Garland, C. W.

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III. SOLID STATE PHYSICSProf. W. P. Allis Dr. W. M. Whitney K. R. DawberProf. S. C. Brown G. Ascarelli T. HigierProf. C. W. Garland R. Dalven R. G. NewburghProf. G. G. Harvey L. J. SlutskyA. FREQUENCY SPECTRUM OF MAGNESIUMApplication of the Born-von Karman theory of lattice dynamics to a central-forcemodel of the hexagonal close-packed metals was recently reported (1). The frequency-distribution spectrum of normal modes for this model has been approximated by a root-sampling technique in which force constants calculated from the elastic constants ofmagnesium (2) at 0O K were used. The frequency spectrum, g(v), is so defined thatg(v)dv is the number of normal modes in the interval from v to v + dv. The spectrummay be approximated numerically by solving the lattice dynamical secular equation ata number of points in reciprocal space and determining the number of roots (properlyweighted according to the degeneracy of the point in reciprocal space) that lie withingiven frequency intervals. This process was carried out on the Whirlwind I computerfor a mesh of 429 points which lie in that twenty-fourth part of the Brillouin zonewhich is irreducible under symmetry. The basic frequency interval chosen was12 -10. 125 X 10 sec ; at low frequency larger intervals were used, and a smooth curvedrawn. The normalized frequency spectrum of this model of magnesium is given in12 -1Fig. III-1, in which both the histogram and the smooth curve up to 3. 75 X 10 secare shown.2.502.00I.Co'I F0.50 -0 125 2.50 375 5.00 6.25 750v IN 10 SEC_'Fig. III-1. Frequency-distribution spectrum, g(v), for a central-forcemodel of magnesium at 0O K.(III. SOLID STATE PHYSICS)o400 oAo A380 -360*\3403200 20T (OK)Fig. 111-2. Equivalent Debye 0 versus temperature for magnesium.O experimental values of Smith* calculated from the data of Craig et al.A obtained from the data of Logan et al.- theoretical result obtained from the frequency spec-trum shown in Fig. III-1.-- based on 217 points in reciprocal space.The lattice heat capacity at constant volume is given by the expressionV LC =kv0og(v) (hv/kT)2 ehv/kT dv/(ehv/kT _ 1)2 (1)where vL is the maximum vibrational frequency, 8. 098 X 1012 sec - 1 for magnesium.The lattice heat capacity was obtained by a numerical integration of Eq. 1 in which thefrequency spectrum given in Fig. III-1 was used. A comparison of the calculated C vwith experimental data in the range from 0* K to 1000 K is presented in Fig. 111-2.This comparison was made, as is customary, in terms of an equivalent Debye temper-ature, 0. The experimental points are those of Logan, Clement, and Jeffers (3);(III. SOLID STATE PHYSICS)Smith (4); and Craig, Krier, Coffer, Bates, and Wallace (5).The dotted curve given in Fig. III-2 represents values of 0 calculated as describedabove from a spectrum derived from solutions of the secular equation at only 217 pointsin reciprocal space. The spectrum agreed in all major features with the spectrumshown in Fig. III-1, and the 0-values calculated from it are seen to be in fairly closeagreement with those calculated from the spectrum based on 429 points. We are there-fore inclined to attribute the disagreement between the calculated curve and the experi-mental points to the inadequacies of the model or inaccuracies in the parameters ratherthan to a root sampling that is not sufficient to characterize the spectrum. It may besaid, however, that the agreement obtained for magnesium is as good or better thanthat which has been recently obtained for other metals.C. W. GarlandReferences1. L. J. Slutsky and C. W. Garland, J. Chem. Phys. 26, 787 (1957); QuarterlyProgress Report, Research Laboratory of Electronics, M.I.T., July 15, 1956,pp. 10-11.2. L. J. Slutsky and C. W. Garland, Phys. Rev. (in press); L. J. Slutsky, QuarterlyProgress Report, Research Laboratory of Electronics, M.I.T., April 15, 1957,pp. 18-20.3. J. K. Logan, J. R. Clement, and H. R. Jeffers, Phys. Rev. 105, 1435 (1957).4. P. L. Smith, Phil. Mag. (7), 46, 744 (1955).5. R. S. Craig, C. A. Krier, L. W. Coffer, E. A. Bates, and W. E. Wallace,J. Am. Chem. Soc. 76, 238 (1954).

Solid State Physics

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Solid State Physics

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