Various Properties of Crystals
Introduction: crystallization energy
The structure of a crystal describes its internal configuration, that is, the disposition of atoms and molecules in the periodic lattice. As far as this is concerned, we can refer to the works of both Rousseau [ROU 99] and Novick [NOV 95]. At present, we will focus on that which is known as crystallization in a vacuum. Applied to a molecule or an atom, this energy is the reduction of potential energy initially taken at infinity and then integrated into the crystal.
The principle of this calculation method consists of accepting that the interaction of two molecules is the total of the atomic interactions of the combined molecules taken two by two. Thus, the interaction potential of molecule k with molecule l is
i: current index of molecule k atoms
j: current index of molecule l atoms
Coefficient 1/2 avoids the need to count the same potential twice. This potential is the total of three terms:
- - the electrical term;
- - the dispersion term (Van der Waals);
- - hydrogen bond.
- 1) The electrical term:
This is due to the charges of atomic nuclei corrected with the masking effect due to the electron cloud that surrounds atoms in a molecule. This effect is often announced by a reduction of atomic (nucleus) charge in the order of 30%. The electron interaction potential [VES 99] is as follows:
r : distance separating nucleus from the point at which we assess potential
r': distance separating nucleus from the current point of the electron cloud
p(r'): electron concentration at point r' of electron cloud
Q: volume of electron cloud
Ze: electron charge of core (Coulomb)
eo: vacuum permittivity
This screening (shielding) effect can be assessed by means of quantum chemistry [VES 99]. The space is split into cubes of equal sizes, the tops of which are the nodes of a three-dimensional net. Using these points, we then calculate both the rigorous potential of the quantum chemistry and the potential corresponding to the first term of Velec. The ratio of these values provides the Ze value of the effective nuclear charge. Note that X-ray diffraction is an alternative method to quantum chemistry.
Ultimately, the Velec potential becomes
2) Dispersion term:
Energy ?ij is the Van der Waals potential well depth, that is, the minimum of this potential, and ц* is the distance corresponding to this minimum. To
be more precise, we can write out [NEM 83]:
Coefficients AiJ and BiJ depend on the nature of the atoms in question.
ai, aj: polarizability of atoms i and j determined by experimentation Ni, Nj: number of electrons effectively surrounding atoms i and j h: Planck’s constant me: Electron mass
Some authors express the VVdWij potential by means of an exponential as
Gavezzotti [GAV 94] provides the value of these parameters according to the nature of the atom in question. This law was proposed by Buckingham and Corner [BUC 47] with the expression:
3) Hydrogen bond:
This is the bond of an atom to a hydrogen atom.
According to Lifson et al. [LIF 79]:
According to Agler et al. [AGL 79]:
4) Global potential between atoms i and j:
If atom j is not a hydrogen atom: p = 0
If atom j is a hydrogen atom: p = 1
For polar molecules, the electron interaction and hydrogen bonds are predominant relative to the Van der Waals forces.
5) Crystallization energy:
For molecule k within a crystal, let us consider a sphere, the radius of which is 3 or 4 mm and centered on molecule k, where n is the number of molecules within this sphere. The bonding energy of molecule k to the crystal is
If the crystal lattice comprises several molecules, we must use the arithmetic mean of the energies corresponding to the molecules of the simple crystal lattice. The lengths of the chemical bonds can be found in the work of Stewart [STE 90].