Binding Energy of Alkali-Halide Compounds and Some Solid Solutions Based on Them

V. N. Belomestnykh, Yu. N. Sukhushin

(Presented by the scientific seminar of the Department of Radiation Chemistry)

Establishing the relationship between the energetics of solids and their elastic properties may be regarded as a method for investigating the binding energy of chemical compounds and solid solutions as a function of composition and external conditions.

The relation between the lattice binding energy and the speed of sound was first considered by Fitzgerald in connection with the particle-wave description (waves were compared with moving particles) of the deformation of solids, and, in part, was used in calculating the rate of their fracture [1, 2]. The binding energy of a monatomic crystal lattice is determined as

\[ D = m C_s^{2}, \tag{I} \]

where \(D\) is the lattice binding energy per atom, \(m\) is the atomic mass, and \(C_s\) is the isotropic speed of sound, which can be obtained by summing three orthogonal sets of velocities: one longitudinal and two transverse.

Good agreement with equation (I) was shown also for 18 metallic crystals with binding energies from 1.13 eV to 10.3 eV.

The indicated relation (I) implies that a certain sum of the speeds of sound in a crystal is invariant; this was proved for the case when small displacements of atoms in crystals can be described by the linear theory of elasticity, and the wavelength is greater than interatomic distances [3]. Thus,

(1) was brought into correspondence with the classical theory of the propagation of sound waves in a solid.

Since relation (1) has been verified for a limited class of substances, it is of definite theoretical and practical interest to extend it to other types of compounds. We have carried out an experimental test of the correspondence (1) for alkali-halide compounds and have shown that it is satisfied for salts containing heavy ions; for salts consisting of light ions, good agreement is achieved by introducing a correction coefficient that is constant for a certain group of salts. An attempt is also made to represent, in qualitative form, the binding energy of solid solutions.

THEORETICAL PART

Let us consider the possibility of extending Fitegerald’s method to more complex structures.

The vibrational energy of crystals containing \(N\) atoms is often expressed [4, 5] as the energy of a system of \(3N\) harmonic oscillators, for the case of linear elastic interactions between atoms. According to Einstein [6], a harmonic oscillator must be regarded as quantized, and the possible energy of the oscillator may be specified as

\[ U_q = \left(q + \frac{1}{2}\right) h\nu_e, \]

\[ q = 0, 1, 2, 3, \ldots \tag{2} \]

where \(\nu_e\) is the frequency of oscillation of the harmonic oscillator of the system.

Adopting the Boltzmann distribution over the possible energy levels, the mean energy per oscillator at a certain temperature \(T\) may be written as

\[ \langle U_q \rangle = \frac{h\nu_e}{2} + \frac{h\nu_e}{e^{h\nu_e/kT} - 1}. \tag{3} \]

where \(K\) is Boltzmann’s constant.

The first term corresponds to the mean zero-point energy of the oscillator (atom), since this term is retained at \(T=0\). This is, naturally, consistent with the concept of quantum mechanics that an atom in a crystal lattice has vibrational energy even at absolute zero.

To each elastic or vibrational frequency level there can be assigned a certain number of phonons [4]. If the energy of such a phonon is defined as \(h\nu_e\), then the number of phonons \(n_e\) that can be associated with a certain vibrational frequency \(\nu_e\) will be

\[ n_e=\frac{\langle U_q\rangle}{h\nu_e}=\frac{1}{2}+\frac{1}{e^{h\nu_e/kT}-1}, \tag{4} \]

The result expressed in (4) suggests an analogy with a quantum of light, or a photon, the usual quantum aspects of which have already been described. From expression (4) it is clear that the lattice contains two types of phonons: intrinsic (non-thermal) and thermal. Intrinsic phonons exist even when \(T=0\) and, consequently, are always present in a constant amount (1/2 phonon per level). Since each atom of the lattice corresponds to three vibrational states associated with it, each atom therefore contains \(3/2\) intrinsic phonons. This is shown schematically in Fig. 1, where two atoms of the lattice are characterized by one longitudinal and two transverse phonons with sound-propagation velocities \(V_e\), \(V_{ta}\), \(V_{tb}\), respectively, because three intrinsic phonons are associated with each two atoms of the lattice [2]. This diagram shows how one longitudinal phonon (velocity \(V_e\)) and two transverse phonons (velocities \(V_{ta}\) and \(V_{tb}\)) can be associated with a diatomic combination in the crystal lattice. Each atom in the combination contributes the mass \(m/2\) of the isolated vibrational level, so that the total phonon mass of such a diatomic system will be \(m\). The advantage of considering three intrinsic phonons associated with each—

... between two atoms of the lattice in that, in this case, one may assert a distinction between longitudinal and transverse phonons (vibrations). The \(3/2\) phonon of one atom, on the other hand, cannot unambiguously determine longitudinal or transverse levels.

Three proper phonons of the type shown in Fig. 1 are sufficient for describing an isotropic material, but, naturally, are insufficient for an arbitrarily anisotropic solid. For the latter, a system of three mutually orthogonal triples of phonons is necessary, as shown in Fig. 2. Thus, in general, a linearly elastic medium, or small displacements of atoms from the equilibrium position in a crystal lattice, can be characterized by a system of nine proper phonons, containing three longitudinal and six transverse phonons. The corresponding velocities of sound propagation will be:

\[ v_{\ell 1}, v_{t1a}, v_{t1b};\quad v_{\ell 2}, v_{t2a}, v_{t2b};\quad v_{\ell 3}, v_{t3a}, v_{t3b}. \]

The nine proper phonons, incidentally, correspond to eighteen vibrational states in the atomic lattice, since each atom has three lattice vibrational states, and each state is equivalent to \(1/2\) of a proper phonon. For an arbitrary direction in the lattice, the vibrational states (and the phonons associated with them) will not be purely longitudinal and purely transverse; however, leaving the preceding arguments in force, in many cases it is necessary to regard them as effectively quasi-longitudinal and quasi-transverse.

Of course, there will be many directions in which purely longitudinal and purely transverse states will be present.

From consideration of the proper phonons in a crystal lattice it is possible to obtain a mass—energy equivalence for the atoms of the lattice. One can arrive at this assertion from the considerations that, first, only three proper phonons are necessary for characterizing a material, and, second,

if only two atoms are separated sufficiently from the lattice in some manner, then three phonons will be disturbed. The value of the mean square velocity of the three phonons may be written as

\[ \overline{C_s^2}=\frac{v_l^2+v_{ta}^2+v_{tb}^2}{3} \tag{5} \]

In some cases the annihilation of three phonons with a velocity value \(\overline{C_s^2}\) may lead to the release of vibrational energy per atom \(\frac{m}{2}\overline{C_s^2}\), or \(m C_s^2\), where \(C_s^2=\overline{C_s^2}/2\). Now, if the lattice is destroyed when thermal energy equal to \(D\) per atom is imparted, the final result will be the destruction of all intrinsic lattice phonons, and the separated atoms have an energy \(D\) per atom greater than their initial energy at \(0^\circ\mathrm{K}\) in the lattice. Consequently, it seems plausible to conclude that the destruction of the intrinsic (zero-point) lattice phonons will be the result of realizing the vibrational binding energy per atom \(D\).

If the ordinary binding energy per atom \(D\) of a crystalline lattice is a substance constant, then relation (1) implies that \(C_s^2\) is also a constant. Thus it follows that the sum of certain three mutually orthogonal components (two transverse and one longitudinal) of the squares of the sound velocities (and, in the general case, nine squares of the sound velocities) may be a constant in an elastic medium. In the case of a cubic crystal or another solid where only three intrinsic phonons are required, calculation of the value of \(C_s^2\) accordingly leads to the result that the sum \((v_l^2+v_{ta}^2+v_{tb}^2)\) may be invariant for any directions in the crystal. The invariant of the sum of the sound velocities can be expressed through the elastic constants:

for cubic crystals,

\[ v_l^2+v_{ta}^2+v_{tb}^2=\frac{c_{11}+2c_{44}}{\rho}, \tag{6} \]

for crystals of hexagonal symmetry,

\[ \sum_{i=1}^{2}\left(v_{li}^2+v_{tia}^2+v_{tib}^2\right) = \frac{ c_{33}+2c_{44}+2\left(\frac{3c_{11}+2c_{44}-c_{66}}{2}\right) }{\rho} \tag{7} \]

where \(\rho\) is the density of the crystal.

An anomalous behavior was observed for Zn and Cd. The lattice structure of these crystals is strongly anisotropic (the \(c/a\) ratio is 1.855 and 1.866, respectively).

The above considerations can also be extended to ionic lattices consisting of diatomic molecules, if the heterodyning in the structure is not very strongly expressed. Indeed, the destruction of \(3/2\) of the intrinsic phonons can be accomplished not only by removing one of the atoms of the lattice, but also by removing the entire molecule. Since the shortest distances between ions in alkali-halide crystals have the property of additivity, then, to a first approximation, the ions of these crystals may be represented as hard spheres, and the removal of a molecule may be considered without changing the nature of the bond. The diagram of vibrational states in this case shows its equivalence to the diagram in Fig. 2. Only in this case the Fitzgerald relation will be:

\[ D = M c_i^{2}, \tag{8} \]

where \(M\) is already the mass of the molecule.

Relation (8) should be well obeyed for crystals with nonionic bonding and a lattice that is not strongly anisotropic. For crystals with a strongly pronounced ionic bond, apparently, a correction coefficient \(n\) should be introduced, characterizing the degree of ionicity of the bond,

\[ D = n M c_i^{2}. \tag{9} \]

The isotropic speeds of sound can be calculated from the elastic constants of single crystals according to one of the theoretical schemes [7], or determined experimentally, as was done by us earlier [8].

EXPERIMENTAL RESULTS AND THEIR DISCUSSION

The elastic constants of single crystals and the isotropic sound velocities were determined or calculated by the method described in [8]; for cesium compounds the data on elastic constants were taken from [9], and for fluorides from [10]. For single crystals the data are given in the table, and for solid solutions in Fig. 3. From the values presented, the binding energy \(D_{\rho 1}\) was calculated by formula (8) through the elastic constants \(C_{11}\) and \(C_{44}\). Comparison with the tabulated values \(D_T\) [11] (for cesium compounds [12]), determined by spectroscopic methods, gives a deviation \(\Delta,\%\), minimal for alkali-metal iodides and maximal for fluorides.

The agreement between the calculated value of the binding energy \(D_{\rho 2}\), obtained from the isotropic sound velocities by the expression

\[ D_{\rho 2} = M \frac{v_l^2 + 2v_t^2}{6} \tag{10} \]

has the same regularity, but the agreement is somewhat worse.

We have drawn attention to the fact that the calculation results are grouped with good accuracy for substances with one and the same anion if the calculation is carried out according to (9) and one takes: for iodides \(n = 1\); for bromides \(n = 1.2\); for chlorides \(n = 1.3\) and fluorides \(n = 1.36\). In this case the maximum deviation of the calculated value of the binding energy from the tabulated one for a group with the same halide also falls on the lightest alkali metal—lithium. One may point to the change of a number of characteristics in going from compounds of light ions to compounds consisting of heavier ions: a decrease in the concentration of ions per unit volume, a decrease in the degree of ionicity of the bond, an increase of the \(n^*\) exponent in the expression determining the potential of the repulsive forces, etc. Thus, this change may be justified. Calculation according to (9) from the characteristics of an isotropic medium (\(D_{\rho 6}\) and \(\Delta_4\)) also gives somewhat worse...

Fig. 1. Scheme of vibrational states for an isotropic solid.

Fig. 2. Scheme of vibrational states for a cubic lattice.

Table No. 1. Binding energy of alkali-halide crystals.

Fig. 3. Elastic constants of single crystals of solid solutions KCl–KBr and KBr–KI.

Fig. 4. Binding energy of solid solutions KCl–KBr and KBr–KI.

correspondence than in the case of an anisotropic medium \((D_{\rho s}\) and \(\Delta_s)\).

Solid solutions of alkali-halide compounds belong to substitutional solid solutions. In the solid solutions of the systems \(KCl - KBr\) and \(KBr - KJ\) studied by us earlier [8], chlorine was replaced by bromine and bromine by iodine, respectively. If one assumes that the coefficient \(n\) introduced by us changes additively when one halide is replaced by another, then the binding energy will have the form shown in Fig. 4. Depending on composition, the binding energy of the solid solutions has a nonadditive character, with the deviation from the additive value being toward a decrease. The maximum deviation of the binding energy from the additive value in both systems is \(3\text{–}4\%\) and occurs approximately at equimolar compositions. This is confirmed by certain properties of such systems. Thus, for example, in [13] it was noted that the system \(50\%\,KBr - 50\%\,KJ\) has the weakest bonding between the structural elements of the lattice.

Conclusions

1. Consideration of the physical model in crystals of alkali-halide compounds in accordance with the Fitzgerald scheme indicates the possibility of using the mass–energy equivalence relation to calculate the binding energy in these compounds.

2. Comparison of the binding energy determined from this relation with spectroscopic measurements confirms it, if a correction factor is introduced that is constant for a group of crystals with the same anion.

3. To calculate the binding energy with allowance for this same coefficient, one may use isotropic values of the sound velocities (longitudinal and transverse); however, the agreement with tabulated values of \(D\) is somewhat worse.

1. For the first time, the elastic properties of single crystals of the solid solution \(KBr - KJ\) have been investigated.

2. Under the assumption of additivity of the correction coefficient, the binding energy in the solid solutions \(KCl - KBr\) and \(KBr - KJ\) has been determined. As a function of composition, the binding energy is not additive, and the maximum deviation from the additive value occurs at equimolar compositions.

REFERENCES

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4. A. J. Dekker, Solid State Physics, Ch. 2, Prentice Hall, N-Y, 1957.

5. M. Blackman, Handbuch der Phys., Vol. VII/1, Spr.-Verl., Berlin, 1956.

6. A. Einstein, Ann. Phys., 22, 180 (I406); 22, 800 (I406); 34, 70 (I4II).

7. H. Wawra, Technik, 23, 2, 93 (1968).

8. V. N. Belomestnykh, E. N. S. Sukhushin. This collection.

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12. Chemist’s Handbook, Vol. I, M.–L., GKhI, 1963.

13. A. A. Vorob’ev, K. A. Vodop’yanov, M. A. Krivov, DAN SSSR, 966, 6, 1135, 1954.