I. Introduction to Ionic Crystals: Background Information
I.1. Fundamental Concepts
According to underlying atomic forces, crystalline solids are roughly classified into five classes: molecular, ionic, covalent, metallic, and hydrogen bonded [9]. These classes tend to blend into each other, but still represent conceptual ideal types. The following discussion will be based soley on the ionic crystal.
Ionic crystals are a class of crystals in which the lattice-site occupants are charged ions,. They are perhaps the simplest type of chemical bonding to visualize since it is almost totally electrostatic in nature [7]. The criterion for such bonding is the difference in electronegativity, and as such occurs between electropositive elements and electronegative elements (generally electropositive elements from Group I or II and electronegative elements from Group VI or VII) [10].
The electronic configurations of all ions of a simple ionic crystal correspond to closed electronic shells, as in the inert gas atoms [7]. For example, consider the lithium fluoride structure in which the configuration of the neutral atoms are: Li: 1s22s, F: 1s22s22p5. Whereas the singly charged ions have the configurations of helium and neon, respectively [7]: Li+: 1s2, F-: 1s22s22p6.
With such a correlation between ionic configuration and the inert gas atoms, we can expect that the charge distributions on each ion in an ionic crystal will have approximately spherical symmetry, with some distortion near the region of contact with neighboring atoms, such as that of the inert gases [7].
To illustrate the distinguishing characteristics of ionic crystals, consider the ideal ionic crystal modeled as a collection of spherical, charged billiard balls. This is most nearly realized by the alkali halides, within which we find NaCl [1]. In the Born-Madelung theory of ionic crystals it is assumed that when neutral atoms of Na and Cl are brought close together, the valence electron of the electropositive sodium atom is easily transferred to the electronegative chlorine atom, as to aqcuire a stable inert gas electron configuration. This formation is schematically represented in Fig. 1 [14].
Figure 1. Schematic representation the formation of an ionic molecule of NaCl.
Such configurations will have nearly symmetric charge distributions around the ions, with some distortion near the region of contact between neighboring atoms, as previously mentioned. This has been confirmed by X-ray diffraction data of the [100] plane of NaCl (see Fig. 2) [1].
Figure 2. Electron density distribution in the [100] plane of NaCl.
Further confirmation of the idea that the alkali halides are composed of slightly distorted localized ions comes from band structure calculations [1]. (see Energy bands in solids and their calculations for more information on crystal structure)
Figure 3. The four highest filled energy bandsof KCl. d is the interionic distance, where the vertical line is at the observed d. Energies of the free ions are indicated as arrows on the right side of the figure.
Fig. 3 [1], shows the energy bands calculated for the alkali halide, KCl, as a function of the lattice constant, compared with the corresponding levels of the free ions. These energies differ by as much as 6.8029 eV from the energies of the isolated ions (even at large separations), due to the interionic Coulomb interactions [1]. It can be seen, however, that the band widths at the observed lattice constant are very narrow, indicating that there is little overlap of the ionic charge distributions [1]. (some information on electron distribution in the alkali halides has also been deduced from nuclear spin resonance measurments [8])
Some typical crystal structures of simple ionic solids are composed of a number of interpenetrating simple cubic or face-centered cubic (FCC) Bravais lattices. Typical crystals with such symmetry are NaCl, CsCl, zincblende, CaF2 Cu2O , etc. A convenient description of an ionic structure is by means of its unit cell and its reciprocal space translation vectors, which is used in the mathematical theories of crystal potential and will be used later in the text. (see Real and reciprocal crystal lattices for more information on crystal structure)
II. Lattice Energy
II.1. Definition of Lattice Energy
A simple definition of the cohesive energy of a crystal is given as the energy that must be added to the crystal to separate its components into neutral free atoms at rest, at infinite separation, with the same electronic configuration [7]. The term lattice energy is often used in the discussion of ionic crystals. This energy is similar to the general definition, with the exception that the component ions are separated into free ions (rather than neutral free atoms) at rest, at infinite separation [7]. It should be noted that the cohesive energy of the crystal is not the same as the bond energy of a molecule (i.e. ionic bonding). This is because in the crystal, each ion interacts with the other ions present, and long-range interactions must be considered.
In the simplest theory of cohesion the assumption is made that the cohesive energy of the crystal is entirely governed by the potential energy of classical particles localized at the equilibrium positions [1]. Because the particles in ionic crystals are electrically charged ions, the largest term in the interaction energy is in fact the interionic Coulomb forces , and can be taken as the sole source of binding in rough calculations [1]. However, the stability of the ionic crystal depends on the balancing of several forces [6]. In addition to the aforementioned electrostatic, or Coulomb forces between the ions, these include: repulsive interaction, arising as a consequence of the Heisenberg uncertainty and Pauli exclusion principles, and very minor contributing effects such as the van der Waals energy, and zero-point energy. The latter terms provide such little contribution to the lattice energy and will only be discussed briefly. For more in-depth information on these terms in detail, refer to Waddington [13].
II.2. Electrostatic Energy and a Simple Example
As stated above, the electrostatic energy of an ionic lattice is the main contributor to the overall observed lattice energy. To validate this point, consider for example, an ideal ionic crystal: NaCl (see figure to the below, with numbers indicating ion separation distances).
This crystal consists of positive and negative ions which can be thought of as rigid spheres that attract electrically until they begin to touch, at which point a repulsive force goes up very rapidly. In this cubic lattice, the ions have been determined by x-ray diffraction to have a spacing of about 2.82 angstroms [1]. The total energy required to separate NaCl ions (cohesive energy) has been experimentally determined to be 7.92 eV per molecular unit [3]. An experimental value for the cohesive energy can be obtained by means of a thermo chemical cycle, originally proposed by Born and Haber for an ionic solid at absolute zero and at atmospheric pressure [12].
It is easy enough to figure out the electrostatic contribution to the experimentally determined cohesive energy for NaCl using e2/r0, where $e^2 = {q_{e}^2 /4 \pi \epsilon_{0}$. This is the energy of an ion with one of its nearest neighbors, where r0 is the center-to-center spacing between the considered ions [3].
By simply plugging in and solving this equation, a value of 5.12 eV is given; this provides the correct order of magnitude, but is incorrect because an infinite sum of terms due to ions throughout the crystal (long-range) have not yet been considered. In taking these long-range interactions into account, rather than just the interactions between two ions, a value much closer to the observed value will be obtained. To begin, consider the cross-section of the cubic NaCl lattice given in the figure below. We can start by looking to a reference ion, such as the one labeled Na in the cross-section, and summing along the horizontal line containing the ion.
From the figure, we see that there are two nearest Cl neighbors a distance r0, two positive ions at 2r0, etc. If we sum these terms, we get an energy corresponding to this line, Wh, of
$W_{h} = {e^2 \over r_{0}}(-{2 \over 1} + {2 \over 2} - {2 \over 3} + ...) = -{2e^2 \over r_{0}}(1 - {1 \over 2} + {1 \over 3} + ...)$
The value inside the brackets is a familiar infinite series known to sum to ln2. This yields:
$W_{h} = -1.386{e^2 \over r_{0}}$
In addition to the calculated horizontal line, we next need to consider other adjacent lines, such as that above, below, in front, in back, and four diagonals. If all these lines are worked through and summed, we arrive at [3]
$W = -1.747{e^2 \over r_{0}}$
Upon substitution for the constants, we obtain a value of $W = 8.94 eV$. This answer is closer to the experimentally observed value for the energy, but is about 10% too high, since repulsive forces, the kinetic energy of crystal vibrations, etc., have not been taken into account. If corrections for these effects are considered, a value in close agreement with the experimental value can be obtained [3]. This does, however, provide insight that the major contribution to the energy of a crystal such as NaCl is, in fact, electrostatic (about a 90% contributor).
II.3. Evaluation of the Electrostatic Energy
As shown in the above example, in typical ionic crystals the electrostatic energy is in fact quite close to the observed cohesive energy in alkali halides [12]. Often times, this energy is written as
$W_{coul} = \alpha {e^2 \over r_{0}}$
where r0 is a characteristic length of the crystal structure, such as the cubic root of the volume of the unit cell, or the nearest-neighbor distance [12]. $\alpha$, which depends only on the crystal structure, is known as the Madelung constant, named so because it was first calculated by Erwin Madelung, a German physicist [12]. This constant is of central importance of ionic crystal theory, and values for common ionic crystals can be found in Table 1 (For more crystal types, see Wyckoff[15]). Often, the electrostatic interaction, $W_{coul}$, also bears this physicists’ name, and is referred to as the Madelung energy.
Table. 1. Madelung constants for some typical ionic structures. The first row of values refer to the nearest-neighbor distance, while the second row refers to the lattice parameter. Both values are correct, but reference to the nearest-neighbor distance is most commonly used.
The NaCl example presented in section II.2 shows a Madelung constant of 1.747, and a characteristic length, a=2.82 angstroms, based on the nearest-neighbor distance. This example showed the nature of the Madelung constant (a large summation of interactions over the entire crystal) and provided a simple look at the electrostatic interactions in an ionic crystal. However, the calculation of these Madelung energies is often not an easy task due to the slow decrease of the Coulomb interaction with distance [2]. It is not possible to write down the successive terms of the Madelung constant by a casual inspection and the series will not converge unless the successive terms in the series are arranged so that the contributions from the positive and negative terms nearly cancel [7]. A powerful general method for lattice sum calculations was developed by Peter Paul Ewald, a U.S., German-born crystallographer and physicist, as well as a pioneer of X-ray diffraction methods. Other methods have also been developed for such calculations. These include those by Bertaut, Evjen, Epstein, and Madelung himself. The Ewald method for the evaluation of the electrostatic potential in a crystal structure of point ions is discussed in the section entitled Ewald Method.
II.4. Repulsive Contribution
Although the Madelung energy (electrostatic) is by far the largest contribution to the ionic crystal lattice energy, a repulsion term is obviously necessary to account for the stability of the crystal. This term arises from the fact that the ions have closed electron shells and resist the overlap of their electron clouds with neighboring ions [7].
The repulsive energy is generally written as a function of the first nearest-neighbors in the crystal (although it could also be expressed in such a way to contain explicitly the contributions of other ions) and is much simpler to determine than the Madelung energy. In this manner, the Born model reports the repulsive contribution for a simple crystal as:
$W_{rep} = Bexp( {-r \over \rho} )$
Here, B is the repulsion constant , $\rho$ is the repulsion exponenet, and r is the distance between two oppositely charged ions. The value of the repulsion constant, B, can be found if the fact that the total potential energy, W is minimum at the equilibrium separation, Re , of the oppositely charged ions, which involves setting the derivative of W equal to zero and solving.
II.5. Overall Lattice Energy
If the contributions of only the electrostatic and repulsive energies are considered (at equilibrium), the total lattice energy of an ionic crystal is [12]
$W_{total} = W_{coul} + W_{rep} = \alpha {e^2 \over r_{0}} + Bexp( {-r \over \rho} )$
This equation is of course that of the static crystal. It could be rewritten to account for the van der Waals contribution and the zero-point correction
$W_{total} = \alpha {e^2 \over r_{0}} + Bexp( {-r \over \rho} ) - ({C \over r^6 } + {D \over R^8}) + {9 \over 4} hv_{max}$
where the third term in the equation accounts for the van der Waals contribution and the last term is the zero-point correction. The $v_{max}$ term in the zero-point correction term is the highest frequency of the lattice vibrational mode and the R-6 and R-8 terms in the van der Waals contribution represent the dipole-dipole and dipole-quadrupole, respectively.
III. Conclusion
The cohesive energy of an ionic crystal is worthwhile to inspect. It makes it possible to address how a crystals choose their equilibrium positions. Various properties, such as melting points, heats of fusion and evaporation, etc. are related to cohesion. Because of the relation to these important properties, Madelung energy computations have generated much interested over the years.
A. Ewald Method[2][5][7][11][12]
The electrostatic potential energy of interaction between point charges $q_i$ at the positions $r_i$ is given by
The final term on the right counts all interactions and divides by 2 to compensate for double counting. For a finite system of charges, this equation can be evaluated directly. This is not the case for an infinite system due to the divergence of summation. However, the Ewald method defines a meaningful interaction energy for the system. The general Ewald approach is to replace a divergent summation with two convergent summations, with the first summation convergent in the form of its Fourier transform, and the second convergent by direct summation. First, we need to define the error function and its complement
Where Ewald noted that
Upon replacement of the divergent summation in the electrostatic potential with two convergent summations, the electrostatic energy would be evaluated using
The parameter $\eta$ needs to be chosen appropriately, so that the second summation above converges quickly and can be evaluated directly. The summation term needs to undergo transformation into Fourier space.
Our goal is to explicitly evaluate the summations in the electrostatic potential. It is therefore necessary to assume a periodic lattice so that every ion can be located by the vector, $r_i = \tau_\alpha + T$, at a location $\tau_{alpha}$ within a unit cell and a periodic translation vector T. By using this notation, the summation becomes
where N is the number of unit cells in the system.
Another important step is to transform the sum over the lattice translation vectors into an equivalent sum over reciprocal lattice translations, G. This transformation may occur due to the identity:
where $\Omega$ is the volume of the unit cell.
We can now rewrite some of the terms in W:
The first term can be written as
The summation of T can be rewritten as (note that $\tau_{alpha} - \tau_{\beta}$ is replaced by $\tau_{\alpha \beta}$ for simplification):
which becomes:
If the last term in the above equation (which comes from the G = 0 contribution) cannot be eliminated, the electrostatic energy would be infinite. Hence, the term can only be eliminated if the system is neutral, and it becomes obvious that it is only meaningful to calculate the electrostatic energy of a neutral, periodic system. Note that although N is an infinite number for the periodic case in consideration, the energy per unit cell, W/N, is still well-defined, and the final Ewald expression can be written as
Here, the prime in the summation is used to omit all self-interaction terms. The expression in parenthesis is the term used to calculate the Madelung constant.
A.1. The CsCl Structure: Electrostatic Energy Using Ewald's Method
The CsCl structure has two sites distinguishable sites housing a cesium atom and chlorine atom. If we take Cs to be located at the origin, we have $- \tau_{Cs} = 0$. With this definition, the chlorine atom, with respect to $\tau$ , is located at $\tau_{Cl} = {a\over 2}( \^{x} + \^{y} +\^{z} )$. We can also define the translation and reciprocal lattice vectors, respectively:
$T_1 = a \^{x} , T_2 = a \^{y} , T_3 = a \^{z}$
and
$G_1 = {2 \pi \over a} \^{x} , G_2 = {2 \pi \over a} \^{y} , G_3 = {2 \pi \over a} \^{z}$
For convenience, we can write W/N as
${W \over N} = {q^2 \over 8 \pi \epsilon_0 a} ( \chi_1 + \chi_2)$
where,
$\chi_1$ = 
and
$\chi_2$ = 
In the above equations, $\eta = { \mu^2 \over a^2 }$ for convenience. In the expressions, the sum over $\alpha$ and $\beta$ has four contributions (with two pairs of identical contributions). W/N can be evaluated with MAPLE (or any computer program). Evaluation gives the value for the Madelung constant in parenthesis:
${W \over N} = -{q^2 \over 4 \pi \epsilon_0 a}(2.035)$
The Madelung constant here matches the value given in Table 1 for the constant with respect to the lattice parameter of the crystal.
