Derive the expressions for the transverse and longitudinal interplanar force constants for neon for wave vectors along the (a) [100], (b) [111], and (c) [110] directions as given in eqns \(8.32,8.33-8.4 .1\). (Hint: for each direction of the wave vector use a set of axes in which the longitudinal direction is along one of the new axes, and express the vectors between an atom and its neighbours as a vector in this new set of axes.)

Understanding the alignment of crystal axes is crucial for analyzing force constants in crystals. Crystals have a regular repeating structure, defined by their axes—often labeled as x, y, and z. The alignment of wave vectors in these directions helps in determining the behavior of atoms within the crystal.
For example, the [100] direction represents a wave vector along the x-axis, [111] represents a wave vector diagonally across all three axes, and [110] lies in the plane of the x and y-axes. Recognizing these alignments allows for the appropriate setup of coordinate systems for subsequent calculations.

Wave vectors are crucial in understanding the propagation of waves through a crystal lattice. These vectors indicate the direction and wavelength of waves.
In crystals, wave vectors can be aligned along specific directions like [100], [111], or [110]. Each of these orientations interacts differently with the crystal's atomic structure and affects the interplanar forces.
For instance, a wave vector along [100] aligns with one of the principal axes, simplifying calculations. In contrast, [111] spans diagonally, impacting all three axes equally, while [110] lies in the plane formed by two axes, making the analysis more complex.

The longitudinal force constant reflects the stiffness of the crystal when stretched or compressed along a certain direction. It helps in understanding how the crystal resists deformation.
Calculation involves projecting the force vectors along the longitudinal direction of the wave. This direction is typically aligned with the wave vector for simplicity.
For example, in the [100] direction, the longitudinal force aligns with the x-axis. This projection requires understanding the interaction of forces between neighboring atoms as they move along this axis.

Transverse force constants measure the resistance against shearing or bending forces perpendicular to the longitudinal direction.
The calculation involves projecting the force vectors along directions orthogonal to the wave vector. Unlike longitudinal forces, these constants consider interactions in the plane perpendicular to the main axis.
For instance, in the [110] direction, while the wave vector lies in the x-y plane, transverse force constants would be calculated in a plane perpendicular to this, often involving multiple axes and their respective components.

Transforming the coordinate system simplifies the calculation of force constants by aligning the longitudinal direction with one of the new axes. This transformation involves translating the positions of all atoms and their interactions into this new system.
For example:
* In the [100] direction, aligning the x-axis with the wave vector simplifies mapping of force vectors.
* In the [111] direction, a diagonal axis transformation helps in evenly distributing the effects into new coordinates.
* For [110], aligning the major diagonal is essential.
Once transformed, the interactions can be easily broken down into longitudinal and transverse components, streamlining the derivation of force constants.