Solid State Physics PHYS 40352

1 Solid State PhysicsPHYS 40352by Mike GodfreySpring 2012 Last changed on May 22, 2017iiContentsPrefacev1 Crystal and basis .. cells .. symmetry .. lattices .. lattices .. cubic crystalstructures.. crystallography .. by a crystal .. reciprocal lattice .. lattice vectors and lattice planes .. Bragg construction .. factor .. geometry of diffraction ..172 Electrons in of free-electron theory, etc.. in a periodic potential .. s theorem .. zones .. odinger s equation in k-space .. periodic potential: Nearly-free electrons .. and insulators .. overlap in a nearly-free-electron divalent metal .. method .. dynamics of Bloch electrons .. velocities .. in an applied field .. mass of an electron .. bands and crystal structure.

2 Of the reciprocal lattice for FCC .. IV elements: Jones theory .. energy of metals .. theory of Group V elements .. of alloys .. resonance .. mass tensor .. of the cyclotron frequency .. resonance in metals .. breakthrough: failure of the semiclassical approximation ..453 in a magnetic field .. in classical mechanics .. Hamiltonian of a charge in a magnetic field .. magnetism in classical Physics .. Hamiltonian of an electron in a magnetic field .. quantities in thermodynamics .. of a gas of free electrons .. spin paramagnetism of an electron gas .. orbital diamagnetism of an electron gas .. magnetic response of the electron gas .. of ions .. s rules .. of closed-shell systems .. of ions with partially filled shells.

3 Magnetic states .. interaction between spins .. interaction .. interaction between ions .. aren t all magnets FM? .. Heisenberg Hamiltonian .. groundstate and excitations .. energy .. excitations and magnons .. theory of the critical point ..66 PrefaceThis document will eventually be a summary of the material taught in the course. In a few places you mayfind that derivations and examples of applying the results are not given, or are very much , it is often convenient to present some material in a different way from in the lectures. A littlecommon sense should therefore be used when reading the notes and a good textbook consulted from timeto would suggest you come back to these notes from time to time, as they are (and are likely to remain)a work in progress.

4 Please let me know if you find any typos or other slips, so I can correct 1 Crystal structureIn preparation: Much of the material in this chapter has been adapted, with permission, from notes anddiagrams made by Monique Henson in are strongly recommended to make sure that you understand and (where appropriate) can solveproblems that involve: the meaning of the termslatticeandmotif[orbasis] simple cubic, face-centered cubic, and body-centered cubic lattices lattice vectorsandprimitive lattice vectors;unit cellsandprimitive unit cells diffraction of X rays by a crystal in terms of the Bragg equationandthe reciprocal lattice vectors the relation between lattice planes and reciprocal lattice vectors be sure you know (and can derive) the reciprocal lattices for the simple cubic, FCC, and BCC lattices[these are useful for the kinds of problems that can be set on nearly-free electron theory and X-raydiffraction] the indexing of X-ray diffraction patterns ( , given the Bragg angles , find plausibleGvectors,or lattice planes you can get additional practice from past paper questions) Lattice and basisA fundamental property of a crystalline Solid is itsperiodicity: a crystal consists of a regular array of iden-tical structural units.

5 The structural unit, which is called thebasis[ormotif] can be simple, consistingof just one atom (as in sodium or ion), or complex, consisting of two or more atoms (as in diamond or inhaemoglobin); see Fig. positions in space of these structural units define the points of any real crystalhas only a finite number of atoms, this number can be very large indeed (1023, say), so that it is oftenuseful to imagine the crystal and its corresponding lattice to be infinite, extending through all space. Theenvironment of every lattice point is identical in all respects, including orientation, so that we can getfrom one lattice point to any other by a simple translation. A vector connecting any two points of thelattice [and hence a possible translation vector] is called alattice vector, and can be expressed in the formR=n1a1+n2a2+n3a3,( )wheren1,n2,n3can take any of the integer values 0, 1, 2.

6 1 Usually named afterBravais, who made a systematic study [ca. 1845] of the lattices possible in two and three 1. CRYSTAL STRUCTUREF igure : An example of a crystal structure. Atoms are represented by grey circles. Three atoms (shadedgreen) make up themotif(orbasis) of the structure. Lattice points are indicated by blue direction in the lattice can be specified by the coefficients[n1,n2,n3]. A set of non-coplanar latticevectorsa1,a2,a3that can be used to generate all of the lattice vectors in accordance with ( ) is said tobeprimitive. The choice of these vectors is not unique; in particular, they need not be the shortest possiblelattice vectors. This is illustrated in Fig. : Two choices of primitive vectors for a 2D lattice. Primitive unit cells are also shown.

7 Theshaded region is a non-primitive cell with twice the area of a primitive Unit cellsAny region of space that contains only one lattice point and can be translated by lattice vectorsRto fillthe whole of space without leaving gaps or forming overlaps is called aprimitive unit cell. For example,the parallelepiped whose edges are the primitive vectorsa1,a2,a3is always a primitive unit many possible choices of unit Fig. primitive vectors and the rectangle they span is a primitive unit cell. Thechoice of vectors is not unique: we could equally well choosea 1anda 2. They are the edges of a primitivecell (a parallelogram) of the same area as the rectangle with edgesa1anda2. These are just two simpleexamples they don t necessarily have to be that the shaded rectangle in Fig.

8 It is still a unit cell, as it could be repeated to fill the whole ofspace, but it is not aprimitivecell as it does not have the smallest possible area. It is not always convenientto work with a primitive unit cell. For example, when discussing the lattices of the cubic system wegenerally use a unit cell that has the shape of a cube, even though for BCC and FCC this conventionalcubic unit cell is non-primitive; see Figs. and parallelepiped is illustrated in Fig. LATTICE AND Crystal symmetryLattice symmetries include translation by a lattice vector, discrete rotations (discussed below), and symmetry (reflections)Mirror symmetry should be familiar enough not to need symmetryFigure : Conventional symbols for axes of pure rotational symmetry. These are shown shaded in thisfigure, as is usual in textbooks of crystallography, but elsewhere in this chapter (and in the lectures) thesymbols have been left notation for axes of rotational symmetry is shown in Fig.

9 These are the only four possiblerotational symmetries that are consistent with the periodicity of a crystal. Why? Consider ann-foldrotation axis A, in two dimensions, such that a rotation through an angle =2 /nabout A maps thecrystal onto itself. This is illustrated in Fig. Now consider a secondn-fold axis, B, that is related toaxis A by a lattice translation, which we suppose to be the shortest possible. Letadenote the length rotation by anticlockwise about A, B maps onto B . Likewise, when the lattice is rotated clockwiseabout B, A maps to the point A . The distance between A and B must be an integer multiple ofa, asaisthe length of the shortest lattice vector. Therefore,A B =pa=a 2acos ,( )wherepis an integer. This can be rearranged ascos =1 p2.

10 ( )This gives the possible values of to be those shown in Table Hence, only rotation axes with ordersn=2, 3, 4 and 6 are possible in a 1. CRYSTAL STRUCTUREF igure : Diagram to illustrate the restrictions on the possible orders of rotational symmetry elements ina 2D crystal. A and B are twon-fold axes of rotation, separated by a shortest lattice vector of lengtha. Therotational symmetry about B and A requires the presence of further axes A and B , and the translationalsymmetry of the lattice (in the direction parallel to AB) requires their separation A B to be an integermultiple n01260 61090 42 12120 33 1180 2 Table : The orders,n, of rotation axes that are consistent with the translational symmetry of a to Fig. forpand =360 /n, and to Eq. ( ) for the relation between them.