Polymerization of vinyl chloride. Ill 53 The authors wish to express their thanks to Professor H. W. Melville, F.R.S., for his sustained interest and invaluable advice during this investigation and to Mr J. Daft for the design and construction of the amplifier.
54 E. W. Elcock, P. Rhodes and A. Teviotdale to electrons distributed in energy bands of any form. Stoner has given a detailed treatment for electrons, or holes, distributed in a single parabolic band of infinite extent, and, more recently, Wohlfarth (1948 a, 1949) has considered the effects associated with overlapping parabolic bands, and has developed the treatment for electrons distributed in an infinite rectangular band (Wohlfarth 1951). The aim of the present paper is to determine the properties associated with a wider variety of band forms and to consider the effects associated with band closure. The properties to be considered are the electronic contributions to the magnetic susceptibility and specific heat of non-ferromagnetics. If exchange interaction effects are neg lected, both of these properties may be determined from the band shape, or electronic density of states relation, i.e, the function giving the number of available electronic states per unit energy range. Moreover, if the exchange interaction can be repre sented by a temperature-independent coefficient, O', the results for the para magnetic susceptibility may easily be modified so as to take into account this interaction.
Magnetic and thermal properties of metals and alloys 55 computed by McDougall & Stoner (1938) enable the treatment to be extended without undue difficulty to other half-integral values of n. The treatment of a rectangular band, as developed by Wohlfarth, corresponds to 0. The obvious generalization of these treatments is to band shapes determined by (1*1) with other positive integral values of n. Methods for evaluating the basic integrals required have been developed by Rhodes (1950). In §2 the temperature variations of susceptibility and electronic specific heat corresponding to n — 1,2,3 are con sidered, and a comparison is made with the already available results for n = 0, Examination of the results shows that in all cases the paramagnetic susceptibility decreases monotonically with increasing temperature, and it may further be shown that this is also true for any density of states of the form (1*1) with a positive, or zero, value of n.
56 E. W. Elcock, P. Rhodes and A. Teviotdale values of the overlap parameter r ^(r 10), and in the present paper (§ 4) calculations are made for overlapping rectangular bands with 1, 2, 5. Here again the bands have been taken to be rectangular for convenience in making the calculations, and since the results may be expected to show the same main features as for other band forms. Moreover, some of the related results for overlapping parabolic bands have been given elsewhere (Elcock 1954). In order to indicate the effects of closure on overlapping bands some results are also given for two closed overlapping rect angular band" The particular results of the theoretical treatments for a single open band, a single closed band and for two overlapping open bands may all be obtained as limiting cases of the general results for two overlapping closed bands, and in order to avoid undue repetition this general treatment is developed first in § 3 and then the detailed calculations for the other cases are considered in § 4.
Magnetic and thermal 'properties of metals and alloys fi being the Bohr magneton. Over the range in which varies linearly with H (that is, the whole of the relevant range for paramagnetics except in high fields at very low temperatures), (2-3) reduces to M = 2pfi'[a0( J c T ) F'(V) + )*+* ]. (2-4) It may be noted that the expression (2-4) does not involve explicitly the number of electrons in the band, but in most applications it is necessary to eliminate the reduced chemical potential, y, by a second equation connecting rj directly or in directly with the number, N, of electrons. The general equation corresponding to (24)is N = 2{a„(kT)Fa(v) + an(kT)^Fn(V)}. (2-5) At absolute zero the band is filled to an energy e0 (e0 being the maximum particle energy in the lowest energy state of the system, often referred to as the Fermi limit), and it is convenient to introduce a reduced temperature, r, given by r = kTje0. Then for r-*0, Fn(i})?jn+1 /(n + 1) and?/->e0/&T = 1/r. To simplify the symbolism, it is further convenient to write , _ \anloa) fco “ so that the band form (2*2) is represented by j/(e) = 2a0{l + a(e/e0)n}.
58 E. W, Elcock, P. Rhodes and A* Teviotdale For 1 the general recurrence relation F'n(rj) = ) may be used in (2*10).
Magnetic and thermal properties of metals and alloys 59 approximate expressions for XlXo appropriate at low temperatures, from which some of the main features of the XlXo> 7 relationship may be deduced. In the limit r -> 0, 7j-> oo and ^ 2 Fn(V) _ _ ^ w+1 + 7LW7w~1 + .,1 2 11 ( - ) K(v) 7) 4- Substituting in (2-6) and putting tij = 1 it is found that 7r2 / net 6 \ 1 +a Y then, from (2*8), X 7T* 5 1+^A t2 + Xo 6 2 12 ( - ) n(n— 1) a / na, \2 where 1 +a \T+a/ \ It can be seen that A < 0 for cc^(n— 1), and it may also be shown that, for a given value of n, A regarded as a function of a has its maximum positive value for a = (»-l)/(n+l). (2-13) The corresponding approximate expression for xlXo appropriate at high tem peratures may be determined from the appropriate approximation for Fn(i]). For 7-»oo, 7)-> — oo, Fn{7j)-^n\eyi. Using (2-6) and (2-8), these give 1+raTl\l (2-14) 1+a / 7* A positive value of A corresponds to x/Xo increasing with at low temperatures, t and since in the high-temperature limit xlXo decreases with r for any positive values of 7i and a, a positive value of A leads to a maximum in the corresponding o>7 curve. The maximum becomes more pronounced as A increases. Conversely, a negative value of A leads to a monotonically decreasing 7 curve. In order to obtain susceptibility, temperature curves exhibiting the full variety of forms which may be associated with bands of the general shape v’ = 2 the values of 7i and a must be chosen so that the corresponding values of A may be both positive and negative. For integral values of n, A can be positive only for 1 (cf. (2-12)), which provides a lower limit to the value chosen for n. A further consideration, which gives an upper limit to the value of ti for which the calculations may be con veniently made, is that in calculating the electronic specific heats associated with these band forms the function Fn+1(7/) is required, and since the functions FJjj) have been tabulated only for ti^ 4 (Rhodes 1950) it is convenient to limit the calcula tions to band shapes with ^ 3. On the basis of these considerations it was decided 71 that results of sufficient generality could be obtained by making detailed numerical calculations of the susceptibility and specific heat for w = 3, i.e. for the band shape v' = 2(a0-t-a3e3), and for a range of values of a. From (2-13), for = 3, A is a maxi mum for a = 0-5, and A = 0 for a = 2. Therefore, in order to show the various.
60 E. W. Elcock, P. Rhodes and A. Teviotdale possible types of X l X o > 7 curves which may occur, computations have been made for a = 0-05 (Amax. > A > 0), 0-5 (A = Amax-), 2 (A = 0) and 5 (A < 0).
Magnetic and thermal properties of metals and alloys 61 where (d^/dr), obtained by differentiating in (2-6) with respect to r, is given by = _ ! \FQ+{n+\)aTnFn\ (2-17) dr r \ + ctrnF'n /* Combining (2-16) and (2*17), and using the relation for (cf. (2-5)), N = (2-18) the final expression for C may be written ) 2F>+ <n + 2> (2-19) Nk {FG + ccTnFn} (i) v' = 2anen In the limit a->oo, a0a->aMeJ, (2-19) reduces to C = {n+2)Fn+l (n+l)*Fn 2 20 Nk F„ ‘ ( - ) The approximations to this expression in the limits of high and low temperatures may be readily derived from the corresponding expressions for and Fn{7f) (cf. (2-11), (2-14)):” r (n + 1 r-» 0, Nk 3 (2-21) T_>00’ N k^n + 1- The high-temperature limiting expression suggests a convenient reduced form in which to represent C.Writing Cmfor and combining (2*20) and (2*21), n + fn+l -(n+1) 2 22 4* n+\\ Fn ( * ) Using the values of rjas a function of r, determined in the susceptibility calculations, {C/Cay)has been evaluated from (2*22) for 0,1,2,3; and the results are shown in figure 3. The curve for n = drawn from the tabulated results of Stoner (19386), is also shown for comparison.
62 E. W. Elcock, P. Rhodes and A. Teviotdale Figure 3. Reduced specific heat, (7/(7^, as a function of reduced temperature, r, for isolated bands, v' — 2 anen.The numbers on the curves give the values of n; = 1) Nk.