Hill”

I

REINVESTIGATION OF LINDEMANN’S RELATION

BETWEEN MEL'flNG POENT AND DEBYE TEMPERATURE

Thesis for {I'm Degree of M. S.

MICHIGAN STATE UNIVERSITY

Ryokan Igei

1956

REINVESTIGATION OF LINDELAEN'S RELATION BETWEEN

MELTING POINT AND DEBYE TEMPERATURE

By

Ryokan Igei

AN ABSTRACT

Submitted to the College of Science and Arts, Michigan

State University of Agriculture and Applied Science

in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

Year 1956

by D J. Mag:

Ryokan Igei

Over half a century ago Lindemann derived a formula

connecting the melting point of a solid of given atomic

weight and given atomic volume with a characteristic tem-

perature derived from specific heat data. He assumed that

at the melting point the atoms touch, treating them as rigid

spheres whose radius is a constant fraction of the atomic

spacing. This assumption is equivalent to stating that at

the melting point the centers of the atoms attain a certain

arbitrary fraction of the interatomic spacing. His expres-

sion gives fairly good agreement with experiment. We have

reexamined his formula, checking its adequacy in the light

of additional data, and have generalized it somewhat to get

a more satisfactory fit of the eXperimental data. The first

part of the generalization is obvious, consisting merely of

_using the actual interatomic spacings as determined from

X-ray diffraction determination of the crystal structure.'

The second part of the generalization is to introduce the

atomic radii determined on the basis of modern crystal-

lography and wave mechanics. To correlate the radii appro-

priate to melting phenomena with the radii appropriate to

crystal binding, it is necessary to introduce another arbi-

trary constant. Improved agreement is then obtained for

closely-related elements, but not for those of different chemi-

cal nature. For alkali halides, the agreement is good for both

the original and the generalized formulation, but a new value of

the constant is required. We conclude that a satisfactory theory

must be based on more profound considerations.

REINVESTIGATION OF LINDEMANN'S RELATION BETWEEN

MELTING POINT AND DEBYE TEMPERATURE

By

Ryokan Igei

A THESIS

Submitted to the College of Science and Arts, Michigan

State University of Agriculture and Applied Science

in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy

1956

’-

ACKNOWLEDGMENTS

I would like to extend my sincere gratitude to

Dr. D. J. Montgomery not only for suggesting the problem

but also for his continuous advice and encouragement

throughout the course of this work.

My appreciation also goes to Dr. Richard Schlegel

and Dr. Alfred Leitner of the Physics Department, and

Dr. J. D. Hill of the Mathematics Department, for be-

coming the members of the examination committee.

Regarding the technical process of production of

this thesis, I would like to thank Mr. N. T. Ban of the

Physics Department for his constructive suggestions.

TABLE OF CONTENTS

PAGE

INTRODUCTION ............................ ..... ......... 1

MELTINC ............................................. 6

SPECIFIC HEAT OF A SOLID .............................. 8

RELATION EETNEEN DYNAMICAL PARAMETERS OF CRYSTAL AND

CHARACTERISTIC TEMPERATORES ........................ 1n

ATOMIC RADII ......................................... 17

INTERATOMIC DISTANCES ............................... 18

DERIVATION OF LINDEMANN'S RELATION ................... 20

TEST OF LINDEMANNiS RELATION ......................... 21

GENERALIZATION OF LINDEMANN'S RELATION ............... 23

COMPARISON WITH EXPERIMENT ' ........................... 26

CONCLUSIONS ........................................... 35

BIBLIOGRAPHY '........................................... 39

LIST OF TABLES

TABLE PAGE

I. Values of Lindemann's Constant ................... 21

II. Crystal Structures of Some Elements and Compounds .. 2h

III. Elements ........................................ 27

Iv. Alkali Halides O0.00....OOOOOOOOOOOOOOO0.0.00.00... 3h

INTRODUCTION

In 1819 Dulong and Petit1 found empirically that for a

great number of elements the atomic specific heat at con-

stant pressure is a constant. This law in those days was

an important tool for determining atomic weights. It was

known that at ordinary temperatures there were serious ex-

ceptions, in particular that some elements of low atomic

weight had too low specific heats. Moreover, by the turn

of the century it had been established that at high temper-

atures all the elements investigated obeyed the Dulong-Petit

law approximately, whereas at low temperatures they had too

low specific heats. Boltzmannz had been able to explain the

Dulong-Petit law and to give a value for the constant, 3R,

by applying the classical-law of equipartation of energy to

the thermal motion of the N atoms considered as independent

harmonic oscillators about their mean positions in the crys-

tal. He considered the atoms as particles with three degrees

of freedom, and obtained 3 x kT/2 x 2 N 8 3RT for the

grampatomic energy. I

In 1906 Einstein3 extended the ideas of quantum.theory

beyond the confines of radiation phenomena as introduced by

Plancku in 1900, and asserted that a vibration of frequency

'gé must have the average energy (kT/ZIx/(ex-l), where

xah‘zé/k‘l' , instead of the average energy kT/é . From this

expression follows Einstein's formula for the specific heat,

6v 8 3RTxg/(ex-1)2 , which gives the type of variation ob-

served experhmentally for all elements, namely, the specific

heat tends to zero as T tends to zero, and to a constant

value 3R as T tends to infinity. Nowadays one frequently

writes the Einstein characteristic frequency as an equiva-

lent temperature, through the relation h1é E kQé .

In 1910 Lindemanns made a connection between the Ein-

stein characteristic frequency 4% and the melting point Tm ,

starting out from an empirical formula proposed by Magnus

and Lindemanné, and modifying it according to the following

considerations. Lindemann assumed that a solid consists of

a set of simple harmonic oscillators arranged in a tetra-

hedral lattice, and that fusion occurs when the amplitude

of thermal vibration of the atoms attains one half the separ-

ation of nearest neighbors diminished by the sum of their

radii; that is, the solid melts when direct contact of

neighboring atoms occurs. Although Lindemann made some

attempt to estimate the atomic size from the dielectric

constant on the basis of the Clausius-Mossotti relation,

he realized the limitations of this formula, and assumed

simply that the atomic radius is some constant fraction of

the atomic spacing. The spring constant b for the equiva-

lent oscillator is obtained from.the Einstein frequency

by taking the mass of the oscillator as the atomic mass M ,

in turn equal to the atomic weight A divided by Avogadro's

number N . If then the melting point Tm. is high enough

that the mean energy of the oscillator is approximately

the equipartition value kT , the root-mean-square displace-

ment may be calculated by setting the mean potential energy

ibiz equal to ikT . The details of this calculation are

reviewed in a later section of this thesis. Thus the fre-

quency1ég(or equivalently, the Einstein temperature @g )

obtained from low-temperature data on the specific heat, is

connected with the melting point Tm.‘

In 1912 Debye7 and Born and von Karman8 extended the

Einstein theory to take into account the fact that a solid

is not a set of independent oscillators, but rather a set of

coupled oscillators. Einstein hhmself had pointed out that

the vibrations do not 311 have the same frequency, and that

one should write the average energy as a sum of terms of the

proper frequencies. Debye used as a model an elastic solid,

and considered that the energy of the elastic waves in it

behave like that of the light waves in radiation. To avoid

the ultraviolet catastrophe, he assumed that there exists an

upper limit to the frequency of waves, determining this

limit by setting the total number of frequencies less than

the maximum equal to three times the number of atoms in the

body. This upper limit is usually expressed as a character-

istic temperature through the relation he; I hub . The

atomic specific heat is then a universal function of the

ratio of the absolute temperature T to the Debye temper-

ature TD . This function, expressed as an integral not

expressible in elementary functions, has the same general

course as the Einstein function x/(ex-l) . Born and von

Karman, on the other hand, used as a model a space lattice

of atoms maintained in the mean equilibrium positions by

the various interatomic forces. Their method is more gen-

eral and more rigorous, and much more complicated,than that

of Debye. We shall make no use of this approach.

. It was natural to use the Debye temperature instead

of the Einstein frequency in Lindemann's relation, and

with this modification reasonably good agreement was ob-

tained between theory and experiment. We have decided to

look into the relation anew, to see what effect the data

made available during the past two decades will have on our

confidence in the formula, and to see if some generalization

can be made to give better agreement. We proceed along two

lines:

1)WWW

nearest-neighbor separation. Lindemann did not have

these data available at the time he proposed his formula,

a year before von Laue in 1911 suggested to Friedrich

and Knipping9 their experiment in 1912 on the diffrac-

.tion of X-ray by crystals.

2) An additional assumption is introduced with respect to

atomic radii. Lindemann's assumption that the radii

are a constant fraction of the spacing, and that upon

the atoms' coming into contact the solid melts, are

equivalent to the assumption that when the centers of

the atoms are displaced a certain constant fraction of

the atom.spacing, the solid melts. We assume instead

that when melting occurs the mean vibrational ampli-

tude reaches a certain fraction of the distance between

the "outer surfaces" of the atoms. This assumption

necessitates some choice of atomic radius, and we have

used one of the conventional radii, the "Pauling

radius," times an adjustable constant. Thus we have

introduced a second constant to be determined by experi-

ment.

In the present work we first set up in detail the back-

ground just mentioned. Then we examine the original Linde-

mann relation, with the Debye temperature in place of the

Einstein frequency. After evaluating the agreement with

experiments we proceed to introduce actual atomic spacings

and the atomic radii. The new results are examined, and con-.

clusions drawn therefrom.

MELTING

If a solid is considered as a crystal consisting of

an array of atoms which are fixed in definite relative

positions in space, the phenomenon of melting can be con-

sidered as a more or less free rearrangement of these atoms.

This rearrangement is postulated to be due to the increasing

amplitude of thermal vibrations of atoms around the equili-

brium position of the atoms. When heat is supplied to the

solid, each atom will gain heat energy, and the amplitude

of vibration will increase. Thus each atom.requires more

room and the whole solid expands. As the temperature con-

tinues to increase, the amplitude becomes greater and greater

until the effect of interatomic forces is lost.10 When the

long range forces become ineffective, the atoms which dis-

place far from the equilibrium positions may not return to

their original points and other atoms may come into these

points.11 If this interchange of atom takes place frequently,

the rigidity of the solid is no longer kept and the melting

sets in.

Therefore quantitative analysis of melting of solid

will be closely connected with the study of vibration of

atqm around its equilibrium position. The analysis of lat-

tice vibrations described later will link the mean displacement

of the atoms, the restoring force (to be expressed through

the Debye temperature), the mass of the atom (to be eXpressed

in terms of the atomic weight), and the melting point of a

solid in a single equation which enables us to find the mean

displacement of atom.st melting point, one of the quantities

necessary in investigating Lindemann's relation.

SPECIFIC HEAT OF A SOLID

The theories of specific heat of solids that we shall

be concerned with are based on the notion that the N atoms

of a gram.atom of a substance are equivalent to a set of

3N harmonic oscillators. The energy of an oscillator of

frequency-sq is quantized with the energy nheg , where

n is an integer and h is Planck's constant. The applica-

tion of Boltzmann statistics, which states that the proba-

bility that a given oscillator will be in the quantum state

n is equal to exp(-nh93/kT) , leads to the following ex-

pression for the average energy of an oscillator

h4/1

shad/{1 . -l

C”

) (I)

where 'N is measured from the zero-point energy.

For the entire solid, the average energy will be given

by multiplying ni , number of oscillators with the frequency

1/i , by the average energy of the i-th oscillator, and

summing over the set:

' ‘- h kTE 32)“? “.“3/(9 W “1):

L.

2‘11 ‘ 3N 'a

(2)

with

If the number of frequencies is large, we may replace the

hi with a number density g(96 defined so that

n1 = 8(4/)d4J, (3)

where n1 is the number of oscillators (or modes of vibra-

tion) with frequencies in the range from 1/ to 4/ + d1}.

In fact, it is convenient to use the continuous formalism

even with discrete distributions. The average energy is then

written:

00

= a hfldutry—eh° (a)

with

Jab/NIH 3N

To obtain the specific heat at constant volume, we differ-

entiate this expression with respect to T:

V/kT

. 3E, h ‘

CV a ’%J%1’ZW(S?

The different forms of the theories of specific heat than

depend on the choice of 3(V).

In the Einstein formulation, the solid is considered

to be equivalent to a set of independent oscillators of the

same frequency 12% . Then we have

33m . 3N Jail-122) . (a)

10

In the Debye formulation the solid is considered as a

set of coupled oscillators, with the frequency distribution

that of an elastic continuum. To avoid divergent integrals,

the assumption is made that a cutoff frequency exists. The

justification for this procedure is that the actual solid

is discrete in structure with. 3N degrees of freedom. The

shortest wavelength would be twice the interatomic distance,

and the total number of modes of vibration would be 3N. We

have 3 2

(9N/1/D )1/ for «u g. V,

3D (v0 3

(7)

O for ¢’>'23

In the Born-von Karman formulation the solid is con-

sidered as a set of coupled oscillators, with the frequency

distribution determined by the crystal parameters and the

interatomic forces. It has not been possible to get many

satisfactory expressions for actual crystals. We refer to

the survey in Born and Huang.12 For the distribution function

we would have I

gBW) x complicated function of 4/. (8)

For the Einstein case, the specific heat, as obtained

from inserting the number density gE of equation (6) into

the general expression for specific heat, equation (5), is:

c -’ 2 hté/kT

V 3 .i2¢{_/le» ° W (Ein tei ) (T3k \ (on air/km _ 1,2 /. s n' 9)

11

Often 4) is replaced by an equivalent temperature @E

defined by

14:93 2 hflE. (10)

For the Debye case, the specific heat as obtained from

inserting the number density gD of equation (7) into the

general expression (5),i s:

c . Nk . :1};my” d1) .(Debye) (11)

It is convenient to change the variable of integration by

the substitution

x 55 hJVkTJ

with the accompanying change for g:

GL2):" 3p (”x/h)

Further let us define the Debye equivalent temperature by

M hflp, (12)

The equation (11) may be written

xh

Q/r

Cv ——- exdm __ (W1 .4.”_) .

For the Born-von Karman formulation the specific heat, in

the few cases for which it has been obtained, is computed

numerically and presented in tabular or graphic form.

12

Expression (9) for the Einstein specific heat can be

evaluated directly. Expression (13) for the Debye specific

heat has been computed most extensively by Beattie13 by

integrating by parts, expanding the integrand in powers of

ex , and summing the resulting integrals. A convenient

nomograph relating absolute temperature, characteristic

temperature, and specific heat as given by expression (9)

and (13), is presented by Eucken.1h From this graph and.

from Beattie's tables data have been obtained and are plotted

in Fig. 1.

To obtain the Debye temperature from the experimental

data on specific heat, the relation (13) is used to deter-

mine @D considered as an unknown, with T and Cv taken

as known. Then the resulting (Eb is plotted against T.

If the Debye temperature is constant, the theory is vindi-

cated. However, in many cases there are discrepancies, as

indeed there must be when the actual lattice structure is

to be taken into account. A treatment of this question,

together with methods of determining Debye temperature, is

given by Kelly and MacDonald.15

Ho - tom/T30 - (‘0) cmntoA quaqsu09 as 4903 oIJtcedg

6----...

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._

__

--

_.7

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A__

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_-—

—_

-_

-k

..

‘4

,7

-.-A

—._

—.-

—-.

-—.—

_-

—-———--—

—'-

‘—

Fig.

1.

Specific

Heat

Cv

__,o

0.2

0.“.AR‘JK'

jI

I14

O.6

008

100

1.2

101*

Rati0

0f

Sample

Temperature

toCharacteristic

Temperature

-«-<——3R

vs

.T/®

..M02

Debye:

:9R

egg-)3'J(::ex

14/):dX‘F(7@)

Einstein:

0.

83R(.£2%)2

.

T'

(£6

13

RELATION BETWEEN DYNAMICAL PARAMETERS OF CRYSTAL AND

CHARACTERISTIC TEMPERATURES

Now let us relate the dynamical parameters of individual

atoms in the crystal to the characteristic temperature.

Suppose first that we have a simple harmonic oscillator of

If x is the displacementmass M’ and spring constant b .

from the equilibrium position, the equation will be

M de/dtz + bx a o ,

which leads to the vibrational frequency given by

.s 1 I b (it)

¢/"'§? FT' 9

or upon solving for the spring constant b ,

z z

b . lflfVM. (15)

The mean potential energy of the oscillator is

(16)E- a (1/2)sie = (l/Z)kT ,

pot .

with 3' the root-mean-square displacement, if we consider

temperatures sufficiently high that the average energy has

its equipartition value. Combining expressions (15) and (16),

(17)

we have

2

H * 1+1:sz Mi” .

15

The next problem.is to associate some frequency of the

crystal with the frequency of the oscillator just considered,

for the purpose of connecting melting point with specific

In the Einstein case it seems clear that oneheat data.

In the Debyeshould choose the characteristic frequency ¢é .

case it is not so clear that one should choose the charac-

teristic frequency yfi . We believe, however, that this fre-

quency is a suitable choice, for the following reasons:

1) The frequency spectrum rises rapidly with 1/, the higher

frequencies constituting the main contribution to the

number density and hence to the energy at high temper-

atures.

The Debye frequency is the frequency at which the rela-2)

tive motion is greatest,adjacent atoms vibrating exactly

out of phase; hence, the tendency to dissociation is

greatest here.

3) In any event, an arbitrary constant remains to be fixed

by experiment, and any constant factor times the Debye

frequency would give the same result.

we choose then, the frequency'e) appearing in equation

(17) as .¢§ , expressing the latter as RGB/h , in accordance

as A/N ,We write the atomic mass Mwith equation (12).

A is the atomic weight and N is again Avogadro'swhere

number. Upon solving the equation (17) with the substitu-

tions mentioned, we have the following expression for the

16

root-mean-square displacement of the atom at the absolute

temperature T :

x * _' — . (18)

271 ‘V k ED ‘V A

Specifically, if we assume that the motion remains harmonic

up to the melting point, we have the following value for xm ,

the root-mean-square displacement at the melting point Tm :

2x - ___m___hNT . (19)3.1....

m an M2 RA

17

ATOMIC RADII

To characterize an infinitude of information, such

as the electron distribution function for an atom, by a

single parameter such as a unique radius involves arbi-

trariness, and we must expect the value assigned to depend

on the property of interest. For atoms or ions built into

a crystal, we should like to deal with a radius such that

tne sum of two radii is equal to the equilibrium distance

between the corresponding atoms or ions.* Tables of such

crystal radii are given by several authors, and we select

those of Pauling16 who combines direct experimental data

with certain considerations from quantum mechanics. The

necessary values are contained in later tables in this thesis.

*On thefother hand, it might be thought that the wave

functions of the isolated atom or ion would give a simple

Usually the radius of an elec-radius useful in our scheme.

tronic orbit is taken as the value of the distance from the

nucleus at which the radial charge density is a maximum, A

1 Unfortunatelytypical set of results is given in Slater. 7

not enough of these values have been calculated and those

for the outer orbits, which are of most utility in the

problem at hand, are known with little precision. Neverthe-

less, we have tried such values, and a factor times them, in

an attempt to get better agreement of Lindemann's formula

But not much success is obtained.with experiment.

18

INTERATOMIC DISTANCES

The phenomenon that a crystal-like rock salt or calcite

splits into fragments all of the same shape or at least with

equal angles is the underlying fact in the development of

theoretical crystallography. The advent of atomic theory

brought the realization that the ultimate units of crystals

are atoms and molecules, the development of X-ray diffrac-

tion giving conclusive evidence on this score. The results

'of the theory of space groups combined with the experimental

data from X-ray diffraction enable us to find the distance

between atoms in crystals. Some structures and the para-

meters thereof are listed in Table II.

At the time when Lindemann proposed the relation between

melting point and mean displacement of the vibrating atoms

of a solid, X-ray diffraction had not been discovered. He

assumed ideal close-packing of spheres (hence, face-centered

cubic or ideal hexagonal close-packed structures), and he

calculated the nearest-neighbor distance under the assump-

tion of.a tetrahedral configuration. Today we are able to

get definite values of atomic spacings. Figures 2 through 5

show a few of the common lattices, together with DC, the

structure constant by which the lattice constant is to be

multiplied in order to get the distance between nearest neighbors.

19

Some Common Crystal Structures

4’2 To) (1;)

,4 /l/ \ .

,« \/ /‘* ‘ ,.

>————amw <2;«~— ~ 1.- a,-—~~ ~ ——..- «WW 0., mm»

Fig. 2. F.c.c. Fig. 3. B.c.c.

o< - (5/2 OK - 5/2

@ g 3 :

/ -_-.@ l /6

,/ /’/ 9 ' ® I

zI I

G}4.____-_,__ a,~---—--~ _—.=, .. _ __ a, ____..__

Fig. 1;. H.C.P. Fig. 5. Tetragonal

0(- 1.00 (Gallium: 0K ' 0.11.33)

2O

DERIVATION OF LINDEMANN'S RELATION

Lindemann's assumption that the mean amplitude of

thermal vibration at the melting point is such that the

atoms touch, coupled with the assumption that the atomic

radius is a constant fraction of the atomic spacing, is

equivalent to the assumption that at the melting point the

mean amplitude of thermal vibrations attains a constant

fraction of the interatomic distance d ; that is,

xm 8 a d . (20)

Under the assumption of a tetrahedral configuration, d is

related to the atomic volume V , defined as the atomic

weight A divided by the density )9 , as follows:

d :- fifz’ ,3/V/N . (21)

Insertion of these values into relation (19) leads to

Lindemann's relation:

‘ NS/éh Tm . 1/Tm Esra

(2D 2 a k3 A V 5 C A

% ’3 A’576 (22)m f .

8 C T

21

TEST OF LINDEMANN'S RELATION

Best agreement would be expected with similar elements

of simple structure. The elements of the first group of

the periodic table, that is, the alkalis, Group IA, (Li, Na,

K Rb, Cs, Fr), with body-centered cubic structure, and the

coinage metals, Group IA (Cu, Ag, Au), with face-centered

cubic structure, are two such sets. We have calculated the

constant C appearing in equation (22) from the most re-

cent experimental data on melting point, Debye temperature,

density, and atomic weight. The results are shown in Table I.

TABLE I

VALUE OF LINDEMANN'S CONSTANT C

Group IA Group IB

Li Na K Rb C s Cu Ag Au

(1211) 115 122 122 121 13).; no 1112

The value for lithium is not very meaningful, since it

is very difficult to assign it a reliable Debye temperature.15

There is considerable difficulty of the same sort with sodium

and potassium, and perhaps with rubidium and cesium. On the

22

other hand, the Debye temperatures for copper, silver and

gold are quite well defined.

One cannot quarrel with the constancy of C shown for

the alkalis, particularly in view of the arbitrariness in

choosing the Debye temperature. The progression of C with

atomdc‘weight-in the coinage metals is somewhat disturbing.

The increase in C in going from Group IA to Group IB is

serious; but in view of our present knowledge we should ex-

pect the difference in structure between the two groups to

be reflected in the change of the constant.

A!“

Us:

11);

23

GENERALIZATION OF LINDEMANN'S RELATION

The first step in generalizing Lindemann's relation is to

take into account the actual nearest-neighbor distance d. At

normal temperatures and pressures the alkalis crystallize in

the body-centered cubic structure (b.c.c.), with the structure

constantcx, equal to the ratio of the atomic spacing to lat-

tice constant, (cf. Figures 2 and 3). The coinage metals

crystallize in the face-centered cubic structure (f.c.c.),

with 0(8 (2/2 . Numerical values for these elements and

some others are given in Table II.

If the average value of C for the alkalis is taken as

120, and for the coinage metals as 137, we have the ratio

137/120 8 1.1h between them. If the corresponding quantities

are calculated taking into account the actual nearest-neighbor

distances, the ratio is 1.20. Hence the agreement is poorer.

But the second step in the generalization, the introduc-

tion of the atomic radius, will remedy the illness. He should

like to avoid the introduction of adjustable constants, but

here it will be necessary to add one. In principle one could

say that no adjustable parameter is being introduced, and that

the atomic radius determined from.melting data is as good as

any other. But in practice we wish to use the atomic radius

derived from.some other property of the substance. If the two

radii are not identical, then we need to assume some relation

TABLE II

CRYSTAL STRUCTURES OF SOME ELEMENTS AND COMPOUNDS

“—_“'-.

Crystal Type of Lattice 0

System. Structure Constant (A) Constant

{342

Structure d 8 0

Substance sou (A

Li Cubic B.c.c.

Na "

K

Rb

Cs

Ba

Cu

As

Au

Ca

Sr

Ha

Al

Co

Ni

Be

M8

Zn

Cd

La

Tl

LiF

LiCl

LiBr

LiI

NaF

NaCl

NaBr

NaI

Kf

KCl

KBr

KI

CsF

RbF

RbCl

RbBr

RbI

CsCl

CsBr

CsI

Ga

In

II

fl

1'

I!

Cubic

N

I!

H

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I!

N

n

N

Hexag.

I!

n

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n

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O‘NQ

WWWWWMWW

NWN

VINO"MNWOENWN

\ONIUI

WNGJOU'IN

between the two. We have found that the Pauling crystal

radius rP times some constant (5 to be determined from

experiment is the least unsatisfactory.

We now make the assumption that m melting _.t_:_1_1_e_ £993

£1332 sguare displacement xm i;M 333 g constant fraction

f 2; _t__h_e distance S between the surfaces 9_1_‘ spheres g;

radius R IFrP centered at the lattice points 2;: £93 crystal:

xm e rs . r [d - (31*RZ)J .-. r d - (ftpP1 + {91¢ng . (23)

Lindemann's assumption is equivalent to taking (3: O , and thus

having just one arbitrary constant f .

26

COMPARISON WITH EXPERIMENT

Instead of examining the equivalent of the constant C

in our new formulation, we shall look at 2f = me/S , the

portion of the distance between atomic spheres used up by

the thermal vibration at the melting point. For comparison

with the original formulation, but with the crystal struc-

tures taken into account, we give also me/d . To compute

S we have assumed that (3‘ 0.35 , a value chosen so as to

give a good fit. The results are given in Table III, which

shows for each element the atomic weight A, the Debye temp

perature EDD , the melting point Tm.’ and twice the root

mean square displacement at the melting point me , as cal-

culated from equation (19); then are given the atomic spacing

d between nearest neighbors, as given in Table II; the Paul-

ing radius rP , the adjusted radius R ' 0.35rP , the Slater

radius r , and the distance S between spheres of radius

S

R with centers d apart. Finally there appear the ratio

me/d , the portion of the distance between centers used up

at the melting point, and the ratio me/S , the portion of

the distance between spheres of radius R used up at the

melting point. All temperatures are in degrees Kelvin, and

all distances in Angstrom units.

Group

I

H

TABLE

III

ELEMENTS

___I

1.01

Group

IA

Li

Na

K Rb

Cs

Fr

6.93

23.0

8525

133.

Group

IB

Cu

A8

Au

63.6

108

197

Group

II

9.01

figoup

IIA

Mg

Ca

Sr

Ba

Ra

2h.3

0.1

7.3

137

226

hOO

160

100

61

315

215

170

1000

290

230

(170)

(113)

#59

370

335

312

299

1356

1233

1336

1623

92h

1083

1073

1123

as

0.278

0.350

0.809

0.u31

0.075

0.20M

0.217

0.21M

0.187

0.297

0.316

0.287

0.352

owl—n00

e e e

MN 00a,

mmddm

moon

moooo

e e e

NNN

NMO‘m

NO‘N M

O C C 0

(“M—3:

rP

OLAMOOO‘

Comm

OOHHH 0.31

0.65

0.99

1.13

l.h3

HMONO‘

mmzmm

O O O O C

00000 0.108

0.23

0.35

0.3

0.5%

cnajgean

CUH\ r~o~

0.0.0

00000

MNO

Md:

0 O .

OOO 00m

0.28

0.56

0.60

0.65

2.h1

3.06

3.57

3.82

8.06

1.87

2.00

1.92

2.06 \0 0m

(figmm

N MMM

0.092

0.098

0.091

0.089

0.091

0.080

0.075

0.07h

0.077

0.092

0.080

0.068

(0.082)

A

elm/S

0.106

0.115

0.115

0.113

0.117

0.109

0.109

0.111

0.091

0.107

0.098

0.081

(0.105)

27

i

A

Group

IIB

Zn

65.8

Cd

112

Hg

201

Group

III_

B'10.8

A1

27.0

Group

IIIA

Sc

5.0

Y8.9

La

139

Ac

227

Group_IIIB

Ga

69e7

In

115

T1

20h

Group

IV

C12.0

TABLE

III

(Cont.)

250

172

96

390

132

152

130

9A

1950

23?.

231.

933

1873

1763

1099

303

579

0.18h

0.157

0.207

0.296

0.192

0.206

0.2h8

d

2.65

2.97

2.99

2.85

3.58

3.72

2.56

3.20

3-85

rP

0.7a

1.11

1.10

0.50

1.15

000‘

e e e

000

R

0.26

0.3

0.3

0.18

0.h0 NCDM

NNM

e ee

000

0.32

0.h1

0.30

0.h0

0.50

2.50

2.92 MNO‘

PRON-

e ee

NNN

2xm/d

0.068

0.062

0.05h

0.073

0.080

0.08h

0.08h

0.071

0.083

0.101

0.090

0.077

0.089

28

A

Group

IVB

81

Ge

Sn

Pb

72.6

119

207

Group_IVA

Ti

Zr

Hf.

91.2

179

Group

V

N P

1h.0

31.0

Group

VB

As

Sb

Bi

78.9

122

209

Group

VI

0 8 Se

Te

P0

16

32.1

79.0

128

210

(90

290

260

88

350

280

213

lhO

100

1232

505

600

2073

2073

.(1973)

90h

Shh

0.198

0.120

0.271

0.265

0.232

0.218

0.271

0.226

TABLE

III

(Cont.)

$.38

3-89

2.92

3-23

3.32

Nv441

Inbdn

0..

000 0.62

0.7h

0.18

0.25

0.29

0.52

0.28

0.22

0.30

0.h8

r-IO‘CD

(DMN

000

e e e

000 0.091

0.072

0.066

0.09h

0.073

0.096

0.052

0.093

0.

1

0.0

7

0.111

0.090

29

A@D

Tm

axm

dr},

ars

sme/d

me/S

Group

VIA

Cr'

52.0'

E85

1888

0.185

2.89

0.52

0.18

2.12

0.078

0.087

MO

96e0

w18h

310

36u3

0.201

2.72

0.66

0.23

2.27

0.07h

0.089

Group

VIIB

'F

19.0

01

35.5

Br

79e9

I127

At

211

Group

VIIA

Mn

5h.9

350

1533

0.211

2.50

0.50

0.18

0.h1

2.15

0.08M

0.098

To

99

Re

186

300

3800

Group

VIII

Fe

55.9

ASB

1808

0.176

2.61

0.75

0.26

0.39

2.05

0.072

0.086

00

58.9

385

1753

0.198

2.51

0.72

0.25

0.36

2.21

0.078

0.090

Ni

58.7

375

1725

0.202

2.u1

0.69

0-2h

0.3A

2.01

0.08s

0.100

Group

VIIIA

Ru

102

MOO

2723

0.182

2.68

0.63

0.22

2.2%

0.063

0.081

Os

190

250

2973

0.222

2.71

0.65

0.23

2.2

0.082

0.098

r.

30

TABLE

III

(Cont.)

ACk)

Tm.

me

dr

Rrs

S2xm/d

2xm/S

GroupVIIIB

0.065

0079

2.25

0.067

0.080

Rh

'103

370

2228

0.175

2.70

0

Ir

193

285

2623

0.180

2.70

0H

N

e

N

€83

OO

GroupVIIIC

Pd

’107

275

1823

0.210

2.72

‘0.076

Pt

195

225

2028

0.200

2.76

0.072

The

data

on

this

table

were

obtained

primarily

from

Seitz20

and

fromForsythe

.Debyg1

temperatures

of

rubidium.and<£sium

were

obtained

from.Dauphinee,

Martin

and

Preston-Thomas.

31

32

An attempt was made to extend the treatment to compounds,

specifically the alkali halides. The calculations are straight-

forward, with 2xm replaced by xml + xm2 . .These quanti-

ties are obtained from equation (19), the same Debye tempera-

ture being used for both ions, but the atomic weight being

changed. Unfortunately values of Debye temperature for most

of the alkali halides are not available. For NaCl, KCl, and

KBr, specific heat measurements have been made and the Debye

temperature calculated (see for example,reference 1h). Mayer

and Helmholtz22 have estimated @D from elastic constants

for all the alkali halides, but not much credence can be

placed in such calculations. Barnes23 has obtained the far

infrared spectrum of most of the alkali halides, and reported

the wavelengths Aw of the principal absorption maximum.

On the simple theory, the equivalent temperature '00 ,

given by 1:00 = hyjo 8 hc/Ao , should coincide with @D .

Actually it is somewhat lower. Hence, to estimate Debye

temperature from his data, we have plotted the observed wave-

length AO against the observed Debye temperature @D for

the three salts for which it is known, and obtained the (3D

for the other salts by interpolation and extrapolation. The

results must accordingly not be taken very seriously. Table

IV summarizes the calculations for the alkali halides.

For the three salts underlined, there are given the

equivalent temperatures 00 obtained from infrared absorp-

tion measurements, and the Debye temperatures (Eb obtained

33

from specific heat measurements. For the majority of the re-

mainder of the salts, there are given the equivalent tempera-

tures 60 and the corresponding Debye temperatures (QB ob-

tained by the §g_hgg correlation procedure just mentioned.

Next in the table are the melting point Tm , xml for the

alkali ion, xm2 for the halogen ion, R1 for the alkali

ion, R2 for the halogen ion, the distance 8 between ions,

and the two ratios, (xml + xm2)/d , (xml + xm2)/S . Again

all temperatures are in degrees Kelvin and all distances in

Angstrom units.

TABLE

IV

ALKALI

HALIDES

RbBr

RbI

CsF

CsCl

CsBr

Cal

168

253

163

111

170

126

111

1&1

107

20

151

133

116

151

112

13

81

820

719

1252

1076

1027

923

1153

10h9

1003

10h6

1033

988

915

955

919

909

89k

0.16h

0.118

0.170

0.208

0.280

0.157

0.200

0.219

0.129

0.175

0.197

0.121

0.163

0.099

0.129

0.137

0.118

0.102

0:165

0.110

0.133

0.200

0.182

0.162

0.23h

0.210

2.01

2.57

2.75

3.00

2.31

2.81

2.97

53 fEiio o

0"

on

NI“ CON

.0.

do

NMMM NMMM MMMM

NO‘MO HONM

OMNO‘

0.21

0.21

0.21

0.20

0.33

0-33

0.33

0.33

0.h7

QMQO com

:hOxON .dKOOb- .d'NO

O C C C O C O O

0000 0000

(DO CDMCDQ COM

HNNN HNNN HNNN

\OF- :3000.0

0000 0000 0000

(Do (DMCDO

~0l\- 3001\-

S

NMOM OlnxO

MNQO IAOJO‘fl

o o o o

Hr-IHN HHHN

00 mo

seas gas. 221.C... 0

d

0.131

(xm1+xm2)

(111114-me

)

S

0.199

0.152

0.152

— 1:

35

CONCLUSIONS

We may discuss the adequacy of Lindemann's relation and

its generalisation by examining the constancy of the parameters

within a group of closely related elements. For this purpose

we select from.Table III the following entries:

Group IA

Elements Zgfid 2Jim/S

Na0.0911

0.115

K0.091

0.115Rb

0.089 0.113Cs

0,0210,112

Average0.091 0.115

The quantity me/d , it will be recalled, is the fraction

of the spacing occupied by the thermal vibrations when the atomm

are assumed to have negligible diameter: the quantity me/S

is that fraction when the atoms are assumed to have the diameter

firp , where P is an adjustable constant, and rp is the Paul-

ing radius. In the table above the value for lithium.has been

emitted because of the ambiguity in choosing a Debye tempera-

ture. For either assumption the constancy of the fraction is

excellent.

The next test is to see how the values of the constant

change from.a subgroup of elements to a closely-related sub-

Sroup. For this purpose we select from Table III the following:

Group 13

Elements”-

Cu.

23

Average

‘We now examine the constancy within the subgroup.

36

2xm/S

0.109

0.109

0.111

0.110

Here

the downward progression for 2xm/d is definite, whereas

there is at most a slight trend upward in me/S . Next we

examine the average value of the constant from.Group IA to

Group IB. For 2xm/d the ratio 1. 0.091/0.076 . 1.20; for

the 2xm/S it is 0.115/0.110 8 1.05. Hence we may say that

the generalized formulation give a better fit. It is to be

recognized, of course, that there has been added a second

parameter which has been adjusted to minimize the progression

within the subgroups and to secure a good agreement fer the

average between subgroups.

we now see how the formulas fit other groups of elements

in the periodic table. So little data are available that we

can say little about progression of the constant within a sub-

group.

we have averaged the fractions over subgroups:

Elements

Group IA

18

IIA

IIB

2xm/a

0.091

0.076

0.086

0.061

% DOVe

+19

0

+13

~20

33:15

0.115

0.110

0.102

0.080

To examine the constancy from.one group to another,

% Dev.

+21

+16

* 7

-18

37

Elements EELS % Dev. 2x S 5% Dev.

Group VII‘ 000861 +10 00098 + 3

VIIIA 0.078 + 2 0.092 - 3

A 0.072 - 6 g 0.089 - 7

B 0.066 ~13 0.080 ~16

C 0.071-‘- " 3 0.089 ' 7

Average 0.076 0.095

There is little to choose between the two assumptions; if

anything, the simpler assumption upon which 2xm/d is the

relevant quantity, leads to a smaller percentage than 2xm/S .

By and large, the two fractions run the same course. Group IA

and Group IB give high values, reflecting a "softness" in the

interatomic forces, corresponding perhaps to the single.mobile

electron per atom. The transition elements in Group VIII, and

more notably the elements of Group IIB, show a "hardness". The

presence of two valence electrons in these elements should re-

sult in some change, but it is doubtful if one would predict

its nature.

In extending our examination of the adequacy of the

formulas to the alkali halides, there is no need to repeat any

of the data from.Table IV. It may be commented that the two

fractions me/d and 2xm/8 are in constant proportion, and

hence need not be discussed separately. This relation follows

from the circumstance that the Pauling crystal radii are deter-

mined primarily from the crystal parameters in the alkali

halides. From the table it is seen that (excepting lithium

fluoride, for which the extrapolation has little significance}

38

the fractions are reasonably constant, there being at most a

slight progression downwards with increasing atomic mass of

the alkali. The effect of the halogen appears negligible.

The value of the fraction is considerably greater than for

the elements, amounting to an increase of 36 percent in

me/d , and 66 percent in 2xm/s .

we conclude then:

(1) Lindemann relation in its original form.retains its

original approximate validity for the additional data obtained

since its formulation in 1910.

(2) The generalization of the Lindemann relation by

taking into account the diameter of the atoms or ions according

to considerations of quantum-mechanical theory or of crystal

structure improves the agreement with experiment for closely

related elements, but does not improve the agreement for ele-

ments of different chemical nature.

(3) It is not worthwhile to attempt a theory of melting

based on the simple picture underlying the formulation of the

Lindemann relation. Its wide validity to an approximate degree

shows that the ratio of amplitude of thermal vibration to some

interatomic distance is of fundamental importance in melting,

but it appears that a more profound approach is necessary to

establish the detailed dependence.

7.

8.

9.

10.

ll.

12.

13.

1h.

15.

16.

17.

39

BIBLIOGRAPHY

P. L. Dulong and A. T. Petit, Ann. chim. et phys. 19,

395(1819); Phil. Mag. 54, 267(1819).

L. Boltzmann, Wien Ber. 91, Abth. 2, 712, 1731(1871).

A. Einstein, Ann. Phys. 22, 800(1907). '

M. Planck, Verh. d. D. Phys. Ges. g, 237(1900).

F. A. Lindemann, Physikal. Zeits. 1;, 609(1910).

A. Magnus and F. A. Lindemann, Zeits. f. Elektrochemie

P. Debye, Ann. Phys. 32, 789(1912).

M. Born and T. von Karman, Physikal. Zeits. 11, 297(1912).

M. Laue, w. Friedrich, and P. Knipping, Ann. Physik‘gl,

971(1913). _

J. Frenkel, Kinetic Theory 2: Liquids, p. 103 (Dover, New

York, 1955). .

J. K. Roberts and A. R. Miller, Heat and Thermodynamics,

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M. Born and K. Huang

p. 17 (Oxford, 19st)

J. A. Beattie, J. Math. and Phys.|§, 1(1926-27).

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A.Eucken in Wien-Harms, Handbuch der Experimentalphysik,

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F. M. Kelly and D. K. C. MacDonald, Canad. J. Phys. 11,

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L. Pauling, Nature 2; the Chemical Bond, 2nd ed., p. 3h}

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J. C. Slater, Introduction 32 Chemical Physics, p. 3H9,

(McGraw-Hill, New York, 1939).

18.

19.

20.

21.

22.

230

to

R. W. G. Wyckoff, The Strggture of Cr stals, 2nd ed.,

Chapt. 10-11, (Chemical Catalog—50., New York, 1931).

W. E. Forsythe Smithsonian Physical Tables, 9th Revised

ed., p. 6u7oh8, (Smithsonian ns itution, Washington,

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F. Seitz, The Modern Theo 23 Solids, p. 110, (McGraw-

Hill, New York, 19H0). .

T. M. Dauphinee, D. L. Martin, and H. Preston-Thomas,

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1932 .

a. B. Barnes, Zeits. r. Physik 15. 723(1932).

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