%)Aside: Although there are numerous possible combinations for constructing alloys, not all metals will form alloys - e.g. pure indium will not form an alloy with pure gallium (ref. Ashcroft & Mermin, "Solid State Physics", Saunders College, 1976). Also under different conditions of temperature, mixing ratio, and pressure the solidified alloy can exist in different phases - e.g. either fcc, bcc, hcp, ordered (true microscopic translational invariance), disordered (no microscopic translational invariance), or even as a composite of microcrystalline domains of different phases.)
2.3) Lattice Points per Unit Cell: For this problem, consider the conventional cubic unit cell. Recall, in a cubic cell, the corner lattice points are shared among eight adjacent cells, and lattice points on the face are shared by two adjacent cells.
%)Aside: There are other ways to define a unit cell for a lattice, the most important being the Wigner-Seitz primitive cell, which has only one lattice site per cell but maintains the full symmetry of the Bravais lattice.
2.4) Nearest Neighbor Distance: In BCC, the nearest neighbor (n.n) is along the body diagonal direction; in FCC the n.n. is along the face diagonal direction; in the diamond structure the n.n. is also along the body diagonal direction. Express the n.n. distance in terms of the length of the sides of a conventional cubic cell - commonly refered to as the lattice constant "a".
%)Note: It is important to distinguish between a (Bravais) lattice and a crystal structure since this distinction is integral to undertanding the concept of a reciprocal lattice to a crystal (the reciprocal lattice is the reciprocal of the Bravais lattice corresponding to a particular crystal structure). EVERY lattice point in a Bravais lattice is exactly identical and indistinguishable. The image of a conventional FCC or BCC unit cell as a cube with either face-center or body-center lattice points superimposed is for visual clarity only - in reality the face-center or body-center point in FCC or BCC respectively are indistinguishable from the corner points. (FCC is just two simple cubic lattices offset by (1/2, 1/2, 0)·a (or any permutation thereof) and it doesn't matter which simple cubic lattice one takes as corresponding to the corner points or the face-center points. Similarly, BCC is just two simple cubic lattices offset by (1/2, 1/2, 1/2)·a). A particular crystal structure is defined by a (Bravis) lattice and the species and location of the constiuent atoms relative to the lattice points. Sometimes, crystals are composed of only one species of atom located at each lattice point in which case Bravais lattice and the crystal structure are the same. The diamond structure is not a Bravais lattice. It is based on the FCC lattice with one atom at each lattice point and another located at (1/4, 1/4, 1/4)·a distance away, sometimes described as two inter- penetrating FCC lattices. These two points are not identical, since although each point of one FCC lattice (e.g. at (0, 0, 0)) may have a neighbor at (1/4, 1/4, 1/4)·a distance away, the points of the other FCC lattice (e.g. at (1/4, 1/4, 1/4)·a) do not ((1/4, 1/4, 1/4)·a + (1/4, 1/4, 1/4)·a = (1/2, 1/2, 1/2)·a which is empty). Similarly, the Bravais lattice corresponding to the NaCl or CsCl structure (see figure 2.6) is the simple cubic structure.
2.8) Orthogonality Between and (hkl) Plane and the [hkl] Direction: Recall that the (hkl) plane intercepts the three axes at (1/h, 0, 0), (0, 1/k, 0), (0, 0, 1/l) and can be expressed mathematically as h·x + k·y + l·z = 1. Also recall that a vector normal to a surface described by f(x,y,z) = const. is given by n = del(f) = (df/dx)·x^ + (df/dy)·y^ + (df/dz)·z^ . Two vectors are parallel if one is a scalar multiple of the other (or equivalently if their cross product is zero).
2.10) Normal Modes of a Five Atom Chain with Fixed Ends: This problem
needs to be modified in order to be correct.
When the problems mentioned "all possble motions of the particles",
it should have stated that one should not include pathalogical
initial conditions which violate the boundary conditions imposed by
Newtonian mechanics. In other words, any arbitrary snap-shot
of the instantaneous displacements of the particless can be realized
by a superposition of the three normal modes, but the
instantaneous velocities of each particle cannot also be arbitrary.
It suffices to show that there are only three physically meaning
normal modes (show that the higher order modes can be simplified into
lower order modes), and that any combination of
instantaneous displacements of the particles can be described by a
superposition of the normal modes (you can show that the
displacements of the three inner particles can be expressed as three
linear equations, with three unknowns - being the three
proportionality constants for each of the normal modes). The
normal modes are un(xm)=Nn·sin(kn·xm),
where un denotes the
(small) displacement from the stationary equilibrium position for mode
n, xm denotes the stationary equilibrium position of particle
m, Nn denotes an arbitrary proportionality constant, and
kn = (n·p)/(4·a) where
a is the spacing between particles when stationary,
and of course n = 1, 2, 3. It is understood that there is a time
dependent term ejwn·t multipling
un (which can be ignored for this problem).