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 kT E 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 hfldu try—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 ) ( T3 k \ (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 ——- exd m __ (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 madamaodeuom. ogmfiaouoahgu on endpwnoQSoB cadgm no :.H m.a o.a m.o c.o :.o IA ‘1 [T d — _ fl .[rdllllflljji 13 Away .nb >0 use: cauaeodm .H .mam k \“ .0 c s. Te a . ,0 .enemv N est. a Em. we in? a .. a as; K 3 as capcm m.o __,o Ho - tom/T30 - (‘0) cmntoA quaqsu09 as 4903 oIJtcedg 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 displacement mass 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)bie = (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 one heat data. In the Debye should 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 M with equation (12). A is the atomic weight and N is again Avogadro's where 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 : 2 x - ___m___h NT . (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 Unfortunately typical 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/ \ . ,« \/ /‘* ‘ ,. >———— awe <2; «~— ~ 1.- are“ ~ ——..- «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 z I 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 K01 KBr KI CsF RbF RbCl RbBr RbI CsCl CsBr CsI Ga In II fl 1' I! Cubic N I! H II I! N n N Hexag. I! n w u Ii Cubic I! N fl n N FeCeCe " n n n I} n n u n C.p. fl eeeoeeeee H U'LNCOUINOJ HU’l-F’OU'LO‘VCIRO HUIN UIU'I \RNH UWICbOOU‘lNU-F'OO‘O‘O-F'EO FNOO‘NN UIUIOWOUIOOUI 000‘ng or)!a OWDFJ ruovq»h&n~umuao¢rhuucmo $EF' #iPfi? -qowNn0waomwvurvun$TOWnUur \uhunHuMJA) buwtrNHmMmF17u> viownvurbo u n I. '_' 4/2/ 2 I! II 6 II II 1| I! v 33% .3333: e e e O‘NN P'U‘LCDO‘NOCDCDU‘I \AJNCDUINO VINO) (Ii-J i=fitowonn» mw (Ibo ODCD-P' NNNH-F—WNNIU rvurrww NWN 0 2:98 B-AS bu N) N e e O‘NQ WWW WWMWW NW N 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 3 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. 27 0mm .0 x000.00 000.0 00.0 00.0 00.0 0:. 00.: 000.0 0000 A0000 000 .0 “000.00 000.0 00.0 00.0 00.0 00.0 00.0 000.0 0000 A0000 0.0 cm 000.0 000.0 :~.0 00.0 00.0 00.0 00.0 000.0 0000 00m 0. .0 000.0 000.0 00.0 00.0 00.0 00.0 00.0 000.0 :00 000 0.00 m: 00H msomu 000.0 000.0 00.0 00.0 000.0 00.0 00.0 000.0 0000 0000 00.0 00 mmnmmmmm 000.0 000.0 00.0 00.0 mm.0 00.0 00.0 :H~.0 0000 000 00a 04 000.0 000.0 00.0 «0.0 .0 0N.H 00.0 000.0 0000 000 000 04 000.0 000.0 00.0 00.0 00.0 00.0 00.0 000.0 0000 000 0.00 so mmnmmmmm as 000.0 000.0 00.: 00.0 00.0 0N.H 40.0 000.0 000 a: .000 00 000.0 000.0 «0.0 00.0 «0.0 0 .0 00.: 000.0 000 00 0.00 00 000.0 000.0 00.0 40.0 03.0 00.0 00.: 000.0 000 000 0.00 a 000.0 000.0 00.0 00.0 00.0 00.0 00.0 000.0 000 000 0.00 .2 000.0 000.0 H:.m 00.0 00.0 00.0 00.0 000.0 00: 00: 00.0 as .«mummmmm 00.0 m H macho 0\sx~ Esau 0 ms 0 ms 0 sum as o® 4 00202040 HHH mumde 28 0000 0.00 0 Muhammmm 000.0 m00.0 00.0 00.0 00.0 00.0 0:. 0:~.0 000 :0 000 00 000.0 000.0 00.m 00.0 00.0 00.0 00.0 000.0 0m: 000 000 :0 000.0 000.0 00.0 00.0 m~.0 00.0 00.0 000.0 000 000 0.00 00 0000100000 000 04 000.0 000.0 m0.m 00.0 00.0 00.0 m0.0 000.0 0000 N00 000 00 00.0 00.0 0000 0.0 s 00.0 0000 0.0 00 «000 00000 000.0 000.9 00.m 00.0 00.0 00.0 00.m 000.0 000 000 0.00 04 00.0 0.00. m MHH macaw 000.0 000.0 «0.0 00.0 0.0 00.0 00.0 000.0 000 00 000 00 000.0 000.0 00.m 00.0 0.0 00.0 00.0 000.0 :00 N00 N00 00 000.0 000.0 00.0 «0.0 00.0 00.0 00.0 000.0 N00 000 0.00 an mHH mfioouw {saw {sum 0. 0.0 0 0.0 0 f0 s0 a® < 0.9coov HHH wands i 29 000 cm 000 00 0.00 .0 0.00 0 00 0 H> md—OHU 000.0 000.0 00.0 00.0 00.0 00.0 000.0 000 000 000 00 000.0 000.0 00.0 00.0 00.0 00.0 000.0 000 000 000 00 00.0 0.00 04 m> @9080 0.00 0 0.00 z Mlmmmmm 000.0 00.0 000.0 000000. 000 000 .00 0 0.0 000.0 00.0 00.0 0w.0 00.0 000.0 0000 000 0.00 00 0 .0 000.0 00.0 00.0 00.0 0 .0 00.0 000.0 0000 000 0.00 _ 00 <>0¢00000 000.0 000.0 00.0 00.0 00.0 00.0 000.0 000 00 000 00 000.0 000.0 00.0 00.0 00.0 0m.0 000.0 000 000 000 00 000.0 000.0 00.0 00.0 00.0 00.0 0 .0 000.0 0000 000 0.00 Mm m>0 0:000 0\ss0 0\su0 0 mu 0 ms 0 s00 ea 000 0 A.pcoov HHH mqm 00000 000.0 000.0 00.0 00.0 00.0 00.0 00.0 000.0 0000 000 0. 02 000n0 000H0 00H0 00.0 00.0 00.0 00.0 000.0 0000 000 0.mm 00 000 0 000 0 00 0 00.0 00.0 00.0 00.0 000.0 0000 000 0.00 00 000> 0:000 0000 000 N00 00 O O O 9 OB 000 0 000 0 00 0 00.0 00.0 00.0 00.0 000.0 0000 000 0.00 as «00> 0:000 000 00 000 H 0.00 .000 0.00 00 0.00 a. m00> 00000 000.0 000.0 00.0 00.0 00.0 00.0 000.0 0000 000 000 3 e e e . Oeoo 02 000 0 000 0 00 0 00.0 00.0 00.0 000.0 0000 000 .0.00 ‘00 900.00 {30.0 0\se0 0 0.0 0 0.0 0 5000 s0. 000 0 0.0c000 000 00000 ‘)|,| 31 .eeEonBaaoueenm one :00002 .eeaandsea Song oea0epoo one: adamflvose.sdaodnsa no meanpeaedaop Hmhoea . enphmaom Scam one ONNpHem aoau 000008000 ooc0euoo one: cane» can» no deep one 000.0 00.0 000.0 0000 000 000 00 000.0. 00.0 000.0 0000 000 000‘ 00 0000000000 000.0 000.0 00.0 00.0 00.0 00.0 000.0 0000 000 000 00 000.0 000.0 00.0 00.0 00.0 00.0 000.0 0000 000 000. 00 0000000000 0 0\sx0 0\sw0 0 m0 0 0 0 000 _00 000 0 0.00000 000 00000 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 la). 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. 0n the simple theory, the equivalent temperature '60 , given by keo = 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 60 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. _ 1: r0:— 00 0 00.0 00.0 00.0 -- -- u- :00 u- u- 000 000.0 000.0 0:.0 00.0 00.0 00.0 000.0 000.0 000.0 000 000 000 0000 000.0 000.0 :0.0 0m.0 00.0 00.0 000.0 :00.0 000.0 000 000 0:0 0000 -- -- :0.0 0 .0 00.0 00.0 n- u- -- 000 -n -u 000 000.0 00.0 00.0 00.0 00.0 0N.0 000.0 000.0 000.0 000 000 000 000 000.0 00.0 00.0 00.0 00.0 0 .0 000.0 000.0 000.0 000 000 000 0000 :00.0 000.0 :0.0 00.0 00.0 00.0 000.0 000.0 000.0 000 :00 000 0000 u- -- 00.0 0:.0 00.0 00.0 .. -- .. 0000 u- -u 000 0.0 mmwnp 00.0 00.0 0:.0 00.0 0 0.0 000.0 0:0.0 0:00 0 0:0 .an 0.0 0 .0 :0.0 m0.0 0:.0 00. 0 0.0 0:0.0 000.0 0000 0 000 0mm. 0 0.0 000.0 :0.0 w.0 0:.0 :0.0 000.0 000.0 000.0 0:00 0 0 000 000 u- .1 00.0 0 .0 0:.0 00.0 n. -u -- 0000 a- n- 00 000.0 000.0 :0.0 00.0 00.0 00.0 0:0.0 000.0 0:0.0 000 :00 000 002 mmwnp. 000.0 00.0 00.0 00.0 00.0 000.0 000.0 000.0 0000 00m 000 0002 000.0 00000 00.0 00.0 00.0 00.0 000.0 000.0 000.0 0000 0 0 000 0000 000.0 000.0 00.0 0:.0 00.0 00.0 0:0.0 000.0 000.0 0000 0 :00 00a .. -- 00.0 00.0 00.0 00.0 n- -- u- 000 n- -- 000 -u s- 00.0 00.0 00.0 00.0 s. n- u- 000 -u -- 0000 -u n. 00.0 0w.0 00.0 00.0 .. u- .. 0mm .. .. 0000 000.0 000.0 00.0 0 .0 00.0 00.0 000.0 000.0 :00.0 0 0 000 0:: 000 0 0 00.05.0005 00250005 0 00 00 0 0.0005000 00» 090 s0. 00.. 00 mmGHddm g4 >H EBB. 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 Na 0.091; 0.115 K 0.091 0.115 Rb 0.089 0.113 Cs 0,021 0,112 Average 0.091 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 me/S 0.109 0.109 0.111 0.110 Here the downward progression for me/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 is 0.09i/o.o76 . 1.20; for the me/S it is O.115/O.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 20/4 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. 21: S 5% Dev. Group VII‘ 000861 +10 00098 + 3 VIIIA 0.078 + 2 0.092 - 3 A 0.072 - 6 , 0.089 - 7 B 0.066 ~13 0.080 ~16 0 000714- " 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 2mm/d is the relevant quantity, leads to a smaller percentage than arm/S . By and large, the two fractions run the same course. Group IA and Group 18 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 sz/B 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. 61, 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. 11, 609(1910). A. Magnus and F. A. Lindemann, Zeits. f. Elektrochemie P. Debye, Ann. Phys. 12, 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: Liguids, p. 103 (Dover, New York, 1955). . J. K. Roberts and A. R. 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