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He is the author of a widely used introductory book on solid state physics. Permissions Request permission to reuse content from this site. Table of contents Mathematical Introduction. Acoustic Phonons. Plasmons, Optical Phonons, and Polarization Waves.

Lecture 3: Introduction to Quantum Theory of Solids

Fermion Fields and the Hartree-Forck Approximation. Many-Body Techniques and the Electron Gas. Polarons and the Electron-Phonon Interaction.

Bloch Funcations--General Properties. Brillouin Zones and Crystal Symmetry. Calculation of Energy Bands and Fermi Surfaces. Semiconductor Crystals: I. Semiconductor Crystals: II. Optical Absorption and Excitons. Electrodynamics of Metals.

Acoustic Attenuation in Metals. Theory of Alloys. Correlation Functions and Neutron Diffraction by Crystals. Recoilless Emission. Ferromagnetic materials such as iron are characterized by the way in which their magnetic properties change dramatically at a particular critical temperature, called the Curie temperature.

Below the Curie temperature a ferromagnetic material can carry a strong remanent magnetization, but above the Curie temperature, its ferromagnetic ordering is broken down by thermal energy and it behaves as a paramagnet. The remanent magnetization of ferromagnetic materials results from the phenomenon of spontaneous magnetization — that is a magnetization which exists even in the absence of a magnetic field.

Spontaneous magnetizations arise from the group- magnetic phenomenon of exchange interactions. Another important property of ferromagnetic materials is that their net magnetic moment is much greater than that of paramagnetic and diamagnetic materials.


This strong magnetic moment of ferromagnets arises because the magnetic exchange interactions between neighboring atoms are so powerful that they are able to align the ferromagnetic atomic moments despite the continual disturbance of thermal agitation. Diamagnetism is a fundamental magnetic property. It is extremely weak compared with other magnetic effects and so it tends to be swamped by all other types of magnetic behaviour. Diamagnetism arises from the interaction of an applied magnetic field with the orbital motion of electrons and it results in a very weak negative magnetisation.

The magnetisation is lost as soon as the magnetic field is removed. Strong magnetic fields tend to repel diamagnetic materials. The spin magnetic moments of electrons do not con- tribute to the magnetisation of diamagnets as all the electron spin motions are paired and cancel each other out. Diamagnetism is for all practical purposes independent of temperature. Many common natural minerals, such as quartz, feldspar, calcite and water, exhibit diamagnetic behaviour. Paramagnetic behavior can occur when individual atoms, ions or molecules possess a permanent elementary magnetic dipole moment.

Quantum Theory Of Free Electron (Sommerfield Theory)

Such magnetic dipoles tend to align themselves parallel with the direction of any applied field and to cause a weak positive magnetization. However, the magnetization of a paramagnet is lost once the field is removed because of thermal effects. In an applied field, para- magnetic materials behave in the opposite way to diamagnetic materials and tend to be attracted to regions of strong field.

Many natural minerals, e. When a field is applied to a paramagnetic substance the spin magnetic moments tend to order and to orientate parallel to the applied field direction. How- ever, the magnetic energies involved are small and thermal agitation constantly attempts to break down the magnetic ordering. A balance is reached between these two competing processes of thermal randomizing and magnetic ordering. The magnetic moment which depends on this balance is thus a function of both the applied field and the absolute temperature. The magnetization of a paramagnetic substance is very weak compared with that of a ferromagnetic, but paramagnetic effects are in turn dominant over diamagnetic effects.

The main natural magnetic minerals we shall be dealing with are special variants of ferromagnets known as ferrimagnets and imperfect antiferromagnets. Ferrimagnetic and antiferromagnetic behavior in natural materials arises from ordering of the spin magnetic moments of electrons in the incompletely filled 3d shells of first transition series elements, particularly iron and manganese, by exchange forces. Ferrimagnetism is outwardly very similar to ferromagnetism; indeed, it is very difficult to distinguish between the two properties even using magnetic measuring techniques.

The magnetic behavior of ferrites ferrimagnets depends on their particular crystal structure. Ferrites are commonly iron oxides with a spinel close-packed face-centred cubic structure containing two types of magnetic sites which have antiparallel magnetic moments of different magnitudes.

Quantum Theory of Solids

Therefore, the elementary magnetic moments of a ferrite are regularly ordered in an antiparallel sense, but the sum of the moments pointing in one direction exceeds that in the opposite direction leading to a net magnetization. The imbalance in lattice moments in ferrites may be due to different ionic populations on the two types of sites or to crystallographic dis- similarities between the two types of magnetic sites. Ferrites have low electrical conductivities and have many industrial applications.

For example, Mn and Zn ferrites are used in radiofrequency cores, while Mn mixture ferrites are used in computer memories. Magnetite is an example of a natural ferrite. In antiferromagnetic materials there are again two magnetic sublattices which are antiparallel, but their magnetic moments are identical, and so the material exhibits zero bulk spontaneous magnetization. Modification of the basic antiferromagnetic arrangement can, however, lead to a net spontaneous magnetization.

Two such imperfect antiferromagnetic forms are parasitic ferromagnetism which may result from heterogeneities due to impurities or lattice defects, and by spin canting, which arises from a slight modification of the true antiferromagnetic anti- parallelism. Spin canting is illustrated in figure below. The mineral hematite is an example of a natural crystal with an imperfect antiferromagnetic structure caused by spin canting.

Arrangement of magnetic moments in ferromagnetic, ferromagnetic, antiferromagnetic and imperfect antiferromagnetic materials.

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The magnetic state of an iron bar depends on both the magnetic field to which it is subjected and the history of the bar. The field dependence of magnetization can be described with the aid of Figure below which plots magnetization on the vertical axis against magnetic field on the horizontal axis. Starting with an unmagnetized piece of iron it is found that its magnetization increases slowly as a small field is applied and that if this field is removed the magnetization returns to zero.

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However, on applying a stronger field, beyond a certain critical field, it is found that an important change in magnetic behavior takes place. The magnetization is now no longer reversible in the straightforward way of the very low fields; instead, on removal of the field, a phenomenon referred to as hysteresis develops.

In short, changes in magnetization associated with the removal of the field now differ from those that occurred during the preceding increase of the field, in such a way that the magnetization changes lag behind the field. Furthermore, it is found that on complete removal of the field, i. At moderate fields magnetization rises sharply with increasing field and at still higher fields saturation of the magnetization sets in and the magnetization curve flattens out.

A complete hysteresis loop is obtained by cycling the magnetic field from an extreme applied field in one direction to an extreme in the opposite direction and back again.

Quantum Theory in Solids - Condensed Matter Physics - Physics

Many of the simple magnetic properties used in later chapters to characterize materials can be classed as hysteresis parameters and the interrelationships between these properties can be best understood in terms of hysteresis loops. Consider five of the most important hysteresis parameters. Upon removal of this field the magnetization does not decrease completely to zero.

The remaining magnetization is called the saturation remanent magnetization, M ps. By the application of a field, in the opposite direction to that first used, the induced magnetization can be reduced to zero. The reverse field which actually makes the magnetization zero, when measurement is made in the presence of the field, is called the saturation coercivity B 0 c.