Cornish–Fisher expansion

Thomas–Fermi screening is a theoretical approach to calculating the effects of electric field screening by electrons in a solid. It is a special case of the more general Lindhard theory; in particular, Thomas–Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit.

The Thomas-Fermi wavevector (in Gaussian-cgs units) is

$k_{0}^{2}=4\pi e^{2}{\frac {\partial n}{\partial \mu }}$

where μ is the chemical potential (fermi level), n is the electron concentration and e is the elementary charge.

Under many circumstances, including semiconductors that are not too heavily doped, n∝e^μ/k_BT, where k_B is Boltzmann constant and T is temperature. In this case,

$k_{0}^{2}=4\pi e^{2}n/(k_{B}T)$

i.e. 1/k₀ is given by the familiar formula for Debye length.

For more details and discussion, including the one-dimensional and two-dimensional cases, see the article: Lindhard theory.

Derivation

Relation between electron density and internal chemical potential

The internal chemical potential (closely related to fermi level, see below) of a system of electrons describes how much energy is required to put an extra electron into the system, neglecting electrical potential energy. A basic fact is: As the number of electrons in the system increases (other things equal), the internal chemical potential increases. This is largely because electrons satisfy the Pauli exclusion principle: Lower-energy electron states are already full, so the new electrons must occupy higher- and higher-energy states. (However, this fact is true quite generally, regardless of the Pauli exclusion principle.)

The relationship is described by a function $n(\mu )$, where n, the electron density, is a function of μ, the internal chemical potential. The exact functional form depends on the system. For example, for a three-dimensional noninteracting electron gas at absolute zero temperature, the relation is $n(\mu )\propto \mu ^{3/2}$. Proof: Including spin degeneracy,

$n=2{\frac {1}{(2\pi )^{3}}}{\frac {4}{3}}\pi k_{F}^{3}\quad ,\quad \mu ={\frac {\hbar ^{2}k_{F}^{2}}{2m}}\quad ,\quad n\propto \mu ^{3/2}.$

(in this context—i.e., absolute zero—the internal chemical potential is more commonly called Fermi energy).

As another example, for an n-type semiconductor at low to moderate electron concentration, $n(\mu )\propto e^{\mu /k_{B}T}$ where k_B is Boltzmann constant and T is temperature.

Local approximation

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The main assumption in the Thomas-Fermi model is that there is an internal chemical potential at each point r that depends only on the electron concentration at the same point r. This cannot be exactly true because of the Heisenberg uncertainty principle. No electron can exist at a single point; each is spread out into a wavepacket of size ≈ 1 / k_F, where k_F is the Fermi wavenumber, i.e. a typical wavenumber for the states at the Fermi surface. Therefore it cannot be possible to define a chemical potential at a single point, independent of the electron density at nearby points.

Nevertheless, the Thomas-Fermi model is likely to be a reasonably accurate approximation as long as the potential does not vary much over lengths comparable or smaller than 1 / k_F. (This length usually corresponds to a few atoms in metals.)

Electrons in equilibrium, nonlinear equation

Finally, the Thomas-Fermi model assumes that the electrons are in equilibrium, meaning that the total chemical potential is the same at all points. (In electrochemistry terminology, "the electrochemical potential of electrons is the same at all points". In semiconductor physics terminology, "the fermi level is flat".)

This requires that the variations in internal chemical potential are matched by equal and opposite variations in the electric potential energy. This gives rise to the "basic equation of nonlinear Thomas-Fermi theory":

$\rho ^{\text{induced}}(\mathbf {r} )=-e[n(\mu _{0}+e\phi (\mathbf {r} ))-n(\mu _{0})]$

where n(μ) is the function discussed above (electron density as a function of internal chemical potential), e is the elementary charge, r is the position, and $\rho ^{\text{induced}}(\mathbf {r} )$ is the induced charge at r. The electric potential $\phi$ is defined in such a way that $\phi (\mathbf {r} )=0$ at the points where the material is charge-neutral (the number of electrons is exactly equal to the number of ions), and similarly μ₀ is defined as the internal chemical potential at the points where the material is charge-neutral.

Linearization, dielectric function

If the chemical potential does not vary too much, the above equation can be linearized:

$\rho ^{\text{induced}}(\mathbf {r} )\approx -e^{2}{\frac {\partial n}{\partial \mu }}\phi (\mathbf {r} )$

where $\partial n/\partial \mu$ is evaluated at μ₀ and treated as a constant.

This can be converted into a wavevector-dependent dielectric function:

$\epsilon (\mathbf {q} )=1+{\frac {k_{0}^{2}}{q^{2}}}$ (cgs-Gaussian)

where

$k_{0}={\sqrt {4\pi e^{2}{\frac {\partial n}{\partial \mu }}}}$

At long distances (q→0), the dielectric constant approaches infinity, reflecting the fact that charges get closer and closer to perfectly screened as you observe them from further away.

Example: A point charge

If a point charge Q is placed at r=0 in a solid, what field will it produce, taking electron screening into account?

One seeks a self-consistent solution to two equations:

• The Thomas–Fermi screening formula gives the charge density at each point r as a function of the potential $\phi (\mathbf {r} )$ at that point.
• The Poisson equation (derived from Gauss's law) relates the second derivative of the potential to the charge density.

For the nonlinear Thomas-Fermi formula, solving these simultaneously can be difficult, and usually there is no analytical solution. However, the linearized formula has a simple solution:

$\phi (\mathbf {r} )={\frac {Q}{r}}e^{-k_{0}r}$ (cgs-Gaussian)

With k₀=0 (no screening), this becomes the familiar Coulomb's law.

Note that there may be dielectric permittivity in addition to the screening discussed here; for example due to the polarization of immobile core electrons. In that case, replace Q by Q/ε, where ε is the relative permittivity due to these other contributions.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.