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In solid-state physics
Solid-state physics

Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism and metallurgy....
, the free electron model is a simple model for the behaviour of valence electron
Valence electron

In science, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. Valence electrons are important in determining how an chemical element reacts chemically with other elements: The fewer valence electrons an atom holds, the less reactivity it becomes and the more likely it is to chemical rea...
s in a crystal structure
Crystal structure

In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice....
 of a metallic solid
Solid

A solid object is in the states of matter characterized by resistance to deformation and changes of volume. In other words, it has high values both of Young's modulus and of shear modulus; this contrasts e.g....
. It was developed principally by Arnold Sommerfeld
Arnold Sommerfeld

Arnold Johannes Wilhelm Sommerfeld was a Germany theoretical physicist who pioneered developments in atomic physics and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics....
 who combined the classical
Classical physics

Classical physics is a general term used to describe the branches of physics based on principles developed before the rise of general theory of relativity and Quantum mechanics, usually including special theory of relativity....
 Drude model
Drude model

The Drude model of electrical conduction was proposed in 1900by Paul Karl Ludwig Drude to explain the transport properties of electrons in materials ....
 with quantum mechanical
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially

Ideas and assumptions
As in the Drude model, valence electron
Valence electron

In science, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. Valence electrons are important in determining how an chemical element reacts chemically with other elements: The fewer valence electrons an atom holds, the less reactivity it becomes and the more likely it is to chemical rea...
s are assumed to be completely detached from their ion
Ion

An ion is an atom or molecule which has lost or gained one or more electrons, giving it a positive or negative electrical charge. According to the Bohr_model this will be from or in the outer shield 'n'....
s (forming an electron gas).






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In solid-state physics
Solid-state physics

Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism and metallurgy....
, the free electron model is a simple model for the behaviour of valence electron
Valence electron

In science, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. Valence electrons are important in determining how an chemical element reacts chemically with other elements: The fewer valence electrons an atom holds, the less reactivity it becomes and the more likely it is to chemical rea...
s in a crystal structure
Crystal structure

In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. A crystal structure is composed of a motif, a set of atoms arranged in a particular way, and a lattice....
 of a metallic solid
Solid

A solid object is in the states of matter characterized by resistance to deformation and changes of volume. In other words, it has high values both of Young's modulus and of shear modulus; this contrasts e.g....
. It was developed principally by Arnold Sommerfeld
Arnold Sommerfeld

Arnold Johannes Wilhelm Sommerfeld was a Germany theoretical physicist who pioneered developments in atomic physics and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics....
 who combined the classical
Classical physics

Classical physics is a general term used to describe the branches of physics based on principles developed before the rise of general theory of relativity and Quantum mechanics, usually including special theory of relativity....
 Drude model
Drude model

The Drude model of electrical conduction was proposed in 1900by Paul Karl Ludwig Drude to explain the transport properties of electrons in materials ....
 with quantum mechanical
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially
  • the Wiedemann-Franz law
    Wiedemann-Franz law

    In physics, the Wiedemann-Franz law states that the ratio of the electronic contribution to the thermal conductivity and the electrical conductivity of a metal is proportional to the temperature ....
     which relates electrical conductivity
    Electrical conductivity

    Electrical conductivity or specific conductance is a measure of a material's ability to electrical conduction an electric current. When an electrical potential difference is placed across a conductor, its movable charges flow, giving rise to an electric current....
     and thermal conductivity
    Thermal conductivity

    In physics, thermal conductivity, , is the List of materials properties of a material that indicates its ability to conduct heat. It appears primarily in Heat conduction#Fourier's law for heat conduction....
    ;
  • the temperature dependence of the heat capacity;
  • the shape of the electronic density of states
    Density of states

    In statistical physics and condensed matter physics, the density of states of a system describes the number of states at each energy level that are available to be occupied....
    ;
  • the range of binding energy values;
  • electrical conductivities.


Ideas and assumptions


As in the Drude model, valence electron
Valence electron

In science, valence electrons are the electrons contained in the outermost, or valence, electron shell of an atom. Valence electrons are important in determining how an chemical element reacts chemically with other elements: The fewer valence electrons an atom holds, the less reactivity it becomes and the more likely it is to chemical rea...
s are assumed to be completely detached from their ion
Ion

An ion is an atom or molecule which has lost or gained one or more electrons, giving it a positive or negative electrical charge. According to the Bohr_model this will be from or in the outer shield 'n'....
s (forming an electron gas). As in an ideal gas
Ideal gas

The ideal gas model is a model of matter in which the molecules are treated as non-interacting point particles which are engaged in a random motion that obeys conservation of energy....
, electron-electron interactions are completely neglected (they are weak because of the shielding effect
Shielding effect

The shielding effect describes the decrease in attraction between an electron and the Atomic nucleus in any atom with more than one electron shell....
).

The crystal lattice is not explicitly taken into account. A quantum-mechanical justification is given by Bloch's Theorem
Bloch's theorem

* For the theorem named after Felix Bloch on wave functions of a particle in a periodic potential, see Bloch wave.* For the theorem in complex variables named after Andr? Bloch, see Bloch's theorem ....
: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass m becoming an effective mass
Effective mass

In solid state physics, a particle's 'effective mass' is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that electrons and electron hole in a crystal respond to electric field and magnetic fields almost as if they were particles with a mass dependent upon the their direction of travel, an ...
 m* which may deviate considerably from m (one can even use negative effective mass to describe conduction by electron hole
Electron hole

An electron hole is the conceptual and mathematical opposite of an electron, useful in the study of physics and chemistry. The concept describes the lack of an electron....
s). Effective masses can be derived from band structure computations. While the static lattice does not hinder the motion of the electrons, they can well be scattered by impurities and by phonon
Phonon

In physics, a phonon is a quantum mode of vibration occurring in a rigid crystal structure, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's thermal conductivity and electrical conduc...
s; these two interactions determine electrical and thermal conductivity (superconductivity
Superconductivity

Superconductivity is a phenomenon occurring in certain materials generally at very low temperatures, characterized by exactly zero electrical resistance and the exclusion of the interior magnetic field ....
 requires more refined theory than the free electron model).

According to the Pauli exclusion principle
Pauli exclusion principle

The Pauli exclusion principle is a quantum mechanics principle formulated by Wolfgang Pauli in 1925. It states that no two identical particles fermions may occupy the same quantum state simultaneously....
, each phase space
Phase space

In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space....
 element (?k)3(?x)3 can be occupied only by two electrons (one per spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 quantum number
Quantum number

Quantum numbers describe values of conserved numbers in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, Spin etc....
). This restriction of available electron states is taken into account by Fermi-Dirac statistics
Fermi-Dirac statistics

Fermi-Dirac statistics is a part of the science of physics, that applies to a system comprised of many particles that obey the Pauli Exclusion Principle....
 (see also Fermi gas
Fermi gas

A Fermi gas, or Free electron gas, is a collection of non-interacting fermions. It is the quantum mechanics version of an ideal gas, for the case of fermionic particles....
). Main predictions of the free-electron model are derived by the Sommerfeld expansion
Sommerfeld expansion

A Sommerfeld expansion is an approximation method developed by Arnold Sommerfeld for a certain class of integrals which are common in condensed matter and statistical physics....
 of the Fermi-Dirac occupancy for energies around the Fermi level.

Technicalities


Effective mass

A band structure computation actually yields a dispersion relation
Dispersion relation

Dispersion relations describe the ways that wave propagation varies with the wavelength or frequency of a wave . This variation has long explained how white light is dispersed into different colors, thus making rainbows possible....
 E(k) between electron wave vector
Wave vector

A wave vector is a vector representation of a wave. The wave vector has magnitude indicating wavenumber , and the direction of the vector indicates the direction of wave propagation....
 k and energy E. An effective mass is obtained by approximating the true dispersion relation in the limit of small k by the free-electron form
(with the free-electron mass m replaced by m*). A lattice electron with a fictitious mass can be seen as a quasiparticle
Quasiparticle

In physics, quasiparticle refers to a group of discrete phenomena whose behaviour is characterised as that of a single particle in a system, coupled with the effect the former has on the latter....
 (though there is a one-to-one correspondence to the real particle which is not the case for other quasiparticles such as phonon
Phonon

In physics, a phonon is a quantum mode of vibration occurring in a rigid crystal structure, such as the atomic lattice of a solid. The study of phonons is an important part of solid state physics, because phonons play a major role in many of the physical properties of solids, including a material's thermal conductivity and electrical conduc...
s).

Relation with other electron models


The assumption of electrons that move freely through a periodic potential should be contrasted with the tight-binding model
Tight binding (physics)

In the tight binding model for a solid-state lattice of atoms, it is assumedthat the full Hamiltonian of the system may be approximated by theHamiltonian of an isolated atom centred at each Bravais lattice....
, which uses the opposite simplification of treating the electrons as tightly bound to the atomic cores. (Coulomb interactions between electrons are still neglected.) The predictions of these two complementary models are reassuringly similar. Taking into account the specifities of the potential in a real, three-dimensional crystal lattice leads to more complicated dispersion relations and to band theory. In fact, the Bethe-Sommerfeld theory generalizes the thermodynamics of free-electron systems also in these respects.

See also

  • Nearly-free electron model
    Nearly-free electron model

    In solid-state physics, the nearly free electron model is a model of electron behavior in solids that enables understanding the electronic band structure of crystalline materials....
  • Free electron laser
    Free electron laser

    A free-electron laser, or FEL, is a laser that shares the same optics properties as conventional lasers such as emitting a beam consisting of Coherence Electromagnetic radiation radiation which can reach high power , but which uses some very different operating principles to form the beam....


External articles and references

  • Ashcroft, Mermin: Solid State Physics.