LOW-TEMPERATURE SPECIFIC HEATS 0F
FACE-CENTERED CUBIC RUTHENIUM‘RHODIUM 7
AND RHODIUM-PALLADIUM ALLOYS '

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
PAUL JA~MIN TSANG.
1968

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6w
LIBRARY

Michigan State
University

       

This is to certify that the

thesis entitled

LOW-TEMPERATURE SPECIFIC HEATS OF FACE-CENTERED

CUBIC Ru-Rh and Rh-Pd ALLOYS

presented by

Paul Ja—Min Tsang
has been accepted towards fulfillment
of the requirements for

AL degree in meL‘LJ-UI'Q

L

f / A / .
y/x’lL/VV'JMA g/kLu! L62 (QC/g,

Major professor

Date April 1;. 1968

e

0-169

 

ABSTRACT

LOW—TEMPERATURE SPECIFIC HEATS OF FACE-
CENTERED CUBIC RUTHENIUM-RHODIUM AND
RHODIUM-PALLADIUM ALLOYS

by Paul J. M. Tsang

The specific heats of a number of face-centered
cubic ruthenium-rhodium and rhodium—palladium alloys were
determined between 1.40 and A.2°K. Taking the Fermi level
of palladium as a reference, the density of states, N(E),
was deduced from the electronic specific heat coefficient y
of each of the alloys, and was plotted against the energy
E. The N(E) versus E curve thus constructed shows features
in qualitative agreement with the total density-of-states
curve calculated for the 3d bands in face-centered cubic
nickel by G. F. Koster, and that for the Ad bands in face-
centered cubic palladium by P. Lenglart e£_al, as well,
both of the calculations using the approximate method of
Linear Combination of Atomic Orbitals, the L.C.A.O. method.
The present results indicate that for the same face-centered
cubic structure the general features of the Ad bands in the
second—long-period transition metals are similar to those
of the 3d bands in the first-long-period transition metals,
and that the L.C.A.O. method for treating the d-electrons

in the transition metals is essentially correct.

LOW-TEMPERATURE SPECIFIC HEATS OF FACE—
CENTERED CUBIC RUTHENIUM-RHODIUM AND

RHODIUM-PALLADIUM ALLOYS

By

Paul Ja—Min Tsang

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of
Metallurgy, Mechanics and Materials Science

1968

ACKNOWLEDGMENTS

My sincere gratitude is due to Professor C. T. Wei,
for his patient guidance and advice. I am indebted to my
colleague Mr. A. V. S. Satya for his assistance in many
experimental works and to Dr. C. H. Cheng of Dow Chemical
Company, Midland, Michigan for lending me drawings of an
arc furnace used in this investigation. Acknowledgments
are also due to Mr. J. w. Hoffman and Mrs. Barbara Judge
of the Division of Engineering Research, College of
Engineering for their cooperative assistance. Finally, I
wish to express my thanks to National Science Foundation

for the financial support of this investigation.

ii

.7

TABLE OF CONTENTS

ACKNOWLEDGMENTS .

LIST OF TABLES,

LIST OF FIGURES .

Chapter

I.
II.
III.
IV.
V.

INTRODUCTION . . . .
EXPERIMENTAL PROCEDURE .
RESULTS AND CALCULATIONS
DISCUSSION . .
CONCLUSIONS.

REFERENCES. . . . . . . . . .

APPENDICES. . .

APPENDIX A--Experimental Data

APPENDIX B--Computer Programs

111

Page
11
iv

in
38
55
7a
76
81
83
97

Table
11.1
II.2a
II.2b

III.1

IV.1

IV.2

LIST OF TABLES

Impurity content of Rh, Ru, and Pd metals.
Components of measuring circuit
Positions of switches for various functions

Values of y and e of Ru—Rh and Pd-Rh alloys
(least square fit: Cv/T - y + 8T2). . .

Coefficients of the least square fit to the
equation: Cv - A + 7T + 8T . . . . .

Coefficients for the equation:

Cv - BT'”2 + {T + 8T3 . .

iv

Page
18
32
33

us

61

62

Figure
1.1

2.3
2.A
2.5

2.6
2.7
2.8
3.1

3.2
3.3
3.A
3.5
3.6

LIST OF FIGURES

Degeneracies in k-space due to the pertur-
bations of the periodic lattice potential

A pattern of energy levels when total
hybridization occurs . . . .

Energy band for 3d Cr. (Asdente et a1.)

Upper part of Ad-band deduced from 7 values
of Rh-Pd and Pd-Ag alloys (Hoare).

Arc furnace. . . .

Debye—Scherrer patterns of Rh and Ru-Rh
alloys; lattice constants of the Ru-Rh
alloys as a function of Ru content in Rh

Cryogenic system . . . .

Cryostat system . . . . . . . . .

(a) Sample Assembly; (b) Heater—Thermometer
copper body. . . . . . . .

Essence of the electrical circuit
The electrical circuit .
Experimental time--temperature curves

Calibration curve for the thermometer used
in sample PT-lS . . . . . . .

Cv/T vs T curves of pure Rh and Ru-Rh alloys
Cv/T vs T curves of Rh-Pd alloys
Cv/T vs T curves of Rh-Ru alloys
Cv/T vs T curve of Pure Pd . . . .
The experimental results of 7 vs T2 of 22at.%

Pd-Rh and pure Rh in two experiments
A and B . . . . . . . . . .

V

Page

15

2O
22

23

27
27
3O
37

HO
A3
an

A5
A6

53

Figure Page
h.l 7 vs e/a curve. . . . . . . . . . . 65

A. 2 To find energy intervals, AE, by numerical
integration . . . . . . . . . 67

4.3 The upper part of the Ad-band deduced from
the y values of the fcc solid solution
alloys of 2nd long period transition metals
i.e., Ru-Rh, Rh-Pd, and Pd-Ag alloys. . . 68

vi

CHAPTER I
INTRODUCTION

Owing to the complexity of the physical and
chemical properties of the transition metals, divergent
theories have been proposed for the electronic structures
of the transition metals since Slater and Stoner's initia-
tion of the search [1, 2]. Among them four major models
stand out, namely, Slater and Stoner's band or itinerant
electron model, Pauling's valence bond model [3], Van
Vleck's minimum polarity model [A], and Mott and Steven's
s-d model [5]. In Slater and Stoner's band model, the d-
electrons are regarded as being itinerant and describable
by Bloch functions. Energy bands form due to the pertur-
bation of the lattice potential. Such properties of the
transition metals as the conduction by d-electrons, a large
low-temperature electronic specific heat, and a high cbhe—
sive energy, can all be explained well by this model.
However, in its earlier form, the theory is not adequate in
interpreting the occurrence of ferromagnetism in Fe, Co,
and Ni. Pauling's valence bond model is an extreme example
of localized d-electrons. In this model a d-electron of one
atom Joins with a d-electron of a neighboring atom forming

a singlet state. They are thus not coupled to other electrons

of their own atoms. By assigning proper spins of the d-
electrons, Pauling was able to explain the saturation
magnetic moments of Fe, Co, and Ni in an empirical way.
However, except the cohesive energy, Pauling's theory does
not explain the other important properties of the transi-
tion metals.

The essence of Van Vleck's minimum polarity model is
the forming of a total energy level by the hybridization of
the various states localized at the individual atoms. For
example, if the ground state of a free atom of a particular
transition element is 3d8bs2, the total ground state of
this element in the metallic state would be a mixture of 3dlo,
3d9, and 3d8 atomic states. This model is essentially a way
of putting a correlation into the itinerant description of
the d—electron; the energy band is formed by the inter-
atomic exchange of the atomic states.

The s—d model was first proposed by Mott and Stevens,
and was modified by Lomer and Marshall [6]. There are three
main features in this model: (a) The ns-electrons in the
outermost orbit of a metal atom are free, part of the
(n-l) d—electrons are free and part of them are bound. The
itinerant d-electrons belong to t2 symmetry, while the
bound ones belong to Eg symmetry. g (b) If an integral occu-
pation occurs, the Eg electrons will carry most of the mag-
netic moment of the metal. (c) The exchange coupling between

the itinerant s-electrons and the bound d-electrons gives

rise to a non—integral magnetic moment of the metal. This
exchange coupling is also ascribed as being the cause of
the large cohesive energy of the transition metals.

In spite of the discrepencies between the above
models, the band or itinerant picture enters into each of
these models in one way or another [7]. Several approxima-
tion models have been used to calculate the energy band of
the transition metals. Krutter [10] and Slater [2] first
used the cellular method to calculate the combined s-d
band of Cu. A narrow d-band with a high density of states
was found to overlap with a broad s-band. Since the high
Speed computer became available, the augmented—plane—wave
(APW) method [8], originated from the cellular method, has
been proved to be one of the most powerful tools for solving
the energy—band problems of solids. The APW method was
first proposed by Slater [ll] in 1937 and modified later by
Saffren and Slater [12, 13]. Using the APW method. the
energy bands of most of the first long period transition
metals have been calculated by various investigators: the
energy band of Cu by Burdick [1A],that of bee Fe by Manning
[15], fcc Fe by Green [16], and Ni by Hanus [l7]. Louks [18],
using the APW method, calculated the Fermi surfaces of Cr,
Mo, and w. In order to see the general trend of the energy
band changing from one element to the next in the iron
series metals, namely,the elements Ar, Co, Ni, Cu, V, Cr,
Fe, Ti, and Zn, Mattheiss [19] used the APW method to calcu-

late the energy bands of these metals along a particular

direction in the first Brillouin zone. He found that a
rigid band model could be applied to alloys of the transi-
tion metals. The energy bands of bee and fcc iron were
recalculated by Wood [20]. The density—of-statescurve
obtained by Wood agrees qualitatively with that deduced
by Wei, Cheng, and Beck [21] from their low-temperature
electronic specific-heat data.

Fletcher first used the tight-binding (T—B) method
to calculate the 3d band in Ni. The method was simplified
by Slater and Koster in 195A [2A]. Since then it has been
used extensively for calculating the d-bands of the transi-
tion metals. The band structures of cubic transition metals
were calculated by Slater and Koster [2A, 25] using their
simplified T—B method. Belding [26] modified the calcula—
tion by introducing the interaction between next nearest
neighbors and found a broadening in the lower part of the
energy band for the bcc structure. The energy band for the
fee structure remained unchanged in Belding's result. Other
transition metals, the energy bands of which have been
calculated by using the T—B method, are Cr by Asdente and
Friedel [27], bcc Fe by Abate and Asdente [28], and Ni by
Yamashita et_al.[29]. The Fermi surfaces and the density—
of-states curves of Pt and Pd were calculated by Friedel et
al. [30, 31], taking into consideration the effect of the
spin—orbital coupling.

Regardless of the approximation methods used in the

calculations, the resulting d bands show some common features.

The five d wave functions of a free atom have two types of
symmetry. Three of the five have the yx, xz, and zy type
of symmetry, the other two have the x2-y2 and y2—22 type.
In both the bcc and fee transition metals the crystal field
causes the five d wave functions to split into two levels
at the origin of the k-space, the two x2—y2 type wave func—

tions become the higher energy states, the P states, and

12
the three xy type wave functions becometfluelower, F25,.

The degeneracies of these states will be removed totally or
partially as k moves away from the origin. Due to the per—
turbations of the periodic lattice potential, the electron
energy states form five overlapping sub-d—bands crossing

each other in the middle of the first Brillouin zone leaving
degeneracies at certain points in the k-Space. As shown in
Fig. 1—1, along certain directions in the first zone partial
or total hybridization between these sub—levels occurs. When
total hybridization occurs, a pattern of energy levels shown
in Fig. l-2 will result. The two lowest bands will be made
of bonding functions with the xy symmetry at the center of
the zone, and of bonding functions with the x2-y2 symmetry

at the boundary. The two highest bands will be made of
antibonding x2-y2 functions at the center, and of antibonding
xy functions at the boundary. The remaining band will be

of bonding type with the xy symmetry at the center of the
zone, and of the antibonding type functions with the xy
symmetry at the boundary. This band will be broader than

the others. The extent of this broadening depends upon the

E5. ' ’

 

 

-——v-k

Fig. l-l.--Degeneracies in k-space due to the perturba-
tions of the periodic lattice potential.

 

 

 

Fig. 1-2.--A pattern of energy levels when total
hybridization occurs.

separation AE of F12 and P25,.

deduced from this general picture would thus be expected to

The density—of—states curve

have a minimum or a "valley" in the middle of the band [32].
Figure l-3 shows the density-of—states curves for the five
3d subdbands of Cr calculated by Asdente and Friedel [27].
This feature of having a "valley" in the middle of the d-
band is prominent in the bcc transition metals calculated by
using the APW or the tight—binding method. For the fcc
metals, although this "valley" is still clearly seen in the
d—band calculated by using the T-B method [25], it is,
however, not clearly shown in that obtained by using the
APW method [1A, 26]. A new interpolation scheme for treating
the electron energy bands particularly in the transition
metals was proposed recently by Hodges et_al. [33], and
Mueller [3A]. They treat the s—p conduction electrons with
the pseudo-potential method and the d-electrons with the
tight-binding method separately, and used the proposed
interpolation scheme to obtain a hybridized total energy
band. Except for a broadened "tail," the features of the
hybridized energy band obtained by them for either the bcc
Mn or foe Cu are similar to those obtained by using the APW
or T-B method.

There are several experimental methods that have been
employed successfully in determining the Fermi surface of
high purity metals, such as de Hass—van Alphen method, high

field magnetoresistance method, Knight shift [5A], and

 

 

 

 

 

 

Fig. 1-3.-—Energy band for 3d Cr. (Asdente et al.[27]).

 

 

 

4.-——-

 

 

no. I (0v: A E!

Fig. l—A.—-Upper part of Ad-band deduced from y
values of Rh-Pd and Pd—Ag alloys (Hoare [53]).

 

ultrasonic attenuation method [35]. For studying the
structure of the electron energy band in metals the soft
X—ray technique and photoemission technique [36] are also
used. However, owing to the effect of impurity scattering,
these methods are not suitable for studying alloys.

The possibility of separating the total Hamiltonian
of a crystal into independent lattice and electronic Hamil—
tonians by using Born and Openheimer's adiabatic approximation
enablesone to regard the total heat capacity of a crystal as
being essentially a linear sum of the various contributions
of respectively the electrons, phonons, magnons, nucleons,
and the interaction effects between them. At elevated
temperatures, as the lattice specific heat is the major
contribution, the heat capacity of simple solids such as the
metals obeys Dulong and Petit law, and has a more or less
constant value of 3H. At temperatures below 6/50, the
total heat capacity of a metal or an alloy can in most cases
be written as a sum of the lattice and the electronic heat

capacities.

CV = 8T3 + yT (1-1)

The T3 term in the above equation is, according to Debye
theory [1], the lattice specific heat. The coefficient 8 is

related to the Debye temperature 8 by

1/3
12W 1
e = <—5——> R <5) (1-2)

10

where R is the gas constant. The term linear in T is the
electronic specific heat. It was first derived by Sommer—
feld [38] based on an energy-band concept, and was found

that y for a free electron gas should be

y = % n2k2N(Ef) (1—3)
where N(Ef) is the density of states at the Fermi surface
at the absolute zero of temperature. The electronic specific
heat of a metal with an arbitrary band shape has been
studied by Stoner [39]. He found that due to the irregular
shape of the band, an additional term dependent on T3 should

be included in the heat capacity. The coefficient Y becomes

. 2
2 2 CuIINEf) N'(Ef)

= :2?" k2N(Ef) {1+6(kT) [6-5- W - (32(m) + ""]}

(l-A)

2
d d
Where N'(E ) = [-— N(E)] _ , N" (E ) = [-—— N(E )] _ :
f dE E-Ef f dE2 E—Ef

2

u
02 = " /12, and Cu = 7" /720.

The above review suggests that if the energy band con—
cept is applicable to metalsand alloys the value of y will
be directly proportional to the density of states at the
Fermi level of the corresponding energy band. The additional
term in T3 will be small unless N'(Ef) and/or N"(Ef) is very

large. Thus the value of y may give a reasonable estimate of

 

ll

N(Ef). Although the slope and the curvature of the density
of states curve might affect the deduced Debye temperature,
6, their effect on y and hence N(Ef) will be small. This
suggests the possibility of using the heat capacity measure-
ment at low temperatures as a meansfor determining the
electron-energy-band structure of metals and alloys in con—
junction with a "rigid-band" model.

The rigid-band model implies that neither the shape
nor the height of the energy band of a matrix metal would
be changed by alloying, provided that the alloy is a solid
solution, and that the crystal structure remains the same
as that of the matrix metal. The only thing that would be
affected by alloying is the number of conduction electrons
and hence the position of Fermi level of the matrix metal.
The addition of an alloy element with a valency higher than
that of the matrix metal will add more electrons to the
energy band of the matrix metal, and hence raise the Fermi
level. On the other hand, the alloying of a matrix metal
with an element of lower valence number will lower the Fermi
level. However, the results of recent specific—heat measure-
ments of alloys of the noble metals [A1, A2, A0] seem to
contradict this model. But theoretical band—structure
calculations [A3, AA] as well as some experimental results
obtained with alloys of the transition metals [A5]
support the validity of the rigid band model in at least
those alloys of neighboring transition elements in the

periodic table.

12

Low temperature specific heats of binary bcc and fcc
solid solution alloys of the first long period transition
metals have been investigated by Wei, Cheng, Gupta and Beck
[21, A5]. The N(E) versus E curve obtained by them agrees
qualitatively with that obtained from the tight-binding
calculation. As for the alloys of the second long period
transition metals the magnetic susceptibilities and the
heat capacities of Pd-Ag alloys and some Pd—Rh alloys have
been measured by Hoare et_§1. [A6, A7], and Montgomery
et_al. [A8, A9]. Their results show that for the Pd-Ag
alloys both the magnetic susceptibility and the electronic
specific heat coefficient decrease with increasing silver
content until the Ag content reaches 60 at.% and will then
level off with small fluctuations. For the Rh-Pd alloys a
peak is observed in both the x versus e/a and v versus e/a
curves at an e/a of approximately 9.95, if the e/a for pure
Pd is taken to be 10.0. A study of the Pd-H system [50]
has also revealed that the leveling off of X or Y at 55
at.% H. It may be concluded that there are about 0.55 holes
per atom in the Ad—band of pure Pd. Lenglart gt_al. [31]
have estimated the number of holes in their calculated Ad—
band for Pd to be just 0.55 per atom. However, according
to their measurement of the de Hass-van Alphen effect of Pd,
Vuillemin and Priestley [51] reported that the number of
holes in Pd should be 0.36 per atom. The results of Kimura

et a1.'s calculation [52] seem to support the last figure.

 

13

By using the rigid—band model, the upper part of the
density-of—states curve of the Ad-band has been deduced by
Hoare et_al. [53] from their experimental y values of Rh-Pd
and Pd-Ag alloys, as shown in Fig. l—A. As can be seen,
the experimental curve is in qualitative agreement with
that obtained theoretically by Burdick [1A] for Cu and that
by Lenglart gt_§l. for Pd [31]. However, Hoare gt_al.'s

work extends only to Rh of the Rh-Pd system, and

0.5Pd0.5
the Ru-Rh system has not yet been investigated. Ru has a
hexagonal close-packed structure at room temperature. It
is expected that Ru would dissolve in fee Rh to a large
extent. The purpose of this work is to extend the investi-
gation of the Ad-band of the second-long-period transition
metals by measuring the low-temperature specific heats of
fee solid solution alloys of the Ru-Rh and Rh-Pd systems

to the extent that such alloys could be found and which had

not been investigated previously.

 

CHAPTER II
EXPERIMENTAL DETAILS

AlloyfiPreparation

 

An arc furnace for melting the alloys was constructed
as shown in Fig. 2-1. It consists of three major parts: a
furnace body, a water cooled crucible, and an electrode with
a tungsten tip. The electrode is held in the furnace by a
"cap assembly," which enables the electrode to swing through
an angle of about 15 degrees. The electrode can be raised
or lowered by turning the cap screw. The crucible can be
rotated at a speed of 1.5 rpm. A a HP motor drives the
crucible.

The power source for the furnace is a Harnischfeger
Model DCR-AOO-HFGW arc welder with a built-in r-f arc starter.
The crucible is connected to the positive pole and is
grounded, while the electrode is connected to the negative.
In so doing, a stable arc can be obtained, and the possibility
of contaminating the alloy by the metal ions which may come
from the electrode is prevented.

The melting operation is first started by evacuating
the furnace until the end pressure of the mechanical pump of
about 10 u is reached, followed by repeated flushing with an

inert gas.

1A

 

15

Legend of Figure 2—1.:

\OCD-QQU‘I

10

Lower Chamber of the Furnace
Lower Plate

Pyrex Furnace Sleeve

Upper Plate

Ball Head

Cap Screw Assembly

Crucible

Driving Pulley

Tungsten Electrode Tip

Water Cooled Electrode

 

16

woterlIN

water
our

 

’/ LucHoHanMO

J

i
I
I/————@
I
I
I

 

 

 

4

 

i+_®

 

 

 

 

 

 

 

 

9
\\ M \ l" <:)
v ‘
TO\MCUUH,.___ ' // ‘——_—_<:)
Pump a ' -
GosSyflem \ ' \\\

,. '\ )33?

water '”

Fig. 2-1.--Arc Furnace.

17

It is not as easy to start an arc with a high frequency
alternating current as with a direct contact method. Among
the many controlling factors, the composition of the furnace
atmosphere, and the distance between the tip of the electrode
and the top of the charge are the most important two. It
was found that the arc was easier to start but had poor
stability in an argon atmosphere, and the opposite was true
in a helium atmosphere. Thus, a 70:30 by volume argon-
helium gas mixture was filled in the furnace and maintained ;

at a positive pressure of about one lb. per square inch

 

throughout the entire melting operation. The best electrode T
gap for the starting arc was found to be 1/8 inch.

The ruthenium, rhodium and palladium metals for making
the alloys were 99.70% pure and were purchased from Gallard-
Schlesinger Chemical Manufacturing Corporation in the form
of rods. Their impurity contents as analysed by the same
company are listed in Table 11-1.

To make the alloys, the metal rods were broken into
small pieces and were washed with hot diluted hydrochloric
acid to eliminate any possible surface contamination caused
by handling. The molten alloy can be made to roll in the
crucible by slowly rotating the crucible and sweeping the are
over it. To ensure the homogeneity of the alloys, each alloy
was melted at least three times. The alloys were weighed be-
fore and after the melting. The maximum loss of the metals due

to evaporation in the entire alloy making process was less than

18

TABLE II—l.--Impurity content of Rh, Ru, and Pd metals.*

 

 

Impurity Rh (%) Ru (%) Pd (%)
Ag __ —- 0.02
Al 0.001 0.001 0.003
Au N.D. N.D.** 0.001
B —— —- 0.001
Cu 0.001 0.005 0.01
Fe 0.01 0.01 0.05
Ir 0.02 N.D. N.D.
Mg 0.0002 0.001 0.001
Mn 0.001 —- 0.001
Mo __ __ <0.001
Ni -- -— 0.005
0s N.D. X N.D.
Pt 0.001 N.D. 0.02
Rh Balance 0.003 0.003
Ru N.D. Balance N.D.
Pd 0.005 0.005 Balance
Si <0.001 —- 0.1

*Analyzed spectrographically by Atomergic Chemetals
Co., Garden City, L. I., N. Y.

**N.D. = element not detected; X = interference

19

0.5% for all the Ru-Rh alloys and less than 1.0% for the
Pd-Rh alloys. Individual checks showed that the rates of
loss for Ru and Rh were almost the same, while it was twice
as much for Pd as for Rh.

The complete equilibrium diagram of neither Ru—Rh nor
Pd-Rh has been reported. However, it is believed that there
is a missibility gap in the middle portion of the Ru-Rh
system. The Ru-rich side is a solid solution with an hcp

structure, and has been eXplored by Hume-Rothery et al.

 

[55a,b] whereas the Rh-rich solid solution has a fcc struc-

~|II'.

ture. The solubility limit of Ru in the fcc Rh-rich solid
solution is believed to be approximately A0 at .% [56]. To
confirm this, a series of test samples of various composi-
tion were made. The powder of each of the samples was
encapsulated in a quartz tube and vacuum annealed at 1100°C
for 2A hours and then water quenched. Debye-Scherrer X—ray
patterns of the powder samples were then taken. The X-ray
patterns confirm that up to A0 at .% Ru, the Ru-Rh alloys
are fcc, single phase alloys, at least at high temperatures.
Beyond that, two—phased alloys are formed. Palladium and
rhodium are both fee and completely intersoluble at high
temperatures. Below 8A5°C, a concentrated alloy may separate
into a phase mixture of Pd and Rh rich solid solutions [57].
However, this transition is very sluggish [58]. Alloys of a
single fcc phase can be obtained by fast cooling through

this temperature range.

20

(a) um (:00) (no) (stanzas) (400)

 

 

 

 

 

: 3.. - /°
5 o
'2
o

8.7 - °
U
I: .x
5

9.: a I l 1 I '

'0 40 8°At$lu 20 IO RH

Fig. 2-2.--(a) Debye—Scherrer patterns of Rh & Ru-Rh alloys.
From top to bottom are: pure Rh, 5at.% Ru-Rh, 30at3; Ru-Rh,
A0at.% Ru-Rh, and 50at.% Ru—Rh. (b) Lattice constants of
the Ru-Rh alloys as a function of Ru content in Rh.

.781“ -

21

Therefore, all the alloys used in this experiment were
encapsulated in quartz tube, vacuum annealed at 1080°C for
at least 2A hours, and then water quenched. Several samples
were examined with an optical microscope, and no second

phase was observed.

 

Cryogenic System f
The main effort of this experiment was to measure
adiabatically the temperature response of each alloy to an

input of thermal energy at low temperatures. The equipment

 

used for this experiment consisted of a cryogenic system

to provide an adiabatic environment for the sample at liquid
helium temperatures, and an electrical system to supply the
thermal energy to the samples as well as to measure its
response.

The entire set up of the cryogenic system was essen-
tially the same as that used by C. T. Wei at the University
of Illinois with only minor changes. The cryogenic system,
consisting of a cryostat, a high vacuum pumping system, a
low vacuum main pumping system, a manometer, and an inert
gas system, is shown in Fig. 2-3. The construction of the
cryostat is shown in Fig. 2-A. The details of the entire
system may be referred to Wei's thesis [59]. Some modifi-
cations made on the cryostat system to suit this experiment
and the capacity that can be achieved by the system are

described in this section.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

23

Legend of Figure 2—A.:

71:20

"U

CH

Di

Do

Liquid Helium Feeding Hole
Connected to High Vacuum Pumping System
Calorimeter Can

Connected to Main Pumping System
Cryostat Head Assembly

Kovar Seal

Safety Valve

Connected to Manometer
Calorimeter Head Housing

Inner Dewar

Outer Dewar

Sample Assembly

2A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2—A.—-Cryosta System

 

 

 

>IW

 

 

/
\
——--r
\ \ \
\ \

 

 

 

 

 

25

The first modification was the elimination of the
liquid nitrogen heat trap; instead, a stainless steel con-
duit connecting the calorimeter can and the head housing
of the calorimeter was soldered directly to the flange of
the head housing. In addition, a safety valve was attached
to the flange so as to prevent any accidental pressure
build—up inside the inner dewar.

Another modification was the insertion of a high
vacuum stOp valve between the manometer and the head
assembly of the cryostat. This enables one to carry out
the calibration of the thermometer alone.

As shown in Fig. 2-3, the pressure in column Ml of the
manometer was kept at about 30p by pump Q, and was measured
by a Stoke's gauge during the thermometer calibration. The
apparent vapor pressure of the liquid helium bath was thus
obtained by subtracting the height of the mercury column in
M2 from that in M1 plus the pressure in column M1 measured
by the Stoke's gauge. The pumping capacity of the main pump
(BP in Fig. 2-3) was about 1.6 cubic meter per minute; it
was large enough to reduce the vapor pressure of the liquid
helium bath to such an extent that a minimum temperature of
about 1.25°K could be reached.

The highest vacuum that could be reached and maintained
in the calorimeter by the high vacuum system in this experi-
ment was 7.0 x 10—6 mm Hg, which was found to be sufficient

for the experiment.

 

26

Sample Assembly

Owing to the difficulty of machining the alloys with
ruthenium content higher than 20 at .%, alloy buttons in
the shape obtained directly from melting were used to make
the sample assembly. A heater—thermometer assembly match-
ing the alloy buttons, was used for supplying the necessary
thermal energy and to measure the change of temperatures of
the sample during the experiment. The sample assembly, as
shown in Fig. 2-5a, was made of two alloy buttons with a
heater-thermometer assembly sandwiched in between. Fig.
2-5b shows the heater-thermometer assembly. A pure copper
disk 1/2 inch in diameter and 1/8 inch thick was cut with
a groove of 1/16 inch wide and 1/16 inch in depth around
its circumference, and a 1/32 inch hole was drilled
radially through its center. The copper disk served as the
heater-thermometer body. No. A0 enamel-coated manganin
wire with a nominal resistance of 3l.A ohm per foot was used
for making the heater. It was wound around the circumfer-
ential groove of the copper disk in such a way that no net
induced magnetic field would result from the heating current.
The total resistance of the heater was approximately 300
ohm at room temperature. A 1/10 watt carbon resistor with
a nominal resistance of 60 ohm at room temperature was used
as the temperature sensing device and was placed in the
radial hole at the center of the copper disk. As conducting
leads, No. 38 double-cotton insulated c0pper wires of about

6 cm. long were connected to the ends of the heater and the

 

27

glzafizaizafiagg.’dlflo s
\\\\\\\"

(a) (b)

Thnnnqmer

 
  
   

  

 

 

H4"
0
Urdu

 

Fig. 2-5.--(a) Sample Assembly; (b) Heater-Thermometer
Copper Body.

 

‘V; IR3IL l-——

 

 

 

 

 

 

A
l<‘3
POTENTIOMETER

‘ _

 

 

Fig. 2—6.--Essence of the Electrical Circuit.

28

carbon thermometer. The flat side of the heat-treated alloy
buttons were ground and polished to a mirror finish. The
surface of the heater-thermometer disk and the polished
surface of the alloy buttons were coated with a thin layer
of silicone grease before assembling. Alloy buttons and
the heater—thermometer assembly were then tied together
with thick copper wires to form the sample assembly.

The four electrical leads of the heater and the ther-
mometer were connected to four corresponding measuring

junctions on a kovar seal through short pieces of manganin

 

wire. It was observed that the insertion of the manganin
wires increasedthe thermal stability and decreased the
temperature drifitng rate of the sample during the measure-

ments.

Electrical Measurement

 

The essential purposes of the electrical circuit
designed for this experiment were: first, to provide a
certain amount of thermal energy to the alloy by sending
a constant current through the heater for a certain period

of time. This was the Joule heat of the heater, i.e.

E = i x R x t

where E is the thermal energy input to the alloy in 10'6 joule,

i is the heating current in milliampere, R is the resistance

h h
of the heater in ohm, and t is the time of heating period in

seconds. Secondly, to record the temperature response of the

29

alloy to this thermal energy. In this experiment, this
temperature responsevmmsmeasured as a resistance change in
the carbon thermometer.

The principle used in measuring the resistance of the
thermometer or the heating current can be illustrated in a
simplified circuit as shown in Fig. 2—6. In Fig. 2-6,

R represents either the thermometer or the heater. R is

l 2

a standard resistor, and R is a variable resistor. By

3

varying R a desired current I can be set up in the circuit.

3
The current is measured by the voltage across the standard
resistor R2. R1 can then be readily known by measuring the

voltage drop Vl across R1. This can be done by using either
a potentiometer or an electrometer.

Fig. 2—7 shows the circuit diagram for the electrical
measurements. It is essentially the same as that used by
Wei, gt_a1. The main features of this circuit are: a Leeds-
Northrup K-3 potentiometer with a d—c null detector for the
standardization of the current I in both the thermometer and
the heater circuits, and to measure the potential drop across
the heater or the thermometer; a Speedomax recorder with a
d—c amplifier for recording the continuous change in the
resistance of the thermometer with time during a heating
period; and a Berkeley Model A10 electronic counter coupled
with a General Time 2001-2P type frequency standard for
measuring the heating time. In addition, the on-off switch

for the heater and that for the counter are synchronized

through an electronic switch.

3O

 

.uasosfio Haofispomam oneuu.sum .wam

 

 

>o__o< I

I6\¢II2I

 

.OATIIIIQ/‘II

 

 

 

 

 

 

 

 

 

 

 

.533 539. —..

5.5.300 0... —.[v

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a on L2” .
43:3. . .4. 5H
f _ I
. L. A IL N
f“ %:€n»:filll.. 2%. Ir? J.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

31

The components of the electrical circuit are listed in
Table II-2(a). The relative positions of the switches for

the various functions are illustrated in Table II-2(b).

Experimental Procedures

 

There were four major steps in the low temperature
specific heat measurements, namely: F1
1. Preparation of the calorimeter system and

first stage cooling.

2. Standardization of the potentiometer as well

.ums—ul‘ I-.—:y.— 1.3- u --

as the heating and thermometer current. Lg.
3. Second stage cooling and the calibration of the

thermometer against the vapor pressure of the

liquid helium bath.
A. Heat capacity measurement.
The calorimeter system, after being set up, was first

evacuated to an end pressure of about 2x10-5

mm. Hg. Any
possible leakage was checked to make sure that the high vacuum
can be sustained during the experiment. The system was then
flushed with dry helium gas for several times. A small amount
of helium gas (at a pressure of about 700 u) was left and
sealed in the calorimeter system for heat exchange. The

inner dewar of the cryostat was also evacuated, flushed with
dry helium gas, and filled with helium gas to a small positive

pressure. For heat conduction, the jacket of the inner dewar

was filled with nitrogen gas at a pressure of about 200 u.

32

 

 

TABLE II-2(a).—-Components of measuring circuit.

R(ohms) B (Volts) C (mfd)
1i 1.6 K B1 to B3; Butt—Sub Model 1 0.01
2i A.7 K BSTC—2l3 constant voltage
3i 16 K supply 3 volts at 2A.A mA.

Ai 16 K
Si 15 K EflICadmiunlLow Temperature
6i 18 K Coefficient Standard Cell;

1.01925.

3T 0-1 Meg (General Radio
Type lA32—M Decade
Resistor; Minimum
Scale 0—100 in ten
steps)

2T 10,000 Standard resistor

T Thermometer (Varies from
800 at A.2°K to 50,000 at
1.25°K)

AT and AH 2.2K

3H O—lO K
(Decade Resistor; Minimum
Scale 0—10 ohm in ten steps)

2H 100 Standard Resistor
H Heater (about 300)

5 1 Meg.
6 10 K

AP 5 K

 

33

 

 

 

TABLE II—2(b).--Positions of switches for various functions.
Function SWltCheS

12 13 1A 15 16 l7 l8 19 20 21
Standardiza-
tion of K-3
potentiometer - — 1 on off off 2 off on -
Standardiza—
tion of Heat-
ing Current off - 1 on 2 off 1 off on on
Standardization
of measuring
current on adj l on 1 off 1 off on off
Measuring EMF
of Heater off - 1 on off 2 2 off on on
Measuring EMF
of Thermometer on adj 2 on off 1 2 off on off
Recording on adj A off off 1 2 on on *
Calibration of
D-C Amplifier &
Recorder off — 3 off off 1 2 on on off

 

*
On for heating, off for stand—by.

3A

This was checked before every eXperiment. Liquid nitrogen
was then gradually filled into the outer dewar which had a
permanent vacuum jacket. As the system was gradually cooled
down, additional helium gas was gradually filled in the inner
dewar to compensate for the contraction of the helium gas
originally in it in order to maintain the positive pressure
within the inner dewar. The maintaining of a positive pres-
sure in the inner dewar was a necessary measure to prevent
the leaking of air into the inner dewar when it was opened

for filling of liquid helium. Five to seven hours were re-

quired to cool the system to the liquid nitrogen temperature.
After the cryostat had been cooled down to the liquid nitro-
gen temperature, the electrical connections and any possible
leakage of the calorimeter were again checked before the
filling of liquid helium. The filling of liquid helium took
less than five minutes to complete. Ten to fifteen minutes
were usually needed for the sample assembly to reach a
temperature equilibrium with the liquid helium bath.

The standardization of the K-3 potentiometer and the
desired currents for the heater and the thermometer were
carried out immediately after the sample assembly had
reached the temperature equilibrium. A cadmium standard
cell which had a low thermal coefficient served as the
standard emf supply.

The calorimeter was further cooled down as the vapor
pressure of the liquid helium in the inner dewar was gradually

reduced by the main pumping system. In the mean time, the

35

R—T relationship of the thermometer was calibrated by meas-
uring the vapor pressure of the liquid helium and the re—
sistance of the thermometer simultaneously. The temperature
of the liquid helium bath could be deduced from its vapor
pressure using the NBS 1958 He“ Scale of Temperature. The
lowest temperature that could be reached in this experiment
was 1.25°K. The resistance settings of the variable resistor . A
R3T or R3H for the desired currents in the thermometer cir- I
cuit or in the heater circuit reSpectively, as well as the

resistance of the heater were all checked again at the

 

lowest temperature for the stability of the instrumentation.
After the lowest temperature had been reached, the
sample assembly was warmed up to about 1.A°K. The measure-
ment of the heat capacity of the sample assembly was then
carried out. The heater was turned on for a certain period
of time while the temperature change of the sample assembly
was automatically recorded by the Speedomax recorder. Fig.
2-8 shows three typical recorded curves. The ordinate of
the figure represents the time while the abscissa is a
measure of the temperature of the sample assembly. As shown

in Fig. 2-8 curve (B), the initial temperature T and the

1
final temperature T2 of the sample assembly are obtained by
extrapolating the initial thermal drifting curve CA and the
final thermal drifting curve BD to the mean time line. The
difference between T1 and T2 is the temperature increment

AT of the sample assembly corresponding to the thermal energy

36

input while their average is taken as the temperature to
which the heat capacity of the sample assembly for the par—
ticular measurement pertains. Curve (A) in Fig. 2-8 with
zero thermal drift before and after the heating is the ideal
one. However, since it was impossible to achieve an absolute
adiabatic environment, some thermal drifting in the sample
would always be present. In order to obtain an acceptable {1
accuracy, it was essential to manipulate the system so as to i ‘

minimize the thermal drift and to have the tendencies of the

 

drift before and after the heating in the same direction, I

Ili"
It

as shown in curve (B) of Fig. 2-8. The most undesirable
one is shown in Fig. 2-8 curve (C). In this case, the
thermal drifts before and after the heating are large and
in opposite directions. A large error may be introduced
in evaluating T1 and T2.

The degree of vacuum in the calorimeter, methods of
connecting the electrical leads to the measuring circuit,
the way of suspending the sample assembly, and the degree
of thermal equilibrium achieved in the sample assembly were
all important factors governing the rate of the thermal
drift. Among them, the degree of vacuum in the calorimeter
was the most important one. It was found that a vacuum of
lower than 5.0 x 10"5 mm Hg becomes undesirable. Usually
the vacuum in the calorimeter was kept in the range of

6

8.0 x 10' mm Hg. to 1.A x 10'5 mm Hg.

37

(A) *2

  
   

 

 

 

 
 

 

 

A.
Ttgtnl), 3 _ / - _.
fl .
8 7] i A \
9 m: or M 1m:
5 c (B) »
p.
(C)

 

 

T (IN TERM OF RESISTANCE)

Fig. 2—8.--Experimental Time-~Temperature Curves.

CHAPTER III
RESULTS AND CALCULATIONS

Calculations
According to Keesom and Pearlman [60], the character-
istics of the carbon thermometer can be expressed in general
as
a N] 1
(ln R/T) = EAi(ln R) (3-1)
1
However, it has been found experimentally that only two terms

of the polynomial are necessary to give enough accuracy [60].

(In R/T)12 = A + Bln R (3—2)

In the temperature range of 1.6°K to A.2°K a total number

of 15 to 20 experimental points were usually taken during
the calibration. Above the A point, the vapor pressure of
the liquid helium was corrected for the hydrostatic pressure
due to its own head above the center of the calorimeter can.
The resulting pressures were then converted to corresponding
temperature values by using the 1958 HeLl Temperature Scale
[61]. These and the corresponding resistances of the ther-
mometer were then fitted with the equation (3-2) using a

Control Data 3600 computer. The coefficients A and B of

38

 

39

(3-2) were obtained by using the least squares method. It
was found in the early stage of this investigation that the
accuracy of measuring the vapor pressure of liquid helium
near its A point is poor and a rather large error could

thus be introduced. It was decided to avoid the neighbor-
hood of the A point in the calibration in the later eXperi-
ments. In general, the maximum error in the calibration was
10.15%. A typical calibration curve is shown in Fig. 3-1.

The thermal energy input into the sample assembly was
shared by the alloy, the c0pper wire used for assembling,
and the heater-thermometer assembly. In order to calculate
the heat capacity of the alloy, it was therefore necessary
to subtract the amount of heat energy absorbed by the copper
wire and the heater—thermometer assembly from the total
heat input.

As described in Chapter II, the heater—thermometer
assembly was mainly composed of a pure copper disk, a small
carbon resistor, about ten feet of number A0 manganin wire,
four short pieces of number A0 double cotton-coated copper
wire, and a small amount of vacuum grease. The total weight
of the assembly amounted to 5.5 grams; out of this, more
than five grams belonged to the pure c0pper disk. There—
fore, it was reasonable to assume that the heat capacity of
the heater-thermometer assembly was essentially the same
as that of pure copper without introducing too much error.
The heat capacity per mole of the heater-thermometer assembly

and the tying COpper wire combined is

.mHan oaasmm Ca com: nopoEoEAonB can now o>nso coaumnndamolt.atm .Mfim

Ni 0.? a.»

\

 

c c.
0.» t.» N.n O.»

ilqll1lqlJ‘. . . ,. . . . q

abomN'Nnmdu n
on_~.o_nN_o._ In 4

mmthOZKNIh mo...
w>¢=o zo....<¢m.._<o

O.N

«.0

1
Q

1
Q

ills
to .

_L
“2

 

 

 

A1

Cv,th = YT + (12/5)w”Rh(T/e)3 (3—3)

where Rh is the gas constant and 6,the Debye temperature.

Substituting y = 1.8 x 10_u cal./mole-deg2 and 0 = 315°K

of pure Cu [62,63], equation (3-3) becomes

_ _ A
c = 1.8 x 10 “T + (1.815 x 10 ”/1u.5973)T3 q
v,th ,
<3-u>
Therefore, the heat capacity of the alloy is A
_J
_ .2
CV — (1 Rt/K — Cv,th x Nl)/AT x N2 (3-5)
where
i = heating current in amperes
R = resistance of heater in ohms
t = time of heating period in seconds
N1 = number of moles of the heater—thermometer
assembly and the tying copper wires combined
AT = the increment of temperature of the sample
assembly during each heating period
N2 = number of moles of the alloy
K = conversion factor from Joules to calories =

A.18A
A correction was further made in the thermal energy input

for the heat dissipated in the inserted manganin wire.

A2

Results

Altogether fourteen samples, including a pure Rh and a
pure Pd, were measured. For each sample, a number of thirty
two to fifty data points were taken, and their specific heat
capacities were calculated according to Eq. (3-5). The
CV/T versus T2 curves of the samples are shown in Figures
3-2 to 3—5, and their detailed eXperimental results are tab-
ulated in Appendix A. As can be seen, some of the Cv/T

versus T2 curves have indeed good linearity whereas others

 

show anomalous curving—up at the lower temperature part. L
These anomalies are obviously either due to experimental
error or to some unknown causes. This will be elaborated
further in the next chapter. However, to the first approxi-
mation, the eXperimental data of each sample were fitted to
the heat capacity equation for metals and alloys at low

temperatures
2
Cv/T = y + 8T (3—6)

and by using the least squares method, the values of y and
B of the alloys were found by extrapolating the linear part
of the Cv/T versus T2 curves to zero temperature and
ignoring the low temperature anomaly. From 8 the Debye

temperature of each of the alloys was then calculated:

9 = (12wu/5)l/3(R)l/3(B)-l/3 (3-7)

A3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I 6
I 4
0 Pure Rn
n
I 2 MO 3000 O om-tD-O-o-OWWOMFC
some lie-Rh - 1.
no
A «.
,2 IcomAu -Rh
% M
§|13 cube:«QCanoao4rorucfiLU'n'5
i
3 I o
O
.3
o I5a:.% sin 4%
S I 2 _0 01.010 ‘—° ‘-
: -mmvwf‘m‘ ”1 °
IO .
L
I 2 200mm -Rh
noaq p oWWo-OO'OCW
I0
0
O 2 4 6 I0 l2 I41 l6 IS 20

T2 8m“)

Fig. 3-2.-—Cv/T vs T2 Curves of,Pure Rh and Ru-Rh Alloys.

AA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l8
'6 wade-Ru A _ a
<b Devoay>o4xxmu1r‘*i
I4
I *r
'6 scone 'Rh
- u w“
N1» 000430000 c We WOW
l4
1‘ J
s ° I
‘3' 6 ° .
‘3 ZZmSSRd-Rh .
000 I‘D—H”..—
S ' 4 ° OID°g_ _ AWN—*M‘W
O
O
V i. <
0»
§ l4 I5aI°/.P 'Rh ~
Do<%>@————‘
>~ WWm—o—‘PW
l2
‘ J
I
l4
0 70l.%Pd'Rh n
d :9
'2 RawmmAfkunmuncookrpn‘°°*T°_P——°°3
IO
b {F
0 2 4 6 8 IO I l4 l6 l8

T2 (°K‘)

Fig. 3—3.——Cv/T vs T2 Curves of Rh—Pd Alloys.

 

7 x Io‘ (cal/mole- dog’)

A5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’1 . . __ pea—-
00604099410940de
,1,
I
. W
e—“ 4‘
dilemmas.Tum-cam:Itaaovcanci'o01:30"?"""""‘I“"'o
i
A
4 Outfit Ru -Rh
1: A
I:
2’ 4 6 8 IO l2 l4 I6 IO 20
T2 («’3

Fig. 3-A.--Cv/T vs T2 Curves,of Rh—Ru Alloys.

.;9'

Y x I o‘ (col/mole-doqz)

A6

 

3O

 

28

PURE Pd °“°‘

 

26

24 W0

 

 

 

N
N

 

N
O

 

 

a

 

 

 

IO

 

 

 

 

 

 

 

 

 

 

 

 

 

t) 2 4 £5 8 IO I2 L4 l6 IS 20
T” (°K2)

Fig. 3—5.-—Cv/T vs T2 Curve of Pure Pd.

 

A7

The calculations were carried out with the Control Data
3600 computer using the program described in Appendix B-l.
The values of Y and 6 thus obtained for all the alloys

as well as pure Pd and Rh are listed in Table III-l.

Accuracy and Reproducibility
of Results

 

The reliability of this investigation depended mainly
upon the accuracy of the experimental results. In this
section the experimental accuracy and the reproducibility

of the results are to be examined.

 

Referring to equation (3-5), it can be seen that there
are three sources of errors to be considered, namely, the
accuracy in supplying the thermal energy i2Rt, the accuracy
in measuring the temperatures of the sample assembly, and
the error introduced in assigning the heat capacity value

and the molecular weight to the heater-thermometer assembly.

A. Inaccuracy in Thermal
Energy Input,gi2Rt

Ai. Sygtematic error in the electricalgsystem.--The
limit of error for both standardizing and measuring in the
lowest range of the K-3 potentiometer is i0.015% of reading
10.5 uV. The d-c null detector has a sensitivity of 10.9
Imicrovolts per division in the maximum sensitivity scale.
Thus the inaccuracy introduced in the standardization of the
LK-3 potentiometer would be at most i0.02%. The maximum

inaccuracy in setting the heater and thermometer currents

A8

TABLE III—1.--Va1ues of Y and 0 of Ru-Rh and Pd—Rh Alloys
(Least Square Fit: Cv/T = y + BT2)

 

 

 

Alloy y x 10“ 2 0 (°K)
(cal./mole—deg )

Pure Pd
This work 23.01 278.28
Rayne* 25.6 :1.3 __
Budworth et al.# 23.1A 27A.0 i3.0

A0 at. % Pd—Rh 1A.9A 397.A6

30 at. % Pd-Rh 1A.l7 AA7.50

22 at. % Pd—Rh 13.99 A98.A1

15 at. % Pd—Rh 12.81 A53.11

7 at. % Pd-Rh 12.01 A51.89

Pure Rh
This work 12.03 528.58
Budworth et al.# 11.56 i0.07 512.0 :17.0
5 at. % Ru-Rh 11.65 A56.61

10 at. % Ru-Rh 11.30 A29.77

15 at. % Ru-Rh 10.89 A27.5A

20 at. % Ru—Rh 10.97 A5A.7A

25 at. % Ru—Rh 10.u8 396.93

35 at. % Ru-Rh 10.A0 AAA.27

A0 at. % Ru-Rh 10.96 527.89

 

*J. Rayne, Phys. Rev. 95, 1A28 (195A).

#0. W. Budworth, F. E.1Rxnxaand J. Preston, Proc.
Roy. Soc. (London) A257 (1960).

A9

was perhaps $0.01% in each case. The combined maximum
possible error in the 12Rt term due to the K-3 potentiometer
and the d-c null detector would thus be i0.05%.

The Beckman/Berkeley Model 5010 electronic counter has
a capability of resolving one-volt pulses of 5p sec. in width
and with a separation of 25p sec., and a counting rate of
0-12500 counts per second. The frequency standard has a I{
frequency of 100 cps and an accuracy of 0.001%. The built-in I
inaccuracy of the counter is thus negligible. The accuracy
of the measured heating time is t0.0005 second. For the
shortest heating period of 10 seconds, the corresponding max-
imum inaccuracy in the measured heating time was thus i0.005%.

Since the time delay in the electronic switch was
negligible, it was thus concluded that the total inaccuracy
in the instrumentation was less than i0.006%.

Aii. Errors introduced during the experiment.—-The
use of the electronic switch caused a deviation in the heat-
ing current of 0.8%, while the resistances of the four
short pieces of manganin wires inserted between the COpper
leads and the measuring junctions on the kovar seal amounted
tc> 1.3% of the total heater resistance. These two factors
enere taken into consideration in the calculation of the heat
calpacity of the sample assembly and hence would not intro-
durze any error in the final results.

The resistance of the heater used in the calculation
was the value measured at A.2°K. However, from A.2°K to

1u1H3K'the resistance of the heater decreased 0.05%. This

50

was the maximum error that could be caused by the variation
in the heater resistance.

Therefore, the maximum total inaccuracy in the thermal
energy input was less than 10.11%.
B. Errors Due to the Inaccuracy in

the Measurement of the Temperatures
of the Sample Assembly

 

The accuracy of measuring the initial and final
temperatures of the sample assembly before and after the
heating period depended upon the accuracy of measuring the
corresponding resistance R1 and R2 of the thermometer and
the accuracy of the thermometer calibration. The following
causes are to be considered.

Bi.--The accuracy of the d-c amplifier, which is
:0.A%, plus the accuracy of the K—3 potentiometer and the
d-c null detector may result in a maximum systematic error
of :0.A005%.

Bii.-—The smoothness and linearity of the branches
of the heating curve corresponding to the thermal drift and
the degree of exactness in extrapolating these branches to
the mean-time line may cause an inaccuracy in the values of
R and R-

1 2

Hence, Bi and B11 may cause a maximum error in R1 and

by 0.15%.

R2 of about 0.A3%.

Biii.-—Inaccuracy of measuring the vapor pressure of

the liquid helium bath.

51

a. The resolution of the cathetometer is £0.01 mm.
Taking the vapor pressure of liquid helium at
1.80K for comparison, i.e., 10 mm Hg, this will
introduce an inaccuracy in the measurement of
10.1%. Due to human error, the inaccuracy can
be as high as 10.2%.

b. The maximum error in the hydrostatic pressure
correction above the A point is i0.25%.

c. The error that may be introduced by the Stoke's
gauge reading for pressure correction is
negligible compared with the errors due to other
sources.

The total inaccuracy in measuring the vapor pressure
of the liquid helium is thus 0.32%. This may give rise to
an error in the temperature of the liquid helium bath of
0.35%.

Therefore, the total uncertainty in the temperature
measurement will be about i0.7%. This will result in a
maximum error of il.A% irrATg the temperature increment of
the sample assembly after each heating, which may in turn
introduce an error in the heat capacity of the sample

assembly of approximately :l.A%.

C. Errors Due to Other Sources

 

The errors due to the inaccuracies in calculating the
molecular weight and the number of moles of each of the

alloysvmneznegligible. Since the combined weight of the

 

52

carbon thermometer and the other added materials in making
the heater-thermometer assembly was only a small fraction

of its total weight and the heat capacity of cOpper was

much lower than those of the alloys, the heat absorbed by
the heater—thermometer assembly as a whole is small. The
error introduced in asigning the heat capacity and the
molecular weight of the heater-thermometer assembly as being
that of pure copper was thus also ignorable.

Q. The uncertainty in the alloy composition was
mainly caused by the metal loss during the melting operation.
In the Ru-Rh alloys this uncertainty was less than 0.5%
while in the Pd-Rh alloys it could have been as large as
1.0%.

As analyzed above, it may be concluded that the
maximum experimental error that might be introduced in the
final results was tl.5%.

The reproducibility of the results were checked for two
different cases: first, the reproducibility of the results
after repeated heating and cooling in one experiment as can
be seen in both of the curves in Fig. 3-6. Secondly, the
reproducibility of the results obtained in different experi-
ments with the same specimen; this is shown by plotting two
sets of experimental data of the same sample in a single

diagram, as also can be seen in Fig. 3—6. In addition, all

 

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53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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5A

the points fall well within the limit of the experimental
error in each case supporting the analyses of the probable

error given above.

 

CHAPTER IV
DISCUSSION

A. Anomaly of the Cv/T versus T2 Curves

 

As described in the previous chapter, the measured
Cv's of each of the alloys were first calculated and fitted
to the usual heat capacity equation, Cv/T = yT + BT2, by a
least-squares method. Accordingly, a straight line with
a lepe related to the Debye temperature 6 through Eq.
(1-2) would be expected. For some of the Ru—Rh alloys
straight lines were indeed obtained. However, conspicuous
Schottky type anomalies were observed in those Cv/T versus
T2 curves of the other alloys, especially of those with
compositions near 22 at .% Pd in Rh. It was intuitively
suspected that experimental mistakes might have caused such
a "curving-up" of the Cv/T versus T2 curve at low tempera-
tures. First of all it could be due to the evaporation of
any condensed helium exchange gas on the surfaces of the
sample assembly during the early period of the experiment.
This possibility is considered unlikely in view of the
following facts. (a) The vapor pressure of the helium
exchange gas was originally 700 u Hg at room temperature,
it reduced to about 8 u Hg at the lowest temperature of

the liquid helium bath of 1.25°K. After being evacuated

55

 

56

continuously for one hour, the vacuum in the calorimeter at
the beginning of the measurement was only 6.0 x 10-5 mm Hg.
It is thus unlikely that there was still an appreciable
amount of condensed helium gas left on the surface of the
sample assembly. It is true that the vacuum in the calori-
meter would eventually be improved and reach an end pres-

sure of 6-8 x 10-6

mm Hg as the experiment went on. This,
however, was proved to be mainly due to the degassing of
the conduit tube connecting the calorimeter can and the
head housing of the calorimeter. (b) Some samples were
first warmed up above the A point of liquid helium, and
left to be cooled gradually without introducing any addi-
tional exchange gas into the system so that most of the
condensed helium, if any, would be evaporated from the sur-
face of the sample assembly. The Cv's of the samples were
then measured. It was found that the CV/T versus T2 curves
obtained in this way were essentially the same as those
obtained without prewarming. (c) No appreciable change of
the vacuum in the system was observed during the heating
period, as would be expected from a burst of gas in such a
high vacuum condition.

The second possible experimental error that could
cause such an anomaly would be the way to calculate the
characteristic parameters of the thermometer. It was first
thought that to fit the thermometer calibration curve to a
polynomial including lnR terms of higher power would improve

the linearity of the Cv/T versus T2 curves. But just to

 

lee-L‘s" "

57

the contrary, the anomaly in the CV/T versus T2 curves was
increased by such a non-linear fit of the calibration curve
of the thermometer.

It seems clear from the above discussion that the
"curving-up" of the Cv/T versus T2 curves of some of the
alloys at low temperatures was not a result of experimental
error but due to some real contributions other than the I:
electronic and lattice specific heats.

Anomaly of this type was first observed by Keesom

et al. [6A] in NinCu6O alloy. Based on the fact that Ni

 

is ferromagnetic they related this anomaly to an effect Of
magnetic in nature, although the alloy itself is paramag-
netic. Similar anomalies were also observed by Wei §£;al.
[65] in Ti-Fe and Fe-V alloys. In these alloys, the
curving up of the Cv/T versus T2 curves at low temperatures
is obviously related to the content of the ferromagnetic
element Fe. The greater the Fe content is the greater is
the anomaly. Schrdder and Cheng [66] suggested that there
are ferromagnetic clusters existing in the alloys, and
each cluster oscillates around an orientation determined
by the energy of the crystal anisotropy. The cluster may
thus contribute an additional term to the specific heat of
the alloy. This "ferro-magnetic-cluster Specific heat" is
roughly independent of temperature in the eXperimental
temperature range, and the low temperature Specific heat

of such an alloy may be expressed by [67]

58

CV/T = y + 8T2 + C'/T (A—l)

Where Y and B have the usual meaning while C' is the
"ferro—magnetic—cluster specific heat" of the alloy. By
assigning appropriate values to the Debye temperature of
the alloys, Schr8der fitted the data of Guthrie et_al.
[68] and that of Wei et_a1. [65] to (0V - C')/T = y + BT2;
obtaining straight (CV - C')/T versus T2 curves.

Anomalies Similar in appearance but of a different
nature have been observed by Arp gt_al. [69, 70] in Co
and Co—Fe alloys that are interpreted, however, by them and
by Marshall [71] as a result of the hyperfine interaction
between the magnetic moment of the nucleus and the
effective magnetic field of the electrons at the nucleus.

In the Pd—Ag and Pd—Rh alloys, investigated by Hoare
et_al., no such anomalies were reported. But recently,
such Schottky tail-type anomalies were observed in Pd-Ag
alloys of compositions near A0 at.% Pd in Ag by Montgomery
et a1. [A8]. Several possible causes were discussed by
them. They seem to favor Schrdder and Cheng's ferromagnetic
cluster picture, but made no conclusion with certainty.

Reviewing the possible contributions to the low
temperature Specific heat other than those of the electronic
and the lattice, it can be found that most of them fall in
either of the following two groups: (a) those depend upon
temperatures with a positive power, and (b) those depend

upon temperatures with a negative power. Contributions

 

59

falling in group (a) are mostly due to magnetic effects,

such as ferromagnetism (C 0°T2/3), antiferromagnetism

ferro.

(C a0T3), etc., while in group (b) nuclear Specific

antiferro.
heat is the most important one. Nuclear specific heat is
attributed to the redistribution of the nuclear Spins

among the Zeeman energy levels [72], the "high-temperature

tail" of which can be expressed as [71]

c /R = l I(I+l)(gnunH/kT)2 + 0(gnan/kr)“ + .... (u—2)

V

LA)

where I is the nuclear Spin angular momentum, gn is the
Lande g factor for nucleus, “n is the nuclear magneton, and
H is the total effective magnetic field.

2

The T- term is due to the hyperfine effect while the

T—u term is due to the dipole-dipole interaction. Generally
only the T-2 term is taken into account. In most rare
earth metals, it is found [73] that the nuclear specific
heat predominates below 1°K.

None of the metals, Ru, Rh, and Pd, investigated in
this experiment is ferromagnetic or antiferromagnetic when
it is pure. Pd and Rh are non-superconducting, and the
transition temperature of Ru is only 0.A7°K, much too low
for Ru to make an appreciable Specific heat contribution in
the temperature range of the present experiment. It is

thus unlikely that the anomaly is due to the superconduc-

tivity of the alloy.

 

60

The curving-up effect itself suggests a contribution
which is either independent of temperature, or dependent
on temperature with a power less than unity. Among these
known effects, the possible causes which may result in such
"Schottky tail" type anomalies would either be the nuclear
specific heat, depending on T-Z, or the specific heat due
to ferromagnetic clusters as suggested by Schrdder, a term I
that is independent of temperature in the eXperimental
temperature range.

In order to check if any of the above effects were

 

Iron-— _

responsible for the low temperature anomaly, the experi-
mental results were first fitted to an equation including

a term of Schrdder's magnetic specific heat:

cv = A + yT + 8T3 (A-3)

and then fitted to an equation with a nuclear specific—
heat term (ignoring the T-u term of the dipole-dipole

interaction)

cv = BT-Z + yT + 8T3 (u-u)

The coefficients A, y, and the corresponding Debye tempera-
tures 6 obtained from Eq. (A—3) are listed in Table IV-l,
while the coefficients B, y, and the Debye temperatures
obtained by fitting to Eq. (A-A) are listed in Table IV-2.

The values of y and 6 obtained by fitting the data to the

61

TABLEZDFl.—-Coefficients of the least square fit to the

equation: Cv = A + yT + 8T3

 

_,__,—.

 

 

 

Alloy A x 10“ y x 10"
at.% Ru-Rh (cal/mole-deg ) (cal/moIe—deg 6(°K)
0.0 3.39 10.51 396.6u
5.0 2.06 10.75 399.10
10.0 2.83 9.88 35A.06
15.0 1.69 10.00 371.AA
20.0 1.21 10.35 A03.10
25.0 1.7A 9.6A 358.87
35.0 1.5A 9.60 38A.89
A0.0 2.AA 9.70 397.55
at. % Pd-Rh
7.0 3.63 10.22 3A9.58
15.0 2.59 11.51 369.99
22.0 6.A7 10.91 328.73
30.0 n.8u 11.95 3A3.A2
A0.0 A.09 12.96 32A.59

 

62

 

 

 

 

TABLE IV—2.—-Coefficients for the equation: Cv = BT-2 + yT
+ 8T3
Alloy B x 10“ y x 10“ 2 O
at.% Ru-Rh (cal-deg/mole) (cal/mole-deg ) 0( K)
0.0 3.66 11.8A A87.99
5.0 2.10 11.57 AA8.52
10.0 3.01 10.97 392.07
15.0 1.78 10.67 399.31
20.0 1.36 10.81 A28.30
25.0 1.8A 10.32 382.53
35.0 1.A6 10.2A A19.82
A0.0 2.6A 10.66 A90.16
at. % Pd-Rh
7.0 3.61 11.86 A06.8A
15.0 2.69 12.53 A13.52
22.0 6.75 13.AA A12.80
30.0 5.62 13.76 A01.53
A0.0 A.35 1A.56 366.17

 

63

conventional equation (Eq. l-l) and those by fitting to
the above two equations (Eqs. A—3 and A-A), together with
the results of Hoare et_§l, are all plotted against the
electron concentrations, e/a, in Fig. A—l. AS can be seen
the results obtained by fitting the data to the equations
(l-l)and(A—A)do not differ appreciably from each other,
Show less scattering, and are more consistent with the
results of Hoare et_al. than those obtained by fitting to
equation(A—3. A smooth continuous y versus e/a curve is
drawn in Fig. A—l that it may, perhaps, represent the
upper part of Ad—band. It seems that the low temperature
anomalies in the present results are not describable by
Schr8der's magnetic-cluster Specific—heat. Although the
results seem to suggest that the anomalous curving-up
effect in the Cv/T versus T2 curves of some of the Rh-Ru
and Pd—Rh alloys would be attributable to the hyperfine
interaction but, because of the experimental temperature
range of this investigation is not low enough, it cannot
be said with certainty that such is the case. Further
investigation at temperatures below 1°K may help in clari—

fying this question.

B. Features of the y Versus e/a Curve

 

As shown in Fig. A—l, the y versus e/a curve embraces
both the results of Hoare et a1. and those obtained in this
investigation. Taking 10 to be the number of the outer

electrons in palladium, the curve Shows that after passing

 

6A

a peak at e/a of 9.95 the y decreases as the electron
concentration, e/a, decreases until a e/a ratio of about
8.7 is reached, it then rises again as the e/a ratio

decreases further.

C. Debye Temperatures of the Alloys

 

The Debye temperatures, 0, of the alloys are also Te
plotted against the electronic concentrations, e/a, in
Fig. A-l. For the Pd-Rh alloys, 0 decreases as the content

of Pd increases, in agreement with the results of Hoare

 

et al. A minimum is observed in the portion of the 0 ' a

1- rs:

versus e/a curve of the Rh-Ru alloy system. The scattering
of 0 is rather large compared with that of the noble metal
alloys. This is probably due to the irregular Shape of

A-d band. According to Stoner [39], the electronic specific
heat of a metal with energy band of irregular Shape depends
not only on the density of states at the Fermi level but
also on the local Slope and curvature of the energy band.
An extra term depending on T3 should be included in the
electronic Specific heat. This T3 term is small compared
with the term linear in T, and is usually ignored. Since
the coefficient of the lattice specific heat is only 0.5%

3

of that of the electronic specific heat, this T term of
the electronic specific heat which cannot be separated from
the lattice specific heat might have affected the correct

evaluation of the Debye temperatures of the alloys.

65

 

 

 

 

 

 

 

 

O
.— \\ £\\ . -SOO
- A: QA++AA* **\+\. o ~4003‘
A A 2,. CD
_ \so‘ "300
O 06.
24 -
O
‘ /
O
20" : \
No
3 P o -
g o/ A
E. I- o/ . °\
'3 ./
3 /9
e 7 9}!
o g” ,
X .. fl 1
0 O ‘
x V ‘1 ‘
IO '- “ ‘ AL I ‘
- ; O
- l o\
I. I I 0‘0-
0 J 1 l l l l 1 I I. | | ! II
80 50 40 20 Rh 80 60 4O 20 Pd 20 4 0

otP/o Ru in Rh

Fig. A—l.--y vs e/a curve.

at.%Rh in Pd at.%Ag in Pd

Soild dots represent results ob-

tained by fittin to C /T = y + BT2, + are results fitted to
C = T 2+y T+BT , A a¥e results fitted to the equation: 0 =
Av+~yT + BT3, while the Open circles are results from Hoar

et al.

1.

 

66

D. N(E) Versus E Curve——the Upper
Part of the Ad-Band

A density of states, N(E), versus energy, E, curve is
more useful than the y versus e/a curve.
The Single spin density of states at the Fermi level

of an alloy is calculated by using

N(Ef) = (3/n2k2I-y. (A-S)

for y expressed in units of 10",]l ca1./mole-deg2, Eq.(A—5)

becomes

N(Ef) = 0.178 v. (A-5')

Where N(Ef) is the single—Spin density of states per eV
at the Fermi level of the alloy. The number of electrons
in a small energy interval AE near the Fermi level is

approximately
An = AE N(Ef). (u—o)

where N(Ef) is the average value of the density of states

of two neighboring alloys, N(Efa) and N(Efb)’ i.e.,

N (Er) = %[N(Efa) + N(Efb)]. (A-7)

Hence

 

). (A—8)

”A

-ku xxx.

 

I“: .\‘M.I'

67

‘8
S
V

v

N (Egg)

 

N(E)

 

 

It \\\\r\\\t\\\\\\\\\\\

 

 

—4>
--|AE |-—

Fig. A-2.--To find energy intervals, AE, by numerical inte-

gration. N(E ) and N(E ) are the density of states

calculated f£8m corresp88ding y's of two neighboring alloys.

N(E ) = a [N(E )+N(E )J. The shaded area under the N(E)

Vs E curve equgI to tAB number of electrons added to the

alloy by increasing the solute content of an amount of Ac.

68

 

 

 

 

 

 

 

 

9
5|-
5 “ 4.
ss~
52_
't
O 1 n 1 1 n 1 1 n
-72 -08 -o.4 o 04
E(in some E.=I.439.VI
4 - d-band in m ( offer Koster“)
3'. o
uI
z:
\ .
2 _ Jo)” :
\, |
I
I
"‘ I
:Rh IPd
o I I I J 1
-414 412 OI) C12
E (eV)

Fig. A-3.-—The upper part of the Ad—band deduced from the
y values of the fcc solid solution alloys of 2nd long
period transition metals,Ii.e., Ru-Rh, Rh-Pd, and Pd-Ag
alloys. Energy range covered corresponds to the A—-A part
of Koster's 3d-band.

 

 

69

Letting 2 be the number of electrons per atom added to the
alloy per unit concentration change, and Ac be the increment

of alloy concentration then,

AB = zAc/NTE'ET. (LI-9)

Strictly Speaking, 2 is a function of c and should be
determined by other experiments. But to a first approxi-
mat ion, 2 can be taken as the relative valency of the two
constituent metals of the alloy. Thus 2 = 1.0 for adding
Pd to Rh or by alloying Rh to Ru. Figure A-2 shows
schematically the numerical method to find the energy
intervals, AE.

Together with the data of Hoare gt_§l., a density of
States versus energy curve is thus constructed, and is
Shown in Fig. A—3. It should be pointed out that the
emergy band shown in Fig. A—3 is the total density of states
of the d—electrons as well as the S-electrons, since the
heat capacity measured experimentally includes the contri—
but; ions from both of them. However, the density of states
of the 5s electrons is much less than that of Ad's; its
Contribution to the heat capacity is, consequently, very
Small'. Furthermore, the energy range covered in this
investigation is small (1.0.8 eV) compared with the entire
Width of the 5s band (% 13 eV). The density of states of
SS band may thus be regarded as being more or less constant

Over” the entire experimental energy range. Therefore, the

 

70

general features of the N(E) versus E curve Shown in Fig.
14—3 can be considered as representing the general features
of the upper part of the Ad-band in the fcc structure. AS
can be seen, to the left of the high peak, the density of
states decreases as the Fermi surface moves toward lower
and lower energy as observed by Hoare [53], and then bends
upward again at about 0.5 ev below the Fermi level of pure
palladium. This last feature may be of significance in
correlating the experimental results to the theoretical Ad—
bands .

The "d-hump" in the total energy band of fcc copper
calculated by Burdick [1A] using the APW method bears a peak
With no "valley" in the middle, whereas a conspicuous "valley"
is Shown in the fcc d-band calculated by using the tight-
binding method. Mott [32] seems to prefer the former picture,
While others regard the tight—binding method Should be the
more appropriate one for treating the d—electrons. To
d€01de which method is more appropriate should be important
for the formulation of the electronic structure of transition
metals, especially in those aspects related to the occur—
rence of ferromagnetism. The "bending—up" feature provides
5‘ C3er to the belief that there is a "Valley" in the middle
Of the d-band for the fee structure, and hence the tight—
binding method might be the more appropriate one.

To compare the Ad-band obtained in the present investi—

gatiOn with those obtained from tight—binding calculations,

 

71

the 3d—band of fcc Ni calculated by Koster [25] is plotted
at the upper left of Fig. A-3. AS can be seen, present
experimental Ad—band bears a fair resemblance with the
upper part of Koster's curve. Resemblance can also be
found between the experimental Ad-band and the Ad-band of
palladium calculated by Lenglart _e_t__§_l_. [31], using also
the tight-binding method, although for accommodating the
same number of electrons the energy range covered in the
band of Lenglart e_t_a_l_. is much smaller while the density
of states in the same energy range is much larger than that
in the experimental band. The introducing of the Spin—
orbital interaction by Lenglart e_t__a_1_. might have caused
these deviations. Due to the discovery of s-electrons in
pure palladium [51], it is thought that the hybridization
0f s—d electrons would alter the total band structure
drastically. This is shown to be perhaps not so in View of
Mueller's band calculation [3A]. By using an interpolation
Scheme, Mueller found that although s-d hybridization does
charige the d-band to some extent yet the main features of a
tiSlat-binding d-band are still retained. The experimental
d“band of this investigation appears to match even better
with Mueller's results than with Koster's in the height of
the band and the energy range covered. Since the comparison
is at best qualitative, it would not be possible to draw a
p08itive conclusion about whether or not Mueller's

interpolation scheme is closer to the truth. Nonetheless.

 

72

trier tight—binding scheme is still employed in constructing
‘tIlE? total d—wave functions in Mueller's method and one may
cc>r1clude that for the d-bands in the transition metals, the
t:fi_gflat—binding approximation is essentially correct,at
Zleeaist fbr'constructing the total d-wave functions. The
lICl-Jsand of palladium and that of silver obtained by Yu and
£3p>ixzer [7A] and Spicer [75] respectively using an optical
nueizliod also Show the features of a tight-binding d-band.
19163 present results support this recognition.

E. Applicability of the Rigid—Band Model
to Alloys of the Transition Metals

 

In calculating the N(E) versus E curve from the
SIDeeczific heat data a rigid—band is implied. There has been
“<3 F>roof of the general applicability of such a model.
PEBI’PlapS the first question to be asked is: would adding an
aCLJ_C>ying element change the band structure drastically?
CcDrlsidering from the first principles applied to the
Erleéxegy band calculations, and confining the case to.binary
aliLCDys of neighboring transition elements in the same
lcnug; period in the periodic table, the answer would be
neflgéltive, since the lattice potentials (self-consistent
field.ofthe element) of the neighboring elements in the
553”“? period are more or less the same, and the periodicity
of 'tlae lattice potential would still be retained in the
allLCDSIS. The wave functions in the alloy would still be of

Bl<3C3I1 type and be more or less the same as that in pure

73

metals. Therefore, alloying such as described will intro—
duce only a small perturbation which may not alter the
essential features of the resulting band. Mattheiss'
calculation [12] provides a theoretical justification for
this point.

It is known that the relative position of the s and d
bands of two neighboring transition metals may vary. Thus B
it is natural to ask a second question : whether or not
the change of the relative position of the S-d bands would

cause a large change in the major features of the total

" W‘MI'

density of states curve of the alloy? The answer is,
according to Mueller [A5], also negative.

The theories are therefore in favor of the applica-
bility of the rigid—band model. Experimentally, the
agreement between Ad—band derived from the Specific—heat
dEtta and that calculated theoretically seems to suggest
that the rigid—band model may indeed have some truth in

aIDDlying to the restricted alloys of the transition metals.

 

 

 

CHAPTER V
CONCLUSIONS

(1). The solubility limit of Ru in fee Rh was
found, by using an X-ray diffraction method, to be
between A0 and A2 at.% Ru.

(2). The specific heats of 1A samples, including

 

12 fee Ru-Rh and Rh-Pd solid-solution alloys and one of
each of pure rhodium and pure palladium, were measured
between 1.A and A.2°K. The coefficients of y of the
Electronic Specific heats of pureRh and pure Pd determined
by the present work agree with those obtained by previous
investigators within the limit of the experimental error.

(3). A "Schottky-tail" type anomaly was observed in
some of the alloys especially in those with composition
near 22 at.% Pd in Rh. The cause of this anomaly has not
been, established unambiguously. However, by fitting the
experimental data to various equations representing the
dij.‘t‘erent solid state phenomena which might cause such an
ahaJ'l'toly, it seems that the hyperfine interaction is the
probable cause.

(A). A plot of the coefficient of the electronic
Specific heat 7 versus the outer electron concentration
e/a for the Samples investigated in the present work joins

smoothly with the existing results of Hoare et al.

7A

75

(5). The uncertainty in the Debye temperatures of
these alloys was found to be large compared with that of the
alloys of the noble metals. A possible eXplanation is that
this uncertainty is due to the effect of the local slope
and curvature of the total density-of—states curves of the
highly irregular Ad—band.

(6). A density-of-states curve constructed numeri- 7L
cal ly for the upper part of the Ad-band from the experi-
mental y values is found to be in qualitative agreement
With the corresponding part of the theoretical 3d-band

calculated for the paramagnetic Ni by G. F. Koster using I

 

.-
T7—

the tight-binding method. It appears that for the fcc
structures the Ad-band of the second long period transition
metals is similar to the 3d-band of the first long period
transition metals, and that the approximate method of Linear
COmbination of Atomic Orbitals is essentially correct for

Treating the d-electrons in the transition metals.

10 .
11.-
12 .

13..

Ill.
155.
163,
17'.

18

19.

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—

 

APPENDICES

81

1‘
r

41.. II._'_'-. “

 

APPENDIX A

EXPERIMENTAL DATA

82

-- _ «FM Q‘S'k -'Fl.‘.':$

 

'83

APPENDIX A:EXPERIMENTAL DATA

A-l Pure Rhodium Eigat Run

Weight of A110y:28.97239m

No. of ”0130:0285

Weight of Heater-Thermometer Auemblyziloa'n No. of Moleuooflfl

Resistance of Heater at 4.2‘K: 279-94 0km

A --1.016970
8 8 0.626236

Thermometer calibration parametera:

 

T2

 

It

[913; ‘0‘ 1 1:5?) ( acv ole-de cal mole de 2
1 1.4814 2.1946 0.0018997 0.0012824
2 1.5441 2.3844 0.0019840 0.0012848
3 1.6306 2.6590 0.0020712 0.0012702
4 1.7193 2.9561 0.0021709 0.0012626
5 1.8047 3.2568 0.0022834 0.0012653
6 1.8811 3.5385 0.0023480 0.0012482
7 1.9532 3.8151 0.0025150 0.0012876
8 2.0295 4.1187 0.0025870 0.0012747
9 2.1308 4.5401 0.0026305 0.0012345

10 2.2213 4.9341 0.0027426, 0.0012347
11 2.2933 5.2593 0.0028342 0.0012359
12 2.3919 5.7210 0.0029653 0.0012398
13 2.4739 6.1200 0.0030641 0.0012386
14 2.6758 7.1598 0.0032790 0.0012254
15 2.7529 7.5787 '0.0033705 0.0012243
16 2.8376 8.0521 0.0034887 0.0012294
17 2.9322 8.5980 0.0036035 0.0012289
18 3.0418 9 2528 0.0037492 0.0012325
19 3.1289 9.7902 0.0038570 0.0012327
20 3.2233 10.3893 0.0039667 0.0012306
21 3.3123 10.9710 0.0040923 0.0012355
22 3.4071 11.6084 0.0042153 0.0012372
23 3.4994 12.2461 0.0043483 0.0012426
24 3.5563 12.6475 0.0044099 0.0012400
25 2.5710 6.6103 0.0031590 0.0012287
26 2.7112 7.3507 0.0033342 0.0012298
27 3.7631 14.1609 0.0047011 0.0012493
28 3.6308 13.1829 0.0045064 0.0012412
29 3. 6999 13.6892 0.0046109 0.0012462
30 3. 7719 14.2273 0.0046751 0.0012394
31 3. 8516 14.8351 0.0048129 0.0012496
32 3.9166 15.3399 0.0049148 0.0012549
33 3.9887 15.9096 0.0049858 0.0012500
34 4.0675 16.5445 0.0051081 0.0012558
35 4.1425 17.1606 0.0052443 0.0012660

Lear: Square Pit of Points from No.20 to No.35
T- 0. 0012028990 c.1./mo1.-d.g§

A II 0. 0000031434 cal. finale-deg
a. 528. 583 OK

°v - 12.0291- + 0.031413 (ca1./mo1e-de3) 1110-4

 

8A '

5.2 S Ac.1 Ru-Rh

 

 

Weight of Alloy: 23.842! gm No. of Molea:o.23|9
Weight of Heater-Thermometer Asaembly:5.262£No. of Molea:o.0828
Resistance of Heater at 4.2 K: 255.195 ohm

A =-1.025886
B = 0.637435

Thermometer calibration parameters:

 

 

. 457

No. of T T3 0 Cv/T 2
Point ("K ) __ (‘K ) (caYZQole-degz Scallmolezdea 1

1 1.4032 1.9691 0.0017186 0.0012248

2 1.4291 2.0424 0.0017346 0.0012138

3 1.4585 2.1274 0.0018313 0.0012556
4 1.4964 2.2392 0.0018607 0.0012435

5 1.5348 2.3556 0.0019086 0.0012436

6 1.5844 2.5104 0.0019493 0.0012303

7 1.6266 2.6459 0.0019722 0.0012125

8 1.6637 2.7679 0.0019801 0.0011902

9 1.6968 2.8792 0.0020657 0.0012174
10 1.7409 3.0306 0.0021141 0.0012144
12 1.8548 3.4403 0.0022342 0.0012045
13 1.9314 3.7304 0.0023308 0.0012068
14 2.0147 4.0590 0.0024352 0.0012087
15 2.1361 4.5629 0.0025863 0.0012108
16 2.1911 4.8007 0.0026429 0.0012062
17 2.2751 5.1762 0.0027487 0.0012081
18 2.3832 5.6796 0.0028830 0.0012097
19 2.5037 6.2685 0.0030215 0.0012068
20 2.5415- 6.4591 0.0030561 0.0012025
21 2.6123 6.8243 0.0031435 0.0012033
22 2.6952 7.2640 0.0032325 0.0011994
23 2.7886 7.7764 0.0033714 0.0012090
24 2.8988 8.4033 0.0034930 0.0012050
25 2.9934 8.9603 0.0035991 0.0012023
26 3.0643 9.3902 0.0037061 0.0012094
27 3.1454 9.8938 0.0038110 0.0012116
28 3.2347 10.4633 0.0039365 0.0012170
29 3.3288 11.0810 0.0040189 0.0012073
30 3.3985 11.5495 0.0041753 0.0012286
31 3.4902 12.2092 0.0042722 0.0012226
32 3.6067 13.0080 0.0044146 0.0012240
33 3.6990 13.6826 0.0045775 0.0012375
34 3.7829 14.3101 0.0046631 0.0012327
35 3.8764 15.0263 0.0048041 020012393
36 3.9794 15.8356 0.0049582 0.0012460
37 4.0916 16.7415 0.0051087 0.0012486
. ------------------------------------------- ----.------.----

 

Lea-t Square fit of points from point No.21 toNo.37

1r-0.0011648541 ca1./mole-de
p -0. 0000048769 cal. /mole-de
9 «.56. 605°K

cva11.649T+0.0488T3 (ca1./mole-deg) x 10-4

8%
a

 

85.

A-3 10 at.%Ru-Rh

Weight of Alloy: 29.4333 9'“ No. of Mole:o.2865
Weight of Heater-Thermometer Assembly:5.2|36 No. of Moles:O-0924

Resistance of Heater at 4.2‘K: 256.145 0km

 

 

Thermometer calibration parameters: A=-1.016366
8:0.6338952
No. of 1 12 cv Cv/T
Point (K)“g (‘K ) (cal/mQIe-deg) (callmole-deg?)
2 1.4217 2.0213 0.0017358 0.0012209
3 1.4379 2.0676 0.0017476 0.0012154
4 1.4852 2.2058 0.0017898 0.0012051
5 1.5444 2.3850 0.0018322 0.0011846
6 1.5883 2.5226 0.0019003 0.0011965
7 1.6511 2.7261 0.0019381 0.0011738
8 1.7270 2.9826 0.0020313 0.0011762
9 1.8023 3.2482 0.0021261 0.0011797
10 1.9390 3.7596 0.0022617 0.0011664
11 2.0358 4.1444 0.0023832 0.0011706
12 2.1500 4.6225 0.0025199 0.0011721
13 2.2470 5.0490 0.0026131 0.0011629
14 2.3245 5.4031 0.0027160 0.0011685
15 2.4443 5.9745 0.0028682 0.0011734
16 2.4900 6.2000 0.0028977 0.0011638
17 2.5549 6.5273 0.0029805 0.0011666
18 2.6253 6.8923 0.0030500 0.0011618
19 2.7055 7.3197 0.0031663 0.0011703
20 2.7955 7.8148 0.0032810 0.0011737
21 2.8949 8.3805 0.0033956 0.0011729
22 2.9994 8.9962 0.0035015 0.0011674
23 3.0682 9.4139 0.0036708 0.0011801
24 3.1469 9.9029 0.0037271 0.0011844
25 3.2286 10.4255 0.0038218 0.0011836
26 3.3263 11.0642 0.0040291 0.0012113
27 3.4388 11.8254 0.0040220 0.0011697
28 3.5493 12.5977 0.0042516 0.0011979
29 3.6387 13.2399 0.0043790 0.0012035
30 3.7214 13.8491 0.0045238 0.0012156
31 3.8341 14.7000 0.0046609 0.0012157
32 3.9284 15.4325 0.0048163 0.0012260
33 4.0588 16.4739 0.0050003 0.0012320
34 4.1517 17.2363 0.0051342 0.0012366
35 4.2403 17.9805 0.0052470 0.0012370

I'm-i"- .

 

Least Square Fit 6f points from No.12 to No.35.
1f. 0.0011291913 ca1./mole-de 2

82
/3= 0.0000058489 ca1./mole-deg
9a 429.766CK

Cv a 11.292T + 0.058513 (ca1./mole-deg) X 10"4

 

'86

 

Weight of Alloy: 28.2757 8" No. of 1.101.342.2155
Weight of Heater-“memometer Assembly:5.l4ql No. of Moleza.0826
Resistance of Heater at 4.2‘K :256JO5 01M

A =I-1.015527
B = 0.633414

Thermometer calibration parameters:

 

 

No. of T ‘1'2 Cv Cv/‘l' 2
Point 00 (K?) (W

1 1.4876 2.2129 0.0016659 0.0011198

2 1.5252 2.3263 0.0017223 0.0011292

3 1.5666 2.4544 0.0017666 0.0011276
4 1.6207 2.6267 0.0018312 0.0011299

5 1.6486 2.7180 0.0018581 0.0011271

6 1.6748 2.8048 0.0018872 0.0011268

7 1.7366 3.0158 0.0019530 0.0011246

8 1.7788 3.1641 0.0019917 0.0011197
9 1.8182 3.3059 0.0020491 0.0011270
10 1.8636 3.4730 0.0020950 0.0011242
11 1.9414 3.7691 0.0021871 0.0011265
12 2.0385 4.1556 0.0023043 0.0011304
13 2.0958 4 3926 0.0073676 0.0011297
14 1.9007 3.6127 0.0021158 0.0011132
15 2.1278 4.5275 0.0023830 0.0011199
16 2.2033 4 8547 0.0024731 0.0011224
17 2.2798 5.1974 0.0025592 0.0011226
18 2.4035 5 7767 0.0027028 0.0011245
19 2.4416 5 9614 0.0027571 0.0011292
20 2.4996 6.2488 0.0028028 0.0011212
21 2.5614 6.5609 0.0028873 0.0011272
22 2.6258 6.8946 0.0029654 0.0011294
23 2.7416 7.5163 0.0030850 0.0011253
24 2.8953 8.3827 0.0033016 0.0011403
25 3.0051 9.0305 0.0034061 0.0011335
26 3.0671 9.4069 0.0034904 0.0011380
27 3.1444 9.8873 0.0036336 0.0011556
28 3.2438 10.5221 0.0037614 0.0011596
29 3.3448 11.1875 0.0038192 0.0011419
30 3.4452 11.8698 0.0039323 0.0011414
31 3.5606 12.6778 0.0041068 0.0011534
32 3.6372 13.2295 0.0042165 0.0011593
33 3.7150 13.8013 0.0042975 0.0011568
34 3.7858 14.3320 0.0044749 0.0011820
35 3.8537 14.8513 0.0046036 0.0011946
36 3.9316 15.4578 0.0046590 0.0011850
37 4.0179 16.1439 0.0048324 0.0012027
38 4.1019 16.8255 0.0048990 0.0011943
39 4.2072 17.7007 0.0050222 0.0011937

N's.- 95.1. (chug!

 

Q
------- ----..---------------------------------.--.----.------. ‘

Least Square Pit of Points from No. 13 to No. 35
”(3 0.0010893089 ca1./mole-deg§
[3 a 0.0000059407 'cal./mole-deg
9- 427.543’K

cv - 10.8932 + 0.059423 (ca1./mole-deg) x 10'“

87.

A-5 2.0 at.% Ru-Rh

height of Alloy: 29.2794 9'“ No. of Nolezo.2855
Weight of Heater-Thermometer Assembly:5'_3897 No. of Mole:o.0648
Resistance of Heater at 4.2‘K :ZSSJG okm 1

 

 

Thermometer calibration parameters: A =-1.027808
B = 0.637368
2
No. of '1‘ T /1‘ .
Point (K) (K2) (cal?¥ole-dcg) (callggle-degz) 5?}
1 1.4325 2.0521 0.0016658 0.0011629 '1
2 1.4623 2.1383 0.0016612 0.0011360
3 1.4967 2.2400 0.0016894 0.0011287
4 1.5300 2.3409 0.0017550 0.0011471
5 1.5528 2.4113 0.0017460 0.0011244
6 1.5723 2.4722 0.0017737 0.0011281
7 1.5901 2.5283 0.0018078 0.0011369'
8 1.6110 2.5954 0.0018242 0.0011323 4
9 1.6314 2.6616 0.0018271 0.0011199 F
1’ 1.6560 2.7423 0.0018740 0.0011317 .9
11 1.6822 2.8299 0.0018622 0.0011070
12 1.7009 2.8931 0.0019019 0.0011182
13 1.7365 3.0155 0.0019595 0.0011284
14 1.7670 3.1222 0.0019880 0.0011251
15 1.7987 3.2353 0.0019979 0.0011107
16 1.8336 3.3619 0.0020660 0.0011267
17 1.8727 3.5070 0.0021031 0.0011230
18 1.9147 3.6660 0.0021501 0.0011229
19 1.9679 3.8727 0.0021921 0.0011139
20 2.0388 4.1566 0.0022858 0.0011212
21 2.1181 4.4865 0.0023759 0.0011217
22 2.1907 4.7992 0.0024507 0.0011187
23 2.2617 5.1152 0.0025533 0.0011289
24 2.3306 5.4315 0.0025994 0.0011154
25 2.3815 5.6715 0.0026805 0.0011256
26 2.4856 6.1781 0.0027988 0.0011260
27 2.5851 6.6829 0.0029107 0.0011260
28 2.6610 7.0811 0.0030123 0.0011320
29 2.7467 7.5444 0.0031092 0.0011320
30 2.8423 8.0786 0.0032523 0.0011442
31 2.9659 8.7967 0.0033915 0.0011435
32 3.0330 9.1989 0.0034512 0.0011379
33 3.1087 9.6643 0.0035805 0.0011517
34 3.1944 10.2041 0.0036548 0.0011441
36 3.2913 10.8329 0.0037786 0.0011480
37 3.3394 11.1514 0.0038439 0.0011511
38 3.3985 11.5495 0.0039257 0.0011551
39 3.4546 11.9341 0.0039825 0.0011528
40 3.5190 12.3834 0.0040890 0.0011620
41 3.5874 12.8692 0.0041652 0.0011611
42 3.6641 13.4260 0.0042471 0.0011591
43 3.7384 13.9757 0.0043442 0.0011621
44 3.7868 14.3399 0.0044144 0.0011657
45 3.8815 15.0663 0.0045334 0.0011679
46 3.9815 15.8523 0.0046816 0.0011758
47 4.0917 15.7420 0.0048585 0.0011874

Least Square Pit of Points from No. 19 to No. 47
’f- 0.0010968688 cal./molc-dagz
[3- 0.0000049371 cal./mole-de3
0- 454.741 °1<

Cv I 10.9691? + 0.0494‘1‘3 (cal./mole-deg) X 10'“

 

88

A-6 25 at-% Ru-Rh

 

weight of Alloy: 29.776I am No. of Mole:O.2907
Weight of Heater-Thermometer Asaembly:5,2538 No. of Mole:'0.0827
Resistance of Heater at 4.2‘K:256.20 ohm

A =-1.012825
B = 0.632941

Thermometer calibration parameters:

 

 

N0. of 1 1% 0 cv/r
£312: (Kl_. CK ) (calllee-desz ‘callmole-degzz
1 1.4539 2.1138 0.0015968 0.0010983
2 1.4838 2.2017 0.0016207 0.0010923
3 1.5443 2.3850 0.0016995 0.0011005
4 1.6219 2.6304 0.0017851 0.0011006
5 1.6876 2.8481 0.0018451 0.0010933
6 1.7794 3.1664 0.0019572 0.0010999
7 1.8626 3.4694 0.0020327 0.0010913
8 1.9371 3.7527 0.0021138 0.0010912
9 2.0319 4.1286 0.0022266 0.0010958
10 2.1594 4.6631 0.0023727 0.0010988
12 1.9461 3.7873 0.0021225 0.0010906
13 2.0334 4.1346 0.0022137 0.0010887
14 2.1503 4.6237 0.0023490 0.0010924
15 2.2705 5 1552 0.0024815 0.0010929
16 2.3626 5 5818 0.0025975 0.0010994
17 2.4845 6.1729 0.0027216 0.0010954
19 2.6235 6.8829 0.0028952 0.0011036
20 2.7233 7.4162 0.0030019 0.0011023
21 2.8434 8.0848 0.0031683 0.0011143
23 2.5888 6.7021 0.0028361 0.0010955
24 2.7848 7.7550 0.0030709 0.0011027
25 2.9026 8.4249 0.0032029 0.0011035
26 2.9928 8.9567 0.0033357 0.0011146
27 3.1150 9.7035 0.0034678 0.0011132
28 3.2250 10.4005 0.0036351 0.0011272
29 3.3230 11.0421 0.0037293 0.0011223
30 3.4315 11.7752 0.0038711 0.0011281
31 3.5432 12.5544 0.0039927 0.0011269
32 3.6075 13.0139 0.0041546 0.0011517
33 3 6882 13.6028 0.0041947 0.0011373
34 3.7737 14.2410 0.0043426 0.0011507
35 3.8629 14.9220 0.0045164 0.0011692
36 3.9655 15.7248 0.0046437 0.0011710
37 4.0762 16.6153 0.0047832 0.0011735
38 4.1957 17.6035 0.0049686 0.0011842
39 4.3158 18.6260 0.0051341 0.0011896

’Leaat Square Fit of Points from No. 15 to No. 39
‘(= 0.0010480844 ca1./mole-deg%
[3: 0.0000074244 ca1./mole-deg
0= 396.925°K

cv = 10.481T + 0.074213 (ca1./mole-deg) x 10-4

 

E39

A-7 35 at.% Ru-Rh

weight of Alloy: 29.5421 3'“ No. of Mole:O.2839

Height of Heater-Thermometer Assembly:5.2476 No. of Mole:O-0826
Resistance of. Heater at 4361025022 aka

A =-1.020427

Thermometer ca librat ion parameters:
8 = 0.635107

 

2

 

No. of 1 1 c CV/T 2
Point 1K) (K2) Qcal/moYe-defi) (cal/mole-deg 1

2 1.3558 1.8382 0.0014789 0.0010908

3 1.4092 1.9858 0.0015234 0.0010810
4 1.4827 2.1985 0.0016057 0.0010829

5 1.5630 2.4430 0.0016820 0.0010761

6 1.6651 2.7725 0.0017962 0.0010787

7 1.7485 3.0573 0.0018724 0.0010708

8 1.8043 3.2555 0.0019251 0.0010670

9 1.8692 3.4939 0.0020039 0.0010720
10 1.9383 3.7570 0.0020800 0.0010731
11 2.0317 4.1279 0.0021919 0.0010788
12 2.0719 4.2929 0.0021965 0.0010601
13 2.1227 4.5057 0.0022806 0.0010744
14 2.1984 4.8330 0.0023540 0.0010708
15 2.2908 5.2478 0.0024494 0.0010692
16 2.3016 5.2975 0.0024564 0.0010673
17 2.3393 5.4721 0.0026084 0.0010723
18 2.3861 5.6935 0.0025578 0.0010720
19 2.4395 5.9511 0.0026080 0.0010691
20 2.4924 6.2122 0.0026848 0.0010772
21 2.5544 6.5252 0.0027286 0.0010682
22 2.6272 6.9019 0.0028297 0.0010771
23 2.6461 7.0017 0.0028608 0.0010812
24 2.6870 7.2197 0.0028926 0.0010765
25 2.7295 7.4502 0.0029619 0.0010851
26 2.7762 7.7075 0.0029804 0.0010735
27 2.8263 7.9881 0.0030719 0.0010869
28 2.8819 8.3051 0.0031279 0.0010854
29 2.9419 8.6548 0.0031796 0.0010808
30 3.0067 9.0405 0.0032307 0.0010745
31 3.0759 9.4614 0.0033556 0.0010909
32 3.1577 9.9713 0.0034462 0.0010913
33 3.2398 10.4961 0.0035536 0.0010969
34 3.3689 11.3496 0.0036836 0.0010934
35 3.4637 11.9971 0.0038193 0.0011027
36 3.5629 12.6941 0.0039386 0.0011055
37 3.6495 13.3191 0.0040295 0.0011041
38 3.7439 14.0171 0.0041852 0.0011179
39 3.8516 14.8344 0.0043015 0.0011168
40 3.9658 15.7274 0.0045046 0.0011359

Least Square Pit of Points from No. 14 to No. 40
0(- 0.0010398129 cal./mole-deg2
/3- 0.0000052946 ca1./mole-deg
9:- 444.269°K

cv - 10.3981 + 0.052913 (cal./mole-deg) x 10"

 

90

A-8 40 at.% Ru-Rh

N0. of Mole:0.3045

Weight of Heater-Thermometer Assembly:5,28&4 No. of Mole:o.083|

weight of A110v331-1085 9'“

Resistance of Heater at 4.2‘K: 297.32 ohm

Thermometer calibration parameters: A =-1.018204

B = 0.627112

 

2

 

 

No. of 1 12 0V cv/1
Point (K) (K 2, (cal/mole-deg), (cal/mole-degzl
1 1.3971 1.9519 0.0016396 0.0011736
2 1.4635 2.1418 0.0016962 0.0011591
3 1.5427 2.3800 0.0017692 0.0011468
4 1.6024 2.5678 0.0018217 0.0011368
5 1.6540 2.7357 0.0018554 0.0011217
6 1.7150 2.9414 0.0019451 0.0011342
7 1.7956 3.2241 0.0020337 0.0011326
8 1.8461 3.4082 0.0020863 0.0011301
9 1.8747 3.5143 0.0021069 0.0011239
10 1.9403 3.7648 0.0021738 0.0011203
11 2.0187 4.0750 0.0022632 0.0011212
12 2.1240 4.5112 0.0023894 0.0011250
13 2.1480 4.6141 0.0024013 0.0011179
14 2.2167 4.9138 0.0024871 0.0011220
15 2.2986 5.3836 0.0025552 0.0011116
16 2.3884 5.7045 0.0026602 0.0011138
17 2.4957 6.2287 0.0027692 0.0011096
18 2.6337 6.9366 0.0029390 0.0011159
19 2.7461 7.5409 0.0030639 0.0011158
20 2.8250 7.9807 0.0031277 0.0011072
21 2.9123 8.4814 0.0032603 0.0011195
22 2.9862 8.9172 0.0033384 0.0011179
23 3.0712 9.4321 0.0034441 0.0011214
24 3.1384 9.8497 0.0035204 0.0011217
25 3.2195 10.3652 0.0036273 0.0011267
26 3.2993 10.8855 0.0037368 0.0011326
27 3.3928 11.5108 0.0037966 0.0011190
28 3.5031 12.2719 0.0039668 0.0011323
29 3.6249 13.1400 0.0041327 0.0011401
30 3.7732 14.2372 0.0043185 0.0011445
31 3.8403 14.7478 0.0043640 0.0011364
32 3.9250 15.4058 0.0044923 0.0011445
33 4.0153 16.1228 0.0046013 0.0011459
34 4.0936 16.7578 0.0047599 0.0011628
35 4.1916 17.5692 0.0048411 0.0011550

Least Square Pit of Points from No. 12 to No. 35

f.

0.0010954537 ca1./mole-de32
/3= 0.0000031560 ca1./mole-de52
9s 527.882‘K

CV = 10.955T + 0.C316T3 (ca1./mole-deg) X 10'“

91

A-9 7 at.% Pd-Rh
0.1311: of Alloy: 30.6927 9!» No. of 1101e:o.2976
Weight of Heater-Thermometer Assembly:49|7’ No. of Mole:0.0774

Resistance of Heater at 4.2%:29122 Ohm

 

 

Thermometer calibration parameters: A 8-1.008480
8 = 0.624752
No. of T T2 C Cv/T 2
Point (K) (K2) (galZmOIe-degl scaIZmoie-deg 2
1 1.3720 1.8825 0.0018150 0.0013229
2 1.3909 1.9346 0.0017858 0.0012839
3 1.4406 2.0753 0.0018621 0.0012926
4 1.4925 2.2276 0.0019443 0.0013027
5 1.5411 2.3751 0.0019853 0.0012882
6 1.5799 2.4962 0.0020119 0.0012734
7 1.6271 2.6475 0.0020750 0.0012753
8 1 6837 2.8349 0.0021416 0.0012719
9 1 7380 3.0208 0.0021795 0.0012540
10 1 7786 3.1633 0.0022287 0.0012531
11 1 8218 3.3189 0.0022746 0.0012486
12 1 8749 3.5154 0.0023504 0.0012536
13 1.9388 3.7588 0.0024343 0.0012556
14 1 9388 3.7588 0.0024160 0.0012461
15 2 0085 4 0340 0.0025081 0.0012488
16 2 1128 4 4639 0.0026589 0.0012585
17 2 2090 4.8799 0.0027385 0.0012397
18 2 2977 5.2794 0.0028426 0.0012372
19 2 3827 5.6773 0.0029323 0.0012307
20 2 4765 6.1330 0.0030604 0.0012358
21 2 5107 6.3034 0.0030974 0.0012337
22 2 5695 6.6023 0.0031634 0.0012311
23 2.6219 6.8743 0.0032387 0.0012353
24 2.6971 7.2744 0.0033326 0.0012356
25 2.7736 7.6928 0.0034443 0.0012418
26 2 8576 8.1659 0.0035414 0.0012393
27 2 9754 8 8530 0.0036677 0.0012327
28 3.0297 9.1791 0.0037468 0.0012367
29 3.1300 9.7968 0.0039254 0.0012541
30 3.2066 10.2826 0.0040088 0.0012501
31 3.2919 10.8369 0.0041053 0.0012471
32 3 3812 11.4327 0.0042480 0.0012563
33 3.4797 12.1083 0.0043518 0.0012506
34 3.6213 13.1137 0.0045887 0.0012671
35 3.6805 13.5461 0.0046689 0.0012686
36 3.7729 14.2345 - 0.0047990 0.0012720
37 3.8700 14.9773 0.0049556 0.0012805
38 3.9553 15.6443 0.0050612 0.0012796
39 4.0522 16.4205 0.0052225 0.0012888
40 4.1410 0.0053813 0.0012995

17.1478

Least Square Fit of Points from No. 17 to No. 40

*(- 0.0012010033 ca1./m01.a-de32
/3- 0.0000050313 ca1./mola-da32
9- 451.035”: .

°v - 12.0101 + 0.050313 (ca1./mole-deg) x 10'“

92

A-10 15 at.% Pd-Rh

Weight of Alloy: 30.3602 91» No. of Mole:0.2935
Weight of Heater-memometer Assembly:5.2507 No. of Mole:0.0826
Resistance of Heater at 4.2‘K: 296.955 01‘"

Thermometer calibration parameters: A --1.017232
8 8 0.627298

 

 

2
No. of T T Cv/T
Point (K) 61(2) (cal/mfie-dag) (cal/mola-dagz)

1 1.3747 1.8898 0.0018776 0.0013659
2 1.4132 1.9972 0.0019079 0.0013500
3 1.4733 2.1707 0.0019912 0.0013515
4 1.5389 2.3681 0.0020679 0.0013438
5 1.5969 2.5499 0.0021179 0.0013263
6 1.6802 2.8231 0.0022338 0.0013295
7 1.7495 3.0607 0.0023296 0.0013316
8 1.8405 3.3874 0.0024366 0.0013239
9 1.8714 3.5022 0.0024736 0.0013218
10 1.9349 3.7440 0.0025551 0.0013205
11 2.0119 4.0478 0.0026481 0.0013162
12 2.1135 4.4668 0.0027923 0.0013212
13 2.1489 4.6178 0.0028208 0.0013127
14 2.2089 4.8791 0.0028825 0.0013050
15 2.2926 5.2561 0.0030072 0.0013117
16 2.3724 5.6283 0.0030985 0.0013061
17 2.4663 6.0824 0.0032285 0.0013091
18 2.5342 6.4222 0.0033098 0.0013060
19 2.5908 6.7124 0.0034055 0.0013145
20 2.6582 7.0662 0.0034925 0.0013139
21 2.7327 7.4679 0.0035918 0.0013143
22 2.8172 7.9365 0.0037133 0.0013181
23 2.9108 8.4727 0.0038291 0.0013155
24 3.0236 9.1419 0.0040079 0.0013256
25 3.1649 10.0163 0.0042243 0.0013348
26 3.2882 10.8120 0.0043935 0.0013362
27 3.6104 13.0353 0.0048480 0.0013428
28 3.7476 14.0444 0.0050533 0.0013484
29 3.8741 15.0086 0.0052257 0.0013489
30 3.9674 15.7405 0.0054092 0.0013634
4.0565 16.4550 0.0055244 0.0013619

32 4.1659 17.3548 0.0057137 0.0013715
33 3.3039 10.9160 0.0043930 0.0013296
34 3.3820 11.4377 0.0044922 0.0013283
35 3.4927 12.1988 0.0046509 0.0013316
36 3.6017 12.9723 0.0048312 0.0013414
37 3.7391 13.9808 0.0050244 0.0013437
38 3.8385 14.7345 0.0051982 0.0013542
39 3.9331 15.4692 0.0053991 0.0013727
40 4.0164 16.1315 0.0055054 0.0013707
41 4.1163 18.9440 0.0056061 0.0013619
42 4.2230 17.8928 0.0058270 0.0013776

Least Square Fit at Points from No. 12 to No. 42
'f- 0.0012806352 cal./mola-da 2

,4- 0.0000049907 cal. finale-clogz
9.7453.108°K

9v - 12.0061- + 0.01.9923 (ca1./mole-deg) x 10"

. 93
1-11 22 at.% man

weight of A110y:25o946' 9‘“ No. of Mole:0.2503
Weight of Heater-Thermometer Assembly:52675 No. of Mole:0.0829
Resistance of Heater at 4.2"K:297.355 91""

A =-1.015841
B 3 0.623333

Thermometer calibration Parameters:

 

2

 

No. of T T . . /T
Point (K) 6K2) (cal/mSYe-deg) (cal/ggle-degz)

1 1.3344 1.7807 0.0022663 0.0016984
2 1.3719 1.8821 0.0022299 0.0016254
3 1.4182 2.0112 0.0022462 0.0015839
4 1.5094 2.2784 0.0023292 0.0015431
5 1.5849 2.5120 0.0024139 0.0015230
6 1.6545 2.7375 0.0024642 0.0014894
7 1.7145 2.9395 0.0024888 0.0014516
8 1.7929 3.2146 0.0026610 0.0014842
9 1.9078 3.6397 0.0028224 0.0014294
10 1.9394 3.7612 0.0028025 0.0014450
11 2.0201 4.0810 0.0029106 0.0014408
12 2.1136 4.4672 0.0031400 0.0014857
13 2.1221 4.5032 0.0030839 0.0014532
14 2.2437 5.0340 0.0032148 0.0014328
15 2.3279 5.4193 0.0033208 0.0014265
16 2.4247 5.8792 0.0034548 0.0014248
17 2.5219 6.3601 0.0035956 0.0014257
18 2.6542 7.0449 0.0037862 0.0014265
19 2.8032 7.8578 0.0039971 0.0014259
20 2.9224 8.5405 0.0041707 0.0014271
21 3.0381 9.2298 0.0043550 0.0014335
22 3.1727 10.0658 0.0045543 0.0014355
23 3.3528 11.2414 0.0048239 0.0014387
24 3.3939 11.5188 0.0048717 0.0014354
25 3.5035 12.2746 0.0050837 0.0014510
26 3.6240 13.1331 0.0052511 0.0014490
27 3.7660 14.1829 0.0054690 0.0014522
28 3.8252 14.6320 0.0055848 0.0014600
29 3.8859 15.1004 0.0056329 0.0014496
30 3.9572 15.6594 0.0057577 0.0014550
31 4.0330 16.2650 0.0059058 0.0014644
32 4.1154 16.9366 0.0060251 0.0014640
33 4.2097 17.7212 0.0061666 0.0014649

 

least Square Fit of Points from No.

1(; 0.0013990152 cal./mole-deg
= 0.0000037496 ca1./mole-deg

6: 498.414°K

c... = 13.9902 + 0.037513 (cal./mole-deg) x 10-4

2
2

16 to No.

33

94

A-12 30at.% Pd-Rh

Weight of Alloy: 224085 9'“ No. of Mole:O.2638
Weight of Heater-Thermometer Assembly:5.2331 No. of Mole30.0824
Resistance of Heater at 4.2‘K: 296.95 01"“

A =-1.020067
B = 0.628324

Thermometer calibration parameters:

 

 

 

No. of T T; GV CV/‘I‘
Point Afix)_ {K ) (cal/mole-deg) (callmole-degi)

1 1.4616 2.1362 0.0022670 0.0015510
2 1.5189 2.3071 0.0023578 0.0015523
3 1.5769 2.4866 0.0024295 0.0015407
4 1.6198 2.6237 0.0024917 0.0015383
5 1.6747 2.8048 0.0025353 0.0015128
6 1.7502 3.0631 0.0026210 0.0014975
7 1.8396 3.3841 0.0027495 0.0014946
8 1.8698 3.4961 0.0027839 0.0014889
9 1.9310 3.7288 0.0028684 0.0014854
10 2.0117 4.0468 0.0029809 0.0014818
11 2.1049 4.4304 0.0031114 0.0014782
12 2.2008 4.8436 0.0032395 0.0014719
13 2.2765 5.1826 0.0033161 0.0014567
14 2.3645 5.5911 0.0034524 0.0014601
15 2.4743 6.1221 0.0036007 0.0014552
16 2.5300 6.4007 0.0036754 0.0014528
17 2.5915 6.7159 0.0037709 0.0014551
18 2.6579 7.0642 0.0038590 0.0014519
19 2.7321 7.4643 0.0039705 0.0014533
20 2.8167 7.9340 0.0041085 0.0014586
21 2.9104 8.4706 0.0042686 0.0014667
22 3.0249 9.1502 0.0044460 0.0014698
23 3.1603 9.9876 0.0046554 0.0014731
24 3.2035 10.2622 0.0046967 0.0014661
25 3.2855 10.7944 0.0048002 0.0014610
26 3.3843 11.5432 0.0049377 0.0014590
27 3.4839 12.1377 0.0051506 0.0014784
28 3.6094 13.0281 0.0053235 0.0014749
29 3.6661 13.4401 0.0054672 0.0014913
30 3.7440 14.0176 0.0055504 0.0014825
31 3.8230 14.6153 0.0057513 0.0015044
32 3.9006 15.2148 0.0057903 0.0014844
33 3.9961 15.9685 0.0060034 0.0015023
34 4.0686 16.5532 0.0061261 0.0015057
35 4.1456 17.1864 0.0062301 0.0015028
36 4.2482 18.0471 0.0064434 0.0015167
37 4.3937 19.3049 0.0066937 0.0015235

Least Square Pit of Phints from No. 15 to No. 37
4: 0.0014166872 cal./mole-deg2
#8 0.0000051808 cal./mole-deg2
93 447.498‘K

cv . 14.167T + 0.0518T3 (ca1./m61e-deg) x 10'“

 

95

A—13 40 at.% Pd-Rh

 

Weight of Alloy: 3|.Iéel am No. of Mole:0.2980
Weight of Heater-Thermometer Assembly:5.233l No. of Mole100831

Resistance of Heater at 4.2“K:297.13 ohm

 

 

Thenmometer calibraion parameters: A =—1.017229
B = 0.627111
No. of T T2 cv Cv/T
Point (5) (K2) (cal/mole-deg;_ (caijmole-degzz
1 1.4058 1.9762 0.0022808 0.0016225
2 1.4380 2.0678 0.0023432 0.0016295
3 1.4751 2.1760 0.0023657 0.0016037
4 1.5270 2.3317 0.0024229 0.0015867
5 1.5797 2.4955 0.0025081 0.0015877
6 1.6529 2.7320 0.0026163 0.0015829
7 1.7101 2.9243 0.0027062 0.0015825
9 1.8769 3.5227 0.0029307 0.0015615
10 1.9353 3.7454 0.0030027 0.0015515
11 2.0127 4.0511 0.0031415 0.0015608
12 2.0490 4.1986 0.0031921 0.0015579
13 2.0990 4.4060 0.0032613 0.0015537
14 2.1339 4.5536 0.0033038 0.0015482
15 2.1799 4.7521 0.0033768 0.0015491
16 2.2556 5.0879 0.0034518 0.0015303
17 2.3331 5.4432 0.0036195 0.0015514
18 2.1121 4.4609 0.0032582 0.0015426
19 2.2110 4.8884 0.0033981 0.0015369
20 2.2902 5.2451 0.0035224 0.0015380
21 2.3737 5.6343 0.0036412 0.0015340
22 2.4659 6.0807 0.0037859 0.0015353
23 2.5429 6.4661 0.0039241 0.0015432
24 2.5972 6.7456 0.0039792 0.0015321
25 2.6596 7.0735 0.0041094 0.0015451
26 2.7270 7.4368 0.0042172 0.0015464
27 2.7758 7.7051 0.0043056 0.0015511
28 2.8477 8.1095 0.0044361 0.0015578
29 2.9188 8.5195 0.0045714 0.0015662
30 3.0045 9.0267 0.0046911 0.0015614
31 3.0677 9.4108 0.0047939 0.0015627
32 3.1382 9.8482 0.0049046 0.0015629
33 3.2083 10.2932 0.0049908 0.0015556
34 3.2940 10.8506 0.0051521 0.0015641
35 3.3860 11.4653 0.0053474 0.0015792
36 3.4914 12.1896 ‘0.0055103 0.0015783
37 3.5856 12.8564 0.0056560 0.0015774
38 3.7186 13.8281 0.0059238 0.0015930
39 3.7980 14.4246 0.0060808 0.0016011
40 3.8810 15.0622 0.0063066 0.0016250
41 3.9591 15.6743 0.0064054 0.0016179
42 4.0653 16.5265 0.0065829 0.0016193
43 4.1449 17.1805 0.0067027 0.0016171
44 4.2284 17.8797 0.0068950 0.0016306

Least Square Pit of Points from No.18 to No. 44
1(2 0.0014941364 cal./mole-deg§
fi- 0.0000073946 cal./mole-deg
0- 397.456?

cv - 14.9411 + 0.073913 (cal./mole-deg) x10-4

..__;.-r ' v

96

A-14 Pure Paladium

.u------------------—‘---n—u---c---------------------------.-.

 

Weight of Alloy: 45.2666 am No. of M01930A254
Weight of Heater-Thermometer Assembly:5.3009 No. of Mole:0.0884
R‘sistance of Heater at 4.2°K :296-97 01““

Thermometer calibraion parameters: A =-1.007676

 

 

B = 0.624920
No of T 12 CV 06/?
. ,2 2
Point (K) .fh ) (cal/mole-dgg) 19217m616-665_2
1 1.3388 1.7924 0.0032276 0.0024108
2 1.3684 1.8726 0.0032737 0.0023923
3 1.4057 1.9759 0.0033966 0.0024164
4 1.4492 2.1001 0.0034784 0.0024002
5 1.4916 2.2249 0.0035664 0.0023909
6 1.5402 2.3722 0.0037277 0.0024203
7 1.5758 2.4831 0.0037602 0.0023862
8 1.6273 2.6482 0.0039035 0.0023987
9 1.6814 2.8271 0.0040452 0.0024059
10 1.7525 3.0712 0.0042081 0.0024012
11 1.8243 3.3279 0.0043519 0.0023856
12 1.9373 3.7530 0.0046207 0.0023852
13 2.0119 4.0479 0.0048149 0.0023932
14 2.1008 4.4135 0.0050419 0.0024000
15 2.2361 4.9999 0.0053899 0.0024105
16 2.3176 5.3711 0.0056132 0.0024220
17 2.4109 5.8122 0.0058555 0.0024288
18 2.4928 6.2142 0.0061196 0.0024549
19 2.5563 6.5348 0.0062681 0.0024520
20 2.6188 6.8582 0.0064235 0.0024528
21 2.6886 7.2283 0.0066041 0.0024564
22 2.7644 7.6420 0.0068230 0.0024682
23 2.8440 8.0883 0.0069659 0.0024493
24 2.9536 8.7239 0.0073185 0.0024778
25 3.0072 9.0431 0.0074645 0.0024822
26 3.0551 9.3338 0.0075997 0.0024875
27 3.1183 9.7237 0.0077913 0.0024986
28 3.1882 10.1645 0.0079727 0.0025007
29 3.2781 10.7458 0.0082102 0.0025046
30 3.3629 11.3088 0.0085571 0.0025446
31 3.4803 12.1125 0.0089174 0.0025622
32 3.5954 12.9266 0.0092858 0.0025827
33 3.6558 13.3651 0.0094211 0.0025770
34 3.8722 14.9941 0.0102153 0.0026381
35 3.9555 15.6463 0.0104713 0.0026472
36 4.0302 16.2425 0.0107221 0.0026605
37 4.1099 16.8914 0.0109890 0.0026738
38 4.2144 17.7612 0.0113432 0.0026915

Least Square Fit of Points from No.11 to No.38.
'(= 0.0023005422 ca1./mole-deg§
{3- 0.0000215464 _cal./mole-deg
9: 278.282°1(

cv=23.005'r + 0.215523 (cal. Amie-.125) x10"

APPENDIX B
COMPUTER PROGRAMS

97

l-H’

 

APPENDIX B

COMPUTER PROGRAMS

B-l. The Main Program

 

This program consists of two major parts:

Part A.

To calculate heat capacity, the characteristics of
thermometer calibration curve, and the values of y, B, as
well as Debye temperature, 0. The least square fit is

carried out by a subroutine "McPals."

Part B.
To plot a Cv/T versus T2 curve from above calculated
data.

The variable assignments are as follows:

R(I) = Resistance of thermometer at each calibra-
tion measurement.

P(I) = Measured vapor pressure of liquid helium.

PI(I) = Chart pressure of liquid helium.

T(I) = Temperature of thermometer at each
calibration measurement.

SR1(I) = Initial resistance of thermometer.

SR2(I) = Final resistance of thermometer.

PR(I) = Resistance of thermometer at each heat
capacity measurement.

FNl = number of moles of heater and thermometer

assembly.

98

I 1
runs?"

 

2O

25
500

200
99

99

FN2 = Number of moles of alloy

FRH = Resistance Of heater at 4.2°K, in ohm.

TM(I) = Heating period (in second).

CIH(I) = Heating current (in ampere)

IC(I) = Sequential number of curves

M = Total number of measurements

N = Maximum power of the polynomial to be fit,
in this case N = 1.

LP = Sequential number of heat capacity measure—
ment. Points of measurement with sequential
number less than LP will not be included
in the least square fit.

ZE = Zero error of the d—c. amplifier.

PROGRAM ESHPLOT

DIMENSION R(NO), P(AO), T(80), PI(80), x<40), Y(40),
cv2<80), _

1 Rl(80), R2(80), Tl(80), T2(80), TM(80). CIH(80), RST(80),

1 TT(80), C(10), TSQ(80), RR(2.80), IC(80), PR(80),
SR1(80),

1 SR2(80), LBL(15), NP0(4), PRM(4,3), XH(101,4),
YV(101,4), LB(14,10),

1 xs0(80), YSQ(80)

SQRTF(LOGF(R)/(T*2.3026))=C(l)+C(2)*LOGF(R)/2.3026+

1 C(3)*(LOGF(R)/2.3026)**2

READ 1001, (LB(MB,l), MB=4.13)

READ 99, NCRV

MLB=l

NL=O

DO 5 L=l,NCRV

D0 5 MB=4.13

LB(MB,L)=LB(MB.1)

READ 100, M, N

IF (M) 20.20.25

CALL PLOT (20., 0.,=1) 8 STOP

READ 1001, (LB(MB,1), M8=1,3)

FORMAT (30x, 3A8)

READ 200, ZE

FORMAT (8F10.7)

FORMAT (I10)

 

100

800

30
1000

35

40
201
202
20”

2030
2029
2031

203
2032

600

100

FORMAT (2I10)

READ 200, (8(1), I=1, M)
READ 200, (PI), I=1,M)
MM= 2*M

READ 200, (PI(I), II=1,MM)
READ 200, (T(I), I=1 MM)

READ 800,LP

FORMAT (I10)

DO 30 J=l.M

I=2*J-l

T(J)=T(I)+(T(I+l)-T(I))/(PI(I+l)ePI(I))*(P(J)—PI(I))

TT(J) = 1.0

PRINT 1000 *~

FORMAT (1H1) 3

PRINT 500,(LB(MB,1), MB=1,3)

PRINT 202

D0 35 I=1,M

R(I)=R(I)-ZE

X(I)=LOGF(R(I))/2.3026

Y(I)=SQRTF(XI)/T(I))

CALL MCPALS(M,N,O.,TT,X,Y,RR,C,LP,IDEG)

N2=N+2

DO 40 I=N2,10

C(I)=0.o

PRINT 201,(R(I),P(I),T(I),X(I),Y(I),RR(1,I),RR(2,I),
I=1,M)

FORMAT (8X,5F15.7,2E20.7/)

FORMAT (18x,*R*, 14x,*P*,1ux.*T*,14x,*x*,14x,*Y*,qu,
*ERRORS*, 12X,*FRAC ERRORS*)

FORMAT(/8x, 5F20.7,15)

{cum

K=N+1 $ L=3
IF (C(L)) 2031,2030,2031
PRINT 2029
FORMAT (14X,*A*,20X,*B*)
GO TO 2032
PRINT 203

FORMAT (14x,*A*,20x,*8*,20x,*C*)
PRINT 300,(C(I),I=1,K)
CALIBRATION PROGRAM ENDS
PRINT 1000

PRINT 500, (LB(MB,l),MB=l,3)
READ 100, M,N

FJ=4.184

READ 200, FNl

READ 200,FN2

READ 200,FRH

FRH=FRH*O.9868

READ 600,(IC(I),I=1,M)
FORMAT (8110)

READ 200,(PR(I), I=1,M)

 

READ 200 ,(SR1(I), I=1,M)
READ 200, (SR2(I), I=1,M)
READ 200, (TM(I), I=1,M)
READ 200, (CIH(I), I=1,M)

READ 800, LP

1001 FORMAT (3A8)
PRINT 302
D0 70 I=1,M
CIH(I)=CIH(I)*0.9913
R1(I)=PR(I)-SR1(I)-ZE
R2(I)=PR(I)4SR2(I)-ZE
x=LOGF(R1(I))/2.3O26
T1(I)=X/(C(1)+C(2)*X+C(3)*(X**2))**2
Y;LOCF(R2(I))/2.3026
T2(I)=Y/(C(l)+C(2)*Y+C(3)*Y**2))**2
T(I)=0.5*(T1(I)+T2(I))
DELT=T2(I)-T1(I)
CVT=CV1/T(I)
CVT=0.0001815/14.5973*T(I)**2+0.00018
RST(I)-CIH(I)**2*FRH*TM(I)/FN2*DELT*T(I)*FJ)—FN1*

CVT/FN2
CV2(I)=RST(I)*T(I)
TSQ(I)=T(I)*T(I)
XH(I,1)=TSQ(I) $ YV(I,1)=RST(I)*10.0**M

70 CONTINUE
RST(I)=C(1)+C(2)*T(I)**2+C(3)?T(I)**u
DO 80 I=1,M
80 TT(I)=1.0
MP=M-LP+1
CALL MCPALS(MP,NJD”TT,TSQ,RST,RR,C,LP,IDEG)
LPP=LP-l
PRINT 301,(Rl(I),R2(I),TM(I),CIH(I),T(I),CV2(I),
TSQ(I),RST(I),
1 IC(I), I=1,LPP)
301 FORMAT (8x,2F10.2,F9.3,F9.5 4F12.7 32x1
PRINT 305,(R1(I),R2(I),TM(I§,CIH(I5,T
1 RR(1,I),RR(2,I),IC(I),I=LP,M)
305 FORMAT (8x,2F10.2,F9.3,F9.5,4F12.7,2E16.7,15/)
302 FORMAT(12x,*R1*,8x,*R2*,8X,*TM§8x,*CIH*,9x,*T*,9x,
CV2*,9X,*TSQ*,

1 9X,*RST*,lOX,*ERRORS*,6X,*FRAC ERRORS*,4X,*C—NO*)
THETA=(234.0*6.0251*1.3803/(4.184*C(2)))**0.3333
K=N+1 $ L=3
IF (C(L)) 3031,3030,3031

3030 PRINT 303
303 FORMAT (/12X,*GAMMA*,qu,*ALPHA*,qu,*TRETA*)
00 TO 3032
3031 PRINT 310
310 FORMAT (/12X,*0AMMA*,14X,*ALPRA*,14X,*BETA*,*THETA*)
3032 PRINT 300,(C*I),I=1,K),THETA
300 FORMAT(//4F20.10//)
PRINT 304

(i), 5C%2(I) ,TSQ(I), RST(I),

 

304 FORMAT (/,14x,*FN1*,15x,*FN2*,15X,*FRR*,15X,*ZE*)

102

PRINT 300, FNl,FN2,FRH,ZE

PRINT 400

400 FORMAT (13x,*PO—RD*,14x,*SM-R—1*,14x,*T-RES—1*,14x,*

SM-R—2*,14X,

1 *T-RES-2*,10X,*C—NO*)

PRINT 204, (PR(I),SR1(I),Rl(I),SR2(I),R2(I),IC(I),

NPC(1)=M
PRM(1,1)+C

LN=O

$ NCOPYS=1

(l)*l0.0**4

DO 9 I=1,NCOPYS

NL=NL + 1

IF (NL.GT.MLB) NL=MLB

DO 6 MB=l,l3
6 LBL(MB)=LB(MB,NL)

NC=NCG
CALL

CALL PLOT
9 CONTINUE
PRINT 1000
GO TO 10
END

GRAPH (YV,XH,NPC,PRM,LBL,MNPC,NCG,NCRV,

(300,X,3)

I=1,M)
$ NCG=1
NL=O $

$ LSQDEG=N

MNPC=101
$ PRM(1,2)=C(2)*10.0**5

PRM(l,3)=THETA

LN,LSQDEG,LP)

$

$

SUBROUTINE GRAPH (YV,XH,NPC,PRM,LBL,MNPC,NCG,NCRV,LN,

LSQDEG,LP)
DIMENSION YV(MNPC,NCG),XH(MNPC,NCG),NPC(4),PRM(u,3),
LBL(15)
COMMON/GRA/LSD
DATA (NCURVE=0)
LSD=LSQDEG
INITIALIZATION
CALL PLOT (0.,0.,0, 80., 80.)
CALL PLOT (0.,-13.75,2)
CALL PLOT (0.,0.,0)
CALL PLOT (2.,3.,2)
CALL PLOT (0.,0.,0)
IUB=(NCRV/NCG)*20+20
CALL PLOT (IUB,X,3)
GRID
FN=NCG-l $ YLBT = FN*0.25+11.0

CALL CHAR (YLBT,0.0,LBL(1),8,0.0,.15,.l
CALL CHAR (YLBT,l.25,LBL(2),8,0.0.,
CALL CHAR (YLBT,2.50,L8L(3),8,0.0,.

CALL PLOT

FLN=10.0

(0.,0.,2)

IF (LN.NE.0) FLN=0.1

103

D0 2 I=2,10,2
A=I—1
CALL PLOT (0.0,A,2)
CALL PLOT (FLN,A,1)
CALL PLOT (FLN,A+1.,2)
CALL PLOT (6.,A+1.,1)
2 CONTINUE
CALL PLOT (0.,0.,1)
DO 3 I=2,10,2
A=I—1
CALL PLOT (A,0.0,2)
CALL PLOT (A,FLN,1)
CALL PLOT (A+1.,FLN,2)
CALL PLOT (A+l.,0.,1)
3 CONTINUE
CALL PLOT (0.,0.,1)
CALL PLOT (0.,0.,2)

SCALE x

CALL CHAR (-1.0,2.00,LEL(4),8,0.0.,15,.1)
CALL CHAR (-1.0,3.25,LBL(5),8,0.0,.15, 1)
CALL CHAR (-1 0,4 50,LBL(6),8,0.0,.15.,1)
XS=0.0

XL=20.0

C=(XL-XS)/10.0
ENCODE (6,4,IS)XS
4 FORMAT (F6.2)
CALL CHAR (-0.25,—0.26,IS,6,0.0,l./8.,1./12.)
B=XS
DO 5 I=1,10
FF=I $ F=FF-0.26 $ G=B+C
ENCODE (6.4,JS)G
CALL CHAR (—0.25,F,JS,6,0.0,1./8.,1./12)
B=G
5 CONTINUE
SCALE Y
CALL CHAR (2.00,—1.5,L8L(7),8,90. . .
CALL CHAR (3.25,—1.5,LBL(8),8,90.,.15,.1)
CALL CHAR (4.50,-1.5,LBL(9),8,90. .
YL=20.0
YS=0.0
C=(YL-YS)/10.0
ENCODE (8,7,KS)YS
7 FORMAT (F8.3)
CALL CHAR (0.0,-l.lO,KS,8,0.0,l./8.,l./l2.)
B=YS
DO 9 I=1.10
F=1 $ G=B+C
ENCODE (8,7,LS)G
CALL CHAR (F,—1.10,LS,8,0.0,1./8.,I./12.)
8:0
9 CONTINUE

100

17

15
20

104

PARAMETER LIST

DO 20 I-1,3

YLB=FN*0.25+10.25 $ XLB=3*(I-l) $ LL=I+9

CALL CHAR ( YLB,XLB,LBL(LL),8,0.0,.lS,.l)

DO 20 NC=l, NCG -

FNl=NCG-NC $ YP=FN1*0.25+10.25 $ XP=XLB+l.25

IF (I.NE,3) 00 TO 17

NCURVE=NCURVE+1

ENCODE (3,100.NCVB)NCURVE

FORMAT (I3)

CALL CHAR (YP,9.870,NCVE,3,0.,l./8.,l./l2.)

IF (NCG.EQ.1) GO TO 17 L
YSY=YP+0.1 $ XSY=9.0 ’
CALL SYMBOL (NC,YSY,XSY,80.,80.)

NCMl=NC—l

IF (NC.GT.1.AND.PRM(NC,I).EQ.PRM(NCM1,I)) GO TO 20

ENCODE (8,15,JP1)PRM(NC,I)

FORMAT (F8.3)

CALL CHAR ( YP, XP,JP1,8,0.,.15,.1) ‘.
CONTINUE I
CALL PLOT (0.,0.,2,80.,80.)

CALL CURVE (YV,XH,NPC,MNPC,NCG,XL,XS,YL,YS,LP)
CALL PLOT (o.,0.,0.80.,80.)

CALL CHAR (12.5,0.,LBL(13),8,0.,0.5,1./3.)
CALL PLOT (20.,0.,2)

END

, page“

SUBROUTINE CURVE (YV,XH,NPC,MNPC,NCG,XL,XS,YL,YS,LP)
DIMENSION YVEMNPC,NCG),XH(MNPC,NCG),NPC(4)
COMMON/CRV/Y 101),X(101),NCURVE
DATA (NCURVE=0)
SY=lOO./((YL—YS)/8.)
SX=lOO./((XL-XS)/8.)

CALL PLOT (YS,XS,0,SY,SX)

DO 25 N0=1,NCG

NCURVE=NCURVE+1

K=NPC(NC)

DO 3 J=1,K

Y(J)=YV(J,NC)

X(J)=XH(J,NC)

IF (K.GE.101) GO TO 9

DO 5 I=1,K

CALL SYMBOL (NC,Y(I),X(I),SY,SX)
CONTINUE

K=NPC(NC)—LP+1

CALL LSTSQ (K,XL,XS,LP)

DO 10 I=l,lOl

IF (Y(I).GT.YL-0.005) Y(I)=YL
IF (Y(I).LT.YS+0.005) Y(I)=YS

 

105

10 CONTINUE
15 CALL PLOT (Y(1) ,X(l), 2, SY, SX)

IF (Y(1). NE. YS. AND. Y(1). NE. YL) CALL SYMBOL (NC,Y(1),

X(l),XY,SX)

NP=K

IF (K.LT.lOl) NP=101

D0 20 I= 2 ,NP $ Jl= 1

11: I+1 $ IM1=I-1

YTl= (Y(I)-YS)*(YL- Y(I)) $ YT2= (Y(IM1)-YS)*(YL—Y(IM1))

IF (YTl. EQ. 0. 0. AND, YT2. EQ. 0. 0) Jl=2

CALL PLOT (Y(I),X(I),X(I),J1,SY,SX)

IF (YT1.EQ.0.0.AND.I.NE.NP) GO TO 17

IF (YT1.NE.0.0.AND.I.EQ.NP) GO TO 10 $ 00 TO 20
17 IF (Y(IM1).EQ.YL.AND.Y(I1).LT.YL) GO TO 19

IF (Y(IM1).LT.YL.AND.Y(Il).EQ.YL) GO TO 19

IF (Y(IMl).GT.YS.AND.Y(Il).EQ.YS) GO TO 19

IF (Y(IMl).EQ.YS.AND.Y(Il).GT.YS) GO TO 10 $ GO TO 20
19 CALL SYMBOL (NC,Y(I),X(I),SY,SX) {
20 CONTINUE J

 

 

2S CONTINUE
CALL PLOT (YS,XS,2,SY,SX)
END

SUBROUTINE LSTSQ (K,XL,XS,LP)
DIMENSION w (101), R(2,101),C(10)
COMMON/CRV/Y(lOl),X(10l),NCURVE
-COMM0N/GRA/N
PRINT 100
100 FORMAT (l3HlLSTSQ 0UTPUT,//)
K=K+LP-l
DO 5 I=l,K
5 W(I)=l.0
KP=K-LP+l
CALL MCPALS (KP,N,0.0,w,X,Y,R,C,LP,IDEG)
PRINT 105, NCURVE,(I,R(1,I),R(2,I),I=LP,K)
105 FORMAT (4x,*CURVE NO.*,13/342x.5HERROR,15x,10HFRAC ERROR,
//,
A (38X,I3,x,2(E12.4,BX)))
IDEGl=IDEG+l
PRINT 110, IDEG,(C(I),I=1,IDGE1)
110 FORMAT (//,X,*IDEG=*,I2,/,X,*LSTSQ COEF*,/,(X,5
(El6.8,X)))

 

D0 10 J=1,101
X(J)=(J-l)*0.0l*(XL-XS)+XS $ Y(J)=0.0
IF (X(J).EQ.0.0) GO TO 9

DO 8 I=1,IDEG1
Y(J)=Y(J)+C(I)*X(J)**(I-l)

CONTINUE $ GO TO 10

Y(J)=C(l)

CONTINUE

END

OKOCD

UTDWNH

12

106

SUBROUTINE SYMBOL (NC,YI,XI,SY,SX)
R=0.o4
CALL PLOT (YI,XI,2,SY,SX)
GO TO (1,2,3,4),NC

CALL CIRCLE (R,RI,XI) $ GO TO 5
CALL TRI (R,YI,XI) $ GO TO 5
CALL SQU (R,YI,XI) $ GO TO 5
CALL DIA (R,YI,XI)

CALL PLOT (YI,XI,l,SY,SX)

END

SUBROUTING CIRCLE (R,YI,XI)
CALL PLOT (YI,XI,2,100.,100.)
CALL PLOT (YI,XI+R,2)

DO 12 I-10,360,10

A=I*3 ,1915926536/180 .

X=R*COSF(A)+XI
CALL PLOT (Y,X,1)

CONTINUE
CALL PLOT (YI,XI,2)

END

$ Y=R*SINF(A)+YI

SUBROUTINE TRI (R,YI,XI)

CALL
CALL
CALL
CALL
CALL
CALL
END

PLOT
PLOT
PLOT
PLOT
PLOT
PLOT

(YI,XI,2,100.,100.)
(-(2./3.)*O.866*R+YI,XI+R,2)
((4./3.)*0.866*R+YI,XI,1)
(-(2./3.)*0.866*R+YI,XI-R,1)
(-(2./3.)*0.866*R+YI,XI+R,1)
(YI,XI,2)

SUBROUTINE SQU (R, YI,XI)

CALL
CALL
CALL
CALL
CALL
CALL
CALL
END

PLOT
PLOT
PLOT
PLOT
PLOT
PLOT
PLOT

(YI,XI,2,100.,100.)
YI-R,XI+R,2)
(YI+R,XI+R,l)
(YI+R,XI~R,1)
(YI-R,XI-R,l)
(YI—R,XI+R,1)
(YI,XI,2)

SUBROUTINE DIA (R,YI,XI)
CALL PLOT (YI,XI,2,lOO.,lOO.)
CALL PLOT (YI,XI+R,2)

 

 

107

CALL PLOT (YI+R,XI,1)
CALL PLOT (YI,XI-R,l)
CALL PLOT (YI-R,XI,1)
CALL PLOT (YI,XI+R,l)
CALL PLOT (YI,XI,2)
END

SUBROUTINE MCPALS(M,N,EPS,w,x,Y,R,C,LP,IDEG)
DIMENSION W(M),X(M),Y(M),A(10,10),SUMXSQ(19),
C(10),R(2,M),B(10)
SUMXSQ(1)=B(1)=0
NMX=N
IF((M-N-1).LT.0) NMX=M—l
NMX1=NMX+1
MN=M+LP-1
DO 1 I=LP,MN
R(2,I)=l.0
B(l)=B(l)+Y(I)+W(I)
SUMXSQ(1)=SUMXSQ(1)+w(I)
R(1,1)=B(1)
NMN=1
IF(EPS.EQ.0) NMN=NMX
DO 10 NN=NMN,NMX
N2=2*NN
N1=NN+l
N21=N2—l
. IF(EPS.EQ.0) N21=1
D0 2 J=N21,N2
J1=J+l
IF(J1.LE.NMX1) B(Jl)=0
SUMXSQ(J1)=O
DO 2 I=LP,MN
R(2,I)=R(2,I)*X(I)
SUM=R(2,I)*W(I)
IF(Jl.LE.NMXl) R(l,Jl)=B(Jl)=B(Jl)+SUM*Y(I)
SUMXSQ(Jl)=SUMXSQ(J1)+SUM
DO 3 I=l,Nl
Jl=I-l
DO 3 J=1,Nl
A(I,J)=SUMXSQ(J1+J)
CALL GAUSS (N1,A,B,C)
DO 4 I=1,Nl
B(I)=R(1,I)
D0 8 I=LP,MN
SUM=C(N1)
DO 5 J=1,NN
SUM=X(I)*SUM+C(N1-J)
SUM=Y(I)-SUM
IF((ABSF(SUM).LT.EPS).OR.(NN.EQ.NMX))GO TO 7

_ ”1;?

 

108

D0 6 J=1,NMX1
R(l,J)=B(J)
GO TO 10
R(1,I)=SUM
CONTINUE
DO 9 I=LP,MN
R(2,I)=R(l,I)/Y(I)
IDEG=NN
RETURN
lO CONTINUE
RETURN
END

\OGDNON

SUBROUTING GAUSS(M,A,B,C)
DIMENSION A(10,10),B(M),C(M)
101 FORMAT (//53X,30H***SINGULAR MATRIX IN GAUSS***//)
DO 6 K=1,M
C(l)=0
IMAX=K
DO 1 I=K,M
T=ABSF(A(1,K))
IF(C(1).GE.T) GO TO 1
C(1)=T
lMAX=I
1 CONTINUE
IF(C(1).NE.0) GO TO 2
PRINT 101
RETURN
2 IF(K.EQ.IMAX) GO TO 4
J=IMAX
T=B(K)
B(K)=B(J)
B(J)=T
DO 3 L=1,M
T=A(K,L)
A(K,L)=A(J,L)
3 A(J,L)=T
4 I=K+l
D0 5 J=I,M
T=A(J,K)/A(K,K)
B(J)=B(J)-B(K)*T
D0 5 L=I,M
5 A(J,L)=A(J,L)-T*A(K,L)
6 CONTINUE
J=M+1
DO 8 K=1,M
I=J-K
T=O
IMAX=I+1
DO 7 L=IMAX,M

 

109

T=T+A(I,L)*C(L)
C(I)=(B(I)-T)/A(I,I)
RETURN
END

 

123

110

B-2. ,Program for Fitting Data to Equation:

CV=ET-‘ + 7T + 8Tj by Least SquaresMethod—

 

READ 200, S
SUMA11=0.0
SUMA12=0.
SUMA13=0.
SUMA22=0.
SUMA23=0.
SUMA33=O.
SUMCl =0.
SUMC2 =0.0

SUMC3= 0.0

DO 123 I=1,M
SUMAll=SUMAll+(l.O/T(I)**S)**2.0
SUMAl2=SUMAl2+T(I)/(T(I)**S)
SUMA13=SUMA13+(T(I)**3.0)/(T(I)**S)
SUMA22=SUMA22+T(I)**2.0
SUMA23=SUMA23+T(I)**4.0
SUMA33=SUMA33+T(I)**6.0

SUMCl =SUMC1+CV2(I)/(T(I)**S)

SUMC2 =SUMC2+CV2(I)*T(I)

SUMC3 =SUMC3+CV2(I)*(T(I)**3.0)
CONTINUE
BAl=SUMA22-(SUMA12*SUMA12)/SUMA11
BA2=SUMA23-(SUMA12*SUMA13)/SUMAll
BB2=SUMA33-(SUMA13*SUMA13)/SUMAll
Dl=SUMC2-(SUMA12*SUMC1)/SUMAll
D2=SUMC3-(SUMA13*SUMC1)/SUMAll
SEW=BB2—(BA2*BA2)/BA1
DEW=D2—(BA2*Dl)/BA1

BETA=DEW/SEW

GAMA=D1/BA1-(BA2/BA1)*BETA
RAT=SUM01/SUMA11

BAU=SUMA13/SUMA11
ALPHA=RAT-BAU*BETA-(SUMAl2/SUMA11)*GAMA

OOOOOO

 

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