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I think this entry in wikipedia explains the Fermi energy and chemical potential in a clear way.
http://en.wikipedia.org/wiki/Fermi_energy
http://en.wikipedia.org/wiki/Chemical_potential
Fermi energy
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In Physics, the Fermi energy (EF) of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. It is equivalent to the chemical potential of the system in its ground state at absolute zero. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The Fermi energy is one of the central concepts of condensed matter physics.
According to quantum mechanics, fermions -- particles with a half-integer spin, usually 1/2, such as electrons -- follow the Pauli exclusion principle, which states that no two particles may occupy the same quantum state. Consequently, fermions obey Fermi-Dirac statistics. The ground state of a non-interacting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied state (or, equivalently, the lowest unoccupied state; the difference is not important when the system is macroscopic in size.)
In the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labelled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.
The Fermi energy of the free electron gas is related to the chemical potential by the equation
\mu = \varepsilon _F \left[ 1- \frac{\pi ^2}{12} \left(\frac{kT}{\varepsilon _F}\right) ^2 + \frac{\pi^4}{80} \left(\frac{kT}{\varepsilon _F}\right)^4 + \cdots \right]
where εF is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature EF/k. The characteristic temperature is on the order of 105 K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in Fermi-Dirac statistics.
Chemical potential
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The precise meaning of the term chemical potential depends on the context in which it is used.
* When speaking of thermodynamic systems, chemical potential refers to the thermodynamic chemical potential. In this context, the chemical potential is the change in a characteristic thermodynamical state function (depending on the experimental conditions, the characteristic thermodynamic state function is either: internal energy, enthalpy, Gibbs free energy, or Helmholtz free energy) per change in the number of molecules. This particular usage is most widely used by experimental chemists, physicists, and chemical engineers.
* Theoretical chemists and physicists often use the term chemical potential in reference to the electronic chemical potential, which is related to the functional derivative of the density functional (sometimes called the energy functional) found in Density Functional Theory. This particular usage of the term is widely used in the field of electronic structure theory.
* Physicists sometimes use the term chemical potential in the description of relativistic systems.
Contents
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* 1 Thermodynamic Chemical Potential
o 1.1 Precise definition
* 2 Relativistic Chemical Potential
* 3 Electronic Chemical Potential
* 4 External links
* 5 See also
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Thermodynamic Chemical Potential
Conjugate variables
of thermodynamics
Pressure Volume
Temperature Entropy
Chem. potential Particle no.
The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle were introduced, with the entropy and volume held fixed. If a system contains more than one species of particle, there is a separate chemical potential associated with each species, defined as the change in energy when the number of particles of that species is increased by one. The chemical potential is a fundamental parameter in thermodynamics and it is conjugate to the particle number.
The chemical potential is particularly important when studying systems of reacting particles. Consider the simplest case of two species, where a particle of species 1 can transform into a particle of species 2 and vice versa. An example of such a system is a supersaturated mixture of water liquid (species 1) and water vapor (species 2). If the system is at equilibrium, the chemical potentials of the two species must be equal. Otherwise, any increase in one chemical potential would result in an irreversible net release of energy of the system in the form of heat (see second law of thermodynamics) when that species of increased potential transformed into the other species, or a net gain of energy (again in the form of heat) if the reverse transformation took place. In chemical reactions, the equilibrium conditions are generally more complicated because more than two species are involved. In this case, the relation between the chemical potentials at equilibrium is given by the law of mass action.
Since the chemical potential is a thermodynamic quantity, it is defined independently of the microscopic behavior of the system, i.e. the properties of the constituent particles. However, some systems contain important variables that are equivalent to the chemical potential. In Fermi gases and Fermi liquids, the chemical potential at zero temperature is equivalent to the Fermi energy. In electronic systems, the chemical potential is related to an effective electrical potential.
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Precise definition
Consider a thermodynamic system containing n constituent species. Its total internal energy U is postulated to be a function of the entropy S, the volume V, and the number of particles of each species N1,..., Nn:
U \equiv U(S,V,N_1,..N_n)
By referring to U as the internal energy, it is emphasized that the energy contributions resulting from the interactions between the system and external objects are excluded. For example, the gravitational potential energy of the system with the Earth are not included in U.
The chemical potential of the i-th species, μi is defined as the partial derivative
\mu_i = \left( \frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}
where the subscripts simply emphasize that the entropy, volume, and the other particle numbers are to be kept constant.
In real systems, it is usually difficult to hold the entropy fixed, since this involves good thermal insulation. It is therefore more convenient to use the Helmholtz free energy A, which is a function of the temperature T, volume, and particle numbers:
A \equiv A(T,V,N_1,..N_n)
In terms of the Helmholtz free energy, the chemical potential is
\mu_i = \left( \frac{\partial A}{\partial N_i} \right)_{T,V, N_{j \ne i}}
Laboratory experiments are often performed under conditions of constant temperature and pressure. Under these conditions, the chemical potential is the partial derivative of the Gibbs free energy with respect to number of particles
\mu_i=\left(\frac{\partial G}{\partial N_i}\right)_{T,p,N_{j\neq i}}
A similar expression for the chemical potential can be written in terms of partial derivative of the enthalpy (under conditions of constant entropy and pressure).
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Relativistic Chemical Potential
For relativistic systems (systems in which the rest mass is much smaller than the equivalent thermal energy) the chemical potential is related to symmetries and charges. Each conserved charge is associated with a chemical potential. Thus, in a gas of photons and phonons, there is no chemical potential. However, if the temperature of such a system were to rise above the threshold for pair production of electrons, then it might be sensible to add a chemical potential for the electrical charge. This would control the electric charge density of the system, and hence the excess of electrons over positrons, but not the number of photons. In the context in which one meets a phonon gas, temperatures high enough to pair produce other particles are seldom relevant. QCD matter is the prime example of a system in which many such chemical potentials appear.
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Electronic Chemical Potential
The electronic chemical potential is the functional derivative of the density functional with respect to the electron density.
\mu(\mathbf{r})=\left[ \frac{\delta E[\rho]}{\delta \rho(\mathbf{r})}\right]_{\rho=\rho_{ref}}
Formally, a functional derivative yields many functions, but is a particular function when evaluated about a reference electron density - just as a derivate yields a function, but is a particular number when evaluated about a reference point. The density functional is written as
E[\rho] = \int \rho(\mathbf{r})\nu(\mathbf{r})d^3r + F[\rho]
where \nu(\mathbf{r}) is the external potential, e.g., the electrostatic potential of the nuclei and applied fields, and F is the Universal functional, which describes the electron-electron interactions, e.g., electron Coulomb repulsion, kinetic energy, and the non-classical effects of exchange and correlation. With this general definition of the density functional, the chemical potential is written as
\mu(\mathbf{r}) = \nu(\mathbf{r})+\left[\frac{\delta F[\rho]}{\delta\rho(\mathbf{r})}\right]_{\rho=\rho_{ref}}
Thus, the electronic chemical potential is the effective electrostatic potential experienced by the electron density.
The ground state electron density is determined by a constrained variational optimization of the electronic energy. The Lagrange multiplier enforcing the density normalization constraint is also called the chemical potential, i.e.,
\delta\left\{E[\rho]-\mu\left(\int\rho(\mathbf{r})d^3r-N\right)\right\}=0
where N is the number of electrons in the system and μ is the Lagrange multiplier enforcing the constraint. When this variational statement is satisfied, the terms within the curly brackets obey the property
\left[\frac{\delta E[\rho]}{\delta\rho(\mathbf{r})}\right]_{\rho=\rho_{0}} - \mu \left[\frac{\delta N[\rho]}{\delta\rho(\mathbf{r})}\right]_{\rho=\rho_{0}}=0
where the reference density is the density that minimizes the energy. This expression simplifies to
\left[\frac{\delta E[\rho]}{\delta\rho(\mathbf{r})}\right]_{\rho=\rho_{0}}=\mu
The Lagrange multiplier enforcing the constraint is, by construction, a constant; however, the functional derivative is, formally, a function. Therefore, when the density minimizes the electronic energy, the chemical potential has the same value at every point in space. The gradient of the chemical potential is an effective electric field. An electric field describes the force per unit charge as a function of space. Therefore, when the density is the ground state density, the electron density is stationary, because the gradient of the chemical potential (which is invariant with respect to position) is zero everywhere, i.e., all forces are balanced. As the density undergoes a change from a non-ground state density to the ground state density, it is said to undergo a process of chemical potential equalization.
The chemical potential of an atom is sometimes said to be the negative of the atom's electronegativity. Similarly the process of chemical potential equalization is sometimes referred to as the process of electronegativity equalization. This connection comes from the Mulliken definition of electronegativity. By inserting the energetic definitions of the ionization potential and electron affinity into the Mulliken electronegativity, it is possible to show that the Mulliken chemical potential is a finite difference approximation of the electronic energy with respect to the number of electrons., i.e.,
\mu_{Mulliken}=-\chi_{Mulliken}=-\frac{IP+EA}{2}=\left[\frac{\delta E[N]}{\delta N}\right]_{N=N_0}
where IP and EA are the ionization potential and electron affinity of the atom, respectively. |
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