1941] Conduction Electrons and Thermal Expansion. 309

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1 1941] Conduction Electrons and Thermal Expansion. 309 In conclusion, the author wishes to express his deep gratitude to Professor H. Yukawa for his kind interest in this work Institute of Theoretical Physics, Kyoto Imperial University. (Received Feb 12, 1941.) (_??_) [Added in Proof: Recently it has been shown by Rarita and Sclwinger (Phys. Rev. 59 (1941), 436) that the nuclear forces derived from the pseudoscalar meson theory agree in sign and spin dependence with the sign and magnitude of the singlet triplet difference and quadripole moment of the deuteron system, if we take the influence of the non-central force into account correctly. In this connection it may also be remark edthat the experimentally observed burst frequency of the meson is in good agreement with the calculation based on the pseudoscalar theory (R, F. Christy and S. Kusaka, Phys. Rev. 59 (1941). 436; See also J. R Oppenheimer, Phys. Rev. 59 (1941), 462)] Contr ibution, of the Conduction Electrons in a Metal to the Thermal Expansion. By Ziro (Read MIKURA. Nov ) ônh ô Abstract ôns ô. The approximate proportionality between the thermal-expansion coefficient and the atomic heat of a metal in the moderate temperature region is well-known. However, it seems that this relation should be changed at those extremely low temperatures, where the contribution of the conduction electrons to the atomic heat is relatively large. It is the purpose of this paper to construct a formula for thermal expansion to cover this temperature region. The result is that in this region the thermal-expansion coefficient is no longer proportional to the atomic heat, and that it consists of two parts, corresponding to the Debye and the electronic terms of the latter respectively. The electronic term of the thermal-expansion coefficient is generally not proportional to that of the atomic heat. The former, however, as well as the latter, is pro portionalto the absolute temperature at low temperatures. From the new formula a possible explanation of the small expansibility of ginvar h has been suggested. ƒì 1. Introduction. The well-known theoretical relation between the thermal-expansion

2 310 ziro MIKURA. [Vol, 23 coefficient and the atomic heat of an isotropic solid is(1) 3ƒ V/ƒÈ=ƒÁCv, ƒá=d logƒ /d logv, (1) where ƒ is the linear expansion coefficient, V the atomic volume, ƒè the compressibility, ƒ the Debye characteristic temperature, Cv the atomic heat at constant, volume. According to Gruneisen, ƒá is a con stant,independent of temperature, and has the value 1-3 for various solids. The agreement between this formula and the experiments with various metals has been shown by Gruneisen in the moderate tempera tureregion. But in applying this formula to a metal it is to be remembered that, as we can easily understand from the form of the equation, in deriving it we have neglected the temperature dependence of the kinetic energy of the conduction electrons. If it is correct to consider the conduction electrons of a metal to be a Fermi gas, the showing of some pressure would be expected, depending on tempera tureand volume. The influence of this pressure upon the thermal expansion of a metal at very low temperatures, may not be over looked,since the specific heat and the thermal-expansion coefficient are closely connected quantities, and the influence of the conduction electrons upon the former has already been found both theoretically and experimentally; at liquid helium temperatures the electronic specific heat is comparable with or larger than Debye's(2). This paper aims to obtain aa formula which holds even at very low temperatures, by taking into account the influence of the conduction electrons. As the experi mentto measure the thermal-expansion coefficient of a metal at liquid helium temperatures is a very difficult one, we wished to know what theoretical interests are involved in the experiment. Further, we thought that this new formula might show whether the discontinuity of the specific heat of a superconducting metal at the normal transition point would be accompanied by a perceptible order of discontinuity of the thermal-expansion coefficient or not, for we supposed that the discontinuity of the specific heat was caused by that of the electronic one. The result of calculations shows that, if we take into account the influence of the conduction electrons, such a simple relation as that 3ƒ V/ƒÈ is proportional to the atomic heat, holds no longer. Moreover, the term which must bo added to the right-hand side of equation (1) is generally not even proportional to the electronic specific heat. In the case of strictly free electrons, however, it is simple and the added (1) E. Gruneisen, Handb. d. Phys. 10 (1926). (2) The experimental results on the electronic specific heat of Ag, Ni, Cu, Al, Sn, Co, Fe, etc, are found in numerous papers on Physica, 1 (1933)-6 (1939) by W.H. Keesom and his collaborators, and by G. Duyckaerts.

3 1941] Conduction Electrons and Thermal Expansion. 311 term is (2/3)Ce, Ce being the electronic specific heat per gram atom of metal. In general cases, the new term depends on the form of the function which represents the density of states of the conduction elec trons.in particular cases, it may be possible that this term is nega tiveand of such a value that it cancels the positive expansion due to the lattice vibrations.ƒì2. Derivation of the New Formula. For the sake of simplicity we considered an isotropic metal built of NA similar atoms, and started from the following two assumptions: 1. The thermal vibrations of the ionic lattice are of the Debye form with a single characteristic temperature ƒ (V), which is a function of the atomic volume alone. The free energy FD per gram atom is then given by(3) FD=F0(V)+FT(V,T)=F0(V)+9NAƒÈT(T/ƒ )3 çƒ T0log(1-c-x) Ex2dx, (2) where F0(V) is the energy in an unexcited state, NA Loschmidt's number, ƒè Boltzmann's constant. 2. The conduction, electrons, which we suppose to form ƒ Fermi gas, and the ionic lattice can be treated as two independent systems of the same temperature; this condition is expressed in the following manner: When the free energy of the lattice vibrations is expressed by equation (2) and that of the conduction electrons alone is given by f=f0(v)+ft(v, T), (3) the free energy of the whole system, F, is given by F=F0(V)+FT(V, T)+fT(V, T). (4) The reason why F is not given by Fp+f is because, F0(V) in equation (2) implicitly involves f0(v) When we know the free energy, we can calculate the pressure p by means of the well-known thermodynamical equationp=-( ÝF/ ÝV)r. (5) As there is a relation )v, (6) and the thermal-expansion coefficient ƒ and the compressibility ƒè are given respectively by the equations (3) E. Gruneisen, loc. cit p. 22.

4 312 Ziro MIKURA. [Vol. 23 3ƒ =1/V( ÝV/ ÝT)ƒË, and 1/ƒÈ=-V( Ýp/ ÝV)_??_, we obtain*3ƒ /ƒè=- Ý/ ÝT( ÝF/ ÝV). (7) Using equations (3) and (4), we obtain from (7) First, we shall calculate the first term of the right-hand side of this equation by substituting FT from (2). We then obtain(3)- Ý/ ÝT( ÝFT/ ÝV)=- Ý/ ÝT{3NAƒÈTƒ '/ƒ D(ƒ /T)}=-3NA The Debye specific heat at constant volume, CD,V, is expressed, with the aid of the function D(ƒ /T), in the form(4) CD,V=3NAƒÈ{D(ƒ /T)-ƒ /TD'(ƒ /T)}. Equation (9) then may be expressed in terms of CD,V in the form ÝV)=-ƒ /ƒ CD,V=ƒÁ/VCD,V. (10) In order to calculate the second term of the right-hand side of equa tion(9) we must know the free energy f of the conduction electrons. It is easily seen that f is given by(5) f=nƒètƒå-ƒètz, (11) where N is the number of conduction electrons per gram atom of metal and Z is an integral given by Z=V ç 0ƒË(ƒÃ)log(1+cƒÅ-ƒÃ/ƒÈT)dƒÃ, (12) ƒë(ƒã) being the density of electronic states per unit volume of the metal as a function of electronic energy ƒã, and ƒå a quantity with which the Fermi distribution function is written in the form tively. * H ere and in the rest of this paper, Ý/ ÝT, Ý/ ÝV always mean ( Ý/ ÝT)V, ( Ý/ ÝV)T respec (4) E. Gruneisen, loc. cit. p. 19. (5) Sec, for example, E.C. Stoner, Phil. Mag 28 (1939), 257.

5 1941] Conduction Electrons and Thermal Expansion /cƒÃƒÈT-ƒÅ+1. The two quantities N and Z are related to each other by the equa tion N=V ç 0ƒË(ƒÃ)/ƒÃ/cƒÈT-ƒÅ+1dƒÃ= ÝZ/ ݃Å. Substituting (11) into the second term (13) of the right-hand side of (9) and referring to (12) and (13), we obtain- Ý/ ÝT( Ýf/ ÝV)=- Ý/ ÝT{NƒÈT ݃Å/ ÝV-ƒÈT ÝZ/ ݃Š݃Å/ ÝV-ƒÈT( ÝZ/ ÝV)T,ƒ For convenience of further calculation. we introduce a function ƒô(ƒã) defined by Then, using this function, we obtain, by means of (12),ƒÈTZ/V=ƒÈT ç 0ƒË(ƒÃ)log(1+cƒÅ-ƒÃ/ƒÈT)dƒÃ=ƒÈT{[ƒÔ(ƒÃ)log(1+cƒÅ-ƒÃ/ƒ The first term of the last expression vanishes, and we get We calculate this integral in the following way: In the energy interval between those values ƒã and ƒã*, which are suitably chosen and satisfy the condition we write the function ƒô(ƒã) in the following form: eitherƒô(ƒã)= <ƒé 0>CƒÉ(ƒÃ-ƒÃ0)ƒÉ where ƒã0<ƒã (17)orƒÔ(ƒÃ)= <ƒé 0>CƒÉ(ƒÃ0-ƒÃ ƒé's m ay be positive or negative, and are not 1imited to integers. The constants ƒé, CƒÉ, ƒã0 must be suitably chosen in order to make the above

6 314 Ziro MIKURA. [Vol. 23 ex pression possible, but, as the number of parameters is large, it seems to be in most cases possible to write the function ƒô(ƒã) in the form (17) or (17'), especially at low temperatures where the interval between ƒã and ƒã* may be taken to be small. Then the function ƒë(ƒã) may be written in the same energy interval in the form ) corresponding respectively to (17) or (17'). Now we separate the integral I= ç 0ƒÔ(ƒÃ)/eƒÈƒÃT-ƒÅ+1dƒÃ into (19) two parts in such a manner as First, accorling to (16) ç0ƒô(ƒã)dƒã and hence we obtain ÝI1/ ÝT=0 (20) Next, from (16) and (17) we obtain and therefore, remembering (20), we get On the other hand, for the electronic specific heat at constant volume per gram atom of metal we obtain, according to (16) and (18),

7 1941] Conduction Electrons and Thermal Expansion 315 The second term of the right-hand side vanishes, because the number of electrons per gram atom of metal with energy greater than ƒã is independent of temperature. Hence we obtain Writing for brevity Cc,v=V Ý/ ÝT{ <ƒé 0>CƒÉƒÉ çƒãƒã(ƒã-ƒã0)ƒé/eƒã/ƒèt-ƒå+1dƒã}. (22) /eƒã/ƒèt-ƒå+1dƒã=cc,v, we obtain from (22) (23) ) From (21), (23) and (24) we see I/ ÝT=1/V <ƒé 0>C_??_,v/ƒÉ. (25) If we use the expression (17') instead of (17), it may easily be shown that the last obtained equation (25) is unaltered. From equations (8), ( 10), (14), (15), (19) and (25) we obtain finally 0>C_??_,v/ƒÉ. (26) If we ignore the second term of the right-hand side, this equation is equivalent to equation (1). ƒì 3. Discussion. We shall discuss in this section what results may be expected from equation (26) in particular cases. (a) Free Electrons. For free electrons, as it is well-known, thedensity of states per unit volume of the metal is given byƒë(ƒã)=1/ 32. In this case the electronic contribution in equation (26) is given by a single term with ƒé=3/2, and accordingly we obtain from (26) 3ƒ V/ƒÈ=ƒÁCD,V+2/3Cc,v. (27) This equation may hold approximately for alkali metals. In the moderate temperature region it is reasonable that the second term of the right-hand side of equation (27) can be neglected, for the constant

8 316 Ziro MIKURA, [Vol. 23 ƒá is of the value 1-3, and CD,V is far greater than Ce,v in this region. At liquid helium temperatures, however, it is found that Ce,v is of the order comparable with, or greater than, that of CD,V, and therefore the influence of the conduction electrons upon the thermal expansion be conesrelatively very large. (b) Low temperature. Both theory and experiment tell us that at low temperatures the electronic specific heat is proportional to the absolute temperature. The theory gives(6) where ƒä0 is the energy value of the highest occupied state at the abs olutezero of temperature. Since in the neighbourhood of the point ƒã=ƒä0 the density of states ƒë(ƒã) can be written in the form (18) or (18'), each term CƒÉ_??_,v in the expression (24) is also proportional to T when Cc,v is given by (28). Then it is obvious from (27) that the same tempe rature dependence holds for the electronic term of the thermal -e xpansion coefficient, provided that the temperature variation of the compressibility ƒè is negligible. On account of this term, the thermale xpansion coefficient of a metal at liquid helium temperatures falls far more slowly than expected from the behaviour at moderate tempera tures,and therefore the measurement of it seems to be possible even at quite low temperatures. Although the question whether the discontinuity of the specific heat of a superconducting metal at the normal transition point is ac companiedby the perceptible order of discontinuity of the thermal -e xpansion coefficient or not cannot be answered until we shall thoroughly understand the supercondueting state of metal, we are tempted to deduce the following conclusion frost equation (27)*: A discontinuity of the atomic heat, which may be either due to the change of CD,V or due to the change of Cc,v, will be accompanied by a discontimity of the thermal-expansion coefficient of nearly the same order of percent, because ƒá has the value 1-3 and the change of ƒè at the normal transition point seems to be neglected õ. McLeman and his collaborators(7) measured the thermal expansion (6) A, S ommerfeld and H.A. Bethe, Handb. d. Phys. 24/2 (1933), 430. * This equation may not be applied for superconducting metals as they have several electrons in the outermost shells, but except in particular cases the conclusion deduced by making use of (27) may hold qualitatively, õ Shoenberg has shown semi-empirically that the discontinuity of the compressi bilityƒè at the normal transition point is negligibly small though that of the thermal - expansion coefficient is comparatively large. See Shoenberg, Superconductivity (193), p. 76. (7) J,C. McLennan, J.F. Allen and J.O, Wilchelm, Trans. Roy. Soc. Canada, III, 25 (1931), 1.

9 1941] Conduction Electrons and Thermal Expansion. 317 of lead down to 4.2 K, but they found no discontinuity of the thermal expansion at the normal transition point. It is to be noted, however, that for lead any attempt(8) to find the discontinuity of the specific heat has also failed, and that McLennan and others' apparatus for measurement of the thermal expansion had only a poor sensibility. It would be desirable to make an experiment by using a more sensi tiveapparatus on another superconducting, metal, whose specific heat has already been found to jump at the transition point. (c) ginvar. h It will be shown that in some particular cases the magnitude and the sign of the electronic thermal expansion are ab normal.for the sake of simplicity, we shall consider an example, in which the function ę(ċ) is represented in the neighbourhood of the Fermi surface by a single term appearing in (17'), nainely The condition that ę(ċ) must satisfy areę (ċ)>0 and dę(ċ)=đ(ċ)>0. In a case, where Ď<o and Că>0, the above conditions are satisfied and it is not an impossible case. It is easily seen from (26) that in this case the electronic thermal expansion should be negative, because both electronic specific heat and the compressibility are always posi tive.this negative expansion due to electrons may have an abnormally large absolute value, when Ď is small but Căă is not too small (that is, the specific heat due to electrons is of the ordinary order of magnitude). In some particular cases it may be possible that the positive expansion due to the lattice vibrations and the negative one due to electrons nearly cancel each other, and that the total coefficient of expansion becomes very small even at moderate temperatures. On the other hand, an alloy called ginvar, h which consists of 64% iron and 36% nickel, is known to have a very small expansion coefficient. Although, as ginvar h is ferromagnetic, there is a thermal expansion due to temperature variation of the spontaneous magnetization, it may not be too absurd to account for the small expansibility of ginvar h in the way mentioned above, namely by eonsidering the effect of alloying to change the functional form of ę(ċ) into one of the above citedparticular forms. In conclusion, I wish to express my hearty thanks to Professor S. (8) See K. Mendelssohn and F. Simon, Zeits. f. Phys. Chem. B, 16 (1932), 72.

10 3 18 Goro HAYAKAWA. [Vol. 23 Aoyama for his kind interest in this work and to Mr. S. Miyahara for his valuable discussions. Research Institute for Iron, Steel and Other Metals, Tohoku Imperial University, Sendai. (R eceived Feb. 20, 1941.) Stark Effect on Hydrogen Molecule (Part I). By Goro HAYAKAWA. (Read April 2, 1939.) The Stark effect on H ôsh ô2 ôss ô-spectrum in the red and yellow region (H ôsh ôa ôss ô 5690 ð) was measured using dispersion abut 8.1 ð/mm. With a field strength 208,000V/em better resolution of components was obtained than in former investigations. The results were compared with the theoretical displacements roughly calculated as due to the mutual in fluencesamong neighbouring terms, specially for antisymmetricat lines of _??_(0, 0) band. The Stark affect on the many-lined spectrum entitted by hydrogen molecule has been investigated by several workers(1). The results hitherto obtained, however, seem to be not very satisfactory, the components being for many lines not fully resolved, and it is difficult to apply theoretical considerations quantitatively to them. Snell made observations on lines in the region between H_??_ and HƒÀ using a high dispersion (3.8 ð/mm), but the field strength obtained was not suflic ientlylarge (90,000V/em). Kiuti and Hasunuma worked with a lower dispersion but larger field strengths. They determined from their observation the relative heights of evergy levels which are to interact with one another in the presence of an electric field. The relative heights of levels are important in explaining and predicting the effects theoretically. (1) M. Kiuti, Jap. J. Phys. 1 (1922), 29; 4 (1925), 13. J. K.L. MaeDonald, Proc. Roy, Soc, 123 (1929), 103; 131 (1931), 146. H. Snell, Phil. Trans. Roy. Soc. London, 234 (1935), 115. W. Rave, ZS. f. Phys. 94 (1939), 72.M. Kiuti and H. Hasunuma, Proc. Phys.-Math. Soc. Japan, 19 (1937), 821