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Law of corresponding states for self-diffusion
PHYSICS
Volume 29A, number 8
LAW
OF CORRESPONDING
30 June 1969
LETTERS
STATES
FOR
SELF-DIFFUSION
T. GESZTI Research Institute for Technical Physics of the Hungarian Academy of Sciences, Budapest
Received 19 May 1969 Empirical rules for the activation energy of self-diffusion are shownto follow from the similarity of interatomic potentials.
The well-known proportionalities between the activation energy of self-diffusion and the melting point T, or the heat of fusion L, respectively : AU--A-Tm=B-Lm
(1)
where A and B are empirical constants roughly independent of the substance, are widely used in processing experimental data on self-diffusion [l]. Still it has been stated even quite recently [2] that the connection of eq. (1) with existing theories of diffusion remains unclear. Interpreting eq. (1) as some direct connection between diffusion and melting (see ref. [l] on such attempts) would lead apparently very far from the physical picture of any of the successful theories. We would like to point out a simple way of interpreting eq. (1) as a consequence of the law of corresponding states [3]. The statement of the law which is of relevance here is the following. If the interatomic potentials of a family of similar atomic substances may be written in the form
aq,...,
r1 r.4 =EU--,...,--)
(
rN
(2)
where ‘1,. . . , rN are the positions of the nuclei, u is the same function of its arguments for all members of the family, while the characteristic energy E and the characteristic length a vary from one substance to the other, then assuming the validity of classical statistical mechanics - the melting points and the heats of fusion of the different materials may be written in the respective forms T,
=ET&;
L,=eL;m
where the reduced quantities T& and Lz
(3)
are the
same for all members of the family. Thus T, and/or Lm are but very convenient measures for the energy scale parameter E. The validity of eq. (3) is independent of the detailed mechanism of melting, and rests solely on the scaling property (2), the only physical requirement being that melting could be described at all in terms of an interatomic potential, which seems to be granted by the fact that melting does not destroy the metallic state and consists in the rearrangement of ions within the given electron gas. On the other hand, evaporation destroys the electron gas, thus the heat of evaporation is much more unlikely to be a good measure for E , in accordance with experience. Now there is good evidence from both highpressure equations of state [4] and phonon dispersion curves [5] that the variation of potentials within groups of crystals of similar structure and electronic properties can be fairly well described by two scaling parameters E and a. Therefore (1) gives evidence for the proportionality of AU to E and for its independence of a. Such a behaviour clearly results, if AU can be represented as a linear homogeneous function of extremal values of the effective atom-atom potential U(q,. . . rN). As a matter of fact, the rate theory of diffusion [l] represents AU as a sum of the migration energy AU, and the vacancy formation energy AUv* AU, is the difference between a minimum and a saddle point of U [6], while AU, is the difference between two minima, namely the equilibrium energies of the crystal with and without vacancy (including the elastic relaxation around the vacancy). Again, all this is true, whenever both migration and vacancy formation can be described in terms of an interatomic potential (like melting). Consequently we can write 425
Volume 29A, number 8
PHYSICS
AU = eA*
LETTERS
(4)
where A* is the same for all members of the family. Comparing eq. (4) with (3) we arrive at the wanted relationship (l), with A = A+/T;$;
B = A*/L;.
The above derivation makes any speculation about the connection between diffusion and melting superfluous. It shows also why A and B are constant only within groups of more or less similar metals, and why they can be sharply different for different electronic structures [2]. For the case of ‘good’ metals the applicability of interatomic potentials for melting and vacancy formation is amply demonstrated in ref. [7]. We mention that the observed proportionality between the activation energy of grain boundary migration and that of vacancy migration in the bulk [8] can also be interpreted as a simple consequence of the similarity of interatomic potentials, both quantities being obviously propor tional to E, which greatly invalidates the conclusion drawn from this proportionality about the connection between the two processes (in the same way as there is no connection between diffusion and melting). We think worth emphasizing as a general moral that measurements of the same quantity on different substances, if connected by a simi-
CAN
ELECTRON
RELAXATION BY HELICON
30 June 1969
larity law, cannot be taken for independent evidences in favour of some theoretical picture. Once the similarity recognized, the scope of a theoretical investigation is shifted towards the more delicate question of explaining the variation of the similarity constant (in our case A or B) from one group of substances to another. This can be regarded as a reduction of the conclusive force of experimental evidence by the action of similarity. The author is indebted to Professor N. H. March, Dr. L. Bartha and Dr. I. Ga.&lfor helpful discussions. References 1. D. Lazarus, Solid State Physics 10 (1960) 116. 2. D. Lazarus, in: Diffusion in body-centered cubic metals (Am.Soc. for Metals, Metals Park, Ohio, 1965) p. 155. 3. J. 0. Hirschfelder, Ch. F. Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1954) Chapter IV. 4. N. Bernardes and C. A. Swenson, in: Solids under pressure (McGraw-Hill, New York, 1963) p. 118. 5. B. N. Brockhouse, in: Phonons in perfect lattices and in lattices with point imperfections (Oliver and Boyd, Edinburgh, 1966) p. 134-141. 6. G.H.Vineyard, J. Phys. Chem. Solids 3 (1957) 121. 7. W. A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966) Chapter VI. 8. M. Feller-Kniepmeier, Z. Metalltide 57 (1966) 862.
TIMES WAVES?
BE
MEASURED
P.HALEVI * Technton - Israel Institute of Technology, Haifa, Israel Received 6 May 1969
We criticize a recent suggestion by Kao that, utilizing helicon waves, it is possible to determine ezperimentally the anisotropy of relazation times in metals, withoutany information on the Fermi surface.
In a recent letter Kao [l] has suggested that electron relaxation times in metals can be de* Based on a thesis to be submitted to the Senate of the Technion -Israel Institute of Technology in partial fulfillment of the requirements for the D. SC.degree.
termined using weakly damped helicon waves. His method is based on the assumption that the damping of the waves is given by the relation Im(q)/He(q)
7) ,
where q is the complex wavevector,
clotron frequency, 426
= l/(Zwc
(1)
wc the cy and 7 the relaxation time.
Volume 29A, number 8
LAW
OF CORRESPONDING
30 June 1969
LETTERS
STATES
FOR
SELF-DIFFUSION
T. GESZTI Research Institute for Technical Physics of the Hungarian Academy of Sciences, Budapest
Received 19 May 1969 Empirical rules for the activation energy of self-diffusion are shownto follow from the similarity of interatomic potentials.
The well-known proportionalities between the activation energy of self-diffusion and the melting point T, or the heat of fusion L, respectively : AU--A-Tm=B-Lm
(1)
where A and B are empirical constants roughly independent of the substance, are widely used in processing experimental data on self-diffusion [l]. Still it has been stated even quite recently [2] that the connection of eq. (1) with existing theories of diffusion remains unclear. Interpreting eq. (1) as some direct connection between diffusion and melting (see ref. [l] on such attempts) would lead apparently very far from the physical picture of any of the successful theories. We would like to point out a simple way of interpreting eq. (1) as a consequence of the law of corresponding states [3]. The statement of the law which is of relevance here is the following. If the interatomic potentials of a family of similar atomic substances may be written in the form
aq,...,
r1 r.4 =EU--,...,--)
(
rN
(2)
where ‘1,. . . , rN are the positions of the nuclei, u is the same function of its arguments for all members of the family, while the characteristic energy E and the characteristic length a vary from one substance to the other, then assuming the validity of classical statistical mechanics - the melting points and the heats of fusion of the different materials may be written in the respective forms T,
=ET&;
L,=eL;m
where the reduced quantities T& and Lz
(3)
are the
same for all members of the family. Thus T, and/or Lm are but very convenient measures for the energy scale parameter E. The validity of eq. (3) is independent of the detailed mechanism of melting, and rests solely on the scaling property (2), the only physical requirement being that melting could be described at all in terms of an interatomic potential, which seems to be granted by the fact that melting does not destroy the metallic state and consists in the rearrangement of ions within the given electron gas. On the other hand, evaporation destroys the electron gas, thus the heat of evaporation is much more unlikely to be a good measure for E , in accordance with experience. Now there is good evidence from both highpressure equations of state [4] and phonon dispersion curves [5] that the variation of potentials within groups of crystals of similar structure and electronic properties can be fairly well described by two scaling parameters E and a. Therefore (1) gives evidence for the proportionality of AU to E and for its independence of a. Such a behaviour clearly results, if AU can be represented as a linear homogeneous function of extremal values of the effective atom-atom potential U(q,. . . rN). As a matter of fact, the rate theory of diffusion [l] represents AU as a sum of the migration energy AU, and the vacancy formation energy AUv* AU, is the difference between a minimum and a saddle point of U [6], while AU, is the difference between two minima, namely the equilibrium energies of the crystal with and without vacancy (including the elastic relaxation around the vacancy). Again, all this is true, whenever both migration and vacancy formation can be described in terms of an interatomic potential (like melting). Consequently we can write 425
Volume 29A, number 8
PHYSICS
AU = eA*
LETTERS
(4)
where A* is the same for all members of the family. Comparing eq. (4) with (3) we arrive at the wanted relationship (l), with A = A+/T;$;
B = A*/L;.
The above derivation makes any speculation about the connection between diffusion and melting superfluous. It shows also why A and B are constant only within groups of more or less similar metals, and why they can be sharply different for different electronic structures [2]. For the case of ‘good’ metals the applicability of interatomic potentials for melting and vacancy formation is amply demonstrated in ref. [7]. We mention that the observed proportionality between the activation energy of grain boundary migration and that of vacancy migration in the bulk [8] can also be interpreted as a simple consequence of the similarity of interatomic potentials, both quantities being obviously propor tional to E, which greatly invalidates the conclusion drawn from this proportionality about the connection between the two processes (in the same way as there is no connection between diffusion and melting). We think worth emphasizing as a general moral that measurements of the same quantity on different substances, if connected by a simi-
CAN
ELECTRON
RELAXATION BY HELICON
30 June 1969
larity law, cannot be taken for independent evidences in favour of some theoretical picture. Once the similarity recognized, the scope of a theoretical investigation is shifted towards the more delicate question of explaining the variation of the similarity constant (in our case A or B) from one group of substances to another. This can be regarded as a reduction of the conclusive force of experimental evidence by the action of similarity. The author is indebted to Professor N. H. March, Dr. L. Bartha and Dr. I. Ga.&lfor helpful discussions. References 1. D. Lazarus, Solid State Physics 10 (1960) 116. 2. D. Lazarus, in: Diffusion in body-centered cubic metals (Am.Soc. for Metals, Metals Park, Ohio, 1965) p. 155. 3. J. 0. Hirschfelder, Ch. F. Curtiss and R. B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1954) Chapter IV. 4. N. Bernardes and C. A. Swenson, in: Solids under pressure (McGraw-Hill, New York, 1963) p. 118. 5. B. N. Brockhouse, in: Phonons in perfect lattices and in lattices with point imperfections (Oliver and Boyd, Edinburgh, 1966) p. 134-141. 6. G.H.Vineyard, J. Phys. Chem. Solids 3 (1957) 121. 7. W. A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966) Chapter VI. 8. M. Feller-Kniepmeier, Z. Metalltide 57 (1966) 862.
TIMES WAVES?
BE
MEASURED
P.HALEVI * Technton - Israel Institute of Technology, Haifa, Israel Received 6 May 1969
We criticize a recent suggestion by Kao that, utilizing helicon waves, it is possible to determine ezperimentally the anisotropy of relazation times in metals, withoutany information on the Fermi surface.
In a recent letter Kao [l] has suggested that electron relaxation times in metals can be de* Based on a thesis to be submitted to the Senate of the Technion -Israel Institute of Technology in partial fulfillment of the requirements for the D. SC.degree.
termined using weakly damped helicon waves. His method is based on the assumption that the damping of the waves is given by the relation Im(q)/He(q)
7) ,
where q is the complex wavevector,
clotron frequency, 426
= l/(Zwc
(1)
wc the cy and 7 the relaxation time.