LATTICEANHARMONICITY AND
THE
3573
THERMAL ACCOMMODATION COEFFICIENT
Lattice Anharmonicity and the Thermal Accommodation Coefficient by P. Feuer School of Aeronautics, Astronautics and Engineering Sciences, Purdue Unizlersity, Lafayette, I n d i a n a (Received M a y 13, 1968)
47907
A theory is presented that takes into account the effect of lattice anharmonicity on the thermal accommodation coefficient, which is a measure of the energy exchange between a beam of gas atoms and a solid. A general expression is derived for the thermal accommodation coefficient, in terms of quantum mechanical transition probabilities associated with the collision of an atom with an anharmonic oscillator. The harmonic and anharmonic cases are compared.
I. Introduction' The quantum theory of the thermal accommodation coefficient, which is a measure of the energy exchange between a beam of gas atoms (or molecules) and a solid, has been considered by many ,v~orkers~-~ within the framework of the harmonic approximation. That is, in these treatments, the oscillations of the atoms of the solid are assumed to be harmonic. It is the purpose of the present work to formulate the theory in such a way as to take anharmonic effects into account. I n a recent note, Shin*has begun a discussion of such effects. It will be shown, however, that the very equation upon which his discussion is based is correct only in the harmonic approximation; thus his analysis is not an appropriate one. I n the collision of an atom with a crystalline solid, the Hamiltonian has the form X =
X g
+ + Vint Xc
(1)
where xgis the Hamiltonian of the gas atom, X, is the Hamiltonian of the crystal, and Vint is the interaction potential between the gas and the solid. I n the harmonic approximation, x,is a sum of harmonic-oscillator Hamiltonians associated with the normal modes of vibration of the crystal. Then X, X, is separable when X, is written in terms of normal-mode coordinates and momenta, and the problem of interaction of the gas atom with the crystal can be treated by perturbation methods . 4 , 5 , I n the anharmonic case, X, involves terms which couple the normal modes t)ogether, and so the Hamiltonian is no longer separable in terms of normal-mode coordinates. The problem of energy exchange with the anharmonic crystal is thus a complex one. It is the intention, in the present work, to formulate the problem for what is perhaps the simplest type of model. The procedure involves an extention of a model of Jackson2 to the anharmonic case. Jackson considered a system in which the crystal surface is represented simply by independently oscillating surface atoms. The model is a one-dimensional one. The Hamiltonian of the system is taken to be
+
where Xg and Vint are the same as in eq 1, and X, is the Hamiltonian of a surface atom oscillating normal to the surface. I n Jackson's work, X, is assumed to be a harmonic-oscillator Hamiltonian. I n the present discussion, it can be the Hamiltonian of an anharmonic oscillator. Using this model, an expression will be derived for the accommodation coefficient in terms of transition probabilities associated with collisions between an atom and an atomic oscillator. The harmonic and anharmonic cases will then be compared.
11. The Accommodation Coefficient The thermal accommodation coefficient, a, is defined in the usual way. Let E2 be the average energy of a beam of gas atoms at temperature Tz falling on a solid surface at temperature Tl. Let E be the average energy of the reflected beam and let El be the average energy that the beam would have if it were to come into thermal equilibrium with the surface. Then (3)
Now consider a one-dimensional Boltzmann distribution of gas atoms falling on the surface. Thus let N(e)de = --eN o -€r/hT2de IC Tz
(4)
represent the flux of gas atoms (at temperature T,), with de, striking the surface. energies between e and e No is the total flux of gas atoms striking the surface. Next, it is convenient to define the transition probabilities p n m ( e ) ,which represents the probability per collision
+
(1) Part of this work was presented a t the American Physical Society Meeting, Berkeley, Calif., March 1968. (2) J. M. Jackson, Proc. Cambridge P h i l . Soc., 28, 136 (1932). (3) J. M. Jackson and N. F. Mott, Proc. R o y . Soc., A137, 703 (1932). (4) J. M. Jackson and A. Howarth, ibid., A142, 447 (1933). (5) A. F. Devonshire, ibid., A158, 269 (1937). (6) R. T. Allen and P. Feuer, J . Chem. Phys., 40, 2810 (1964). (7) R. T. Allen and P. Feuer, ibid., 43, 4500 (1965). (8) H. K. Shin, J . P h y s . Chem., 71, 1540 (1967). V o l u m e ?t?2, Number 10 October 1968
P. FEUER
3574 that a gas atom with energy E, in colliding with the surface, changes the vibrational quantum number of the surface atom from n to m. (These transition probabilities can be calculated from the quantum mechanical perturbation theory for the system whose Hamiltonian is given by eq 2.) Then
where the quantity Wnvrepresents the energy of the surface atom in the nth vibrational state and a, is the fraction of surface atoms in the nth vibrational state. Thus
-e1 - Wnv/kTl
an =
Z V
where T1is the temperature of the surface and Z, is the partition function 2, = Ce- Wn"/kh
(7)
n
For small temperature differences, AT between the gas and the solid
e- I ~ W n " - W n " ~ / k l ~ ~ ~ / T ~ ~ - ~ ~ / h ~ l ---f
W n v - W," AT . . . (13) k TJ'z Then since AEequil = NokAT, it follows from eq 3, 12, and 13 that in the limit as T1-+ T2-P T
+
1-
n>m
111. Comparison of the Anharmonic and Harmonic Cases First it will be shown that eq 14 reduces to the result of Jackson in the harmonic case. In the harmonic approximation, in which the surface atoms behave as simple harmonic oscillators of frequency v
wny= (n + 2, =
Ce-(n+1/2)hv/kT
= ,-hv/2kT
n
It is convenient to write AI3 as
an = (1 -
(15)
l/&v
(1 -
,-hv/kT
1-
1
,-hv/kT),-nhv/kT
(16) (17)
Thus in the harmonic case n>m
[anPnm(e)
- am~mn(e)l
(8)
The transition probabilities must obey (principle of detailed balancing) the equation ~m,(e)
= Pnnz(E
- W,'
+
(9)
wm")
Further, the assumption will be made that pmn(E) = 0
(E
< W,'
- WmV)
(10)
which implies that trapping of the gas atom is neglected. Next, using eq 9 and 10, it is seen that
I = c d e e-f/kTzp mn (e ) =
Jrn
de e - e / k T z
Wn" - Wm"
= e - (Wnv-
W m ~ ) / k Tc z
Thus eq 8 can be written
Plant(€
-
Wn'
+
wm")
- ,)2phV/kT
PnmH(e)
(18)
n,m
n>m
which is the result of J a c l t s ~ n . ~ ~ ~ The superscript H on the transition probabilities indicates that they are calculated with the use of harmonic-oscillator wave functions. (Transition probabilities of this type have been obtained by Jackson and Motta for a repulsive exponential potential interaction and by Devonshird for a Morse potential interaction between the gas and the solid.) If, in addition it is assumed that only single quantum transitions are imH portant, then n - m = 1 ande pn,n-1 (E) = np1oH(e). The sum over n is then easily carried out, to yield
d e e-f/kT2p,m(e) When a l H ( v )is integrated over a Debye frequency distribution, the equation is obtained upon which Shin8 bases his discussion of anharmonic effects. However, it is to be emphasized that eq 19 and also the procedure of integrating over the normal-mode frequency distribution of the crystal lattice are correct only within the limits of the harmonic approximation, and hence the analysis of Shin is not valid. With regard to multiple-quantum transition in eq 18, one can expect these to become increasingly important (9) This expression for aH(v) differs from that of Jackson by a factor of a/a; the need for the correction has been indicated by Devonshire.6
The Journal of Physical Chemistry
VARIATIONAL PRINCIPLE FOR
THE
3575
POISSON-BOLTZMANN EQUATION
as the temperature rises. Shin* has discussed twoquantum processes briefly and concludes, incorrectly, that such processes become important only at low temperatures. I n his argument he states that the n +n - 2 transition probability, like the n + n - 1 transition probability, is proportional to p l O ( e ) , which is not the case.3 Two-quantum processes and the accommodation coefficient associated with them have been discussed by Jackson and Mott3 and by Allen and Feuer .la Next, let us examine eq 14 for the more general anharmonic case. Transition probabilities for the collision of an atom with an anharmonic (Morse) oscillator have been obtained by i\Iiesl1 in connection with work on the collision of atoms with diatomic molecules. lllies writes pnm(e) in the form
a, (eq 6 and 7 ) is more complex than for the harmonic
case, as is W,". form
For a Morse oscillator, W," has the
+ '/d
Wn" = A [ ( n
- B(n + ' / z ) ~ ]
(22)
where A and B are constants which depend on the mass of the oscillator and the two parameters associated with the Morse potential. Thus eq 21 (together with eq 6 and 7 ) represents a general expression for the accommodation coefficient of a beam of gas atoms colliding with a solid surface for which anharmonic effects are taken into account. This Pnm(4 = Pnm(4Anm(% (20) equation, rather than eq 19, should be used to obtain numerical results for the anharmonic case and to obtain and discusses it in detail for a repulsive exponential information regarding the conditions under which anpotential interaction between the atom and the oscillator. The function Pnm(e)is proportional to the square harmonic effects would become important. Previous considerations' of the effect of surface impurities on of a nondiagonal matrix element V,, associated with the accommodation coefficient suggest that this model the perturbing interaction potential. Anm(e,X) is an anharmonicity function which i\Sies has expressed in could be particularly applicable to the collision of gas atoms with light surface impurities on heavy lattices. terms of hypergeometric series and X2 ( = Vnn/Vmm), the ratio of diagonal matrix elements associated with the perturbing potential. (In the harmonic case, all (10) R. T. Allen and P. Feuer, Advan. AppZ. Mech. Suppl., 1 , (4), A n m ( E , X ) = 1 and Pnm(e) = pnmH(e).) Thus eq 14 109 (1967). becomes (11) F. H. Mies, J . Chem. Phys., 40, 523 (1964).
On the Variational Principle for the Poisson-Boltzmann Equation by A. D. MacGillivray and J. D. Swift Department of Mathematics, State University of New York at Buffalo, Buffalo, New York
(Received May 16, 1968)
Variational principles for two forms of the Poisson-Boltzmann equation are considered, particular attention being paid to boundary conditions. It is shown that the functional used in a paper by Dresner is not generally applicable, although it is used correctly for the example he treats. The problem of two charged flat plates separated by an electrolytic solution is considered, and it is shown that an approximate solution obtained with the calculus of variations agrees well with the exact solution of Verwey and Overbeek.
I. Introduction Although the behavior of electrolytic and polyelectrolytic solutions has recently been studied using 'luster theory,' Of the Boltzmann equation continues to provide a reasonable qualitative and to some extent quantitative basis for the understanding of the diverse phenomena peculiar to polyionic solutions.
I n the past few years, approximate solutions of the Poisson-Boltzmann equation have been obtained analYticab'j3~4 and by direct methods of (1) G. 5. Manning and B,H. Zimm, J. Chem. Phys., 43,4250 (1965). (2) L. Kotin and M. Nagasawa, ibid., 36, 873 (1962). (3) Z. Alexandrowicz and A. Katohalsky, J. Polymer Sci., A I , 3231 (1963). (4) A. D. MacGillivray and J. J. Winkleman, Jr., J. Chem. Phys., 45, 2184 (1966). Volume 78. Number 10 October 1968
