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Polar model of order-disorder
Ralph J. Harrison, Arthur Paskin
To cite this version:
Ralph J. Harrison, Arthur Paskin. Polar model of order-disorder. J. Phys. Radium, 1962, 23 (10), pp.613-615. �10.1051/jphysrad:019620023010061300�. �jpa-00236648�
613.
POLAR MODEL OF ORDER-DISORDER
By RALPH J. HARRISON and ARTHUR PASKIN,
Materials Research Laboratory, U. S. Army Materials Research Agency Watertown, Massachusetts.
Résumé. 2014 Paskin a montré que l’introduction des concepts les plus récents de l’effet d’écran dans les métaux applique aux alliages de type AB conduit à un modèle dipolaire semblable à celui de Mott. Les atomes A et B portent des charges d’écran opposées dues surtout à leur différence de valence.
L’énergie d’ordre évaluée à partir des paramètres d’écran théoriques pour le laiton 03B2 est en
accord raisonnable avec l’expérience, ce qui justifie la validite de ce modèle. On passera en revue les hypothèses de base et les conséquences de ce modèle en ce qui concerne les énergies d’ordre et
on décrira une extension de ce modèle au calcul de la contribution électronique au spectre des
vibrations élastiques de l’alliage.
Abstract. 2014 It has been pointed out [2] that the introduction of the more recent concepts of
electronic screening in metals applied to the AB alloys leads to a polar model resembling that
of Mott [1]. The A and B atoms have opposite charges mainly related to their differences in valence. The ordering energy estimated from the theoretical screening parameters for 03B2 brass
is in reasonable agreement with experiment, lending support to the physical validity of this model.
The basic assumptions and consequences of the model with respect to the ordering energy will be reviewed and in addition an extension of this model to the computation of the electronic contri- bution to the elastic spectrum of the alloy will be described.
LE JOURNAL DE PHYSIQUE ET LE RADIUM TOME 23, OCTOBRE 1962,
An important step towards the understanding of
lhe origin of the ordering energy in binary alloys
was taken in 1937 by N. F. Mott [1] with the deve-
lopment of the " polar model ". Mott discussed this model with particular référence to the ordered beta brass (Cu Zn) alloy. The polar model may briefly be described as follows :
As a first approximation one considers the two
types of atoms as point charges in a " sea " of conduction electrons. The magnitudes of these point charges are taken to be the respective nuclear charges diminished by the average of the electronic
charge contained within the atomic polyhedra sur- rounding the respective ions. In this first approxi- mation the conduction electrons are considered to be distributed uniformly throughout the alloy, i.e.,
for Cu Zn there are assumed to be one and a half
conduction electrons per polyhedron. The point charges in this first approximation are then, (1/2)e
at the Zn site and - (1/2)e at the Cu site. If one
computes the elertrostatie energy of this ordered
point charge configuration relative to the disor-
dered case, one finds it to be an order of magnitude greater than the observed ordering energy. The
major cause of this large discrepancy is that the
conduction electrons, far from being uniform,
screen the ion charges quite strongly. Mott com- puted the magnitude of this effect by the use of a
linearized form of the Thomas-Fermi method. The conduction electron distribution thus obtained re-
presents the change in électron distribution from the case where all sites are occupied by fictitious
atoms having the average nuclear charge. The
Mott-Thomas-Fermi screening model led to the
result that the net charge within an atomic poly-
hedron is about 1/6 of the point ion charge of + (1/2)e. In its final approximation then, the polar model represents the ordering energy‘ by the
electrostatic energy of a point charge distribution of -f- .075 e located at the Zn sites and - .075 e
cloated at the Cu sites. After correcting this elec-
trostatic energy for the estimated contribution of terms associated with the ion-core repulsion, Mott
found that the result was in fairly good agreement
with experimental determinations of the ordering
energy (about 0.04: eV) in the beta brass (Cu Zn) alloy.
In recent years there have been further deve-
lopments in the theory of electronic çcreening in
metals that have led us to reexamine the problem
of the electronic structure of ordered alloys [2].
We have attempted to keep within the framework of the original Mott polar model. The screening
model we have used is one [3] that leads to an
asymptotic expression for the potential at large
distances from a screened point charge Ze of the
form
Kohn and Vosko [3] and Blandin and,Friedel [4]
have regarded the constants A and cp as empirical parameters to be determined with the aid of data on
resistivity of dilute alloys. The validity of the asymptotic form of the potential given by Eq. (1)
is somewhat more general than the free electron
model, since it includes the case of Bloch electrons
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010061300
614
in a periodic potential under restrictions of sphe-
rical energy surfaces and isotropic scattering.
Langer and Vosko [5] derived a non-empirical
form for the free electron gas potential and charge density valid for closer distances. We have shown [6] that to within the nearest neighbor dis-
tance this charge density is closely approximated by
where the first term is the usual Thomas-Fermi term in the linearized form given by Mott, and in
the second term the constants A and p are given
definite values.
In Mott’s original calculation on the polar model
the value of the shielding parameter q (= 2kp oc)
was taken as a semi-empirical parameter adjusted
to the resistivity data, whereas in current semi-
empirical screening models it is the second term in
E . (2) which is adjusted.
e have estimated the degree of polarity by tintez grating Eq. (2) from the atomic radius, ra, to infi-
nity, finding the total charge outside ra and thence the charge inside ra screening the point charge at
the center. To estimate the amount of electronic
charge inside ra arising from screening of charges
outside ra, we take as first approximation the assumption that all the screening charges outside a
distance ra from a point charge divide themselves equally among nearest neighbor cells.
The result of the integration of Eq. (2) is to give a screening charge in beta Cu Zn equal to
0.24 Z, where Zi is the assumed ionic charge (= + 1/2e).
The Madelung energy, EM, computed as if these charges were located at the lattice points is equal
to 0.07 electron volts. The estimate of energy by
the Madelung method disregards any information
provide by the screening model regarding the dis-
distribution of charge within the unit cell. One
may if oné wishes, utilize this information by taking as the interaction energy of pairs of ions
where V is the potential at n of the screened charge
at m. This expression seemingly neglects the elec-
trostatic interaction of the electronic screening charge at m with the screening charge at n. Howe-
ver it is the correct expression, and results from consideration of electrostatic and kinetic energy terms as may be shown in several ways [7]. The
total interaction energy is then given by
where
The phase factor y has been introduced in view of the fact that Kohn and Vosko have found that
although the coefficient of the cosine term is well
apuroximately by the Langer and Vosko expres- sion (5), the phase factor is not.
Carrying out the summations for first neighbors only in the ordered Cu Zn lattice of parameters a :
Adding on the next neighteen neighbor shells gives
Thus the estimate of energy becomes
It may be pointed out that most of this comes
from the first shell alone, although there are fluc-
tuations in value even quite far out. The experi-
mental indications are that the ordering energy is about 0.04 electron volts per atom.
We have also been attempting to evaluate the contribution of the conduction electron part of the
interactions in beta brass to the vibration spec- trum. Our oscillation bas generally followed the procedure of Toya [8]. Values of the elastic cons- tants for the acoustic branch of the spectrum indi-
cate that the ion-core repulsions play a major role.
With regard to the optical branch, one similarly expects the core repulsion exchange forces to play
a major role. However there is some preliminary expérimental evidence [9] from neutron scattering
the frequency of the longitudinal optical mode is
such as to indicate tbat forces other than the nea- rest neighbor ion-core exchange must be involved.
The observed frequency for Cu Zn is 1.3 to 1.4 X 1013 sec-1. The frequency to be expected
in the case that short range ion-core forces are dominant can be shown to be
for beta brass. This figure would be slightly redu-
ced were a correction made for a possible electronic
contribution to C44. Another estimate, using the Born-Mayer constant for copper is 5 . 5 X 1012 sec-le An estimate of the electronic contribution alone on
the basis of the free électron screening model
would give less than 2 X 1012 sec-1. Since the squares of the contributions are additive, the elec-
tronic contribution is only a small fraction.
One may further estimate possible long-range
contributions to the optical frequencies for example by computing the frequencies corresponding to a completely ionic model of the lattice still arise.
One may use the sum rule [11] discussed by Brout
as a generalization of the Szigeti relation for lattice frequencies According to which the sum of squares
615
of longitudinal and transverse frequencies is related
to the compressibility modulus. Using this, and
even assuming zero frequency for transverse optical
modes in the long wave length limit the experi-
mental value for the frequency of the longitudinal optical mode is still somewhat too large. The experimental value is incorrect, or assumptions
involved in Brout’s sum rule must be violated.
One such assumption is that of nearest neighbor
forces alone for the ion core repulsion. Another is
the adiabatic assumption. In any case, whatever the mechanism leading to the observed high optical frequency, the polar character of the lattice in the
sense of Mott’s model is not in itself an explanation ;
insofar as long range electrostatic forces continue
to be significant, the unshielded point charges
seem more relevant than the uncompensated charges within each atomic polyhedron.
REFERENCES
[1] MOTT (N. F.), Proc. Phys. Soc., London, 1937, 49, 258.
[2] PASKIN (A.), Bull. Amer. Phys. Soc., Ser. II, 1962, 7,
216.
[3] KOHN (W.) et VOSKO (S. H.), Phys. Rev., 1960, 119,
912.
[4] BLANDIN (A.) and FRIEDEL (J.), J. Phys. Chem. Solids, 1960,17,170.
[5] LANGER (J. S.) and VOSKO (S. H.), J. Phys. Chem.
Solids, 1959, 12, 196.
[6] HARRISON (R. J.) and PASKIN (A.), J. Phys. Soc., Japan,1960,15,1902.
[7] e. g., SILVERMAN (B. D.) and WEISS (P. R.), Phys.
Rev., 1959, 114, 989. More directly it follows from
considerations paralleting those involved in the
Hellman-Feynman theorem.
[8] TOYA (T.), J. Res. Inst. for Catalysis, Hokkaido Uni-
versity, 1968; 6,161, 183 ; Progr. Theor. Phys., 1958, 20, 973, 94 ; also WOLL (E. J., Jr.) and KOHN (W.), Phys. Rev., 1962,126,1963.
[9] ANTAL (J.), Private communication experimental re-
sults quoted are only tentative at the present time.
[10] However, it has been pointed out to us by Friedel
that corrections for the Bloch character of the con-
duction electrons can be quite large.
[11] BROUT (R.), Phys. Rev., 1959, 113, 43.
