Diffusion of Chromium
51 into Copper
toy
Wendell L. Seitz
A Thesis Submitted to the Faculty, of the
DEPARTMENT OF PHYSICS
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 6 3
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of re­
quirements for an advanced degree at The University of Arizona and
is deposited in The University Library to be made available to bor­
rowers under rules of the Library.
Brief quotations from this thesis are allowable without special
permission, provided that accurate acknowledgment of source is made.
Requests for permission for extended quotation from or reproduction
of this manuscript in whole or in part may be granted by the head of
the major department or the Dean of the Graduate College when in
their judgment the proposed use of the material is in the interests
of scholarship. In all other instances, however, permission must
be obtained from the author.
SIGNED:
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
^
CARL T. TOfUZUKA
Professor of Physics
---------
Z
Date
ABSTRACT
Measurements of the diffusion of chromium 51 into single
crystals of copper grown from 99.999% pure copper were performed
for four specimens.
The activation energy and frequency factor
were determined to be 53,4 db 2,0 kcal/mole and 3.1 ± 2,9 cm3/sec
respectively,
A calculation of the difference in activation energies be»
tween impurity diffusion and self-diffusion in copper, using the
Lagarus-Le Claire theory and the valance difference, yielded poor
agreement between theory and experiment.
However, a reasonable
value of the effective excess charge for chromium dissolved in
copper is predicted using the Lazarus-Le Claire theory and the
experimentally determined difference in activation energies.
ACKNOWLEDGMENTS
X wish to express my appreciation to my advisor,
Dr, Carl T, Tomizuka, for his invaluable guidance, advice, and
assistance with this project,
I would like to thank Mr, David Styris for his
assistance with the chemical etching technique used for the
sectioning of specimen no, 6,
I wish to thank Mr, Denial Albrecht for furnishing the
copper single crystals,
I would like to thank Mr, Palmer Weir for his assistance
in maintaining the furnace temperature equipment,
I would also like to thank Mr. Carl Perceny for his
assistance with the alignment of the lathe cutting tool.
This work was supported in part by the United States
Atomic Energy Commission through Contract AT(11<*1)»1041.
support of this agency is gratefully acknowledged.
The
TABLE OF CONTENTS
Page
ABSTRACT
ixi
ACKNOWLEDGMENTS
xv
Chapter,
X»
II.
XXI0
IV0
Introductxon
^
Theory
3
Oh jec txve .^.00000.09$9.000.
o##-000.
Experimental Procedure
10
V.
Experxmenta1 ResuItSA.o.ao.........*.#.....*..*
17
VI.
Dxscussxon o^ Results e..©.....#.........©.....
23
VIl.
Conclusxons '©©©©©.©©©.©.©©^.©..©©©©©a©****#**^*
25
REFERENCES ©©.©©©©©©©©©o©.©.©©..©©©©©©©©©©©©©©©©©©©©©.©©©
v
26
CHAPTER I
INTRODUCTION
Considerable attention has been given to the study of chem.-*
ical diffusion in the last decade due to the availability of very
pure radioactive isotopes of high specific activity.
Previous to the availability of radioactive isotopes chem»
ical diffusion measurements were made using standard chemical ana~
lytical techniques.
The data obtained by chemical analysis yielded
conflicting results due to the large inherent concentration gradients.
Measurements by use of radioactive tracers in several of the
noble metals (e.g. see references 1 and 2) have revealed that the
activation energies and frequency factors differ significantly for
self"diffusion and impurity diffusion.
Several theories have been proposed to explain this differ­
ence.
The two most prominent are those proposed by Swalin (3) and
Lazarus (4).
Swalin attributes the difference in activation ener­
gies, AQ, to the atomic size difference and elastic properties of
the solute and solvent,
diffusion in Ag.
Swalin is fairly successful for solute
Lazarus ignores size differences and attributes
AQ to a screened electrostatic interaction between solute atoms
and vacancies.
This latter theory is fairly successful in
accounting for AQ for solute diffusion,in noble metals.
Le Claire (5)
2
later modified Lazarus1 theory by considering a solute atom to be
flanked at the saddle point* by two half vacancies with ~%e charge
localized at the centroids of the half vacancies,
Le Claire also gives
detailed consideration to the correlation effects between successive
jumps of an atom by considering the rates of exchange of solute atoms
and vacancies,
Le Claire is quite successful in estimating AQ for solutes
which lie to the right of the noble metal in the periodic table but not
so successful for solutes that lie to the left of the noble metal.
The
success of these theories is based on a limited number of experimental
results.
The values of AQ for impurity elements to the right of copper,
such as zinc, gallium, and arsenic diffusing in copper are *1.54(6),
*1,24(7), and *5,14(7), respectively.
The values of AQ for impurity
elements to the left of copper, such as iron, cobalt, and nickel
diffusing in copper are 4,66(8), 6,96(8), and 9.36(8), respectively.
*
The saddle point is that point at which the energy is a
maximum along the diffusing path and a minimum along directions
perpendicular to the diffusing path.
CHAPTER II
THEORY
The mass flow for one dimensional flow of a solute in an
isotropic solvent material is given by
3c/dt s D cf c/3%P
as described by Fick (9) in 1855.
(1)
c is the concentration of solute at
position x., at time t; D is the diffusion coefficient.
For the case of a
semi-infinite medium, with an impermeable plane boundary (10) where the
diffusing atoms are concentrated at time t = 0, the solution of the above
equation is
3-“I
c(x, t) = c(0,0)/(TOt)^
exp(»x3/4Dt)»
—
(2)
The diffusion coefficient is found to vary with temperature
according to the Arrhenius equation
D =$ A expC-Q/RT)
where A is the frequency factor and
(3)
is the activation energy.
It is found experimentally that the frequency factor and
activation energy in general differ for self.««diffusion and solute
diffusion.
Lazarus proposed a theory in 1954 to account for the
difference in activation energies.
By considering the electrostatic
screening interaction between an impurity ion and a nearly solvent ion
4
he calculated the reduction of the energy of formation of a vacancy.
He
calculated the energy of motion by using the same potential and using the
Fuchs' type elastic interaction.
Le Claire published a paper in 1962
proposing a more direct calculation of the energy of motion by ignoring
the elastic interactions and considering the impurity ion to be flanked
at the saddle point by two half vacancies.
Le Claire also gives detailed
consideration to the correlation effect.
Le Claire starts with the two Arrhenius type equations
D3 = Aa exp(-Qs/RT)
(4)
D0 = Aq exp(~Q0/RT)
(5)
and
that have been found to be accurately obeyed by the experimental data.
He has used the subscript "2" to refer to solute diffusion and the
subscript "o" to refer to self-diffusion.
He then gives the theoretical
expressions
(6)
and
(7)
for the diffusion coefficients for face centered cubic structures where
a is the interatomic or jump distance,
is the effective vibrational
frequency of a migration atom, Hq is the activation energy for exchange
of a solvent atom with a vacancy, E0 is the energy to form a vacancy in
pure solvent, f0 is the correlation factor for self-diffusion by vacancy
mechanism, vs is the effective vibrational frequency of an impurity atom.
5
Hg is the activation energy for the exchange of an impurity atom with a
vacancy, Es is the energy to form a vacancy next to an impurity atom, and
fs is the correlation factor for impurity diffusion.
f0 is purely a
geometrical factor which depends only on the crystal structure; therefore
d &
f0/a(l/T) = 0
(8 )
and
Q0 = H0 + E0 .
fs isdependent on the relative
’ (9)
rates of exchange of the vacancy with the
impurity atom and with neighbor and near-neighbor solvent atoms.
Lidiard
and Le Claire (11) use the relation
W^ere
f3 = (wx + 7w 3/ 2)/ (w1 + ws + 7ws/2)
(10)
w1 =
exp(-'H1/RT),
(11)
w2 = u3 exp(-%/RT),
(12)
Ws = V3 exp(-Hg/RT);
(13)
and
Wj is the rate of exchange of a vacancy neighboring a solute atom with
any of the four solvent atoms that are also neighbors of the solute atom;
w3 is
the rate ofexchange of a vacancy with a neighboring solute atom;
ws is
the rate ofexchange of a vacancy neighboring a solute atom with
any one of the seven solvent atoms adjacent to the vacancy but not
neighbors of the solute;
and u3 are the vibrational frequencies
associated with Wj and w3 jumps; and
activation energies.
and Hg are the respective
Le Claire then takes the logarithm of both sides
6
of equation (7) and differentiates with respect to l/T to obtain
Qs = %
Where
+ 53
Qa = -R3
f3/d(l/T)
(14)
Dg/ad/T),
He then defines AQ = Q3 « Q0 '} AHS = Hs ” Bq , ASa = Es - E0 ^ and
C = R3 ^2 fa/d(l/T).
Using these definitions, he is able to write
AQ = AHg + AEg - C.
By equating the ratio Ug/Uo
(15)
equations (4) and (5) to the ratio
D3/ d0 from equations (6) and (7), he obtains
Vg/uq = (Agfo/Aofa) exp(C/RT).
The assumption is then made that Pj = p3 = p0 since
refer to vibrations of solvent atoms»
(16)
, us and p0 all
By making use of equation (10)
and the assumptions above, he obtains the final expression for C to be
C
+ 7/2(AHg
= (Agfo/Ao) exp (C/RT)
(AHs «
A % ) exp (-AHg-
AHX )/RT (17)
- AHg)exp(-A% - Aiy/RT / exp(~AH1/RT) +7/2 exp(-AH3/RT)]3
He then explains that the calculation of AQ can be accomplished
by calculating AE, AH1 , AH3, and AHg.
The assumption is made that the
energy differs for a metal containing solute atoms from that of the pure
metal only by a term due to the interaction of the vacancy with the
screened effective charge difference between solute and solvent ion
cores.
The interaction is calculated by use of the Thomas«Fermi
approximation,
A free electron model is assumed for the pure metal.
7
The adding of a solute atom is regarded as replacing a solvent ion with
another ion core of different charge.
The difference in charge, Ze, is
considered to be a point charge concentrated at the center.of the solute
ion.
The extra conduction electrons that enter with the solute are con­
sidered to go into the conduction band and maintain the Fermi energy
constant, provided the solute concentration is very small.
For positive
Z Le Claire uses the leading term,
V(r) = (dfZe/r) exp(-qr),
(18)
of an exact solution of the Thomas-Fermi equation to calculate the.
potential at a distance jr from the excess point charge where
is the
screening parameter given by
qs = (4me3/hs) (3n0/fi')V'>
(19)
a is a constant dependent on jS, m is the mass of an electron, _e is the
charge of an electron, n0 is the number of electrons per unit volume, and
h is Plank,s constant.
No exact solution to the Thomas-Fermi equation for
negative Z has been published.
For negative Z there exists a sphere of
radius r about the ion in the Thomas-Fermi model in which no electrons
c
,
are allowed and on which V(r) = -E
where
is the Fermi energy,
Alfred
and March (12) have assumed equation.(18) to be valid for r > r , have
determined V(r) for r < r , and obtained approximate values of a for
negative Z of -1, -2, and -3,
The interaction energy of a vacancy with the potential field
of an impurity of excess charge Ze can thus be given by
AE = -(Ze3a/a) exp(-qa),
(20)
8
By considering the electrostatic forces at the equilibrium
position and saddle point for a solute atom, AHg can be derived as
AHg = (-16Ze30f/lla) exp(**llqa/l6) + (Ze3or/a) exp(«qa)„
(21)
The last term on the right is «AEj therefore
£H3 + AE = (»16Ze3Q'/lla) exp(-llqa/l6).
Calculation of C is accomplished by substituting the estimated
values of AH^, AE2 , and AHg and the experimental values of As and Aq
into equation (17),
Using equation (10) and assuming %
value of 9/ll is obtained for f0 .
.
= w3 = ws , a
CHAPTER III
OBJECTIVE
It was the objective of this investigation to determine
experimentally the activation energy and frequency factor for the
impurity diffusion of chromium 51 into single crystals of copper
and compare the experimentally determined difference in activation
energy between chromium impurity diffusion in copper and self"
diffusion in copper with that predicted by the Lazarus-Le Claire
theory.
It was the intention of the author to assist in determining
the limits of validity of the Lazarus-Le Claire theory for the
diffusion of impurities in the noble metals.
9
CHAPTER XV
EXPERIMENTAL PROCEDURE
The radioactive isotope, chromium 51 (half life of 27,8 days),
was obtained from the Radioisotope Division of Oak Ridge National
Laboratory in the form of CrCl3 in 1,0 N HC1 solution.
A copper single crystal grown from 99.999% pure copper in the
form of a rod 5/8 inches in diameter and 6 inches in length was
centered inside of a Lucite tube and secured by pouring plaster of
Paris around it.
The copper single crystal was then sawed into
specimens l/2 inches in length by use of a Carborundum blade.
After sawing, each specimen was etched in a 50% HN0s solution
to remove most of the worked metal, mechanically polished to give an
even, flat surface, and etched again in a 50% HN03 solution to remove
all of the worked metal.
The specimens were then annealed for 24 hours
at a temperature of 900°C, and etched again in a 50% HN0g solution in
order to check for grain boundaries.
boundaries were ready for plating.
Specimens with no grain
Four specimens 9/16 inches in
diameter and l/2 inches in length, grown by the same method from
"spectrascopically pure" copper, were also used.
The specimens were plated by wrapping a copper wire around
each one and placing them one by one into a plating solution pre­
pared by putting three or four drops of the radioactive solution
10
into 50 ml, of distilled water.
A platinum wire was placed in the
solution to be used as the anode.
A small battery (~22 volts) was
placed in the circuit between the platinum and copper wires providing
a potential difference sufficient to produce a current of approximately
20 ma.
The specimens were plated until a reading of 30,000 cpm was read
on a geiger counter when the specimens were placed l/2 of an inch from
the geiger tube window.
Each specimen was then sealed inside indi­
vidual evacuated vycor capsules in preparation for annealing.
Each specimen was annealed in a diffusion furnace for a pre­
determined time at a given temperature.
The specimens were annealed
at temperatures ranging from 1069 to 1308°K,
The temperature of the
furnaces was maintained to within ± 0.5eC by use of a platinum
resistance thermometer that was one leg of an a.c. resistance bridge.
The temperature of one furnace was monitored with a platinum versus
platinum-107, rhodium thermocouple that had been standardized by the
National Bureau of Standards.
The temperature of the other furnace
was monitored with a chromel versus alumel thermocouple that had been
standardized against the platinum versus platinum-10% rhodium thermo­
couple,
The thermocouples were placed as close as physically poss­
ible to the location of the specimens.
Thermal emf rs of the thermo­
couples were read by a precision potentiometer.
After the anneal time had elapsed the specimens were removed
from the furnaces and quenched in ice water.
The anneal times were
taken to be the difference between the time of quenching and the time
at which the specimens were within 5 G® of the stable temperature.
12
The anneal times were determined by estimating the diffusion
coefficient and calculating the time required for the penetration depth,
2(Dt)2, to become approximately 6 mils.
After the specimens were removed from the capsules, all speci­
mens except number 6 were placed in a collet and mounted in a specially
designed chuck on a Rivett lathe.
Specimen number 6 was sectioned using
a chemical etching technique developed by David Styris (13).
The
specially designed chuck allowed for the aligning of the specimen face
relative to the axis of rotation.
Specimens were aligned so that a normal'
to the specimen face was within two minutes of being parallel to the
axis of rotation as determined by a dial gauge.
In order to eliminate the effect of any diffusion from the
sides, the specimens were turned down an amount equal to five times
the estimated penetration depth before the sectioning process was
begun.
After the turning down of the sides, the alignment of the
specimens was rechecked and corrected if necessary.
The
cutting tool was electrically insulated from the other
lathe parts
enabling one to determine the origin of the sectioning
process by determining the point of electrical contact between the
cutting tool and the face of the specimen.
using an ohmmeter on the
contact and
It was found that by
one megohm scale the point of electrical
the point of mechanical cutting were in agreement to
within ± 0 . 1 mils.
13
Each specimen was sectioned from 10 to 18 cuts ranging in depth
from 0.2 to 1.0 mils, depending on the penetration depth.
Most specimens
were annealed the proper time to allow for a 0.5 mils cut.
Several
cutting tools and positions of the cutting tool were tried.
It was
found that a Vasco Supreme or equivalent cutting tool gave the most
satisfactory results.
It was also found that the most critical
adjustment of the cutting tool was the height and should be so that the
cutting tool is level or sloping slightly upward toward the cutting
point.
The shavings from each cut were collected inside a specially
constructed plexiglass incasement.
After each cut the shavings were
transferred to a small glass vial that had been previously washed with
soap and water, rinsed with distilled water, dried in an oven at 100°G
for approximately 20 minutes, cooled in a desiccator, and weighed.
vials were always handled with tongs after being washed.
The
The vial
containing the shavings was weighed and then placed in the well of a
well scintillation counter to determine the activity.
The counting
system consisted of a scintillation probe, a gain«of”one preamplifier,
a single channel pulse height analyzer and a transistorized scalertimer.
In order to improve the accuracy in determining the cut depth
the following procedure was used.
The sum of the masses of all the
cuts except number 1 was divided by the difference in the dial reading
for the back face of the last cut and the dial reading for the front
face of cut number 2, and this quotient called
.
The sum of the
masses of all the cuts except numbers 1 and 2 was divided by the
14
difference in the dial reading for the back face of tljxe cut and the
dial reading for the front face of number 3, and this quotient called Qfg.
Following the same procedure as ,
Ofg, •••
were calculated.
The
average of these ffi's was taken as the proportionality constant between
mass and distance.
The cut depths were then determined by dividing the
mass of each cut by the average a.
The distance of the mid-point of
each cut from the front face of the specimen was used as the distance in
making the graph of the logarithm of specific activity versus distance
squared.
The specific activity of each cut was determined by dividing
the activity of each cut by the mass of each cut.
Considerable experimental difficulty was encountered.
Measure­
ment of the diffusion coefficient was attempted for nine specimens.
The specific activity for specimens 1, 1, and 9 was very high for the
first cut and then dropped almost to zero for all cuts thereafter.
Therefore, no usable data were obtained from the specimens.
It is be­
lieved that the sudden drop in specific activity can be explained by
the formation of an oxidized layer on the specimen surface during the
plating and sealing process.
Specimens 2 and 4 showed extremely high
specific activity for all 18 cuts.
The specific activity for the
second cut from these two specimens was approximately 270 cpm/mg as
compared to 25.cpm/mg for the second cut from specimens 3, 5, 6, and 8.
The diffusion coefficients obtained from specimens 2 and 4 were also
high compared to those obtained from specimens 3, 5, 6, and 8.
A
black glaze was observed on the inside of the vycor tubing for these
two specimens at the time of removal from the furnaces.
There was also
15
(
a small quantity of black material on the surface of the specimens.
It is believed that these impurities inside the vycor tubing caused the
extremely high specific activities and diffusion coefficients.
It is
also believed that these impurities arose from the fact that the
acetylene torch was allowed to burn inside the vycor tubing while sealing
off the open end before the specimens were placed inside.
Due to the
above difficulties only four points are given on the graph of log D3
versus l/l.
Difficulty was also encountered in trying to obtain desirable
penetration depths.
It was desirable to have a penetration depth of
10 mils in order that the specific activity would be sufficiently high
to give a smooth graph of the logarithm of the specific activity verstis
the distance squared.
For penetration depths of less than 10 mils the
specific activity, after the second or third cut, was so low that only
a very small contamination of the shavings caused a large error in the
specific activity.
Due to the limited time available for the experiment and the
short half life of chromium 51, it was impossible to leave the speci­
mens in the furnaces long enough to obtain a penetration depth of 10
mils for all of the specimens. A time of approximately 45 days would
have been required for specimen number 5 and even a longer time for
specimen number 6,
In. a period of 45 days the radioisotope would
have almost gone through two half lives.
The increase in specific
activity gained by the extra penetration depth would have been lost
by the decay of the radioisotope.
It was therefore necessary to use
16
penetration depths of less than 10 mils for all specimens except number
8 and be extremely careful not to contaminate the shavings.
Although
great care was used in handling the shavings, considerable fluctuation
of the specific activity was present for specimens 5 and 6.
It is
believed that the error could be reduced considerably by using the
chemical"'sectioning technique (13) (14),
A graph of the logarithm of specific activity versus distance
squared was made, and the best fit straight line was determined by use
of the least squares method.
diffusion coefficient.
The slope of this straight line gave the
After the diffusion coefficient for all of the
specimens had been determined, a graph of the logarithm of the diffusion
coefficient versus the reciprocal of temperature was made, and the best
fit straight line was determined by use of the least squares method.
The slope of this straight line gave the activation energy, and the
ordinate intercept gave the frequency factor.
CHAPTER V
EXPERIMENTAL RESULTS
Plots of the logarithm of the specific activity versus the
square of the penetration distance are shown in Figs. 1«4.
Diffusion
coefficients obtained from Figs, 1«4 are given in Table I,
A plot of
the logarithm of the diffusion coefficient versus the reciprocal of
temperature is shown in Fig, 5,
The activation energy, Q3 , determined from Fig, 5 is 53,4 ±
2.0 kcal/mole.
The frequency factor, Ag, was determined to be
3.1 ± 2,9 cm3/sec,
TABLE I
Diffusion of Chromium into Copper
Specimen No.
Temperature
(°K)
6
1069.64
4.09 X ICT11
.5
1190,90
3,75 X 10"10
8
1282.00
2,60 X 10"9
3
1308.34
3.88 X lO"9
Diffusion Coefficient
(cm3/sec)
^45
Lf)g <5 vs
Specimen No* 6
T * 1669164^::
1*727 x 10^ sec
1.25
1.05
0*6
40.55
-0.75
0
6170
20.10
26.80
40.20
Stas
1.50
W e
1.20
Specimen NoJ 5
W o
7 # : ^ TF 101 s#c
1.00
0.90
0.80
Ow70
0.60
0.40
0.30
10.20
0.10
W e
0
21.0
42.0
63.0
X2 % lO^
84.0
(cm2)
105.0
126.0
0.70
0.60
pdciden Mo. 8
0.50
ec
0.25
0*20
0.^5
0.05
-0.05
0
Umo
ffix J.06i ftanS
1392
j m
^b57
t«0|g: C Vsir^T
Spacilieii Nol:i3
-T_4JL3Q8i34 K
ec
1,37
JKw
e^97
0.87
d
180
270
360
■7.8
8.2
■
8.6
•
9.0
9.6
•
(N
9.8
m m
te-a-
tedt
te*
ii#
8.2
8*6
l/T x 104- (QK>“f
910
9^8
CHAPTER VI
DISCUSSION OF RESULTS
The probable errors of %
and Ag were determined using the
method outlined by Worthing and Geffner (15)„
The diffusion equation
resulting from these experimentally determined quantities is
Dg = (3.1 ± 2,9) exp *(53.4 ± 2„0)/RT cmP/sec,
It will be recalled that the Lazarus*Le Claire theory for
impurity diffusion is based on the valance difference between solute
and solvent ions.
It is customary to assume the valance difference for
chromium dissolved in copper to be "*5,
le Claire calculates £Q for the
diffusion of solutes in copper with a valance difference of -1, *2, and
*3; he does not calculate 4Q for a valance difference of *5 in copper,
A calculation of
for chromium diffusing into copper, using a valance
difference of *5, yielded a value of 124.0 kcal/mole,
4Q equal to
124,0 kcal/mole is in poor agreement with the 4Q of 3.8 kcal/mole
experimentally determined in the present work*
Le Claire discusses the plausibility of using an effective
excess charge rather than the valance difference for the transition
metals.
An estimate of the effective excess charge can be made from
magnetic susceptibility and electron spin resonance experiments.
Le Claire, using the estimated effective excess charge, has calculated
23
24
4sQ for nickel, cobalt, and iron diffusing in copper and has found the
results to be considerably improved.
The 4Q*s calculated, in this way
differed from 4sQ (experimental) by 6.7%, 6.6%, and 21.9% for nickel,
cobalt, and iron, respectively.
Unfortunately insufficient data are available at present to
estimate the effective excess charge for chromium dissolved in copper.
However, using
experimentally determined and calculating an effective
excess charge yielded a value of -0.5.
Comparing the estimated effective
excess charge of -0,5 for chromium with those of -0.75, -0,50, and
-0.25 for nickel, cobalt, and iron, respectively, seems to reveal that
it is at least a reasonable value.
It will also be recalled that the calculation of ZsQ requires
the use of the experimentally determined frequency factors which
normally have a rather large probable error.
It is believed that a-
major improvement in the theory would be to provide for a direct
theoretical calculation of ZsQ without the necessity of using the
experimentally determined frequency factors.
CHAPTER VII
CONCLUSIONS
The activation energy and frequency factor for the diffusion
of chromium into copper were determined to be 53,4 ± 2,0 kcal/mole
and 3,1 ± 2,9 cmP/sec, respectively,
A theoretical calculation of
using the valance difference
between chromium and copper resulted in poor agreement between
theory and experiment,
A value of *0,5 for the effective excess
charge is obtained by using the value of AQ determined by the present
experiment and performing a calculation using the Lagarus-Le Claire
theory,
A revision of the theory is needed to provide for a
theoretical calculation of AQ independent of the experimentally
determined frequency factor*
More experiments of the type described herein should be
performed in order to determine further the limits of validity of
the theory.
25
REFERENCES
L, Slifkin, D„ Lazarus, and C, Tomizuka, Je Appl. Phys, 23,
1032 (1952),
E. Sender, L. Slifkin, and C„ Tomizuka, Phys, Rev, 93, 970
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