The low temperature electrical resistivity of lattice defects in deformed tungsten single crystals

THE

LOW

TEMPERATURE

DEFECTS

IN

ELECTRICAL

DEFORMED

H. B. SHUKOVSKY,t:

RESISTIVITY

TUNGSTEN

SINGLE

R. M. ROSE,t

OF

LATTICE

CRYSTALS*

and J. WULFFt

The generation of vacancies and dislocations during plastic deformation of tungsten single crystals has been studied using electrical resistivity measurements at low temperatures. An etch pit technique has been used to determine the dislocation density ND as a function of strain. The dislocation resistivity at8 low temperatures is found t,o obey the followmg relation:

The density of dislocations produced has been found to be proportional to cl.8 and also to the work done cr order where or and er are the true stress and true strain per unit volume in deforming the crystal s0 respectively. The vacancy resistivity has been found to vary as c 0.9for strains of less than 2 percent and as &9 between 2 and 8 percent. At higher strains the vacanoy resistivity increases less strongly with increasing strain. The vacanoy resistivity, and hence the vaoancy concentration variation with strain, is consistent with both the moving jog and dislocation dipole recombination mechanisms of vacancy production if one considers the energy balance, fixed barrier, and growing barrier models. The tempera. ture variation of the ideal resistivity of a typical unstrained crystal is in agreement with Berthel who reports a T5 dependence for tungsten. LA RESIST~ITE

ELECT~IQUE A BASSE TEMPERA~RE DE DEFAUTS RETI~LAIRES DANS DES ~O~OCRISTAUX DE TUNSGTE~~ DEFORMES

La production de lacunes et de dislocations au tours de la deformation plastique de monocristaux de tungstitne a 6% etudihe au moyen de mesures de resistivite Blectrique aux basses temperatures. Une technique de piqures de corrosion a et& utilisee pour determiner la densite de dislocation LV, en fonetion de la deformation. La resistivite aux basses temperatures due aux dislocation est donnee par la relation suivante: apD a&

6,7 x 10-r’ ,n&cm8

Les auteurs trouvent Qgalement que la densite de dislocations produites est proportionnelle a &a ainsi CT arderou oret lrsont respectivement qu’au travail effect& par unite de volume en deformant le cristal s0 la tension et la deformation r&&s. La resistivite due aux lacunes varie suivant ls.%pour des deformation inferieures a 2% et suivant t*.* entre 2 et 8%. Pour des deformations plus Blevees la resistivite due aux lacunes augmente mains fort pour des deformations croissantes, La r&ktivittt due aux lacunes, et done la variation de la concentration de laeunes en fonction de la d&formation, eat en accord avec les m&anismes de production de laeunes, cran en d&placement et reeombin~sol~ de dipoles de dislocation, si l’on oonsidere les modeles de bilan Onergetique, de barriere fixe et de barriere en croissance. La variation en fonction de la temperature de la resistivite id&de d’un eristal type non-deform8 est en accord avec Berthel, qui indique une dependance en T5 pour le tungstene. DER

ELEKTRISCHE WIDERSTAND WOLFRAMEINKRISTALLEN

VON GITTERFEHLERN IN VERFORMTEN BE1 TIEFEN TEMPERATUREN

Die Erzeugung vonleerstellen und Versetzungen wiihrendder plasticshen Verformung van Wolframeinkristallen wurde mit Messungen des elektrischen Widerstandes bei tiefen Temperaturen untersucht. Die Versetzungsdichte Ns wurde in Abhiingigkeit van der Dehnung mit einer dtzgriibchenmethode untersucht. Der Versetzungswiderstand bei tiefen Temperaturen folgte der Beziehung

af- _ 6,7 -_ ah

x

10-11$&cm3

Die Dichte der erzeugten Versetzungen war proportional zu +,

ebenso die ~~erformung~rbeit

“T or&r

s pro ~‘ol~lmeinheit, wo or und er die wahre Spannung bzw. die wahre Abgleitung ist. Der Widerst~nd van Leerstellen geht mit Eo.e fur Dehnungen unterhalb 2% und mit e1.s fur 2-$76 Dehnung. Bei hiiheren Dehnungen nimmt der Leerstellenwiderstand weniger stark mit der Dehnung zu. Die Variation des Leerstellenwiderstandes und folglich such der Leerstellenkonzentration mit der Dehnung ist konsistent mit den Mechanismen der Leerstellenbildung auf Grund sieh bewegender Versetzungsspriinge und der Rekombin&ion van Versetzungsdipolen, wenn man Modelle iiber Energiebilanz, feste Hindernisse und wachsende Hindernisse in Betracht zieht. Die Temperaturabhangigkeit des idealen Widerstandes eines typischen unverformten Kristalles stimmt mit Berthel tiberein, dor eine !fS-Abhangigkeit fur Wolfram anpegeben hat. * Received August 26, 1965. t Department of Metallurgy, Massachusetts Institute of Technology, $ Now at: Western Electric Corporation, Princeton, New Jersey. ACTA METALLURGICA,

VOL.

14, JULY

1966

821

Cambridge, Massachusetts.

ACTA

a22

METALLURGICA,

INTRODUCTION

This paper describes of plastic

of high purity tungsten

and an attempt

at separation

plastic

elec-

single crystals

of the contributions

of the various defects generated

to

by the

deformation.

Previous

of the

tivity have been chiefly concerned copper, silver and gold.(1-3) more

few

dislocations

by

Buck.(5)

that

to dislocations

for

gation

that

according

Thus the expansion

indicates

that

generated

by plastic

vacancy

resistivity,

3

of interdensities.

for columbium,

work of Schultz(g)

due

to

dislocations

is greater

than

has found a linear

concentration

and

Friedelo’)

formation,

consideration.

Stacking

faults

can also be omitted

energy

is large compared

f.c.c.

in b.c.c.

metals(il)

their effect, cations

metals

so that their concentration,

should be small.(12)

and vacancies

of the resistivity

of dislocations.(13) Read

fault

to that

in

and hence

Thus we expect dislo-

to be responsible

changes

been deformed. We shall consider

is larger

of all, the stacking

for almost all

in pure tungsten

We have selected tungsten little

work

of this

three

that has

models for the generation Frank-

sources under an applied stress which generate

dislocation loops that expand freely until they reach the crystal surface, and then uses an “energy balance” method to calculate increase in dislocation density

nature

of two slip

Seitz(16) and as opposed

for three reasons. has

been

crystals.

Tungsten

to

electron-beam tivity to

on tungsten

zone melting,

observe

a relat,ively at

introduced. direct

low

single

and the residual

made quite small, and therefore

resistivity

First,

done on b.c.c.

can be purified to a high degree by

large

fractional

change

when

tungsten

lends itself

Finally,

measurement

of the

resis-

one can expect

temperatures

defects

dislocation

in are

well to a density

by

means of an etch pit technique.

The investigation crystals

PROCEDURE

was organized

measurements

as follows.

of (100) axial orientation, ratio.

which initially

Measurements

as well as after 500°C anneals,

of which was the removal

in varying the purpose

Tensile strain

of vacancies.

was used in all cases, and was determined

by direct

measurement.

electrical

After

completion

measurements

the dislocation

using

pit technique.

an etch

dislocation on the

density

by counting

shoulder

the

density The

density was obtained

undeformed

of

was obtained

initial

as-grown

by counting etch pits of each

after deformation

sample.

The

was determined

etch pits in the deformed

The details of the work are described

gauge section. below.

The single crystals of tungsten were seeded for a (100) axial orientation and grown by a floating-zone electron-beam

techniqueos)

in vacua of 2 x 1O-5 torr

and 2 x 10~~ torr at a rate of 2 mm/min. material

was + in. dia. polycrystalline

obtained

from Electronic

energy stored due to lattice

a quoted purity

Accord-

had

were per-

formed before and after plastic deformation amounts,

Elec-

were made on tungsten

with strain; the gain in strain energy due to the expansion of dislocation loops is balanced by the defect formation.

non-

lie in parallel

metals and none. to our knowledge,

dislocation Van Bueren(3s14) considers

annihilation

mechanism

EXPERIMENTAL

and since they were

First

the

moves

recombination

low

we shall omit them from

on the basis of energy.

the

sign that

favor the latter

the same resistivity

by Schultz,(g)

segment is

involves

mecha-

when a dislocation

the moving jog mechanism.

energy

not observed

jog

there

of plastic

to several

planes, one or two atomic spacings apart.

trical resistance

than that for vacancy

if the

Also

We shall consider the possibilities of the creation of interstitials, and stacking dislocations, vacancies, faults, by plastic deformation. Since the activation of self-interstitials

as a result

can be produced

a jog moves:

of

density.

loops becomes more

of vacancies

Vacancies

once again for tungsten.

for the formation

dislocation

has been attributed

conservatively.

finds

than

and Schultz(*)

flow in tungsten Schultzdo)

In

for a given elon-

than

Later

resistivity

that due to vacancies. between

change

greater

and molybdenum. the

greater

deformation with

dislocation

strain.

of opposite

and

are believed

to Seeger(15) the number

of dislocation

which

of

barriers”

with increasing

dislocations

Gregory(‘)

strains

wires Krautz

the resistivity

is considerably

temperature

increases

mechanism

on strain.

to the migration

tantalum

relation

Here,

barriers

been

vacancies

at higher dislocation

In the case of tungsten observe

to

columbium

in resistivity

which he attributes

stitials

model.

metals.

resistivity

a linear dependence

the case of polycrystalline percent

due

have

b.c.c.

the electrical

et a1.c4)

“fixed

The third model is called the “growing barrier”

nisms.

on copper

There

of the

molybdenum

exhibits

a decline

loops.

The production

resis-

with polycrystalline

Investigations

investigations

has found

polycrystalline

model

to limit the motion of the stress generated

of plastic

electrical

have been carried out by Blewitt

recently

relatively Peiffer?

effect

on the low temperature

single crystals

1966

limited with increasing

investigations

deformation

and

of the effect

on the low temperature

trical resistivity the resistivity

14,

ing to a second

an investigation

deformation

VOL.

Space Products,

of 99.99 percent.

The starting tungsten

rod

Inc. having

The single crystals

RHUKOVSKY

GERMANIUM

VOLTAGE

THERMISTOR

LEADS

RESISTIVITY

et al.:

- \

OF

DEFECTS

IX

DEFORMED

tor was imbedded. \

\

\

,,,,.. /

\\

-

In addition,

from the thermistor

---y

,/.‘.

anchored

The temperat~ure was varied the

apparatus

approxima~

SAMPL

the electrical

leads

as well as the current leads to the

sample were thermally

\

823

W

from

to the copper block.

by slowly

a liquid

withdrawing

helium

bath.

The

rate of heating was 20°K per hr.

The

voltage across both the thermistor and sample was measured with two Keithley model 149 milli-microvoltmeters. welded

Current leads to the sample

and

the

voltage

leads

were

were spot-

ultrasonically

soldered with indium. Tensile tests were conducted using an Instron employed

/ ‘\-

STYROFOAM

THERMAL

INSULATOR

Leitz mjcrollardness

electrical resistivitg as a functiorl of temperature.

were

then

centerless

ground

to

shouldered

specimens having a gauge section diameter of0.060 Two

to three

were

removed

aqueous

by

solution

an applied worked

thousandths

electropolishing

of 2 percent

voltage

surface

of an inch

of 6 volts

layer.

in an

sodium

agitated

hydroxide

to remove

It is probably

in.

of material

mation

before and after deforThe strain ranged from

0.3 percent to 10.9 percent. After completion urements,

graphic observation to the tensile axis. polished

of the electrical

the samples

at to

through

The metallographic 4/O metallographic

1 1~ alumina slurry on a polishing electropolished

and Wall~~rork(19)and JacqueP)

note disturbed zones In

reagent

view

for

Berlec ;(25)

layer for

ferricyanide,

the

rather

low

dislocation

mobility

tungsten,‘2*) one would expect the disturbed t)ungsten to be no deeper. of only one thousandth(22) some

apparent

On the other hand, removal has been criticized’23)

justification.

surface deformation

is readily

hydroxide,

metallography

by

potassium and

320 ml

to determine

the areas

contamination pump

The apparat,us of Fig. 1 was used to measure the resistance

as well

as the

RESULTS

The measured stress and strain for each sample are

(see S~hultz~9)), a two-

with a mechanical

EXPERIMENTAL

for

before being sealed off. electrical

employed

32.7 g

under the microscope.

and back filled with argon or helium gas five times

point

4.8 g sodium

Quantitative

that

was

chosen randomly

each sample was plaeed in a vycor tube with titanium

ice

to

At any rate, residual

To prevent

chips and the t’ube evacuated

similar

composit.ion

shown up by the etch

by the deformation was used.

was

its

was carried out over twenty

All samples were annealed at 500°C for one hour to To remove vacancies grown-in vacancies. anneal

water.

used

etch pit> density

remove hour

wheel before being

in a 5 percent agitated sodium hydrox-

with

pits ; (23,24) no such etch pit evidence was found the specimens used in this investigation.

produced

samples were paper and in a

ide solution at an applied voltage of 8 V. The etch pit

of a few thousandt,l~s in other, similar materials. of

meas-

for metallo-

to view (100) planes perpendicular

remove material to at least such a depth to eliminate Samuels

resistance

were mounted

the disturbed

material left by the grinding;

stage of a

As a check, the over-all

with a micrometer.

the cold-

necessary

The strain

with the micrometer tester.

sample length was measured

tensile

The strain rate

by measuring the gauge length before

and after deformation

FIG. 1. Sehemstic of apparatus for measurement of the

at room temperature

machine.

was 0.002 in. per in. per min.

was determined

\

testing

low

shown

as a point

in Fig.

engineering stress-strain crystals.

2. which

is a composite

curve for this group of single

Figure 3 is a conversi~)n of Fig. 2 to true

stress-strain

coordinates,

and

indicates

that

the

relation oT = hlETm IS . obeyed up to 5% percent strain, with na = 0.33. In Table 1, the ratio of the electrical resistance at 4.2”K to that at 273°K is tabulated for the crystals in the unstrained

condition,

after

being

strained

after the vacancy anneal following deformation.

temperature electrical resistance between 4.2’K and 40°K. A calibrated germanium thermistor was used

the data given in Table

to measure the t,emperat~~re of the specimen. To insure that the sample and the thermistor were at t,he

difference

temperature

resistivity

1, one can obtain

and Using

the low

due to vacancies by taking the

in low temperature

resist,ance between

the

same ten~perature, the sample was placed in t,hermal

strained crystal and the strained and annealed crystal. The variation of the low temperat~lrc resistivity due

contact with the copper block into which the thermis-

to vacancies

as a function

of strain is given in Fig. 4.

ACTA

824

~~~ALLUR~I~A, TABLE

1.

VOL.

2B 4c 3B 13D 4A 13E 13F

1.05 1.00 1.04 1.09 1.20 9.05 9.25

x x x x x x x

1966

Electrical resistivity data

p 4.2 p 273 k-1 Unstrained

Sample

14,

and annealed

Strained

10-4 lo-” 10-4 IO-” IO-4 IO-5 IO-5

0.3 1.0 1.7 3.1 5.3 8.0 10.9

3.38 1.03 1.89 3.98 8.40 1.35 1‘70

x x x x x x x

2133 7.33 1.35 2.62 4.78 6.60 8.05

10-a 10-a 1O-3 1O-3 IO-$ 10-z 10-z

x x x x x x x

10-e 10-4 10-Z 10-z 10-s 1O-3 IO-3

/

0

t/

IO0 ‘=

,-

:: Y

v n 0 0 x + X

90

b 80

60

50

NO. NO NO NO NO NO NO

28 4c 38 130

/

4A

I

13E 13F

‘IY

I L_

I

L.

0

2

4

I

6 8 E t % )

I 10

/ I2

._ 1 14

FIU. 2. Composite engineering stres+strain curve, drawn through points for all specimens.

7 NO, 2B D NO 4C 3 NO 38 0 130

NO

t

FIU.

x NO 4A + NO 13E S NO l3F

3.

True

stress-strain

curve, from data of Fig. 2.

0

01

,3/!.;;

/

.7 ? NO 28

-I -/

.- NO 4c NO 38

i

Y NO 130 x NO 4A c NO 13E l NO 13F

J -!

i

/-

,111

,I

,!/

I

10

-1

l5-iO

E(%l

FIG. 4. Vacancy resistivity versus strain.

The vacancy resistivity is proportional to e”.s for strains up to 2 percent and proportional to P for strains between 2 and 8 percent. Above 8 percent the vacancy resistivity increases less strongly with strain. One can also take the difference in low temperature resistivity between the strained and annealed crystal and the unstrained crystal to obtain the dislocation resistivity as shown in Fig. 5. The dislocahion resistivity is proportional to .z1,3for strains up to 5.3 percent. For larger strains the resistivity increases less strongly with strain. In Figs. 6 and 7 photomicrographs are shown of etch pits in the unstrained and annealed crystals. As the numerical data for etch pit density were determined from twenty observations on each specimen, Figs. 6 and 7 are not quantitatively representative ; they were selected only as illustrations. The change in etch-pit dislocation density as a result of deformation is plotted against strain in Fig. 8. For strains up to 3.1 percent the dislocation density is proportional to EI.~. At higher strains the dislocation density increases less strongly with strain. In Fig. 9 the dislocation resistivity is plotted

SHUKOVSKY

RESISTIVITY

et al.:

OF

DEFECTS

IN

DEFORMED

825

W

agree well with e2 dependence predicted by the “energy balance”

model using a moving

also that

of the “growing

jog mechanism

barrier”

model

and

using

a

recombination mechanism. In the last region the number of vacancies produced per unit strain decreases because of the lower increase in dislocation

/

0

density at

higher strains (Figs. 5 and 8). In addition, annihilation of vacancies appears to be important v b 0 0 Y +

/

n /

l

NO NO NO NO NO NO NO

because the probability

28 4C 36 13D 4A 13E 13F

to the density tivity

of dislocations.

is seen to

at higher strains

of annihilation

is proportional

The dislocation

linearly proportional to the density (Fig. 9). The dislocation resistivity

dislocation

varies as &,3 for strains up to 5.3 percent. percent

/

resis-

be

the

resistivity

(and

Above

dislocation

5.3

density)

increases less strongly with increasing strain.

None of

the models described heretofore predict an &3 dependence of the dislocation exponent

FIG. 5. Dislocation resistivity verms strain.

density.

For a work-hardening

of 0.33, the tensile work per unit volume

depends on the strain in a very similar manner, that is, against the dislocation mation.

density

s

change due to defor-

CT

It is found that the relation is given by

uT

= (const.)

A,

l,r1.33

(2)

0

= 1.4 x lo-11 AN, The variation

(1)

of the low temperature

resistivity

typical

curve for an unstrained

sented.

The ideal resistivity

ture is presented

in Fig. 10. A

sample

is also pre-

as a function

where E,

of the stress-strain

curves of Figs.

2 and 3 suggests that the seven samples of comparable have

essentially

similar

mechanical

properties.

The value of the strain hardening exponent

YUis 0.33.

Ferriss(26) reports values from 0.26 to 0.32

and Schadler

and Low@‘)

report values of 0.378 and

0.463 for (100) crystals. The variation (Fig. strains

of the vacancy

4) appears below

linear following

to

consist

2 percent

of

8 percent

resistivity of three

the variation

an eo.g dependence.

percent an l1,g dependence and

higher

with strain

regions.

For

is essentially

Between 2 and 8

is observed the

is the increase

in dislocation

density.

is the line energy of the dislocation,

the fraction

DISCUSSION

ratio

AN,

Further, we suggest the energy balance

to the

results of other investigators.

resistance

where

of tempera-

in Fig. 11 and is compared

The smoothness

therefore propose the relation

of

the strained and annealed crystals in the temperature range from 4.2”K to 40°K is presented

We

and at strains

vacancy

in the form of dislocations. the dimensionless

The line energy E,

and

variable f must depend on the strain

in the same way, if AN, proportional.

and the integral are to be

If f is a constant,

energy is not strain-sensitive; interaction

and f

of the total tensile work which is stored

or nearly so, the line this may be due to an

energy of the dislocations

which is either

insensitive to strain or negligible compared

to the self

energy.

on copper

Buckc5) has done an experiment

single crystals

in the easy glide region in which the

stress is given by an expression u, + KEY. Equation

of the form,

oT =

(3) predicts that the dislocation

density in this case should be given by

resistivity

increases less strongly with strain. In terms of the moving jog mechanism, this behavior suggests that

Buckc5) has only found

the jog density increases with strain between 2 % and

resistivity

8% strain. The linear dependence

quadratic relation held throughout both easy glide and the linear work-hardening region. For the linear work-

“fixed barrier” of the vacancy

model predicts a concentration on

strain for both moving jog and recombination mechanisms. In the second region the experimental results

on strain.

a linear dependence

Blewitt

et

aZ.c4) found

hardening region equation (3) predicts ence of the dislocation density.

of the that a

an e2 depend-

826

ACTA

METALLURGICA,

VOL.

14,

1966

SAMPLE 2B

SABIPLE

PIG. 6. Photomicrographs

The self energy per unit length of a dislocation tungsten Rose

has been given

as 9

x

et CZZ.(~~)Using this value for and equation

in

E,, the experi-

mental

data,

(4), one finds that the

fraction

of the total tensile work that contributes

to

=

se

1.7%

sections.

If one compares

the total resistivity

due to vacancies plus dislocations to that of Schultz(g) elongation

.3

E; =

3B

of strained and unstrained

10s eV per cm by

E

ratio change

at 10 percent strain

for the equivalent

one finds values of 1.6

x

percentage

1OW’ and 1.8 x

1O-2 respectively. The absolute change in the residual resistivities

can

dislocation generation is approximately 1 percent. The fraction of the energy usually stored in a cold-

be simply

worked crystal is generally between 1 and 5 percent.(28) Since most of the energy stored is presumably due to

change in the ice point resistivity, and assuming the latter to be constant. From the data in Table 1, it is

dislocations, equation (4) appears to be consistent with direct stored energy measurements.

be of the order of one percent, and in the majority

calculated

by ignoring

clear that the maximum

the accompanying

error introduced

thereby will of

SHUKOVSKY

el

al.:

Fia. the cases considerably point resistivity

less.

RESISTIVITY

7 P~ot~micr~~aphs

Using the measured

value of 4.8 $2.cm,(2s)

OF

equation

has a lower dislocation

(1)

worked

AN, 1.9

x

-

and

coworker.G2)

lo-l2 for this quantity

x

10-rl @2-cm3

(6)

heavily

deformed

copper. a factor

fraction

a

value

of

by using Schultz’s datacg)

The discrepancy

between

of 35, can be in~rpreted

either as evidence that severely cold-worked

tungsten

827

W

density

etch

than severely

cold-

or that the etch pits reveal only (e.g.

3%)

of the

The first interpretation

as the

documented. obtained

and estimating the dislocation density in Schultz’s material by assuming that it is the same as that in the two figures,

copper,

density.

ap

~ = 6.7 aND

DEFORMED

of strain sections.

small Ap4.2 _

IN

ice

becomes

Basinski

DEFECTS

pit

techniques

for

The dislocation

actual

is probably tungsten

density

a

dislocation better, are well

data reported

here are in good agreement with that of Scbadler and Low, who used a different reagent,(27) and checked their results by quantitative

correlation with measured

slip step heights, and also by the Nye bending formula. It also appears that dislocations in tungsten make relatively tivity.

large contributions

to the electrical

resis-

There are a number of reasons why one would

828

ACTA

METALLURGICA,

VOL.

14,

1966

-

Fra. 8. Dislocation density change versus strain.

-.

expect dislocations to be very effective scatterers of electrons in tungsten. One such is the particul&r nature of the metallic bond for Group VI of the transition series,(m) which may lead to very large perturbations of the electronic structure in the vicinity of structural defects.t31) The resistivity versus temperature curves for the

strained and annealed crystals are presented in Fig. 10. The difference between the curves for the strained and unstrained crystals shows that, in the temperature range studied the dislocation resistivity is essentially independent of temperature. In Fig. 11 the ideal resistivity, i.e. the lattice contribution pi(T) of 8 typical unstrained crystal is

SHUKOVSKY

et al.:

RESISTIVITY

OF

DEFECTS

IK

DEFORMED

and 2.6 x 1O-2 ,&-cm

respectively

a residual

of

present

resistivity

investigation.

investigation

as compared

4.8 x 10-4,&-cm

The

agree quite

reports that pi(T)

829

W

results

of

well with

with

in the

the

present

Berthelc32) who

K T5 in agreement with the Bloch-

Gruneisen theory.caQ

The difference between the four

curves can be attributed

to the fact that deviations

from

become

Matthiesson’s

rule

residual

resistivities.

observed

deviations

sten.

Krautz

larger

and

from Matthiesson’s

In addition,

for

higher

Schultz(35) have rule in tung-

in the case of large residual resis-

tivities and low temperatures,

where pr > p,(T),

it is

hard to obtain an accurate value for pi(T) by subtraction of the residual resistivity from the total resistivity. CONCLUSIONS

The following

conclusions

may be drawn from the

present investigation. 1. The dislocation

i

resistivity

is linearly proportional

at low temperatures

to the dislocation

aPD ___ = 6.7 x lo-ii

@-cm3

ah

1

100

10

T (“K) FIG. 10. Electrical

resistivity

versus temperature.

density N,

and is given by

2. The density of generated dislocations single crystals is proportional unit volume in deforming

in tungsten

to the work done per

the crystal and is given by

ANDa

lT or dell u&3 s0 resistivity (and hence the vacancy

3. The vacancy concentration)

variation with strain is consistent with

both the moving of vacancy

jog and recombination

production

if one considers

mechanisms the energy

balance, fixed barrier, and growing barrier models. 4. The temperature of a typical

variation of the ideal resistivity

unstrained

crystal is in agreement

Berthel who reports a T5 behavior

with

for tungsten.

ACKNOWLEDGMENTS

The authors appreciate Science 2069. Many

Foundation stimulating

A. Shepard

Backman FIG. 11. Ideal resistivity

plotted

as a function

versus temperature.

of temperature

with the results of Berthel,(32) Vanden

and compared Berg,(33) and

White and Woods(29) whose crystals had residual resistivities of 1.4 X 10e5 @-cm, 2.2 X lop3 $-cm, 3

would

contract

discussions

and Dr. Harvey

acknowledged. authors

the support of the National

through

For

number

GP-

with Dr. Lawrence

E. Cline are gratefully

experimental

like to thank

assistance

Mr. I. Puffer,

the

Mr. E.

and Mr. G. Arndt. REFERENCES

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