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The other, more satisfactory, way to think about diffraction from lattices
is the method due to von Laue, which does emphasise scattering from atoms
and their constituent electrons and nuclei. The scattering angle is
2q. The planes of the crystal are
characterised by the Fourier coefficients of the electron density
r(**r**) (for X-ray scattering) or of the
interaction potential V(**r**) (for electron scattering). In either case we
can write the scattering agent in terms of a Fourier series based on the
reciprocal lattice **g**, i.e.

r) = Σ_{g}
V_{g}exp(ig.r).
(3.1) |

We can visualise this relation using the following figure 3.1, which shows
the traditional Bragg's law picture (on its side, figure 3.1(a)) and the von
Laue picture, leading to the Ewald sphere construction (figure 3.1(b)). The
incoming wave is described by a vector **k** and the outgoing wave by
**k'**, where |**k**| = |**k'**| =
2p/l.
From the geometry of the Ewald sphere there is a one-to-one relation between
the two pictures provided that
d = 2p/|**g**|, and that the plane described
by **g** is drawn as a vector normal
to the planes in the Bragg picture. This picture can be filled out, for any
*real* crystal lattice, by expressing **g** as the hkl *point*
of the *reciprocal* lattice, such that

g =
hb + k_{1}b + l_{2}b.
(3.2)_{3} |

where h, k, and l are the Miller indices, and the **b**'s are the unit
vectors of the reciprocal lattice. Just to keep you on your toes, **g**
is written **G** (Sutton, Pettifor sometimes) or capital **K**
(Ashcroft and Mermin, chapter 6). The last possibility is a nightmare for lecturers who
have to repeat sentences like *k-prime equals k plus K* many times!

Note that the traditions in the crystallography and solid state
literature differ, in that 2p enters the
equations at different points. The solid state literature typically uses
exp(i**g.r**) as above, with
|**k**| = 2p/l and
|**g**| = 2p/d. The crystallography literature
has exp(2pi**g.r**), but with
2p absent from the other definitions.

b = 2p
_{k}a x _{i}a/t,
where t is the volume of the unit cell, i.e.
t =
_{j}a x _{i} . a_{j}a.
(3.3)_{k} |

Here **.** represents the scalar and x the vector product, and the
indices i, j and k have to be taken cyclically. Again we have to
be consistent about the 2p, which is needed
for the 'solid state' definition of **b _{k}** used here, but
not for the 'crystallography' definition where the
2p is in the exponential.
If you are not familiar with these ideas, check out for yourself the
relation between the two spaces via problem 3.4.1.

A case which is particularly instructive is when only one **g**-vector (one
reciprocal lattice point) is involved, as this reduces to a 2x2 matrix
which can be solved analytically, as in problem 3.4.3.
This case is discussed in some detail by Sutton and by Pettifor.
When V_{g} is small relative to the electron energy E,
this corresponds to the Nearly Free Electron (NFE) model, which we will
discuss in the lecture. The model is widely used, not only for conduction
electrons, but also in (transmission) electron microscopy, where it was
instrumental in providing the first understanding of contrast from crystal
defects in
thin crystalline samples. In the latter case, the **k**-vectors are much
longer, corresponding to energies of 100 keV or more, but the angle
q is correspondingly smaller, so that the same
low order **g**-vectors are involved.

The *plane* in the reciprocal lattice perpendicular to **g**
through the point **g**/2 is known as the Brillouin zone boundary,
sometimes called the BZB, and any **k**-vector which ends on the BZB
satisfies the condition for Bragg scattering via
**k'** = **k** + **g**. In 2D or 3D a series of such planes for
the shortest **g**-vectors forms a polygon called the first Brillouin
zone (BZ). Because of way the reciprocal lattice has been set up, all
electron states can be referred back to the first BZ, which is called the
*reduced* as opposed to *extended* zone scheme. The mathematics
behind this was outlined in section 2.3,
and is described by both Sutton and Pettifor, amongst many others.

Show, for *one* of these lattices, that the intensity
of diffraction I_{g}, in the weak scattering (kinematic) limit, is
proportional to |V_{g}|^{2}.

If you haven't studied this model before, now is a good time to do so. You could produce a 'working model' which allows you to vary the parameters and see the result, maybe viewable on the web, or on the computer projection system. See me if this appeals to you. The 1D programs developed by Edward Hernandez a few years ago may be studied in the computation sessions if there is time.

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