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Thermodynamic properties of zirconium-cadmium and certain other solid solutions
THERMODYNAMIC
PROPERTIES OF ZIRCONIUM-CADMIUM OTHER SOLID SOLUTIONS*
J. H. FRYE,
JR.,? J. 0. BETTERTON,
AND
CERTAIN
JR.t$ and D. S. EASTON?
The experimental work reported here has to do with the zirconium-rich zirconium-cadmium alloys. The phase diagram has been determined up to a concentration of 20 at. % Cd. Lattice parameters have been measured for the hexagonal phase. The vapor pressure of cadmium in equilibrium with the solid solution has been measured for both the hexagonal and cubic phases as a function of temperature and concentration. These data are considered together with earlier data on the Zr-Ag, Zr-In, Zr-Sn and Zr-Sb primary substitutional solid solutions. Empirical relations are found between the hexagonal (c/a) ratio, the temperature of the transformation from cubic to hexagonal zirconium and the electronic specific heat coefficient on the one hand and the concentration of principal quantum number 5 electrons on the other. These simple relations suggest the possibility that the principal quantum number 5 electrons are entering a rigid conduction band in zirconium. In order to investigate this possibility, we have calculated the partial molal energy of cadmium in zirconium from the vapor pressure measurements. We have also done this for certain other solutions for which vapor pressure data have been published. We conclude that the rigid band model is a bad approximation and point out that this is in accord with recent quantum mechanical calculations. Furthermore, the effect of cadmium on the transformation temperature of zirconium is the resultant of two opposite and nearly balanced effects-an absolute zero energy effect which tends to stabilize the cubic form and an entropy effect which tends to stabilize the hexagonal form. Finally, it is shown that, for most of the solutions considered here, the energy of a Wigner-Seitz polyhedron is independent of concentration within experimental error. PROPRIETES
THERMODYNAMIQUES AUTRES
DU ZIRCONIUM-CADMIUM SOLUTIONS SOLIDES
ET
DE
CERTAINES
Le travail experimental present6 ici est relatif aux alliages zirconium-cadmium riches en zirconium. Le diagramme des phases a et6 determine jusqu’a une concentration de 20 at. % Cd. Les parametres du reseau ont et6 mesures pour la phase hexagonale. La pression de vapeur du cadmium en Bquilibre avec la solution solide a et6 mesuree, a la fois pour la phase hexagonale et pour la phase cubique, en fonction de la temperature et de la concentration. Ces resultats sont compares avec les resultats anterieurs relatifs aux solutions solides primaires de substitution Zr-Ag, Zr-In, Zr-Sn et Zr-Sb. Les auteurs ont trouve des relations empiriques entre le rapport hexagonal (c/a), la temperature de transformation du zirconium cubique en zirconium hexagonal et le coefficient de la chaleur specifique Qlectronique, d’une part, et la concentration des electrons de nombre quantique principal 5, d’autre part. Ces relations simples suggerent la possibilite que, dans le zirconium, ces electrons entrent dans une bande de conduction rigide. Pour etudier cette possibilite, les auteurs ont calcule l’energie moleire partielle du cadmium dans le zirconium B partir des mesures de pression de vapeur. 11sont fait ceci egalement pour d’autres solutions pour lesquelles les resultats des mesures de pression de vapeur ont et6 publies. Les auteurs concluent que le modele de bande rigide est une mauvaise approximation et montrent que ceci est en accord avec les calouls ¢s de la mecanique quantique. En outre, l’influence du cadmium sur la temperature de transformation du zirconium est la r&&ante de deux effets opposes et a peu p&s Bquilib&s-un effet d’energie au zero absolu qui tend a stabiliser la forme cubique et un effet d’entropie qui tend a stabiliser la forme hexagonale. Finalement, les auteurs montrent que, pour la plupart des solutions Qtudiees ici, l’energie dun polyedre de Wigner-Seitz est independante de la concentration, aux erreurs experimentales p&s. THERMODYNAMISCHE
EIGENSCHAFTEN ANDERER
VON ZIRKON-KADMIUM LEGIERUNGEN
UND
EINIGER
Die vorliegende Arbeit behandelt zirkonreiohe Zirkon-Kadmium-Legierungen. Das Phasendiagramm wurde bis zur Cd-Konzentration von 20 At. ‘A bestinnnt. Fur die hexagonale phase wurden Gitterwurde sowohl fiir die hexagonale Phase parameter gemessen. Der Kadmium-Gleichgewichtsdampfdruck als auoh fur die kubisohe Phase als Funktion der Temperatur und Konzentration gemessen. Diese Daten werden mit friiheren Daten tiber Zr-Ag, Zr-In, Zr-Sn und Zr-Sb verglichen. Es werden empirische Beziehungen gefunden zwischen dem (c/a)-Verhiiltnis der hexagonalen Phase, der Umwandlungstemperatur vom kubischen in hexagonales Zirkon und dem Koeffizienten der elektronischen spezifischen Warme einerseits und der Konzentration der Elektronen mit der Hauptquantenzahl 5 andererseits. Die einfachen Zusammenh5nge sind ein Hinweis fur die Moglichkeit, da5 die Elektronen mit der Hauptquantenzahl 5 in ein starres Leitungsband gehen. Urn diese Mogliohkeit zu untersuchen, haben wir aus Dampfdruckmessungen die partielle molare Energie von Kadmium in Zirkon berechnet. Ahnliche Rechnungen haben wir fur einige andere Legierungen angestellt, fur die Dampfdruckdaten veroffentlicht sind. Wir kommen zu dem Ergebnis, da13das rigid band model eine sohlechte Niiherung ist und weisen darauf hin, da13wir damit in Ubereinstimmung mit neueren quantenmechanisohen Rechnungen sind. AuDerdem ist der EinfluD des Kadmiums auf die Umwandlungstemperatur von Zirkon das Ergebnis zweier entgegengesetzter und sioh nahezu aufhebender Effekte: der Nullpunktsenergie-Effekt stabilisiert die kubische und der Entropie-Effekt die hexagonale Struktur. Schlieljlich wird gezeigt, da8 fur die meisten der hier untersuchten Legierungen die Energie einer Wigner-Seitz-Zelle innerhalb des experimentellen Fehlers unabhiingig von der Konzentration ist. * Received November 26, 1971; revised February 17, 1972. Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. t Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830. 2 Now at: Department of Physics, Louisiana State University at New Orleans, New Orleans, Louisiana 70100. ACTA
METALLURGICA,
VOL.
20, OCTOBER
1972
1183
1184
ACTA
METALLURGICA,
VOL.
INTRODUCTION
20,
1972
24
This is the last of a series of papers(14) having to do with the properties of primary substitutional solid solutions with zirconium as the solvent and silver, cadmium, indium, tin and antimony as solutes. Previous work had shown interesting relationships between certain properties of these solutions and the number of electrons having a principal quantum number of 5 in each solute atom. Data were lacking, however, for the effect of cadmium on lattice parameters and on the temperature of transformation between cubic and hexagonal zirconium. In the present work we have measured the lattice parameters of zirconium at different concentrations of cadmium. We have also established the phase diagram to 20 at.% Cd. These data have been used together with previously published data to draw the graphs of Figs. 1 and 2. Relationships between the hexagonal c/a ratio, the transformation temperature and y, the electronic specific heat coefficient, on the one hand and the concentration of quantum number 5 electrons on the other, are summarized in Figs. 1-3.‘1-6) It is tempting to try to explain these relationships in terms of the electronic band energies at absolute zero. In order to do this we need to know the energy of an atom in solution against a reference state of its ions and electrons at infinite separation. This is the reference state for the potential energy operator in the Schroedinger equation. As will be shown later this can be obtained from vapor pressure data. With this in mind we have also measured the vapor pressure of cadmium in zirconium-cadmium alloys as
t6
0 0 H
-3%
4
0
:
-h? +
r_” -8 9
-32
-t”
0
2
3
4
5
6
quantum number 5 electrons
FIN. 2. The derivative with respect to concentration, at zero concentration, of the average of the temperature boundaries of the u + /l region as a function of the number of quantum number 5 electrons.
a function of temperature and concentration. In the last part of this paper these data are analyzed and their possible physical significance is discussed. EXPERIMENTAL
PROCEDURES
AND RESULTS
Materials A particular bar of Westinghouse grade I iodide zirconium of 99.96% purity was selected on the basis of
0.004
0.003
quantum number 5 electrons
Fro. 1. The derivative with respect to concentration of the axial ratio (c/a) as a function of the number of quant,um number 5 electrons in the solute atom.
FRYE,
JR.
et al.:
THERMODYNAMIC
PROPERTIES
OF
1185
ZIRCONIUM-CADMIUM
f
{
-I 1 i
2.8 4
0
6
(2
46
quantum number 5 electrons FIG.
9OC,
of trace
5 Cu, 65 Fe,
impurities
lOHf,
5 K,
as follows: 123 0,
10 Al,
1 MO, 29N,
1 Nb, 12 Ni, 7 Si, 2 Ta and 2 Zn (in at. ppm). 40 ppm H in the as-received than
1 ppm
resistivity The
by
vacuum
24
American
The
bar was reduced to less
degassing.
The
was of 99.99+%
of CdO.
analysis of cadmium
Both
X-ray
in method 2 were solved by hold-
boiler
of pure liquid
thermally
was held at some higher temperature end.
four samples in a single capsule were held at different
the
Principal
and chemical
temperatures gas.
Each
according
3 was modified
with a constant sample produced
to its individual
Reaction
between
tantalum analysis was done at each stage of the to ensure that no contamination
occurred
or
distillation
distillation
The preparation of zirconium-cadmium alloys is very difficult due to the high vapor pressure of cadwere unsuccessful,
attempts
to
produce
alloys
the following
by
arc
techniques
were developed that proved to be effective preparation of alloys up to 20 at.% Cd:
in the
1. Cadmium was sealed within an evacuated zirconium container and annealed at elevated temperatures. 2. Vapor diffusion between cadmium gas and zirconium
was achieved
in evacuated
and sealed silica
capsules in the temperature range lOOO-1200°C; however, several capsules ruptured due to uncontrolled cadmium pressures. 4
of cadmium composition
the
zirconium foil.
and
and the protective
was introduced
foil.
into the capsules by
from an attached
In addition
in was
reaction
side arm.
This
a small quantity
oxide present from unavoidable
contamination.
quartz
the zirconium No
was desirable to remove
of cadmium
Alloy preparation
pressure a different
by wrapping
molybdenum
The cadmium
so that as many as
temperature.
noted between the cadmium double
due to the heat treatments.
After
at the opposite
from
ation.
melting
iso-
2 atm.
taken from the walls of capsules
mium.
where a
Cadmium pressures were not permitted to exceed
capsule was prevented
Extensive
gradient,
was maintained
at one end of the capsule and the sample
after anneals showed no evidence of silicon contamin-
experiment
cadmium
electrical
in atoms per million were 27 Cu, 5 Pb and
a small amount
(X ro--( 1
3. The difficulties
Later, method purity
Smelting and Refining Company.
impurities
32
of solute
ing the capsules in a temperature
at 4.2”K was 0.28 @&cm.
cadmium
28
x mole fraction
3. The coefficient of the electronic specific heat term, y, as a function of the number of quantum number 5 electrons.
Neutron activation, spectroextensive analyses. graphic and chemical analyses were used to determine the amounts
20
to the amount
surface of cad-
mium necessary to provide a certain alloy composition, extra cadmium was added to each capsule to compensate for the volume
of the system
according
to the
ideal gas law. Phase diagram The phase boundaries were determined by optical examination of isothermally annealed and quenched samples, a method that has been used extensively in other zirconium systems.‘1-6) Alloys prepared by diffusion in the p region were sealed in evacuated quartz
capsules
(along
with additional
cadmium
to
1186
ACTA
METALLURGICA,
prevent alloy depletion) and annealed at intermediate temperatures for periods up to 77 days. Furnace temperatures were controlled by platinum resistance thermometer controllers to within +0.3”C. The Pt vs. Pt-10% Rh thermocouples used for temperature measurement were frequently calibrated against the melting points of palladium, gold, silver, aluminum, zinc and ice, or in the case of some later experiments against a standard thermocouple which was periodically calibrated against these melting points. A typical homogeneous a alloy containing 12.9 at.% Cd is shown in Fig. 4. The b.c.c. p phase could not be retained in this system, but could be recognized by the appearance of serrated grain boundaries in contrast to the smooth boundaries as seen in Fig. 4. The “basket weave” or martensitic transformation structure common to other zirconium systems(1-3) was not found in this study.
VOL.
20,
1972
FIG. 5. Ziroonium-oadmium alloy containing 13 at. % Cd after annealing 76 days at 701’C. 750 x .
pressure measurements. The temperature range of the a//l transformation shown for the pure zirconium and the existence of a three-phase (a + /? + y) region are due to minor impurities as found in previous work.‘1-3) Earlier work(‘) showed the existence of at least two intermediate phases of the approximate compositions Zr,Cd and ZrCd,, both having f.c.c. structures. Lattice parameter The lattice parameters of alloys annealed in the a region were determined for compositions up to 13
F
I _I
FIG. 4. Zirconium-cadmium alloy containing 12.9 at. % Cd after annealing 33 days at 845°C. 250 x .
A microstructure of an a + y alloy containing 13.0 at.% Cd is shown in Fig. 5, illustrating characteristic Widmanstiitten and pearlite structures. Platelets of Zr,Cd form on the basal plane of the hexagonal a phase in the same manner as was observed in the zirconium-indium system.(l) The zirconium-rich portion of the zirconiumcadmium phase diagram is shown in Fig. 6. The a/j3 transformation temperature rises only gradually with an unusual maximum occurring at approximately 13 at. y. Cd. This maximum was later confirmed by
0
5
40 CAOMlUM
15
20
(at. %I
FIQ. 6. The ziroonium-rioh portion of the zirconiumcadmium phase diagram. (Note: Deviations from binary equilibrium are due to the presence of minor impurities.)
FRYE.
JR.
THERMODYNAMIC
et al.:
at.% Cd by the Debye-Scherrer were first annealed,
method.
the u structure
PROPERTIES
Samples
OF
cadmium,(lO)
was confirmed
C, = 5.31 + 2.94 x 1O-3
T
Cal/mole “K
optically and then swaged or hammered into wires. The wire samples were given a 3 min vacuum anneal
and values for the heat of vaporization
in the cc region
the solid metal.
The samples
to remove
were then chemically
0.010 in. for the constants
the cold-work
X-ray
etched
measurement.
can be represented
structure. to about
The
lattice
by the following
equa-
tions :
Alloys
were
0.2892,
(1)
c0 = 5.1473 -
0.0302,
(2)
c/a = 1.5923 + 0.1382,
(3)
where x is the atomic fraction
ium
was
first
by the boiler
rolled
to
foil
(0.10-0.28
Cadmium pressure measurements cadmium
cadmium method.@)
pressures
over
the
zirconium-
by the Hargreaves
the specimen
is held at
temperature T, in equilibrium with a liquid drop of pure cadmium at T, (observation window). The vapor pressure of the alloy at from published
log,, P(mm)
T, is
then determined
values of the vapor pressure of pure
liquid cadmium.
The
experimental
distribution
in the furnace
A thin planar
5709 -
1.1283 log,,
T,,
to represent the pressures of pure
liquid cadmium. The vapor pressure of solid cadmium = 9.7822 -
5715 - T -
This formula
is illustrated
throughout
observation
in
temperature the experi-
window
of silica
at the end of a 1 mm capillary
was taken as
0.2214 log,,
3.212 x lo4
is based on the heat capacity g
1050
e
1000
g
950
2 z
900
f
850
5
800
+
750
of
in a
a tubular
furnace.
A
slight
temperature
in the system,
was required
T.
to concen-
was placed around the window
and the resistance
remained
window
along the axis of the capsule
monotonically
to the specimen,
increasing
for solid
from the
which was held at the flat By adjustments
of the shunts, it was possible to reduce the gradient normal to the window to any desired extent. The
zirconium-cadmium
at a high temperature be detected
alloy
was held
initially
until no further changes could
in the temperature
at which the liquid window.
This period
of time was necessary to establish equilibrium (5)
of
the electric furnace windings was adjusted with shunts
drops formed in the observation
T
temperature
trate the liquid drops upon the observation window for a sensitive observation. A nickel conductor block
between
the gas phase and the alloy, since changing the temperature of the sample from high to low results in a transfer
DISTANCE ALONG t:::.1___....,...... i*+:J’. ;...._:..:. ......’ ... _: :. ‘_ ,_,i_, :..;.,>:. .,
p:
O__--_
axis
maximum in the center of the furnace. (4)
log,, P(mm)
and
gradient,, to ensure that the window was at the lowest
always
T, due to Lumsden(g)
arrangement
so that the gradient
We have selected the formula,
= 11.7750 -
mm)
position normal to the direction of heat flow along the
alloys were determined In this method
de-
spacer strips
Fig. 7, which also shows the sigmoidal ment.
of cadmium.
method
with the exception that the zircon-
between each layer.
was attached The
prepared
scribed previously,
(6)
and entropy of
loosely rolled into a coil using zirconium
a, = 3.2326 -
1187
ZIRCONIUM-CADMIUM
of cadmium
FURNACE
(28 in.) *
from the gas to the alloy foil
.._.__-- ._........ r’_‘_.__... :‘r: ,_.__.. “\\a i FIRE
TELESCOPE SURROUNDED BY RESISTANCE WINDING AND INSULATION
FIQ. 7. Dew-point apparatus.
ACTA
1188
(a 1 758.4 OC
(b) 757.2
6 min
7.5
COOLING
METALLURGICA,
‘C
VOL.
(cl 756.5
min
20,
‘C
(d) 760.5
MAXIMUM
“C
(e) 760.9
63/4 min
CONDENSATION
CONDENSATION
1972
“C
7 3/4 min EVAPORATION
HEATING
FIQ. 8. Photographs of window showing various stages of heating and cooling.
which enriches the surface of the foil. Sutlicient time must be allowed for diffusion to make the alloy foil composition uniform at each shift in specimen temperature. Testing of the equilibrium is done periodically by lowering the window temperature slightly until several liquid drops appear, then reheating until the drops disappear. A sequence of photographs representing the typical appearance of the liquid drops of cadmium during condensation and evaporation is shown in Fig. 8. For greater precision it was necessary to determine both condensation and evaporation temperatures of the liquid drops, since nucleation was delayed on a clear window on cooling and complete evaporation was delayed on heating. Further, the temperature inside the quartz capillary was somewhat higher than in the nickel block due to the need for a small temperature gradient across the window to concentrate the initial condensate. The extent of these effects was investigated by use of a high-purity liquid cadmium bath in place of the specimen, care being taken to reproduce exactly the gradient conditions in the vicinity of the window. A typical compensation chart, or calibration, is shown in Fig. 9, illustrating the effect of heating and cooling rates (taken at
a constant temperature gradient) on the condensation and evaporation temperatures. Based on these results, all the experiments were done at a standard heating and cooling rate of 0.8”C/min. The zero of the curve represents the temperature of the molten bath of cadmium at the opposite end of the capsule. It will be noted that an average of the temperature of condensation and the temperature of evaporation at the rates 0.8’C/min was slightly greater than zero. The deviation of this average from the pure cadmium bath was proportional to the temperature gradient across the quartz window. A correction curve, Fig. 10, was established by means of the pure cadmium to allow empirically for the temperature gradient across the window. All of the window temperatures for the zirconium-cadmium alloys were determined by averaging condensation and evaporation temperatures, observed after a slight cooling and reheating of the window at 0.8’C/min, and then correcting the average temperature by means of Fig. 10. Thermal effusion, or pressure drop along the temperature gradient in the gas phase, is not expected to contribute a large error in these experiments. This is based on thermal effusion relations given by Bichowsky and Wilson.(lz) The atomic weight of cadmium is large, and we assume that the viscosity and Sutherland constant of cadmium gas are not much different than helium. These relations show that thermal effusion effects should be small at high temperatures and in large tubes and should increase as the temperature
_I___~I . ;l.+E,K l-
-to
1
0
I
I
I
I
I
t
2
3
4
5
RATE
OF HEATING
AND
COOLING
(deghinl
FIG. 9. The temperature of evaporation or condensation less the temperature of a high-purity cadmium bath as a function of the heating or cooling rate, respectively.
6
,L 0
xi
i ~~~ ___ t
2 WINDOW
3
4
TEMPERATURE
5 GRADIENT
_._ 6
7
(deg/in.)
Fro. 10. Correction curve to adjust for window temperature gradient.
E
FRYE,
et al.: THERMODYNAMIC
JR.
PROPERTIES
drops and the tube diameter becomes small. For this reason a short length of 1 mm i.d. tubing was used at the window end of the capsule where the temperature gradient was smallest, and the capsule was expanded to at least 4 mm i.d. in the more rapidly rising gradient region. The partitioning of the fixed amount of cadmium between the gas phase and the foil sample occurred relatively quickly for specimens in the cubic ,Bregion of the zirconium-cadmium system at temperatures of 1160-1327’K, but in the hexagonal ccregion, the diffu. sion of cadmium is much slower and surface enrichment or depletions of the alloy occur. Early attempts to estimate the size of this effect by utilizing different approaches to equilibrium temperatures in the a region are illustrated in Fig. 11 where (a) represents approaching the desired sample temperature from a previously established equilibrium at some higher temperature and (b) approaching from a lower temDifficulties in establishing equilibrium perature. were at least partially overcome in the following ways. First, alloys were held in the a region for periods of time as long as S~O-10,000 hr. Second, the temperature range of the a-phase determinations was restricted to a narrow region, 109%1180°K, just under the a/B transformation temperature. Third, the alloy was taken slowly through the (a + /I) region, allowing the major portion of the transfer of cadmium from the gas phase to the alloy specimen to take place in /I grains. Finally, design changes were made in the capsule, where the uniform 25 mm silica tube shown in Fig. 7 was replaced over half its length by 4 mm tubing at the lower temperature end. This second
OF
ZIRCONIUM-CADMIUM
1189
type of capsule minimized the gas phase volume and by also increasing the surface area of the foil by both increasing the mass of the specimen and reducing its thickness, greater alloy surface-to-gas phase volumes were made possible. The log,, cadmium pressures (in mm) over the closepacked hexagonal (a) and the b.c.c. (p) forms of zirconium with up to 15.4 at.% Cd and in the temperature ranges 1093-1327’K are shown in Fig. 12. The break in the curves is associated with the twophase a + /I region. The a/,5 phase boundaries observed by this method are shown by dotted lines in Fig. 6. It can be seen that the two-phase region found by the pressure measurements was much narrower than indicated by the quenched alloy method. This may be due partly to the fact that the lines in the phase diagram are rotated in these alloys due to the small amount of impurities present. There was no certain order (in regard to temperature) in which the experiments were conducted, and it was found that equilibrium could be reestablished with essentially the same pressure for the specimens held in the p region. This result is independent of how long the sample was held at high temperature, thus confirming the absence of import.ant contamination of the zirconium-cadmium alloy during the experiment. Similar results have been found by spectroscopic analysis of the final samples. Neutron activation analyses revealed only slight contamination by silicon and tantalum. The average silicon cont’ent was 42 ppm with a range 4-91 ppm and the average tantalum content was 23 ppm with a range 0.3-98 ppm.
J
0
200
400
600 TIME
800
1000
1200
(hr)
Fra. 11. Variation of the mean evaporation and oondensation ~mperature with time for a 5.2 at.7 Cd alloy. (a) Cooled to 850°C. (b) Efsated to 856°C.
FIG. 12. Experimentally determined log,, cadmium pressures as a function of lIToK.
1190
ACTA
METALLURGlCA,
The cadmium compositions were determined by chemical analysis of at least three locations on the coil of alloy foil. For compositions at temperatures other than the temperature just prior to quenching the alloy, the amount of cadmium in the gas phase was calculated and the alloy composition was obtained by difference of the fixed total cadmium and that in the gas phase. The variations of the alloy composition for sequence of temperature were small, but they were still large enough to warrant consideration in the development of thermodynamic properties. DISCUSSION
In this section of the paper we will firstly calculate certain partial molal quantities, with particular emphasis on the partial molal energy at absolute zero. This will enable us to draw certain conclusions as to the effect of cadmium on the transformation temperature and as to the variation of the partial molal energy with concentration. Secondly, we will examine the implications of these findings for the energies of Wigner-Seitz polyhedra in primary substitutional solid solutions. Thirdly, we will discuss the appropriateness of the rigid band model for such solutions. Calculation of the energy a8 CPK When the alloy is in equilibrium with its gas, we can write
where C, = C, = NI = iVs = P = T =
Gibbs free energy of the gas ; Gibbs free energy of the alloy; No. of atoms of the solvent element; No. of atoms of the solute element; pressure ; absolute temperature.
There are data that indicate that cadmium gas at the temperatures and pressures observed here is nearly ideal and monatomic.(ll) From the temperature and vapor pressure data presented above, we can calculate the energy. This will consist of the kinetic energy of atoms and the product of the pressure multiplied by the volume. The kinetic energy is zero at absolute zero when the atoms are infinitely separated. If the atoms are confined in a finite volume, they will have a finite energy which can be calculated by use of the Schroedinger equation. The pressure-volume product will also be finite. Thus the reference state for zero energy is that in
VOL.
20,
1972
which the atoms are infinitely far apart and at absolute zero. Fo~unately, the energy at absolute zero for the gas densities of interest here is negligible compared with the energy at elevated temperatures. Thus we can use an equation for the free energy of each species which neglects this energy and which contains other approximations, but which is simple to use and quite accurate enough for our purposes.
The logarithmic fraction is the partition function for an ideal monatomic gas.(is) T
1143 C,==ES_
C,dT/T
fl,dT-k
s0 -
T
C,dT
s 1143
-
T
1143 s 0
T
s
C, dT/T 1143
(9) where R k m N, N, V h E C,
= = = = = = = = =
gas constant; Boltzmann’s constant; mass of one atom of solute; No. of solvent atoms; No. of solute atoms in the gas ; volume of the gas ; Planck’s constant; energy of the alloy at O*K; heat capacity of the alloy at constant pressure.
Equation (9), which is due to Jones,fi4) implicitly assumes completely random mixing of solute and solvent atoms. This is not perfectly achieved in real systems. Thus, the last term will not exactly calculate the configurational entropy. The error will, however, approach zero as the concentration approaches zero. The reference state for E is that of the atoms at infinite separation at absolute zero. In other words, it is the energy released when these atoms are brought together from infinity at absolute zero to form the alloy. Later in this discussion we will be concerned with the quantum mechanics of solid solutions. Here the reference state for zero energy must be the nuclei and electrons at infinite separation at absolute zero. This is readily obtained from the present reference state by subtracting the ionization energies of the solute and solvent atoms.
FRYE.
JR.
et al.:
THERMODYNAMIC
PROPERTIES
C, is zero at absolute zero. Upon taking the partial of equations (8) and (9) with reference to N,, equating and rearranging, we get ln
h3fW, + N,)
a T
1
(2n-m)3/2(kT)5/2N2 - %? G T C,dT 1 a ' aN, f1143 T = RT
-
OF
a2E
axaN
+
ZIRCONIUM-CADMIUM
s
a2
1143
o
axaN
aT
+
s1143'
‘p dT = axaN
a2E -= axaN
--R aN, s o
9. T
(10)
The literature has been searched, although not exhaustively, for vapor pressure data in other primary substitutional solid solutions. These data(15-22) together with our experimental data have been used to calculate the left side of equation (10). This quantity has then been plotted against l/T for constant x for these various systems, where x is the atomic fraction, N,/(N, + N,). From these plots have been evaluated 1143
C,dT
(11)
and
1143C dT 2,. T
(12)
Values of these functions have been plotted against the atomic fraction, x, in Figs. 13 and 14. For most of the solutions it can be seen (Figs. 13 and 14) that
FIG. 13. [g
+
= 0. (13)
(14)
O.
It is not possible to obtain aE/aN, 1143C dT
T
a
From the Debye theory it follows that
+ -&jo1143CDdT] 1 a
s
C &T
1 R
1143
a2
8 c
1191
a[ Jo1143cp a,]
without knowing
/aa,.
The latter can be calculated from a[ [1143C, dT/Tl I-J”
/aN, -II
by use of the Debye model. Nernst has provided tables which we have used for this purpose.(B) Similar tables can be found in Lewis and Randall.(2Q) Values of aE/aN, are listed in Table 1. These calculations ignore differences between C, and C,. As a check on accuracy we have also calculated values for the pure metals and compared these with experimental measurements.(25) It can be seen that the errors are small and consistently in the same direction, thus lending confidence to values calculated for the various solutes. It would seem to be a safe conclusion that aElaN, is more negative for cadmium in /l than in a zirconium. Thus, we conclude that cadmium tends to stabilize p at absolute zero, but that this is opposed by the
Cd
--
Cd Cd Zn-Zn Cr Ag
~ -~ ___ __ ~--
Ag /SZr aZr Ni cu Fe Au
--
& /;143Cp dTIT,.( l/ Ill011.3 0x3 ) as & function of concentration.
1192
ACTA
0
0.1
METALLURGICA,
0.2
0.3
x
VOL.
0.4
At 1143°K the latter overbalances between
0 and
the
0.11
mole
fraction
Duhem
follows
from
that a2E/axaN,
= 0.
=
equation
0,
it
tions apply to most of the solutions
the
Gibbs-
These condi-
depicted
in Figs.
13 and 14. We can write for these solutions
E = (1 - x)aE/aN1 + xaE/aN2 a2E
solute (see Henry’s
Crystal
law).
-26,800
- 145,500 -143,500 -68,000 0.76 -87,200 -93,800 0.50 -97,000 -31,100 0.32 0.28
* From Nernst tables.
(15)
molal energy of the pure
0.11 0.15 0.37
- 102,200 -80,600
a new reference state for the energy.
define energy,
L, of the crystal
at absolute
We
zero as
being the energy which results from bringing together the nuclei and electrons form the crystal.
out that aE/aN, is not neces-
TABLE 1. Comparison of measured and calculated values of aElaN, aE/ aN, (Cal/mole) Maximum Calculated* Measured zz
Cd Cd in a-Zr Cd in @-Zr Cd in Ag Ci-Zr tAg in Au AU Cr Cr in Fe Fe Zn Zn in Ni Zn in Cu Ni cu
the nuclei and the electrons in the crystal and there-
from infinite
Furthermore,
separation
to
it is convenient
at
this point to express L in terms of the energy/atom.
(16)
axaN, -==2=O.
It should be pointed
polyhedra*
L is related to E by the equation
a2E
sarily equal to the partial
of concentration.
Energy of Wigner-Seitz
fore need
atqaxaN,
0.7
We shall now be concerned with the energy of both of
cadmium. If
0.6
as a function
former and there is a small increase in transformation temperature
1972
0.5
Fm. 14.
entropy effect.
20,
-26,000 -27,800 -30,100 -31,200 -144,500 -67,800 -68,900 -86,600 -92,300 -93,000 -96,100 -30,200 -46,800 -37,300 - 100,900 -80,100
L = E/(N, + N,) -
(1 -
x)1, -
x12,
(17)
where I, = ionization
energy of the solvent atom;
I, = ionization
energy of the solute atom.
It can be seen from the data presented earlier that EI(N, + N,) is a linear function of concentration. Thus, L is a linear function of concentration. The
quantum
mechanical
virial
theorem(26,27)
enables us to express the total energy of the crystal in terms of either the expectation energy (K) or the expectation energy, (W). The equation is L = -(K)
value of the kinetic
value of the potential
= 0.5(W).
(18)
This theorem has been known since 1930. It is now often used in a priori quantum mechanical total energy calculations.(29~30) The fact that we can express total
energy
in terms of either kinetic
or potential
* “The Wigner-Seitz cell is obtained by drawing the planes bisecting the link joining an atom to its neighbors until they enclose a volume. “w) Thus the crystal will be divided up into polyhedra such that there will be one polyhedron surrounding each atom.
FRYE,
JR.
et al;.:
TRERMODYNAMIC
PROPERTIES
energy will be useful in the qualitative discussion which follows. Consider any random substitutional solid solution composed of numbers 1 and 2 atoms. We divide the crystal into feigner-Seitz polyhedra. The expectation value of the kinetic energy of the entire crystal can be written as
OF
ZIRCONIUM-CADMIUM
1193
suggest that (K,) and (K,) in equation (19) are approximately independent of concentration and not greatly different for the values in the pure crystals.
Relationships between the hexagor~al c/a ratio, the transformation temperature and TX,the electronicspecific-heat, coefhcient on the one hand and the concentration of quantum number 5 electrons on the other for the solutions Zr-Ag, Zr-Cd, Zr-In, Zr-Sn and (K,) = expectation value of kinetic energy of solZr-Sb are summarized in Figs. 1-3.(1-s) These simple vent polyhedra ; relationships suggest that the quantum number 5 (K2> = expectation value of kinetic energy of electrons may be entering the conduction band of solute polyhedra. zirconium without altering that band. If we assume The expeetation value of the potential energy of such a rigid baud model, f-5’ecan calculate t>hedependthe entire crystal can be written as ence of the energy of these electrons on concentra0.5(W) = 0.5ffl - z)(W,> + x(Wz> + {I - %)2(lY11) tion by use of the data in Fig. 3. This is of particular + 2X(1 - $(W& + za(Wa,>] = L, f2OY interest for the Cd-Zr alloys, since we know how their total energy varies with concentrat,ion. where @‘i,>, (11’12>and (W,,) are the expectation The density of electronic states is proportional to values of the potential energy due to interaction the low-temperature, electronic specific heat, tl, between solvent and solvent polyhedra, between provided there is uo coupling between electrons and solute and solvent polyhedra and between solute and phonons. We ignore such coupling : but, as we shall solute polyhedra, respectively, see later, the conclusion which we draw would be Many years ago, Mott and Jones@i) argued that strengthened if it had been taken into account,. (W,,) should be small in a pure metal. The data The data of Fig. 3 can be represented by a straight presented above suggest that it and (VI,) and (Wz,) line with suf%ient accuracy for present purposes. are small in some solid solutions. The arguments are We write as follows : ~~~~~ = a “r ~~~~~, f21) 1. It was pointed out above that & is a linear funcwhere tion of concentration. One way in which equat8ion A -2 No. of states; (20) can be realized is for (W,,), (W& and ( W,,) to U = Fermi energy; be small in comparison with E/(X, + N,) [see equa2 = atomic fraction ; Lion (17)]. y -= No. of quantum number 5 electrons. 2. For most of the solutions which we have cousidFor cadmium ered here, the energy required to remove a solute y-_2
atom from a dilute solution is little different from
that required to remove it from its pure crystal even though the solvent atoms are much more tightly bound (see Table 1). This is hardly possible if the binding is due solely to interaction between palyhedra. Thus, it would appear that ( WX1>,( Wiz) and (W& are small in comparison with Ej(.iV, + .N& These arguments are not conclusive, but they do * Equation (19) contains no t,erms for interaction between
polyhedra. Equation (20) does. This difference stems from the difference between the kinetic energy and potential encrgy operators in the Hamiltonian. The kinetic energy operator is of the form - (ha/2na)X,B,e, where ala= mass of t,he particle. The potential energy aperator is of the form
&&a =
drtt.
assume that the two quautum number 5s electrons enter a rigid conduction band in zirconium. Then
We
U=
i
0G$
(221
+ tj’ll
where U, = Fermi energy in pure zirconium, jrJ
=2 hl I1 + b
W)xl +
u
1’
(231
Let D = energy, per atom of cadmium, of the electrons lying above U,. where Ba and 2~ are the charges on the cc and @particles, respectively, and R?. and Rg are t,he position vectors of the ct and /? pa&i&s, respectively.
ACTS
1194
METALLURGICA,
The integral gives the energy in terms of the energy~ atom of alloy. We divide by x to get the energy/atom of cadmium. We do this in order to have an easy comparison with aElaN,.
11 + WMlln Cl+ W44
i7 ,2
+ 2u
- (VG
b”/u
X1
I
1'
P-3
In zirconium, 1 + (b/a)x has a maximum value of 1.13 for 11% Cd. Thus, ln [l + (b/a&] firr(b/a)s. i7 Gw2x/a + 2lJ,,
(26)
where a is equal to 0.6 statesjev at. Then U = (3.32 i_ 2U,) eV~atom = (‘77,000~ + 2U,) cal/mole,
(27)
when x = 0.11, 0 = 0.36 eV/atom + 2U, = 8300 Cal/mole + 2U1.
(28)
Since aE~aN~ plus the appropriate ionization energy for the free cadmium atom is the total energy of one mole of cadmium in zirconium at absolute zero, it must contain 0. The variation of 0 with concentration is sufficiently great to give an observable positive slope to the curve for cc zirconium in Fig. 13. No such slope is observed. Furthermore, an even greater effect would be indicated if coupling between electrons and phonons had not been ignored. Analogous calculations for zinc in copper and for chromium in iron-assuming zinc adds one electron and chromium subtracts two electrons-yield g&l-zn = (78,400~ + 39,200G) callmole + V;,
(29)
=QJFe-Cr = (-46,000~
(30)
+ 23,000~~) cal/mole -
ZU,.
Again, no such variation is observed in the appropriate curves of Fig. 13. Using a free-electron gas model, analogous calculations have been made for zinc in nickel, cadmium in silver and cadmium in cubic zirconium. It is assumed that these solutes, respectively, increase the number of conduction electrons from one to two, from one to two and from four to six. The resulting equations are Dx;i-zn = (92,0002 + 137,000) cal/mole,
(31)
O,,,, -
= (83,400~ + 125,000) Cal/mole,
(32)
(170,000~ + 508,000) cal/mole.
(33)
UZr_Cd =
VOL.
20,
1972
Again, no such variation is observed in the appropriate curves of Fig. 13. The curve in Fig. 13 for silver in gold, where no valence effect would be expected, shows no appreciable variation. For these systems, it must be concluded that the rigid band approximation is not a good one. It is possible to be more general. Darken and Gurry(s2) say of Henry’s law : “‘it is found empirically that, if the concentration of the solute (conventionally designated component 2) of any binary solution is sufficiently low, then at constant temperature the partial pressure of the solute is proportional to its mole fraction.” They also state ‘<. . . the gas phase behaves ideally, as it usually does at low concentration and hence at low partial pressure of component 2, . . . .” When these conditions are met, then the energy and the temperature dependent entropy of the solute are independent of concentration. The energy at zero Kelvin is independent of concentration. The equation for the rigid band model is Da = 2
“dU “&& -ad?&+ s0 [S 0 dn
1
U, .
(34)
The term in square brackets is merely the Fermi energy. Henry’s law would predict ~)~=s=
($j),=,=
0.
(35)
This is in contradiction to the rigid band model. These conclusions are in accord with recent density of state calculations. For example, Stocks et c~1.f~) have calculated the densities of states of the nearly random alloy system copper-nickel for a range of concentrations of constituents across the complete alloy diagram. They conclude that: “for all Cu-Ni alloys the rigid band model and the virtual crystal approximation are quite inappropriate.” CONCLUSIONS
Lattice parameter measurements are in agreement with other solutes with quantum number 5 electrons in that the proportionality of the axial ratio to the number of such electrons is due to a change in the c spacings with the a dimension sharing a common curve. Slopes of c/a, ([d{e/a)/dz],=,), increase linearly with the number of quantum number 5 electrons in the solute for Ag, Cd, In, Sn and Sb. Phase boundaries determined by the dew-point method agree quite well with those obtained by the isothermal anneal and quench method. There is a
FRYE.
rather
JR.
unusual
solid solution
et al.:
maximum at about
THERMODYNAMIC
which
13 at.%
found in the lead-thallium
occurs
PROPERTIES
in the
a
Cd, similar
to that
Solubility
as high
system.
as 17 at.yb Cd exists in both the a and b phases. Slopes of the a//3 phase boundaries,
increase
monotonically
number
5 electrons
and Sb.
transformation This
is the
with the number
balanced
from
of quantum
the temperature
hexagonal
resultant
effects-an
to b.c.c.
of two
opposite
absolute
zero
zirconium. effect entropy
an
alloy at zero Kelvin
an
form.
is given by
the equation,
E =
-_(l
-
z)[(K,)
The data which for
some
have
primary
(K,)and (K,)are and
are
-
II]
-
s[(K,)
(36)
I,].
been considered here imply that, substitutional
solid
nearly independent
not very
-
different
from
solutions,
of concentration
the
corresponding
considered
here.
for the
Furthermore,
it is
incompatible
with Henry’s law which is widely obeyed
by metallic
solid solutions.
This
accord with recent band-energy
conclusion
is in
calculations.
ACKOWLEDGEMENTS
The authors Faulkner,
are grateful
G. S. Painter
to J.
Brynestad,
and G. M. Stocks
J.
S.
for many
helpful discussions.
of Technology, -_
10. 11.
12. 13.
::: 16.
S. EASTON and J. 0. BETTERTON, JR., Metallurgy Division Semi-annual Progress Report ORNL-2217, p. 216 (1956). R. HARGREAVES, J. Ilast. Metals 64, 115 (1939). J. LUMSDEN. !Chermodunamics of Allows, D. 134. Institute of Metals (1952). ” ” ” _ K. K. KELLEY, U.S. Bureau of Mines, Bulletin No. 476 (1949). R. HULTGREN, Selected Values of the Thermodynamic Pronerties of Metals and Allovs. Mineral Research Laborato&, University of Californfa i1956). F. RUSSELL BICHOWSRY and C. W. WILSOS, Phys. Rev. 33,851 (1929). E. U. CONDON and HUGH ODISHAW, Handhook of Physics, pp. 5-11. McGraw-Hill (1958). H. JONES, Proc. Phys. Sot. 49, 243 (1937). C. L. MCCABE, H. M. SCHADEL, JR. and C. 13. BIRCHENALL, J. Metals 5, 709 (1953). G. SCATCHARD and R. H. BOYD. J. Am. Ch.em. Sot. 78,
3889 (1956).
17. C. E. BIRCHENALL and C. H. CHENG, Trans. metall. Sot.
A.I.M.E.
185, 428 (1949).
18. E. E. UNDERWOOD and B. L. AVERBACH, J. Metals 3. 1198 (1951). 19. G. SCATCHARD and R. A. WESTLUND, JR.. J. .4m. Chem. sot. 75, 4189 (1953). 20. 0. KUBASCHEWSKI and G. HEYMER, Acta Met. 8, 416 21. A. W. HERBENAR, C. A. SIERERT end 0. S. DUFFENDACK, Trans. Metall. Sot. A.I.M.E. 188, 323 (1950). R. HARGREAVES, J. Inst. Metals 64, 115 (1939). W. NERNST, The New Heat Theorem. E. P. Dutton (1926). ;:: LEWIS and M. RANDALL, Thermodynamics. 24. G. N. McGraw-Hill (1961). 25. R. R. HULTGREN, R. L. ORR, P. D. ANDERSON and K. K. KELLEY, Selected Values of Thermodynamic Properties of Metals and Alloys. John Wiley (1963). 26. V. FOCK. 2. Phusik 63. 855 (1930). J”. Mol. Spect&. 3(i), 46-66 (1959). 27. P-O. L&DIN, and Phonons, p. 4. Oxford Univer28. J. M. ZIMAN, Electrons sity Press (1960). and K. W. JOHNSON. Phus. Rev. Third 29. M. Ross Series, B, 2, 4709 (1970). 30. A. ROTHWORF, Phys. Rev. Lett. 29A (1969). 31. N. F. MOTT and J. H. JONES, The Theory of the Properties of Metals and Alloys. Dover (1958). 32. L. S. DARKEN and R. W. GURRY, Physical Chemistry of Metals, p. 256. McGraw-Hill (1953). 33. G. M. STOCKS, R. W. WILLIAMS and J. 8. FAULKNER, Phys. Rev. B4, 4390 (1971). ”
REFERENCES I. J. 0. BETTERTON, JR., J. H. FRYE, JR. and D. S. EASTON, Phase Diazmm Studies of Zirconium with Silver, Indium and Anti
Sot. A.I.M.E.
Institute
(1960).
energies in the pure crystals. The rigid band model is not appropriate solid solutions
8. 9.
nearly
energy
effect which tends to stabilize the hexagonal of
of the
and
which tends to stabilize the cubic form and The energy
Illinois
7. D.
in the solute for Ag, Cd, In, Sn
slightly increases
Foundation.
p. 32 (1952).
The relation is linear for Cd, In, Sn and Sb.
Cadmium
1195
ZIRCONIUM-CADMIUM
3. D. S. EASTON and J. 0. BETTERTON, JR.. Metall. Trans. 1, 3295 (1970). 4. J. 0. BETTERTON, JR. and J. H. FRTE, JR., Acta &Zet. 6, 205 (1958). 5. G. D. KNEIP, JR., J. 0. BETTERTON, JR. and J. 0. SCARBROUOH, Phys. Rev. 131, 2425 (1963). 6. M. HANSEN and D. J. MCPHERSON, Report COO89 Armour Research
+ Tp)/2~~I,=&
WT,
OF
212,470 (1958).
PROPERTIES OF ZIRCONIUM-CADMIUM OTHER SOLID SOLUTIONS*
J. H. FRYE,
JR.,? J. 0. BETTERTON,
AND
CERTAIN
JR.t$ and D. S. EASTON?
The experimental work reported here has to do with the zirconium-rich zirconium-cadmium alloys. The phase diagram has been determined up to a concentration of 20 at. % Cd. Lattice parameters have been measured for the hexagonal phase. The vapor pressure of cadmium in equilibrium with the solid solution has been measured for both the hexagonal and cubic phases as a function of temperature and concentration. These data are considered together with earlier data on the Zr-Ag, Zr-In, Zr-Sn and Zr-Sb primary substitutional solid solutions. Empirical relations are found between the hexagonal (c/a) ratio, the temperature of the transformation from cubic to hexagonal zirconium and the electronic specific heat coefficient on the one hand and the concentration of principal quantum number 5 electrons on the other. These simple relations suggest the possibility that the principal quantum number 5 electrons are entering a rigid conduction band in zirconium. In order to investigate this possibility, we have calculated the partial molal energy of cadmium in zirconium from the vapor pressure measurements. We have also done this for certain other solutions for which vapor pressure data have been published. We conclude that the rigid band model is a bad approximation and point out that this is in accord with recent quantum mechanical calculations. Furthermore, the effect of cadmium on the transformation temperature of zirconium is the resultant of two opposite and nearly balanced effects-an absolute zero energy effect which tends to stabilize the cubic form and an entropy effect which tends to stabilize the hexagonal form. Finally, it is shown that, for most of the solutions considered here, the energy of a Wigner-Seitz polyhedron is independent of concentration within experimental error. PROPRIETES
THERMODYNAMIQUES AUTRES
DU ZIRCONIUM-CADMIUM SOLUTIONS SOLIDES
ET
DE
CERTAINES
Le travail experimental present6 ici est relatif aux alliages zirconium-cadmium riches en zirconium. Le diagramme des phases a et6 determine jusqu’a une concentration de 20 at. % Cd. Les parametres du reseau ont et6 mesures pour la phase hexagonale. La pression de vapeur du cadmium en Bquilibre avec la solution solide a et6 mesuree, a la fois pour la phase hexagonale et pour la phase cubique, en fonction de la temperature et de la concentration. Ces resultats sont compares avec les resultats anterieurs relatifs aux solutions solides primaires de substitution Zr-Ag, Zr-In, Zr-Sn et Zr-Sb. Les auteurs ont trouve des relations empiriques entre le rapport hexagonal (c/a), la temperature de transformation du zirconium cubique en zirconium hexagonal et le coefficient de la chaleur specifique Qlectronique, d’une part, et la concentration des electrons de nombre quantique principal 5, d’autre part. Ces relations simples suggerent la possibilite que, dans le zirconium, ces electrons entrent dans une bande de conduction rigide. Pour etudier cette possibilite, les auteurs ont calcule l’energie moleire partielle du cadmium dans le zirconium B partir des mesures de pression de vapeur. 11sont fait ceci egalement pour d’autres solutions pour lesquelles les resultats des mesures de pression de vapeur ont et6 publies. Les auteurs concluent que le modele de bande rigide est une mauvaise approximation et montrent que ceci est en accord avec les calouls ¢s de la mecanique quantique. En outre, l’influence du cadmium sur la temperature de transformation du zirconium est la r&&ante de deux effets opposes et a peu p&s Bquilib&s-un effet d’energie au zero absolu qui tend a stabiliser la forme cubique et un effet d’entropie qui tend a stabiliser la forme hexagonale. Finalement, les auteurs montrent que, pour la plupart des solutions Qtudiees ici, l’energie dun polyedre de Wigner-Seitz est independante de la concentration, aux erreurs experimentales p&s. THERMODYNAMISCHE
EIGENSCHAFTEN ANDERER
VON ZIRKON-KADMIUM LEGIERUNGEN
UND
EINIGER
Die vorliegende Arbeit behandelt zirkonreiohe Zirkon-Kadmium-Legierungen. Das Phasendiagramm wurde bis zur Cd-Konzentration von 20 At. ‘A bestinnnt. Fur die hexagonale phase wurden Gitterwurde sowohl fiir die hexagonale Phase parameter gemessen. Der Kadmium-Gleichgewichtsdampfdruck als auoh fur die kubisohe Phase als Funktion der Temperatur und Konzentration gemessen. Diese Daten werden mit friiheren Daten tiber Zr-Ag, Zr-In, Zr-Sn und Zr-Sb verglichen. Es werden empirische Beziehungen gefunden zwischen dem (c/a)-Verhiiltnis der hexagonalen Phase, der Umwandlungstemperatur vom kubischen in hexagonales Zirkon und dem Koeffizienten der elektronischen spezifischen Warme einerseits und der Konzentration der Elektronen mit der Hauptquantenzahl 5 andererseits. Die einfachen Zusammenh5nge sind ein Hinweis fur die Moglichkeit, da5 die Elektronen mit der Hauptquantenzahl 5 in ein starres Leitungsband gehen. Urn diese Mogliohkeit zu untersuchen, haben wir aus Dampfdruckmessungen die partielle molare Energie von Kadmium in Zirkon berechnet. Ahnliche Rechnungen haben wir fur einige andere Legierungen angestellt, fur die Dampfdruckdaten veroffentlicht sind. Wir kommen zu dem Ergebnis, da13das rigid band model eine sohlechte Niiherung ist und weisen darauf hin, da13wir damit in Ubereinstimmung mit neueren quantenmechanisohen Rechnungen sind. AuDerdem ist der EinfluD des Kadmiums auf die Umwandlungstemperatur von Zirkon das Ergebnis zweier entgegengesetzter und sioh nahezu aufhebender Effekte: der Nullpunktsenergie-Effekt stabilisiert die kubische und der Entropie-Effekt die hexagonale Struktur. Schlieljlich wird gezeigt, da8 fur die meisten der hier untersuchten Legierungen die Energie einer Wigner-Seitz-Zelle innerhalb des experimentellen Fehlers unabhiingig von der Konzentration ist. * Received November 26, 1971; revised February 17, 1972. Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. t Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830. 2 Now at: Department of Physics, Louisiana State University at New Orleans, New Orleans, Louisiana 70100. ACTA
METALLURGICA,
VOL.
20, OCTOBER
1972
1183
1184
ACTA
METALLURGICA,
VOL.
INTRODUCTION
20,
1972
24
This is the last of a series of papers(14) having to do with the properties of primary substitutional solid solutions with zirconium as the solvent and silver, cadmium, indium, tin and antimony as solutes. Previous work had shown interesting relationships between certain properties of these solutions and the number of electrons having a principal quantum number of 5 in each solute atom. Data were lacking, however, for the effect of cadmium on lattice parameters and on the temperature of transformation between cubic and hexagonal zirconium. In the present work we have measured the lattice parameters of zirconium at different concentrations of cadmium. We have also established the phase diagram to 20 at.% Cd. These data have been used together with previously published data to draw the graphs of Figs. 1 and 2. Relationships between the hexagonal c/a ratio, the transformation temperature and y, the electronic specific heat coefficient, on the one hand and the concentration of quantum number 5 electrons on the other, are summarized in Figs. 1-3.‘1-6) It is tempting to try to explain these relationships in terms of the electronic band energies at absolute zero. In order to do this we need to know the energy of an atom in solution against a reference state of its ions and electrons at infinite separation. This is the reference state for the potential energy operator in the Schroedinger equation. As will be shown later this can be obtained from vapor pressure data. With this in mind we have also measured the vapor pressure of cadmium in zirconium-cadmium alloys as
t6
0 0 H
-3%
4
0
:
-h? +
r_” -8 9
-32
-t”
0
2
3
4
5
6
quantum number 5 electrons
FIN. 2. The derivative with respect to concentration, at zero concentration, of the average of the temperature boundaries of the u + /l region as a function of the number of quantum number 5 electrons.
a function of temperature and concentration. In the last part of this paper these data are analyzed and their possible physical significance is discussed. EXPERIMENTAL
PROCEDURES
AND RESULTS
Materials A particular bar of Westinghouse grade I iodide zirconium of 99.96% purity was selected on the basis of
0.004
0.003
quantum number 5 electrons
Fro. 1. The derivative with respect to concentration of the axial ratio (c/a) as a function of the number of quant,um number 5 electrons in the solute atom.
FRYE,
JR.
et al.:
THERMODYNAMIC
PROPERTIES
OF
1185
ZIRCONIUM-CADMIUM
f
{
-I 1 i
2.8 4
0
6
(2
46
quantum number 5 electrons FIG.
9OC,
of trace
5 Cu, 65 Fe,
impurities
lOHf,
5 K,
as follows: 123 0,
10 Al,
1 MO, 29N,
1 Nb, 12 Ni, 7 Si, 2 Ta and 2 Zn (in at. ppm). 40 ppm H in the as-received than
1 ppm
resistivity The
by
vacuum
24
American
The
bar was reduced to less
degassing.
The
was of 99.99+%
of CdO.
analysis of cadmium
Both
X-ray
in method 2 were solved by hold-
boiler
of pure liquid
thermally
was held at some higher temperature end.
four samples in a single capsule were held at different
the
Principal
and chemical
temperatures gas.
Each
according
3 was modified
with a constant sample produced
to its individual
Reaction
between
tantalum analysis was done at each stage of the to ensure that no contamination
occurred
or
distillation
distillation
The preparation of zirconium-cadmium alloys is very difficult due to the high vapor pressure of cadwere unsuccessful,
attempts
to
produce
alloys
the following
by
arc
techniques
were developed that proved to be effective preparation of alloys up to 20 at.% Cd:
in the
1. Cadmium was sealed within an evacuated zirconium container and annealed at elevated temperatures. 2. Vapor diffusion between cadmium gas and zirconium
was achieved
in evacuated
and sealed silica
capsules in the temperature range lOOO-1200°C; however, several capsules ruptured due to uncontrolled cadmium pressures. 4
of cadmium composition
the
zirconium foil.
and
and the protective
was introduced
foil.
into the capsules by
from an attached
In addition
in was
reaction
side arm.
This
a small quantity
oxide present from unavoidable
contamination.
quartz
the zirconium No
was desirable to remove
of cadmium
Alloy preparation
pressure a different
by wrapping
molybdenum
The cadmium
so that as many as
temperature.
noted between the cadmium double
due to the heat treatments.
After
at the opposite
from
ation.
melting
iso-
2 atm.
taken from the walls of capsules
mium.
where a
Cadmium pressures were not permitted to exceed
capsule was prevented
Extensive
gradient,
was maintained
at one end of the capsule and the sample
after anneals showed no evidence of silicon contamin-
experiment
cadmium
electrical
in atoms per million were 27 Cu, 5 Pb and
a small amount
(X ro--( 1
3. The difficulties
Later, method purity
Smelting and Refining Company.
impurities
32
of solute
ing the capsules in a temperature
at 4.2”K was 0.28 @&cm.
cadmium
28
x mole fraction
3. The coefficient of the electronic specific heat term, y, as a function of the number of quantum number 5 electrons.
Neutron activation, spectroextensive analyses. graphic and chemical analyses were used to determine the amounts
20
to the amount
surface of cad-
mium necessary to provide a certain alloy composition, extra cadmium was added to each capsule to compensate for the volume
of the system
according
to the
ideal gas law. Phase diagram The phase boundaries were determined by optical examination of isothermally annealed and quenched samples, a method that has been used extensively in other zirconium systems.‘1-6) Alloys prepared by diffusion in the p region were sealed in evacuated quartz
capsules
(along
with additional
cadmium
to
1186
ACTA
METALLURGICA,
prevent alloy depletion) and annealed at intermediate temperatures for periods up to 77 days. Furnace temperatures were controlled by platinum resistance thermometer controllers to within +0.3”C. The Pt vs. Pt-10% Rh thermocouples used for temperature measurement were frequently calibrated against the melting points of palladium, gold, silver, aluminum, zinc and ice, or in the case of some later experiments against a standard thermocouple which was periodically calibrated against these melting points. A typical homogeneous a alloy containing 12.9 at.% Cd is shown in Fig. 4. The b.c.c. p phase could not be retained in this system, but could be recognized by the appearance of serrated grain boundaries in contrast to the smooth boundaries as seen in Fig. 4. The “basket weave” or martensitic transformation structure common to other zirconium systems(1-3) was not found in this study.
VOL.
20,
1972
FIG. 5. Ziroonium-oadmium alloy containing 13 at. % Cd after annealing 76 days at 701’C. 750 x .
pressure measurements. The temperature range of the a//l transformation shown for the pure zirconium and the existence of a three-phase (a + /? + y) region are due to minor impurities as found in previous work.‘1-3) Earlier work(‘) showed the existence of at least two intermediate phases of the approximate compositions Zr,Cd and ZrCd,, both having f.c.c. structures. Lattice parameter The lattice parameters of alloys annealed in the a region were determined for compositions up to 13
F
I _I
FIG. 4. Zirconium-cadmium alloy containing 12.9 at. % Cd after annealing 33 days at 845°C. 250 x .
A microstructure of an a + y alloy containing 13.0 at.% Cd is shown in Fig. 5, illustrating characteristic Widmanstiitten and pearlite structures. Platelets of Zr,Cd form on the basal plane of the hexagonal a phase in the same manner as was observed in the zirconium-indium system.(l) The zirconium-rich portion of the zirconiumcadmium phase diagram is shown in Fig. 6. The a/j3 transformation temperature rises only gradually with an unusual maximum occurring at approximately 13 at. y. Cd. This maximum was later confirmed by
0
5
40 CAOMlUM
15
20
(at. %I
FIQ. 6. The ziroonium-rioh portion of the zirconiumcadmium phase diagram. (Note: Deviations from binary equilibrium are due to the presence of minor impurities.)
FRYE.
JR.
THERMODYNAMIC
et al.:
at.% Cd by the Debye-Scherrer were first annealed,
method.
the u structure
PROPERTIES
Samples
OF
cadmium,(lO)
was confirmed
C, = 5.31 + 2.94 x 1O-3
T
Cal/mole “K
optically and then swaged or hammered into wires. The wire samples were given a 3 min vacuum anneal
and values for the heat of vaporization
in the cc region
the solid metal.
The samples
to remove
were then chemically
0.010 in. for the constants
the cold-work
X-ray
etched
measurement.
can be represented
structure. to about
The
lattice
by the following
equa-
tions :
Alloys
were
0.2892,
(1)
c0 = 5.1473 -
0.0302,
(2)
c/a = 1.5923 + 0.1382,
(3)
where x is the atomic fraction
ium
was
first
by the boiler
rolled
to
foil
(0.10-0.28
Cadmium pressure measurements cadmium
cadmium method.@)
pressures
over
the
zirconium-
by the Hargreaves
the specimen
is held at
temperature T, in equilibrium with a liquid drop of pure cadmium at T, (observation window). The vapor pressure of the alloy at from published
log,, P(mm)
T, is
then determined
values of the vapor pressure of pure
liquid cadmium.
The
experimental
distribution
in the furnace
A thin planar
5709 -
1.1283 log,,
T,,
to represent the pressures of pure
liquid cadmium. The vapor pressure of solid cadmium = 9.7822 -
5715 - T -
This formula
is illustrated
throughout
observation
in
temperature the experi-
window
of silica
at the end of a 1 mm capillary
was taken as
0.2214 log,,
3.212 x lo4
is based on the heat capacity g
1050
e
1000
g
950
2 z
900
f
850
5
800
+
750
of
in a
a tubular
furnace.
A
slight
temperature
in the system,
was required
T.
to concen-
was placed around the window
and the resistance
remained
window
along the axis of the capsule
monotonically
to the specimen,
increasing
for solid
from the
which was held at the flat By adjustments
of the shunts, it was possible to reduce the gradient normal to the window to any desired extent. The
zirconium-cadmium
at a high temperature be detected
alloy
was held
initially
until no further changes could
in the temperature
at which the liquid window.
This period
of time was necessary to establish equilibrium (5)
of
the electric furnace windings was adjusted with shunts
drops formed in the observation
T
temperature
trate the liquid drops upon the observation window for a sensitive observation. A nickel conductor block
between
the gas phase and the alloy, since changing the temperature of the sample from high to low results in a transfer
DISTANCE ALONG t:::.1___....,...... i*+:J’. ;...._:..:. ......’ ... _: :. ‘_ ,_,i_, :..;.,>:. .,
p:
O__--_
axis
maximum in the center of the furnace. (4)
log,, P(mm)
and
gradient,, to ensure that the window was at the lowest
always
T, due to Lumsden(g)
arrangement
so that the gradient
We have selected the formula,
= 11.7750 -
mm)
position normal to the direction of heat flow along the
alloys were determined In this method
de-
spacer strips
Fig. 7, which also shows the sigmoidal ment.
of cadmium.
method
with the exception that the zircon-
between each layer.
was attached The
prepared
scribed previously,
(6)
and entropy of
loosely rolled into a coil using zirconium
a, = 3.2326 -
1187
ZIRCONIUM-CADMIUM
of cadmium
FURNACE
(28 in.) *
from the gas to the alloy foil
.._.__-- ._........ r’_‘_.__... :‘r: ,_.__.. “\\a i FIRE
TELESCOPE SURROUNDED BY RESISTANCE WINDING AND INSULATION
FIQ. 7. Dew-point apparatus.
ACTA
1188
(a 1 758.4 OC
(b) 757.2
6 min
7.5
COOLING
METALLURGICA,
‘C
VOL.
(cl 756.5
min
20,
‘C
(d) 760.5
MAXIMUM
“C
(e) 760.9
63/4 min
CONDENSATION
CONDENSATION
1972
“C
7 3/4 min EVAPORATION
HEATING
FIQ. 8. Photographs of window showing various stages of heating and cooling.
which enriches the surface of the foil. Sutlicient time must be allowed for diffusion to make the alloy foil composition uniform at each shift in specimen temperature. Testing of the equilibrium is done periodically by lowering the window temperature slightly until several liquid drops appear, then reheating until the drops disappear. A sequence of photographs representing the typical appearance of the liquid drops of cadmium during condensation and evaporation is shown in Fig. 8. For greater precision it was necessary to determine both condensation and evaporation temperatures of the liquid drops, since nucleation was delayed on a clear window on cooling and complete evaporation was delayed on heating. Further, the temperature inside the quartz capillary was somewhat higher than in the nickel block due to the need for a small temperature gradient across the window to concentrate the initial condensate. The extent of these effects was investigated by use of a high-purity liquid cadmium bath in place of the specimen, care being taken to reproduce exactly the gradient conditions in the vicinity of the window. A typical compensation chart, or calibration, is shown in Fig. 9, illustrating the effect of heating and cooling rates (taken at
a constant temperature gradient) on the condensation and evaporation temperatures. Based on these results, all the experiments were done at a standard heating and cooling rate of 0.8”C/min. The zero of the curve represents the temperature of the molten bath of cadmium at the opposite end of the capsule. It will be noted that an average of the temperature of condensation and the temperature of evaporation at the rates 0.8’C/min was slightly greater than zero. The deviation of this average from the pure cadmium bath was proportional to the temperature gradient across the quartz window. A correction curve, Fig. 10, was established by means of the pure cadmium to allow empirically for the temperature gradient across the window. All of the window temperatures for the zirconium-cadmium alloys were determined by averaging condensation and evaporation temperatures, observed after a slight cooling and reheating of the window at 0.8’C/min, and then correcting the average temperature by means of Fig. 10. Thermal effusion, or pressure drop along the temperature gradient in the gas phase, is not expected to contribute a large error in these experiments. This is based on thermal effusion relations given by Bichowsky and Wilson.(lz) The atomic weight of cadmium is large, and we assume that the viscosity and Sutherland constant of cadmium gas are not much different than helium. These relations show that thermal effusion effects should be small at high temperatures and in large tubes and should increase as the temperature
_I___~I . ;l.+E,K l-
-to
1
0
I
I
I
I
I
t
2
3
4
5
RATE
OF HEATING
AND
COOLING
(deghinl
FIG. 9. The temperature of evaporation or condensation less the temperature of a high-purity cadmium bath as a function of the heating or cooling rate, respectively.
6
,L 0
xi
i ~~~ ___ t
2 WINDOW
3
4
TEMPERATURE
5 GRADIENT
_._ 6
7
(deg/in.)
Fro. 10. Correction curve to adjust for window temperature gradient.
E
FRYE,
et al.: THERMODYNAMIC
JR.
PROPERTIES
drops and the tube diameter becomes small. For this reason a short length of 1 mm i.d. tubing was used at the window end of the capsule where the temperature gradient was smallest, and the capsule was expanded to at least 4 mm i.d. in the more rapidly rising gradient region. The partitioning of the fixed amount of cadmium between the gas phase and the foil sample occurred relatively quickly for specimens in the cubic ,Bregion of the zirconium-cadmium system at temperatures of 1160-1327’K, but in the hexagonal ccregion, the diffu. sion of cadmium is much slower and surface enrichment or depletions of the alloy occur. Early attempts to estimate the size of this effect by utilizing different approaches to equilibrium temperatures in the a region are illustrated in Fig. 11 where (a) represents approaching the desired sample temperature from a previously established equilibrium at some higher temperature and (b) approaching from a lower temDifficulties in establishing equilibrium perature. were at least partially overcome in the following ways. First, alloys were held in the a region for periods of time as long as S~O-10,000 hr. Second, the temperature range of the a-phase determinations was restricted to a narrow region, 109%1180°K, just under the a/B transformation temperature. Third, the alloy was taken slowly through the (a + /I) region, allowing the major portion of the transfer of cadmium from the gas phase to the alloy specimen to take place in /I grains. Finally, design changes were made in the capsule, where the uniform 25 mm silica tube shown in Fig. 7 was replaced over half its length by 4 mm tubing at the lower temperature end. This second
OF
ZIRCONIUM-CADMIUM
1189
type of capsule minimized the gas phase volume and by also increasing the surface area of the foil by both increasing the mass of the specimen and reducing its thickness, greater alloy surface-to-gas phase volumes were made possible. The log,, cadmium pressures (in mm) over the closepacked hexagonal (a) and the b.c.c. (p) forms of zirconium with up to 15.4 at.% Cd and in the temperature ranges 1093-1327’K are shown in Fig. 12. The break in the curves is associated with the twophase a + /I region. The a/,5 phase boundaries observed by this method are shown by dotted lines in Fig. 6. It can be seen that the two-phase region found by the pressure measurements was much narrower than indicated by the quenched alloy method. This may be due partly to the fact that the lines in the phase diagram are rotated in these alloys due to the small amount of impurities present. There was no certain order (in regard to temperature) in which the experiments were conducted, and it was found that equilibrium could be reestablished with essentially the same pressure for the specimens held in the p region. This result is independent of how long the sample was held at high temperature, thus confirming the absence of import.ant contamination of the zirconium-cadmium alloy during the experiment. Similar results have been found by spectroscopic analysis of the final samples. Neutron activation analyses revealed only slight contamination by silicon and tantalum. The average silicon cont’ent was 42 ppm with a range 4-91 ppm and the average tantalum content was 23 ppm with a range 0.3-98 ppm.
J
0
200
400
600 TIME
800
1000
1200
(hr)
Fra. 11. Variation of the mean evaporation and oondensation ~mperature with time for a 5.2 at.7 Cd alloy. (a) Cooled to 850°C. (b) Efsated to 856°C.
FIG. 12. Experimentally determined log,, cadmium pressures as a function of lIToK.
1190
ACTA
METALLURGlCA,
The cadmium compositions were determined by chemical analysis of at least three locations on the coil of alloy foil. For compositions at temperatures other than the temperature just prior to quenching the alloy, the amount of cadmium in the gas phase was calculated and the alloy composition was obtained by difference of the fixed total cadmium and that in the gas phase. The variations of the alloy composition for sequence of temperature were small, but they were still large enough to warrant consideration in the development of thermodynamic properties. DISCUSSION
In this section of the paper we will firstly calculate certain partial molal quantities, with particular emphasis on the partial molal energy at absolute zero. This will enable us to draw certain conclusions as to the effect of cadmium on the transformation temperature and as to the variation of the partial molal energy with concentration. Secondly, we will examine the implications of these findings for the energies of Wigner-Seitz polyhedra in primary substitutional solid solutions. Thirdly, we will discuss the appropriateness of the rigid band model for such solutions. Calculation of the energy a8 CPK When the alloy is in equilibrium with its gas, we can write
where C, = C, = NI = iVs = P = T =
Gibbs free energy of the gas ; Gibbs free energy of the alloy; No. of atoms of the solvent element; No. of atoms of the solute element; pressure ; absolute temperature.
There are data that indicate that cadmium gas at the temperatures and pressures observed here is nearly ideal and monatomic.(ll) From the temperature and vapor pressure data presented above, we can calculate the energy. This will consist of the kinetic energy of atoms and the product of the pressure multiplied by the volume. The kinetic energy is zero at absolute zero when the atoms are infinitely separated. If the atoms are confined in a finite volume, they will have a finite energy which can be calculated by use of the Schroedinger equation. The pressure-volume product will also be finite. Thus the reference state for zero energy is that in
VOL.
20,
1972
which the atoms are infinitely far apart and at absolute zero. Fo~unately, the energy at absolute zero for the gas densities of interest here is negligible compared with the energy at elevated temperatures. Thus we can use an equation for the free energy of each species which neglects this energy and which contains other approximations, but which is simple to use and quite accurate enough for our purposes.
The logarithmic fraction is the partition function for an ideal monatomic gas.(is) T
1143 C,==ES_
C,dT/T
fl,dT-k
s0 -
T
C,dT
s 1143
-
T
1143 s 0
T
s
C, dT/T 1143
(9) where R k m N, N, V h E C,
= = = = = = = = =
gas constant; Boltzmann’s constant; mass of one atom of solute; No. of solvent atoms; No. of solute atoms in the gas ; volume of the gas ; Planck’s constant; energy of the alloy at O*K; heat capacity of the alloy at constant pressure.
Equation (9), which is due to Jones,fi4) implicitly assumes completely random mixing of solute and solvent atoms. This is not perfectly achieved in real systems. Thus, the last term will not exactly calculate the configurational entropy. The error will, however, approach zero as the concentration approaches zero. The reference state for E is that of the atoms at infinite separation at absolute zero. In other words, it is the energy released when these atoms are brought together from infinity at absolute zero to form the alloy. Later in this discussion we will be concerned with the quantum mechanics of solid solutions. Here the reference state for zero energy must be the nuclei and electrons at infinite separation at absolute zero. This is readily obtained from the present reference state by subtracting the ionization energies of the solute and solvent atoms.
FRYE.
JR.
et al.:
THERMODYNAMIC
PROPERTIES
C, is zero at absolute zero. Upon taking the partial of equations (8) and (9) with reference to N,, equating and rearranging, we get ln
h3fW, + N,)
a T
1
(2n-m)3/2(kT)5/2N2 - %? G T C,dT 1 a ' aN, f1143 T = RT
-
OF
a2E
axaN
+
ZIRCONIUM-CADMIUM
s
a2
1143
o
axaN
aT
+
s1143'
‘p dT = axaN
a2E -= axaN
--R aN, s o
9. T
(10)
The literature has been searched, although not exhaustively, for vapor pressure data in other primary substitutional solid solutions. These data(15-22) together with our experimental data have been used to calculate the left side of equation (10). This quantity has then been plotted against l/T for constant x for these various systems, where x is the atomic fraction, N,/(N, + N,). From these plots have been evaluated 1143
C,dT
(11)
and
1143C dT 2,. T
(12)
Values of these functions have been plotted against the atomic fraction, x, in Figs. 13 and 14. For most of the solutions it can be seen (Figs. 13 and 14) that
FIG. 13. [g
+
= 0. (13)
(14)
O.
It is not possible to obtain aE/aN, 1143C dT
T
a
From the Debye theory it follows that
+ -&jo1143CDdT] 1 a
s
C &T
1 R
1143
a2
8 c
1191
a[ Jo1143cp a,]
without knowing
/aa,.
The latter can be calculated from a[ [1143C, dT/Tl I-J”
/aN, -II
by use of the Debye model. Nernst has provided tables which we have used for this purpose.(B) Similar tables can be found in Lewis and Randall.(2Q) Values of aE/aN, are listed in Table 1. These calculations ignore differences between C, and C,. As a check on accuracy we have also calculated values for the pure metals and compared these with experimental measurements.(25) It can be seen that the errors are small and consistently in the same direction, thus lending confidence to values calculated for the various solutes. It would seem to be a safe conclusion that aElaN, is more negative for cadmium in /l than in a zirconium. Thus, we conclude that cadmium tends to stabilize p at absolute zero, but that this is opposed by the
Cd
--
Cd Cd Zn-Zn Cr Ag
~ -~ ___ __ ~--
Ag /SZr aZr Ni cu Fe Au
--
& /;143Cp dTIT,.( l/ Ill011.3 0x3 ) as & function of concentration.
1192
ACTA
0
0.1
METALLURGICA,
0.2
0.3
x
VOL.
0.4
At 1143°K the latter overbalances between
0 and
the
0.11
mole
fraction
Duhem
follows
from
that a2E/axaN,
= 0.
=
equation
0,
it
tions apply to most of the solutions
the
Gibbs-
These condi-
depicted
in Figs.
13 and 14. We can write for these solutions
E = (1 - x)aE/aN1 + xaE/aN2 a2E
solute (see Henry’s
Crystal
law).
-26,800
- 145,500 -143,500 -68,000 0.76 -87,200 -93,800 0.50 -97,000 -31,100 0.32 0.28
* From Nernst tables.
(15)
molal energy of the pure
0.11 0.15 0.37
- 102,200 -80,600
a new reference state for the energy.
define energy,
L, of the crystal
at absolute
We
zero as
being the energy which results from bringing together the nuclei and electrons form the crystal.
out that aE/aN, is not neces-
TABLE 1. Comparison of measured and calculated values of aElaN, aE/ aN, (Cal/mole) Maximum Calculated* Measured zz
Cd Cd in a-Zr Cd in @-Zr Cd in Ag Ci-Zr tAg in Au AU Cr Cr in Fe Fe Zn Zn in Ni Zn in Cu Ni cu
the nuclei and the electrons in the crystal and there-
from infinite
Furthermore,
separation
to
it is convenient
at
this point to express L in terms of the energy/atom.
(16)
axaN, -==2=O.
It should be pointed
polyhedra*
L is related to E by the equation
a2E
sarily equal to the partial
of concentration.
Energy of Wigner-Seitz
fore need
atqaxaN,
0.7
We shall now be concerned with the energy of both of
cadmium. If
0.6
as a function
former and there is a small increase in transformation temperature
1972
0.5
Fm. 14.
entropy effect.
20,
-26,000 -27,800 -30,100 -31,200 -144,500 -67,800 -68,900 -86,600 -92,300 -93,000 -96,100 -30,200 -46,800 -37,300 - 100,900 -80,100
L = E/(N, + N,) -
(1 -
x)1, -
x12,
(17)
where I, = ionization
energy of the solvent atom;
I, = ionization
energy of the solute atom.
It can be seen from the data presented earlier that EI(N, + N,) is a linear function of concentration. Thus, L is a linear function of concentration. The
quantum
mechanical
virial
theorem(26,27)
enables us to express the total energy of the crystal in terms of either the expectation energy (K) or the expectation energy, (W). The equation is L = -(K)
value of the kinetic
value of the potential
= 0.5(W).
(18)
This theorem has been known since 1930. It is now often used in a priori quantum mechanical total energy calculations.(29~30) The fact that we can express total
energy
in terms of either kinetic
or potential
* “The Wigner-Seitz cell is obtained by drawing the planes bisecting the link joining an atom to its neighbors until they enclose a volume. “w) Thus the crystal will be divided up into polyhedra such that there will be one polyhedron surrounding each atom.
FRYE,
JR.
et al;.:
TRERMODYNAMIC
PROPERTIES
energy will be useful in the qualitative discussion which follows. Consider any random substitutional solid solution composed of numbers 1 and 2 atoms. We divide the crystal into feigner-Seitz polyhedra. The expectation value of the kinetic energy of the entire crystal can be written as
OF
ZIRCONIUM-CADMIUM
1193
suggest that (K,) and (K,) in equation (19) are approximately independent of concentration and not greatly different for the values in the pure crystals.
Relationships between the hexagor~al c/a ratio, the transformation temperature and TX,the electronicspecific-heat, coefhcient on the one hand and the concentration of quantum number 5 electrons on the other for the solutions Zr-Ag, Zr-Cd, Zr-In, Zr-Sn and (K,) = expectation value of kinetic energy of solZr-Sb are summarized in Figs. 1-3.(1-s) These simple vent polyhedra ; relationships suggest that the quantum number 5 (K2> = expectation value of kinetic energy of electrons may be entering the conduction band of solute polyhedra. zirconium without altering that band. If we assume The expeetation value of the potential energy of such a rigid baud model, f-5’ecan calculate t>hedependthe entire crystal can be written as ence of the energy of these electrons on concentra0.5(W) = 0.5ffl - z)(W,> + x(Wz> + {I - %)2(lY11) tion by use of the data in Fig. 3. This is of particular + 2X(1 - $(W& + za(Wa,>] = L, f2OY interest for the Cd-Zr alloys, since we know how their total energy varies with concentrat,ion. where @‘i,>, (11’12>and (W,,) are the expectation The density of electronic states is proportional to values of the potential energy due to interaction the low-temperature, electronic specific heat, tl, between solvent and solvent polyhedra, between provided there is uo coupling between electrons and solute and solvent polyhedra and between solute and phonons. We ignore such coupling : but, as we shall solute polyhedra, respectively, see later, the conclusion which we draw would be Many years ago, Mott and Jones@i) argued that strengthened if it had been taken into account,. (W,,) should be small in a pure metal. The data The data of Fig. 3 can be represented by a straight presented above suggest that it and (VI,) and (Wz,) line with suf%ient accuracy for present purposes. are small in some solid solutions. The arguments are We write as follows : ~~~~~ = a “r ~~~~~, f21) 1. It was pointed out above that & is a linear funcwhere tion of concentration. One way in which equat8ion A -2 No. of states; (20) can be realized is for (W,,), (W& and ( W,,) to U = Fermi energy; be small in comparison with E/(X, + N,) [see equa2 = atomic fraction ; Lion (17)]. y -= No. of quantum number 5 electrons. 2. For most of the solutions which we have cousidFor cadmium ered here, the energy required to remove a solute y-_2
atom from a dilute solution is little different from
that required to remove it from its pure crystal even though the solvent atoms are much more tightly bound (see Table 1). This is hardly possible if the binding is due solely to interaction between palyhedra. Thus, it would appear that ( WX1>,( Wiz) and (W& are small in comparison with Ej(.iV, + .N& These arguments are not conclusive, but they do * Equation (19) contains no t,erms for interaction between
polyhedra. Equation (20) does. This difference stems from the difference between the kinetic energy and potential encrgy operators in the Hamiltonian. The kinetic energy operator is of the form - (ha/2na)X,B,e, where ala= mass of t,he particle. The potential energy aperator is of the form
&&a =
drtt.
assume that the two quautum number 5s electrons enter a rigid conduction band in zirconium. Then
We
U=
i
0G$
(221
+ tj’ll
where U, = Fermi energy in pure zirconium, jrJ
=2 hl I1 + b
W)xl +
u
1’
(231
Let D = energy, per atom of cadmium, of the electrons lying above U,. where Ba and 2~ are the charges on the cc and @particles, respectively, and R?. and Rg are t,he position vectors of the ct and /? pa&i&s, respectively.
ACTS
1194
METALLURGICA,
The integral gives the energy in terms of the energy~ atom of alloy. We divide by x to get the energy/atom of cadmium. We do this in order to have an easy comparison with aElaN,.
11 + WMlln Cl+ W44
i7 ,2
+ 2u
- (VG
b”/u
X1
I
1'
P-3
In zirconium, 1 + (b/a)x has a maximum value of 1.13 for 11% Cd. Thus, ln [l + (b/a&] firr(b/a)s. i7 Gw2x/a + 2lJ,,
(26)
where a is equal to 0.6 statesjev at. Then U = (3.32 i_ 2U,) eV~atom = (‘77,000~ + 2U,) cal/mole,
(27)
when x = 0.11, 0 = 0.36 eV/atom + 2U, = 8300 Cal/mole + 2U1.
(28)
Since aE~aN~ plus the appropriate ionization energy for the free cadmium atom is the total energy of one mole of cadmium in zirconium at absolute zero, it must contain 0. The variation of 0 with concentration is sufficiently great to give an observable positive slope to the curve for cc zirconium in Fig. 13. No such slope is observed. Furthermore, an even greater effect would be indicated if coupling between electrons and phonons had not been ignored. Analogous calculations for zinc in copper and for chromium in iron-assuming zinc adds one electron and chromium subtracts two electrons-yield g&l-zn = (78,400~ + 39,200G) callmole + V;,
(29)
=QJFe-Cr = (-46,000~
(30)
+ 23,000~~) cal/mole -
ZU,.
Again, no such variation is observed in the appropriate curves of Fig. 13. Using a free-electron gas model, analogous calculations have been made for zinc in nickel, cadmium in silver and cadmium in cubic zirconium. It is assumed that these solutes, respectively, increase the number of conduction electrons from one to two, from one to two and from four to six. The resulting equations are Dx;i-zn = (92,0002 + 137,000) cal/mole,
(31)
O,,,, -
= (83,400~ + 125,000) Cal/mole,
(32)
(170,000~ + 508,000) cal/mole.
(33)
UZr_Cd =
VOL.
20,
1972
Again, no such variation is observed in the appropriate curves of Fig. 13. The curve in Fig. 13 for silver in gold, where no valence effect would be expected, shows no appreciable variation. For these systems, it must be concluded that the rigid band approximation is not a good one. It is possible to be more general. Darken and Gurry(s2) say of Henry’s law : “‘it is found empirically that, if the concentration of the solute (conventionally designated component 2) of any binary solution is sufficiently low, then at constant temperature the partial pressure of the solute is proportional to its mole fraction.” They also state ‘<. . . the gas phase behaves ideally, as it usually does at low concentration and hence at low partial pressure of component 2, . . . .” When these conditions are met, then the energy and the temperature dependent entropy of the solute are independent of concentration. The energy at zero Kelvin is independent of concentration. The equation for the rigid band model is Da = 2
“dU “&& -ad?&+ s0 [S 0 dn
1
U, .
(34)
The term in square brackets is merely the Fermi energy. Henry’s law would predict ~)~=s=
($j),=,=
0.
(35)
This is in contradiction to the rigid band model. These conclusions are in accord with recent density of state calculations. For example, Stocks et c~1.f~) have calculated the densities of states of the nearly random alloy system copper-nickel for a range of concentrations of constituents across the complete alloy diagram. They conclude that: “for all Cu-Ni alloys the rigid band model and the virtual crystal approximation are quite inappropriate.” CONCLUSIONS
Lattice parameter measurements are in agreement with other solutes with quantum number 5 electrons in that the proportionality of the axial ratio to the number of such electrons is due to a change in the c spacings with the a dimension sharing a common curve. Slopes of c/a, ([d{e/a)/dz],=,), increase linearly with the number of quantum number 5 electrons in the solute for Ag, Cd, In, Sn and Sb. Phase boundaries determined by the dew-point method agree quite well with those obtained by the isothermal anneal and quench method. There is a
FRYE.
rather
JR.
unusual
solid solution
et al.:
maximum at about
THERMODYNAMIC
which
13 at.%
found in the lead-thallium
occurs
PROPERTIES
in the
a
Cd, similar
to that
Solubility
as high
system.
as 17 at.yb Cd exists in both the a and b phases. Slopes of the a//3 phase boundaries,
increase
monotonically
number
5 electrons
and Sb.
transformation This
is the
with the number
balanced
from
of quantum
the temperature
hexagonal
resultant
effects-an
to b.c.c.
of two
opposite
absolute
zero
zirconium. effect entropy
an
alloy at zero Kelvin
an
form.
is given by
the equation,
E =
-_(l
-
z)[(K,)
The data which for
some
have
primary
(K,)and (K,)are and
are
-
II]
-
s[(K,)
(36)
I,].
been considered here imply that, substitutional
solid
nearly independent
not very
-
different
from
solutions,
of concentration
the
corresponding
considered
here.
for the
Furthermore,
it is
incompatible
with Henry’s law which is widely obeyed
by metallic
solid solutions.
This
accord with recent band-energy
conclusion
is in
calculations.
ACKOWLEDGEMENTS
The authors Faulkner,
are grateful
G. S. Painter
to J.
Brynestad,
and G. M. Stocks
J.
S.
for many
helpful discussions.
of Technology, -_
10. 11.
12. 13.
::: 16.
S. EASTON and J. 0. BETTERTON, JR., Metallurgy Division Semi-annual Progress Report ORNL-2217, p. 216 (1956). R. HARGREAVES, J. Ilast. Metals 64, 115 (1939). J. LUMSDEN. !Chermodunamics of Allows, D. 134. Institute of Metals (1952). ” ” ” _ K. K. KELLEY, U.S. Bureau of Mines, Bulletin No. 476 (1949). R. HULTGREN, Selected Values of the Thermodynamic Pronerties of Metals and Allovs. Mineral Research Laborato&, University of Californfa i1956). F. RUSSELL BICHOWSRY and C. W. WILSOS, Phys. Rev. 33,851 (1929). E. U. CONDON and HUGH ODISHAW, Handhook of Physics, pp. 5-11. McGraw-Hill (1958). H. JONES, Proc. Phys. Sot. 49, 243 (1937). C. L. MCCABE, H. M. SCHADEL, JR. and C. 13. BIRCHENALL, J. Metals 5, 709 (1953). G. SCATCHARD and R. H. BOYD. J. Am. Ch.em. Sot. 78,
3889 (1956).
17. C. E. BIRCHENALL and C. H. CHENG, Trans. metall. Sot.
A.I.M.E.
185, 428 (1949).
18. E. E. UNDERWOOD and B. L. AVERBACH, J. Metals 3. 1198 (1951). 19. G. SCATCHARD and R. A. WESTLUND, JR.. J. .4m. Chem. sot. 75, 4189 (1953). 20. 0. KUBASCHEWSKI and G. HEYMER, Acta Met. 8, 416 21. A. W. HERBENAR, C. A. SIERERT end 0. S. DUFFENDACK, Trans. Metall. Sot. A.I.M.E. 188, 323 (1950). R. HARGREAVES, J. Inst. Metals 64, 115 (1939). W. NERNST, The New Heat Theorem. E. P. Dutton (1926). ;:: LEWIS and M. RANDALL, Thermodynamics. 24. G. N. McGraw-Hill (1961). 25. R. R. HULTGREN, R. L. ORR, P. D. ANDERSON and K. K. KELLEY, Selected Values of Thermodynamic Properties of Metals and Alloys. John Wiley (1963). 26. V. FOCK. 2. Phusik 63. 855 (1930). J”. Mol. Spect&. 3(i), 46-66 (1959). 27. P-O. L&DIN, and Phonons, p. 4. Oxford Univer28. J. M. ZIMAN, Electrons sity Press (1960). and K. W. JOHNSON. Phus. Rev. Third 29. M. Ross Series, B, 2, 4709 (1970). 30. A. ROTHWORF, Phys. Rev. Lett. 29A (1969). 31. N. F. MOTT and J. H. JONES, The Theory of the Properties of Metals and Alloys. Dover (1958). 32. L. S. DARKEN and R. W. GURRY, Physical Chemistry of Metals, p. 256. McGraw-Hill (1953). 33. G. M. STOCKS, R. W. WILLIAMS and J. 8. FAULKNER, Phys. Rev. B4, 4390 (1971). ”
REFERENCES I. J. 0. BETTERTON, JR., J. H. FRYE, JR. and D. S. EASTON, Phase Diazmm Studies of Zirconium with Silver, Indium and Anti
Sot. A.I.M.E.
Institute
(1960).
energies in the pure crystals. The rigid band model is not appropriate solid solutions
8. 9.
nearly
energy
effect which tends to stabilize the hexagonal of
of the
and
which tends to stabilize the cubic form and The energy
Illinois
7. D.
in the solute for Ag, Cd, In, Sn
slightly increases
Foundation.
p. 32 (1952).
The relation is linear for Cd, In, Sn and Sb.
Cadmium
1195
ZIRCONIUM-CADMIUM
3. D. S. EASTON and J. 0. BETTERTON, JR.. Metall. Trans. 1, 3295 (1970). 4. J. 0. BETTERTON, JR. and J. H. FRTE, JR., Acta &Zet. 6, 205 (1958). 5. G. D. KNEIP, JR., J. 0. BETTERTON, JR. and J. 0. SCARBROUOH, Phys. Rev. 131, 2425 (1963). 6. M. HANSEN and D. J. MCPHERSON, Report COO89 Armour Research
+ Tp)/2~~I,=&
WT,
OF
212,470 (1958).