known}

approximately :

d ⁼

(1 +0.1)

^{mm. If T is the} transit time of an ion in ^a field E ⁼

v/d,

^the

mobility

^is

Given the

uncertainty

^{in the values}

of d,

^{the fol-}

lowing

measurements should be

interpreted

^{as relative}

measurements

(absolute

values of the

mobility

^have

20

%

relative

error).

The

high voltage V (positive

^or

negative)

^can vary from 10 to 1 000

V, giving

^an electric field of

102

to

104 V/cm ; i

^varies

typically

^{from 100 ms to 30 s.}

The outer electrode is set to the

high voltage,

^the

inner electrode is

grounded

^{via the}

detecting

^{device :}

the

signal

^is

amplified

^{in a current}

amplifier (rise

^time

about 5 ms and

gain ^{1 O1 °} V/A),

then passes

through

a multichannel

analyzer

whose memory is read ^{on a} X-Y

recorder,

^which

gives

^the

intensity

^{versus the}

time.

The currents measured varied from

10 -12

to

10-9 A,

and the

signal

^to noise ratio was

always

^{better than}

3 for ^a

single

weep.

A

typical signal

^{is shown on}

figure 2, together

with the

X-ray pulse

^{and the} theoretical

signal.

^The

FIG. 2. ^- Ionic current signals. ^{1 :} Experimental signal ^for positive

ions in bcc 3He. V ⁼ 20.40

cm3/mole ;

^{T = 1.60} K ; E ⁼ 6 000 V/cm.

2 : X-ray pulse. ^{3 :} Theoretical signal.

theoretical

signal

^is

trapezoidal : during time io

^ions

are

produced

at constant rate and start

drifting,

^the

current increases

linearly;

^then

at t >, -c,,

^{the whole} cloud of

charge

carriers drifts inside the bulk

leading

to a constant

intensity.

^{At time t =} r, the ions start

reaching

the inner electrode and the

intensity

^decreases

linearly during

ro.

3.

Experimental

^{results. -}

First,

^we have checked the _presence of a definite

mobility (for positive ions ;

for

negative

^{ions see}

below),

^{i. e.} the transit time T is

inversely proportional

^to the electric field E. This relation i

proportional

^to

1 /E

^{was verified to within}

10 %

for field variations from 100 to 1 000

V/cm,

and

103

^to

104 V/cm

^{for both}

^’He

^and

^’He crystals (the

whole range

102

to

104 V/cm

^{could not be inves-}

tigated

^{for a}

given crystal

^{at a}

given temperature

^and ionic

density).

An increase in the ionic

density (pro-

duced

by

^{an increase of the}

X-ray power) changes

the

shape

^{of the}

signal (indicating

^an ionization inside the

bulk),

^{but does not affect the}

mobility.

This

mobility

^{was found to be}

reproducible (except

for

negative

^{ions in}

hcp ’He,

^see

below)

^{to better}

than 50

%

^{from one}

crystal

^{to another of same molar}

volume.

After

cooling

^or

warming

^the

crystal

^{to a certain}

temperature,

the drift

mobility

^was

always

^instan-

taneously

^reached

(i.

^{e. in a} time less than ^a few

minutes).

The

cylindrical shape

^{of the}

probe

^{allowed us to}

vary the drift direction relative to the

crystal

^axes,

by rotating

^the

probe

^{relative to the}

X-ray

^beam.

Except

^for

negative

^{ions in}

hcp ^’He (see below),

we did not detect _any

anisotropy

within 10

%.

At least from a temperature ^{0.1 K below the}

melting point,

^{down to 1.35}

K,

^the

mobility

^{seems to} vary

exponentially

with inverse

temperature, suggesting

an activated process. The diffusion coefficient is fitted to :

giving

^a

mobility

For each

crystal,

^{we have}

plotted log MT versus 1/r,

and from this

straight

^{line we} have deduced the activation _{energy d} and the

prefactor Do.

Figure

^{3 shows a}

typical plot.

^The

mobility

^at

each

temperature

^is determined

by measuring

^the

transit time in three different electric fields.

For

crystals

of low molar

volume,

^{the low}

tempe-

rature measurements were limited

by

^the very small

mobility (we

^{did not measure} mobilities less than 5 x

10-7 cm2 V -1 s -1 ) ;

the measured variation

of p

was over 2 or 3 decades.

For

crystals

^of

great

^molar

volume,

^{we were limited}

by

^our

cryostat

^{to a lower}

temperature

of 1.35 K and the variation

of p

^{does not exceed one}

decade, giving

^{less accurate} values of L1 and

Do.

FIG. 3. ^- Mobility multiplied by temperature ^{as a} function of inverse temperature. T,. is ^the melting temperature ^{of the} crystal.

3. 1 TRAPPING OF NEGATIVE IONS. ^- In both solid

’He

and

’He,

^{we have observed}

trapping

^{of the}

electrons in the bulk of the solid. This has been seen

by

^{several means.}

First,

^the

negative

^ion

signal

^{does not have the}

nearly

^ideal

trapezoidal shape

^{of the}

positive

^ion

signal.

^See

figure

^{4 : on this}

signal,

^{there is still a}

discontinuity

^{in the}

slope (point A)

^which

yields

^a

transit time T

inversely proportional

^{to the electric}

field.

But,

^{for some}

crystals,

^the

negative

^ion

signals

would not show _any transit

time, indicating

^{that the}

whole cloud of electrons is

trapped

^before

reaching

FIG. 4. ^- Signal of negative ^{ions in} hcp ^{4He. V = 18.24}

cm3/mole ;

T ⁼ 2.50 K ; E ^{= 6 000} V/cm. ^The transit time i is measured at

point A, where there is an abrupt change ^{in the} slope.

the inner electrode. In any case, the

shape

^{of the}

negative

^ion

signal

makes much more uncertain the measurements of

negative

^ion mobilities than for

positive

^ions.

The

following experiment gives

another indication of this

trapping : applying

^a

negative voltage

^{on the}

outer

electrode,

^{we let}

negative

ions drift inside the bulk for a few

minutes ;

^{then we}

stop

^the

X-ray

irradiation and set the

voltage

^{to zero. Now if we}

send a new

pulse

^of

X-rays,

^{we observe an inverse}

signal,

^{i.e. a}

positive

^ion

signal

which is due to the presence of an inverse electric field

produced by

space

trapped negative charges.

This inverse field

frequently

reached 100-500

V/cm, giving

^a

trapped

electrons

density

^of

^109-101° electrons/cm3.

^{Since we}

did not

study quantitatively

^this

trapping phenome-

non, ^{we can}

only give qualitative

features : the

detrap- ping

^time exceeds 10

minutes;

^{and the}

importance

of

the’ trapping

varies from one

crystal

^{to another.}

Trapping

^of

positive

^{ions has never} been observed.

We shall now

present

^the

experimental

^results

for both

signs

of ions in

hcp,

^bcc

^’He

^and

hcp, ^{bcc 3He.}

3.2 HCP

4 HELIUM.

^- 3.2.1 1

Negative

^{ions. -} Besides the

trapping phenomenon, negative

^ions

exhibit a

mobility varying exponentially

with inverse

temperature,

^thus

yielding

^an activation energy ^L1 and

prefactor Do.

^{But these two}

quantities

^{d and}

D,

are not

reproducible

^{from one}

crystal

^{to another}

(grown

^{at the same}

pressure).

^Seven

experiments,

at a molar volume of 18.80

cm3/mole,

gave for

J/A;

(in degrees Kelvin) : 17.7; 25; 25.8 ; 27.5; 32; 35 ; 52 ;

^whereas

Do

varied from

10-5

to 5 x

10 -1 cm2/s-1 !

^!

Then,

^{for a}

given crystal,

^the

mobility

^is

highly anisotropic (multiplied by

^{3 for a 90°}

rotation) ;

this

anisotropy only

affects the

prefactor Do

^{and not}

the activation energy 4 ,

All these

phenomena suggest

^{that the}

negative

ion

mobility

^is

governed by crystal imperfections,

such as

dislocations,

^{which can} vary from ^one

crystal

to another.

3.2.2 Positive ions. ^- 31

crystals

^were grown ^at pressures

varying

^{from 27 atm to 120 atm. For a}

given

^molar

volume,

^{L1 was}

reproducible

^{to within}

12

%,

^and

D,

^{to within 50}

%.

FIG. 5. ^- Activation _energy of the diffusion coefficient in hcp ^4He

versus molar volume : § : positive ions, ^this experiment, ^{+ :} positive ions, ^measures of Sai-Halasz and Dahm [7], ^{d : 3He} atoms,

in a mixture of 1.94 % ^{’He in} hcp ^’He [8].

FIG. 6. ^- Diffusion constant Do ⁱⁿ hcp 4He, ^{versus molar volume.}

1 : positive ions, this experiment, ^{+ :} positive ions, ^measures of Sai-Halasz and Dahm [7], ^{4 : 3He atoms in a} mixture of

1.94 % ^{3He in} hcp ^4He [8].

Figures

⁵ and 6 show the variation of J and

D.

with molar volume. The error bars ^are estimated from the

reproducibility only.

^The

points

^{are some-}

what lower than those measured

by

Sai-Halasz and Dahm

[7].

^For

comparison,

^{we have}

plotted

^activation

energies

for atomic diffusion

(believed

^to be vacancy activation

energies)

^measured

by

^{NMR in a mixture}

of 1.94

% ^’He

ⁱⁿ

hcp ^’He [8] (1).

The activation energy J rises

monotonically

^as molar volume decreases

except

between 18.30 and 19.50

cm3/mole : J/A:

suddenly jumps

^{from 14.4 K to 23.3 K} then remains

roughly

constant to a molar volume of 19.30

cm31

mole.

An

analogous

^{effect occurs for}

Do :

^see

figure

^{6 :}

Do

^{has a}

discontinuity

^{at 19.46}

cm3/mole

and exhibits

a maximum around 19.90

cm3/mole.

3.3 BCC

4HELIUM.

^- The

mobility

^{of the}

negative

ions is about 50 times

higher

^{than in}

hcp ’He,

^while

the

mobility

^of

positive

^{ions is}

roughly

^{the same as}

in

hcp ^’He (within

^{a factor}

2).

^{The small extent of}

the bcc

region

makes it difficult to measure an acti- vation _energy.

3 .4 BCC

3 HELIUM.

^- Both

negative

^and

positive

ions exhibit a

mobility

which is well

reproducible.

3.4.1 1

Negative

^{ions. -} The external electric field

was set much

higher

than the electric field

generated by

^the

trapped

electrons in order to be able to measure

mobilities.

For a

given sample,

^{when the}

temperature

^is

lowered,

^the

mobility

first increases

by

^a factor 2-3 at 0.1-0.2 K from the

melting point,

then it decreases

exponentially (see Fig. 7).

V.f

FIG. 7. ^- Mobility ^{of ions versus inverse} temperature ^{in bcc} 3He.

V ⁼ 20.98

cm3/mole.

^The right ^hand parts ^{of the} curves are straight lines, ^the left hand parts ^are simply ^{an aid to} the eye. Tm is the

melting temperature.

3.4.2 Positive ions. ^- The

mobility

first remains

roughly

constant to 0.1-0.2 K from the

melting point,

^then it decreases

exponentially (see Fig. 7).

Figure

⁸ shows the activation

energies

^deduced

from the

exponential part

^{of the curve}

log MT

^versus

IIT. J+

varies almost

linearly

with molar

volume,

whereas J _ exhibits ^a

sharp

^maximum.

The same

qualitative

^{features occur for}

Do (see Fig. 9).

3.5 HCP

3 HELIUM.

^- We have

just

started mobi-

lity

measurements in this

phase

^{and we cannot} yet

give

extensive results.

The variation of the

positive

^ion

mobility

^{is still}

exponential, but,

^{for a}

given sample,

^it passes

through

a

discontinuity (of

^{about a factor}

2)

^{when the}

tempe-

(1)

Polycrystalline-blocked capillary ^method.

FIG. 8. ^- Activation _{energy of} the diffusion coefficient in bcc 3He,

versus molar volume. § : ions, this experiment, ^{+ : 3He atoms.}

Reich [20].

FIG. 9. ^- Diffusion constant Do ^{in bcc}

3He,

^{versus molar volume :}

! : positive ions, ^this experiment, : negative ^{ions, this} experiment,

J : 3He atoms. Reich [20].

rature is lowered

(see Fig. 10).

^This

discontinuity

is crossed

reversibly

^{and whitout any} visible relaxation time when the temperature ^is

increased,

^{and there}

is a small

temperature

^{area where two different} mobilities coexist

(see Fig. 11).

^{Table 1 shows the}

variation of the transition

temperature T,

^{with molar}

volume.

FIG. 10. ^- Mobility ^of positive ^{ions versus inverse} temperature in hcp ^{3He. V = 18.74}

cm3/mole.

FIG. 11. ^- Double mobility signal ^of positive ^{ions in} hcp ^3He.

V ⁼ 18.92

cm3/mole ;

^{T = 2.38} K ; ^{E = 8 000} V/cm.

TABLEAU 1

Variation

of Tt,

^the temperature

of

^the

discontinuity of

y,, ^versus molar

volume,

ⁱⁿ

hcp ^{3 He}

Several facts make ^us think

strongly

^{that this}

transition cannot be the bcc ^H

hcp

transition :

- The transition

temperature

^and pressure do not

correspond

^{to the bcc H}

hcp

transition

(see Fig. 12) ; by comparison

of our data for the bcc ^H

hcp

transition with those of

Grilly

^{and Mills}

[9],

^{we see}

that our pressure would be

slightly

underestimated

(or

^the

temperature overestimated) ;

^but

taking

account of this error will

displace

^the

points

^{of the}

transition of _{y, away} from the bcc ^H

hcp

^line.

- The bcc ^H

hcp transition,

which has been observed on two

samples,

^{does not have the same}

features ^as the

previous

transition.

3.6 CONTINUOUS CURRENT MEASUREMENTS. ^- We have also

performed

continuous current measure-

FIG. 12. ^- The transition points of,u+ plotted ^{on a} phase diagramm of 3He : + : transition points ofy + ; ^this experiment, ^{0: bcc H} hcp transition points : ^this experiment, 0 : ^{bcc H} hcp transition points : Grilly ^{and Mills} [9]. ^{We have not taken into account} the decrease of the pressure of the crystal ^{due to a} decrease of the temperature

at constant volume. For a molar volume of 18.74

cm3/mole,

^this

decrease is 1.65 atm/K, ^{near the} melting point, ^{which can be} neglec- ted on a pressure of 140 atm.

ments,

using

^a permanent

X-ray

beam instead of a

pulse.

The characteristics i

= f ( V )

^{do not show a} square law

dependence

^{of i on V}

(which

characterizes a

space

charge

^limited

current).

Our characteristics have the form : i

proportional

^to

vn,

where n is about 1.5. This is not

clearly explained :

^it may show that the drift current is intermediate between a space

charge

^{limited current and an « ohmic » current}

(i proportional

^to

V).

But it is in contradiction with the measurements of Sai-Halasz and Dahm

[7],

^who

observed a law i

proportional

^to

V2, though

^the

density

^{of ions is}

thought

^to be lower than in this

experiment.

4. Comments ^on

expérimental

^{results. - The fact}

that the mobilities decrease with

decreasing tempe-

rature leads us to

adopt

^{a localised ion as a}

starting point.

^{On this}

basis,

^{we shall in turn : comment on a} list of

questions

^{we feel} should be

asked ; suggest possible

^{clues to the answers from the}

experimental data ;

^{describe a} naive model of

phonon

^activated

quantum

diffusion ;

remark upon the

positive

^ion

mobility discontinuity

ⁱⁿ

3He ;

^{show that}

trapping

may result from the interaction between ions and dislocation lines.

4. 1 SOME QUESTIONS, ^{SOME COMMENTS AND A FEW} ANSWERS. ^- 4. 1. 1 What are the lattice excitations

of the

pure solid ? ^- Phonons and vacancies are the established low _energy excitations. Phonons describe

quite

well the vibrational

motion, despite

^the

nasty potential

^and

large

^zero

point

^movement

[10], [11].

The vacancies appear ^to be of the

Schottky type existing

^{either as}

non-propagating

^{modes or} vacancy

waves

[12], [13].

For

4He,

^the

phonon dispersion

^{has been}

^mapped by

^neutron

scattering

^at

and in

hcp

^and

in bcc

phases. Complementary

^{data are}

given by

^the

sound

velocity, specific heat,

^thermal

conductivity [17]

and Raman

scattering [18]

measurements. The whole

seems to indicate that

phonons

^are

quite good

^elemen-

tary excitations, although

^certain very broad neutron

peaks

^{do not seem to} fit into the usual

phonon

^des-

cription.

The existence of vacancies is inferred from the behaviour of the

spin

diffusion of

3 He impu-

rities

[8].

^{The vacancy} excitation

energies

^{so obtained}

are

superimposed

^on

figure

^5.

For

’He,

^the

high absorption

cross-section excludes neutron

scattering. Specific heat,

thermal conduc-

tivity,

^sound

velocity [17]

^Raman

scattering [18]

^and

comparison

^{with the}

corresponding symmetry phases

of

’He

must be used to leam about the

phonons.

There has been some

difficulty

ⁱⁿ

correlating

^{all the}

data on the bcc

phase,

^{but a}

reasonably

^coherent

picture

^is

emerging [17], [19].

The evidence for vacan-

cies is rather more extensive in bcc

’He,

^due

largely

to their rather low

énergies ; spin

diffusion and relaxation

[20],

^excess

specific

^heat

[21]

^and

lately

mean lattice

parameter [22]

measurements

give

^a

reasonably

consistent

picture.

^The vacancy excitation

energies

^{so inferred are}

superimposed

^on

figure

^8.

4 .1. 2 Structure

of ions

and associated excitations. ^- The

primary

^{notions on ion structure are outlined}

in the introduction. In what follows we

try

^{to make} these ideas ^a little more

explicit.

Negative

^{ion. One}

imagines

^{that at the moment of}

formation of its bubble the ion is surrounded

by interstitials,

but that these ^are

rapidly

^relaxed

by

vacancies

migrating

from the bulk to leave a hole of radius R ;:t 9

Á (displacing

^{some 90}

atoms)

^sur-

rounded

by

^an

elastically

distorted lattice. The latter carries both dielectric

polarization

and elastic defor- mation

energies

associated with the bubble. If r is the distance from the bubble centre and ^a the atomic

polarizability,

^the

polarization

energy per ^atom is 1

2

1 ote

^{1.6 K at r =}

^{10 Â.}

^{In an}

elastically 2 r4

isotropic

^continuum

approximation,

the elastic defor- mation is described

by

^a radial strain _urr ^{= -} 2 £R

’Ir’

and its

corresponding

^{stress. e} is the fractional

change

in the radius of the surface

bounding

the bubble from its radius in the

unperturbed

solid. We may

put

^a

rough

upper bound

on 1 B 1 by arguing

^{that if}

the

strain 1 u,,.(R) 1 ;; ^2,

^{non linear effects}

^(e.

^g.

interstitial

formation)

would relax it.

Thus 1 E 1 ;$ 4-

One _may

locally

excite the bubble

by displacing

an atom from its surface into its volume. We can

estimate the _energy

W, required by supposing

^{it to}

consist of two

parts WD

⁼

WE

⁺

WS ; WE

^{is the}

increase in electronic energy when ^{an atom} is

present

inside and

Ws

^{is the} energy

required

^{to remove an}

atom from the surface in the absence of electronic forces.

By constructing

the Fermi

pseudo-poten-

tial

[23]

from the low energy ^s-wave electron-He

scattering length [24]

^a,

- Ws

is of the order of the

ground

^state energy per

particle

of the solid.

Typically W,

^{+ 5 K for}

hcp 4He

^at

Vrn

^{= 19}

cm3/mole-l [26], Ws -

^{1 K}

for bcc

3He

at

Vm

^{= 21}

cm3/mole-1 [27].

^{This is}

a

possible

^{mechanism for}

producing

the surface vacancies

proposed by

Sai-Halasz and Dahm

[7]

to

explain

^their values for the

negative

ion mobilities in

’He.

Positive ion. ^- One

expects

the formation of ^a

molecular

^ion which should be treated ^as an

entity

surrounded

by

^a distorted lattice. The small molecular ion

He’ [28]

^is

thought [29], [30]

^to be linear and

symmetrical

^{with an} internuclear

spacing

^{of 1.2}

^A

and a

binding

^{energy of N2 000 K over He +}

He ) . He2

itself has an internuclear

separation

^{of 1.1 1}

^À

^{and a}

binding

^energy of about 20 000 K

[31 ] over He + He + .

It is not inconceivable that even

He’

fits into an

interstitial

position

^{and is able to diffuse}

by

^thermal

activation to

equivalent sites ;

^{it is more difficult to}

imagine

^{how it}

might translationally

^{diffuse if one}

of its constituents forms a lattice

point,

^{for an}

exchange

must then be made with ^a

neighbouring

^{atom or}

vacancy. But without a detailed

knowledge

^{of the}

He - Hej

interaction

potential

^{it seems}

pointless

^to

speculate

^on detailed models.

4.1.3 What is the interaction between the

charge defects

^{and the}

crystal

excitations ? ^- If we had

good

^{answers to} the ionic structure

problem,

^we

would be able to guess ^at the interaction

potential,

calculate the modifications to the excitations induced

by

^{the defect}

(e.

^{g. localized nature} of certain

modes)

and

finally

deduce the diffusive behaviour of the defects.

4.2 POSSIBLE CLUES TO THE DIFFUSION MECHA- NISMS. ^- As we lack the

knowledge

^{to make an}

a

priori calculation,

^{we shall}

try

^{to extract some} clues from the

experimental

^results.

Shikin’s

suggestion

^of

displacement by

^collision

with vacancies is _very

appealing,

^{but we do not}

think that it can be the most

important

^mechanism.

This mode is

basically

^{the same as} that for the atomic diffusion of

3He

where an encounter leads to a dis-

placement

^{of one nearest}

neighbour length

^{a, whereas}

for ^a defect of dimension R > a the step ^{for the}

centre of

charge

^must be rather less. Continuum

theory [3] suggests

^{that the defect diffusion constant}

Dd z (aIR)3 Cv Dv

^where

Cv

^{in the} fractional _vacancy concentration and

D,

^the vacancy diffusion constant.

(alR)’ lo

^{for a} bubble and z 1 for

3He.

The vacancy activation energy

entering

ⁱⁿ

C,

^{is taken}

at the defect

boundary

and should thus be

augmented by

^the

polarization

^{energy over its bulk}

value ;

^this additional

polarization

barrier will

probably

^also

reduce

D,

from its bulk value. On

comparing Dd

with

D3, ^then,

^we

expect

the activation _energy to be

higher

^{and the}

pre-exponential

^{factor to} be smaller

on their

model ; Dd

^should

always

be smaller than

D3.

In what follows ^a

super-prefix

denotes the

isotope

number of the host and the infra-suffix the

impurity,

e. g.

4D 3

^{is the diffusion constant of}

^{3 He}

ⁱⁿ

^{a 4He}

^host.

(9 ^and e ^denote

positive

^and

negative

^ions.

We _compare the values for

4Do

^and

4D3, bearing

in mind that if the

positive

^{ion in a}

He’

^molecule

and the _vacancy must

approach

^its centre to within about a lattice constant

plus

^an internuclear

spacing,

it will cost - 20 K in

polarization

^{energy. A}

glance

at

figures

5 and 6 shows that

only

^for

do the

equalities

^{come near to}

being

satisfied. The

pre-exponential

^for

4Do

is about 50 times

less,

^but

the

activation

energy is ^a little lower. For

there are no measurements for

4D3,

^{but one} expects its

pre-exponential

^{to show a}

slight decrease,

^whereas

the

pre-exponential

^of

4Do

^increases

by

^a factor of 50.

The

comparison

^of

3D Ef)

^and

3D e

^with

3D 3

^is

more

striking.

^From

figure

^{8 we see that the activation}

energies

^for

3D +

^{are the}

requisite

^{10 K as more}

higher

^{than for}

’D3,

^{but that the}

pre-exponential

factor is 10 to 100 times

higher.

^For

3Dg

^{the acti-}

vation

energies

^are

again

^{10 K or so}

higher

^whereas

we would

expect

the difference to be reduced to some 2 K due to the

larger radius ;

^the

pre-exponential

factor is

again ^1-103

^times

higher.

We conclude that the contribution from bulk vacancies to the ion diffusion ^never dominates in the _{range of} our measurements.

Surface diffusion

[31].

^{We can also}

envisage

^that

the surface atoms can be excited into the bubble at a rate

1/ïg

exp -

0/ T where 1 /,r,

^{is a factor}

dependent

on interaction with thermal bath

(e.

g.

phonons)

and the 0

represents

^an activation _energy for this process. Once in the

bubble,

^{the mean free}

path

before recondensation is R and the

consequent step

of the bubble 1 ^’"

Â-a3 4n (

³

^R

^.

^Assuming

^g

the

evaporation

^{not to be} bottlenecked

by

the conden- sation rate, ^{we are} led to an upper limit of

Unlike the bulk vacancy ^{case, we} have no other measurements with which to compare

it,

^{but we}

can

put

^a numerical limit on the

pre-exponential