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Magnetic Properties


This gives the increase in exchange energy when two equal spins are rotated from exact parallelism to make some small angle f with each other.

Now, let f0 denote the total change of angle between two domains and the change occurs in N equal steps, so that the change of angle between neighbouring spins is f0 /N. Then, the exchange energy between each pair of neighbouring atoms is




(DEex)pair = JeS 2



and the total energy of the array of atoms

J S 2 f 20

(DEex)total = e



This shows that the exchange energy decreases when N increases. Now, one may argue that why does not the wall become infinitely thick (to increase N)? This is explained by the concept of anisotropy energy.

Since the spins within the wall are nearly all directed away from the easy direction, an anisotropy energy is associated with the wall which is roughly proportional to the thickness of the wall. This energy also needs to be minimum. Thus, the actual thickness and the energy of the wall is, therefore, the result of compromise between the two energies, viz., the exchange energy and the anisotropy energy.

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Crystal Imperfections


The concept of an ideal crystal with a perfect arrangement of atoms is, strictly speaking, valid only at the absolute zero temperature because then there is no entropy contribution. However, at a finite temperature, a certain native configurational disorder is introduced into the structure (a direct consequence of laws of thermodynamics) and solid becomes structurally imperfect. In fact, in almost all cases, a crystal just cannot be grown without an initial imperfection. Thus, imperfections are mistakes in the crystallographic structure of crystals.


Defects and imperfections affect especially the structure-sensitive properties of solids. As, for instance, it is well known that electrical and thermal conductivities are greatly reduced due to scattering of electrons and phonons by lattice defects. The semiconducting properties of solids are also considerably influenced by the presence of impurities in the lattice, which are responsible for creating localised (defect) levels in the energy gap between the valence band and conduction band. Dielectric behaviour of materials depends largely on the state of polarisation, temperature and perfection. Optical properties are also related to energy gaps (as in semiconductors) and different point defects (which is more apparent in ionic crystals), e.g. colour of some materials and sensitivity of photographic emulsions (to light) are due to imperfections.

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Transport Properties


During the nineteenth century, there was no precise concept known for atomic structure. The electron was discovered by Thomson in 1897, and this discovery had a vast and immediate impact on theories of structure of matter. Three years later (i.e. in 1900) Drude gave his theory of electrical and thermal conduction by considering the metals to be containing free electrons, and thereby applying the kinetic theory of gases to metals.

Accordingly, it is assumed that when atoms of a metallic element are brought together to form a metal, the valence electrons get detached and wander freely within the metal, whereas ions remain intact and play the role of immobile particles. Surrounding the nucleus are Za electrons of total charge –eZa. Out of these, there are Z relatively weakly bound valence electrons and remaining (Za – Z) are relatively tightly bound and are known as core electrons. When atoms condense to form the material, core electrons remain bound to the nucleus, but valence electrons detach from their parent atoms and are called conduction electrons. They are akin to atoms in a gas moving against a background of heavy immobile ions (see Fig. 6.1).

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Dielectrics, Plasmons, Polarons and Polaritons 505


Dielectrics, Plasmons,

Polarons and polaritons


In insulating materials (dielectrics), electrons are very tightly bound to the atoms (and forbidden gap in the energy band picture is comparatively larger). Consequently, electrons cannot be made free, however they can only be displaced a bit within the molecule (under the application of an external electric field) and their cumulative effect accounts for the characteristic behaviour of dielectric materials. Thus, under the action of a strong electric field, the centres of the positive charges are displaced slightly in one direction (i.e. the direction of the field) and that of negative charges in the opposite direction. This gives rise to a local electric dipole moment throughout the crystalrelative to lattice deformation. The internal distortion in the lattice can be studied in detail if one has the knowledge of the dielectric constant.


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Crystal Structure


On the basis of structure, solids may be divided into two broad categories – crystalline and amorphous. In crystalline solids, the atoms are stacked in a regular manner, forming a three-dimensional pattern which may be obtained by a three-dimensional repetition of a certain pattern unit. When the periodicity of the pattern extends throughout a certain piece of material, one speaks of a single crystal. In polycrystalline materials, the periodicity of structure is interrupted at the so-called grain boundaries such that the structure is periodic within a single grain and the size of the grains within which the structure is periodic may vary from macroscopic dimensions to several Angstroms*. When the size of the grains becomes comparable to the size of the pattern unit, one can no longer speak of crystallinity, rather one speaks of amorphous substances. Due to regularity in structure, there is a long-range order in single crystals. There remains no periodicity in structure in an amorphous substance and long-range order diminishes to a very short-range order.

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