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i.e. an electron with this wave vector G would have a wave function equal at all points in real space separated by a Bravais lattice vector R. Therefore:
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which implies that Now learning from the form for the Bravais lattice vectors R given earlier in equation (1.22), it might be expected that the reciprocal lattice vectors G could be constructed in a similar manner from a set of three primitive reciprocal lattice vectors, i.e. With these choices then, the primitive reciprocal lattice vectors can be written as follows:
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It is possible to verify that these forms do satisfy equation (1.30):
Now bi is perpendicular to both a2 and a3, and so only the product of b1 with ai is non-zero, and similarly for b2 and b3; hence:
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and in fact, the products b1.a; = 2T; therefore:
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Clearly B1a1 + B2a2 + B3a3 is an integer, and hence equation (1.30) is satisfied. Using the face-centred cubic lattice vectors defined in equation (1.21), then:
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which gives:
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Therefore, the first of the primitive reciprocal lattice vectors follows as:
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A similar calculation of the remaining primitive reciprocal lattice vectors b2 and b3 gives the complete set as follows:
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which are of course equivalent to the body-centred cubic Bravais lattice vectors (see reference [1], p. 68). Thus the reciprocal lattice constructed from the linear combinations: is a body-centred cubic lattice with lattice constant 4h/A0. Taking the face-centred cubic primitve reciprocal lattice vectors in equation (1.45), then:
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The specific reciprocal lattice vectors are therefore generated by taking different combinations of the integers B1, B2, and B3. This is illustrated in Table 1.1. It was shown by von Laue that when waves in a periodic structure satisfied the following: then diffraction would occur (see reference [1], p. 99). Thus the 'free' electron dispersion curves of earlier (Fig. 1.5), will be perturbed when the electron wave
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Table 1.1 Generation of the reciprocal lattice vectors for the face-centred cubic crystal by the systematic selection of the integer coefficients B1, B2, and B3
B22 B3 (B B3
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G(2T/A 0 ) (0,0,0) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (0,0,2) (0,2,0) (2,0,0) (2,2,2)
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0 0 0 1 0 0 0 1 0 0 0 1 - 1 0 0 0 - 1 0 0 0 -01 1 1 1 1 -1 -1 -1 1 1 0 1 0 1 0 1 1 -1 1 0 (etc.)
1 1 1 (1,1,1) -1 -1 -1 (1,1,1)
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vector satisfies equation (1.49). Along the [001] direction, the smallest reciprocal lattice vector G is (0,0,2) (in units of 2T/Ao). Substituting into equation (1.49) gives:
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This then implies the electron will be diffracted when:
Fig. 1.14 illustrates the effect that such diffraction would have on the 'free-electron' curves. At wavevectors which satisfy von Laue's condition, the energy bands are disturbed and an energy gap opens. Such an improvement on the parabolic dispersion curves of earlier, is known as the nearly free electron model. The space between the lowest wavevector solutions to von Laue's condition is called the first Brillouin zone. Note that the reciprocal lattice vectors in any particular direction span the Brillouin zone. As mentioned above a face-centred cubic lattice has a body-centred cubic reciprocal lattice, and thus the Brillouin zone is therefore a three-dimensional solid, which happens to be a 'truncated octahedron' (see, for example reference [1], p. 89). High-symmetry points around the Brillouin zone are often labelled for ease of reference, with the most important of these, for this work, being the k = 0 point, referred to as T', and the < 001 > zone edges, which are called the 'X' points.
Figure 1.14 Comparison of the free and nearly free electron models
2.1 THE INFINITE WELL The infinitely deep one-dimensional potential well is the simplest confinement potential to treat in quantum mechanics. Virtually every introductory level text on quantum mechanics considers this system, but nonetheless it is worth visiting again as some of the standard assumptions often glossed over, do have important consequences for one-dimensional confinement potentials in general. The time-independent Schrodinger equation summarises the wave mechanics analogy to Hamilton's formulation of classical mechanics [22], for time-independent potentials. In essence this states that the kinetic and potential energy components sum to the total energy; in wave mechanics, these quantities are the eigenvalues of linear operators, i.e.
where the eigenfunction w describes the state of the system. Again in analogy with classical mechanics the kinetic energy operator for a particle of constant mass is given