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| [[Image:hfspec1.jpg| 400px | right | thumb | Computed regular (non-chaotic) [[Rydberg atom]] energy level spectra of hydrogen in an electric field near n=15 for [[magnetic quantum number]] m=0. Each [[Principal quantum number|n-level]] consists of n-1 [[Degenerate energy level|degenerate sublevels]]; application of an [[electric field]] breaks the degeneracy. Note that energy levels can cross due to underlying symmetries of dynamical motion.<ref name=Courtney1995>{{cite journal |last=Courtney |first=Michael |coauthors=Neal Spellmeyer, Hong Jiao, and Daniel Kleppner |title=Classical, semiclassical, and quantum dynamics of lithium in an electric field |journal=Physical Review A |year=1995 |volume=51 |issue=5 |pages=3604-3620 |doi=10.1103/PhysRevA.51.3604|bibcode = 1995PhRvA..51.3604C }}</ref>]]
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| :''"Stark shift" is not to be confused with [[Stokes shift]].''
| | Here is my web-site :: [http://circuspartypanama.com clash of clans hack cydia] |
| The '''Stark effect''' is the shifting and splitting of [[spectral line]]s of atoms and molecules due to presence of an external [[electric field]]. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects. The first-order effect is linear in the applied electric field, while the second-order effect is quadratic in the field.
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| The Stark effect is responsible for the pressure broadening (Stark broadening) of spectral lines by charged particles. When the split/shifted lines appear in absorption, the effect is called the '''inverse Stark effect'''.
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| The Stark effect is the electric analogue of the [[Zeeman effect]] where a spectral line is split into several components due to the presence of a [[magnetic field]].
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| The Stark effect can be explained with fully quantum mechanical approaches, but it has also been a fertile testing ground for semiclassical methods.
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| [[Image:lfspec1.jpg| 400px | right | thumb | Computed chaotic [[Rydberg atom]] energy level spectra of lithium in an electric field near n=15 for m=0. Note that energy levels cannot cross due to the ionic core (and resulting quantum defect) breaking symmetries of dynamical motion.<ref name=Courtney1995 />)]]
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| ==History==
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| The effect is named after [[Johannes Stark]], who discovered it in 1913. It was independently discovered in the same year by the Italian physicist [[Antonino Lo Surdo]], and in Italy it is thus sometimes called the '''Stark-Lo Surdo effect'''. The discovery of this effect contributed importantly to the development of quantum theory. Ironically, soon after their discoveries, both Stark and Lo Surdo rejected developments in modern physics and allied themselves with the political and racial programs of Hitler and Mussolini.<ref>{{cite journal |last1=Leone |first1=M. |last2=Paoletti |first2=A. |last3=Robotti | first3=N. |year=2004 |title=A Simultaneous Discovery: The Case of Johannes Stark and Antonino Lo Surdo |journal=Physics in Perspective |volume=6 |issue=3 |pages=271–294 |publisher=Springer |doi=10.1007/s00016-003-0170-2 |accessdate=29 May 2013|bibcode = 2004PhP.....6..271L }}</ref>
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| Inspired by the magnetic [[Zeeman effect]], and especially by Lorentz's explanation of it, [[Woldemar Voigt]]<ref>W. Voigt, ''Ueber das Elektrische Analogon des Zeemaneffectes'' (On the electric analogue of the Zeeman effect), Annalen der Physik, vol. '''4''',
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| pp. 197-208 (1901).</ref> performed classical mechanical calculations of quasi-elastically bound electrons in an electric field. By using experimental indices of refraction he gave an estimate of the Stark splittings. This estimate was a few orders of magnitude too low. Not deterred by this prediction, Stark<ref>J. Stark, ''Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien I. Quereffekt'' (Observations of the effect of the electric field on spectral lines I. Transverse effect), Annalen der Physik, vol. '''43''', pp. 965-983 (1914). Published earlier (1913) in Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss.</ref> undertook measurements on excited states of the hydrogen atom and succeeded in observing splittings.
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| By the use of the Bohr-Sommerfeld ("old") quantum theory [[Paul Sophus Epstein|Paul Epstein]]<ref>P. S. Epstein, ''Zur Theorie des Starkeffektes'', Annalen der Physik, vol. '''50''', pp. 489-520 (1916)</ref> and [[Karl Schwarzschild]]<ref>K. Schwarzschild, Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss. April 1916, p. 548</ref> were independently able to derive equations for the linear and quadratic Stark effect in [[hydrogen]]. Four years later, [[Hendrik Anthony Kramers|Hendrik Kramers]]<ref>H. A. Kramers, Roy. Danish Academy, ''Intensities of Spectral Lines. On the Application of the Quantum Theory to the Problem of Relative Intensities of the Components of the Fine Structure and of the Stark Effect of the Lines of the Hydrogen Spectrum'', p. 287 (1919);''Über den Einfluß eines elektrischen Feldes auf die Feinstruktur der Wasserstofflinien'' (On the influence of an electric field on the fine structure of hydrogen lines), Zeitschrift für Physik, vol. '''3''', pp. 199-223 (1920)</ref> derived formulas for intensities of spectral transitions. Kramers also included the effect of [[fine structure]], which includes corrections for relativistic kinetic energy and coupling between electron spin and orbit. The first quantum mechanical treatment (in the framework of Heisenberg's [[matrix mechanics]]) was by Wolfgang Pauli.<ref>W. Pauli, ''Über dass Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik'' (On the hydrogen spectrum from the point of view of the new quantum mechanics). Zeitschrift für Physik, vol. '''36''' p. 336 (1926)</ref> Erwin Schrödinger discussed at length the Stark effect in his third paper<ref>E. Schrödinger, ''Quantisierung als Eigenwertproblem'',
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| Annalen der Physik, vol. '''385''' Issue 13, 437-490 (1926)</ref> on quantum theory (in which he introduced his perturbation theory), once in the manner of the 1916 work of Epstein (but generalized from the old to the new quantum theory) and once by his (first-order) perturbation approach.
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| Finally, Epstein<ref>P. S. Epstein, ''The Stark Effect from the Point of View of Schroedinger's Quantum Theory'', Physical Review, vol '''28''', pp. 695-710 (1926)</ref> reconsidered the linear and quadratic Stark effect from the point of view of the new quantum theory. He derived equations for the line intensities which were a decided improvement over Kramers' results obtained by the old quantum theory.
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| While first-order perturbation effects for the Stark effect in hydrogen are in agreement for the Bohr-Sommerfeld model and the [[quantum mechanics|quantum-mechanical]] theory of the atom, higher order effects are not.{{Citation needed|date=December 2011}} Measurements of the Stark effect under high field strengths confirmed the correctness of the quantum theory over the Bohr model.
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| == Mechanism ==
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| ===Overview===
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| An electric field pointing from left to right, for example, tends to pull nuclei to the right and electrons to the left. In another way of viewing it, if an electronic state has its electron disproportionately to the left, its energy is lowered, while if it has the electron disproportionately to the right, its energy is raised.
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| Other things equal, the effect of the electric field is greater for outer [[electron shell]]s, because the electron is more distant from the nucleus, so it travels farther left and farther right.
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| The Stark effect can lead to splitting of [[degenerate energy level]]s. For example, in the [[Bohr model]], an electron has the same energy whether it is in the [[electron shell|2s]] state or any of the [[electron shell|2p]] states. However, in an electric field, there will be [[Orbital hybridisation|hybrid orbitals]] (also called [[quantum superposition]]s) of the 2s and 2p states where the electron tends to be to the left, which will acquire a lower energy, and other hybrid orbitals where the electron tends to be to the right, which will acquire a higher energy. Therefore, the formerly degenerate energy levels will split into slightly lower and slightly higher energy levels.
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| ===Classical electrostatics===
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| The Stark effect originates from the interaction between a charge distribution (atom or molecule) and an external [[electric field]]. Before turning to quantum mechanics we describe the interaction
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| classically and consider a continuous charge distribution ρ('''r''').
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| If this charge distribution is non-polarizable its interaction energy with an external [[Electrostatic#Electrostatic potential|electrostatic potential]] ''V''('''r''') is
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| :<math> E_{\mathrm{int}} = \int \rho(\mathbf{r}) V(\mathbf{r}) d\mathbf{r}^3</math>.
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| If the electric field is of macroscopic origin and the charge distribution is microscopic, it is reasonable to assume that the electric field is uniform over the charge distribution. That is, ''V'' is given by a two-term [[Taylor expansion]],
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| :<math>
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| V(\mathbf{r}) = V(\mathbf{0}) - \sum_{i=1}^3 r_i F_i </math>, with the electric field: <math>F_i \equiv -\left. \left(\frac{\partial V}{\partial r_i} \right)\right|_{\mathbf{0}}</math>,
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| where we took the origin '''0''' somewhere within ρ.
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| Setting ''V''(0) as the zero energy, the interaction becomes
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| :<math>E_{\mathrm{int}} = - \sum_{i=1}^3 F_i \int \rho(\mathbf{r}) r_i d\mathbf{r} \equiv
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| - \sum_{i=1}^3 F_i \mu_i = - \mathbf{F}\cdot \boldsymbol{\mu}</math>.
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| Here we have introduced the [[dipole|dipole moment]] '''μ''' of ρ as an integral over the charge distribution. In case ρ consists of ''N'' point charges ''q''<sub>j</sub> this definition becomes a sum | |
| :<math>\boldsymbol{\mu} \equiv \sum_{j=1}^N q_j \mathbf{r}_j</math>.
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| ===Perturbation theory===
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| Turning now to quantum mechanics we see an atom or a molecule as a collection of point charges (electrons and nuclei), so that the second definition of the dipole applies. The interaction of atom or molecule with a uniform external field is described by the operator
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| :<math> V_{\mathrm{int}} = - \mathbf{F}\cdot \boldsymbol{\mu}.
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| </math>
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| This operator is used as a perturbation in first- and second-order [[perturbation theory]] to account for the first- and second-order Stark effect.
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| ====First order====
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| Let the unperturbed atom or molecule be in a ''g''-fold degenerate state with orthonormal zeroth-order state functions <math> \psi^0_1, \ldots, \psi^0_g </math>. (Non-degeneracy is the special case ''g'' = 1). According to perturbation theory the first-order energies are the eigenvalues of the ''g'' x ''g'' matrix with general element
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| :<math>
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| (\mathbf{V}_{\mathrm{int}})_{kl} = \langle \psi^0_k | V_{\mathrm{int}} | \psi^0_l \rangle =
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| -\mathbf{F}\cdot \langle \psi^0_k | \boldsymbol{\mu} | \psi^0_l \rangle,
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| \qquad k,l=1,\ldots, g.
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| </math>
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| If ''g'' = 1 (as is often the case for electronic states of molecules) the first-order energy becomes proportional to the expectation (average) value of the dipole operator <math>\boldsymbol{\mu}</math>,
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| :<math>
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| E^{(1)} = -\mathbf{F}\cdot \langle \psi^0_1 | \boldsymbol{\mu} | \psi^0_1 \rangle =
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| -\mathbf{F}\cdot \langle \boldsymbol{\mu} \rangle.
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| </math>
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| Because a dipole moment is a [[Polar and axial vectors|polar vector]], the diagonal elements of the perturbation matrix '''V'''<sub>int</sub> vanish for systems with an [[parity (physics)|inversion center]] (such as atoms). Molecules with an inversion center in a non-degenerate electronic state do not have a (permanent) dipole and hence do not show a linear Stark effect.
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| In order to obtain a non-zero matrix '''V'''<sub>int</sub> for systems with an inversion center it is necessary that some of the unperturbed functions <math> \psi^0_i</math> have opposite parity (obtain plus and minus under inversion), because only functions of opposite parity give non-vanishing matrix elements. Degenerate zeroth-order states of opposite parity occur for excited hydrogen-like (one-electron) atoms. Such atoms have the principal quantum number ''n'' among their quantum numbers. The excited state of hydrogen-like atoms with principal quantum number ''n'' is ''n''<sup>2</sup>-fold degenerate and
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| :<math>
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| n^2 = \sum_{\ell=0}^{n-1} (2 \ell + 1),
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| </math>
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| where <math>\ell </math> is the azimuthal (angular momentum) quantum number. For instance, the excited ''n'' = 4 state contains the following <math>\ell</math> states,
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| :<math>
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| 16 = 1 + 3 + 5 +7 \;\; \Longrightarrow\;\; n=4\;\hbox{contains}\; s\oplus p\oplus d\oplus f.
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| </math>
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| The one-electron states with even <math>\ell</math> are even under parity, while those with odd <math>\ell</math> are odd under parity. Hence hydrogen-like atoms with ''n''>1 show first-order Stark effect.
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| The first-order Stark effect occurs in rotational transitions of [[Rotational spectroscopy#Classification of molecules based on rotational behavior|symmetric top molecules]] (but not for linear and asymmetric molecules). In first approximation a molecule may be seen as a rigid rotor. A symmetric top [[rigid rotor]] has the unperturbed eigenstates
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| :<math>
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| |JKM \rangle = (D^J_{MK})^* \quad\mathrm{with}\quad M,K= -J,-J+1,\dots,J
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| </math>
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| with 2(2''J''+1)-fold degenerate energy for |K| > 0 and (2''J''+1)-fold degenerate energy for K=0.
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| Here ''D''<sup>''J''</sup><sub>''MK''</sub> is an element of the [[Wigner D-matrix]]. The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings
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| in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent [[electric dipole moment]] of the symmetric top molecule.
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| ====Second order====
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| As stated, the quadratic Stark effect is described by second-order perturbation theory.
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| The zeroth-order problems
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| :<math>
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| H^{(0)} \psi^0_k = E^{(0)}_k \psi^0_k, \quad k=0,1, \ldots, \quad E^{(0)}_0 < E^{(0)}_1 \le E^{(0)}_2, \dots
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| </math>
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| are assumed to be solved. It is usual to assume that the zeroth-order state to be perturbed is non-degenerate. If we take the ground state as the non-degenerate state under consideration (for hydrogen-like atoms: ''n'' = 1), perturbation theory gives
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| :<math> | |
| E^{(2)} = \sum_{k>0} \frac{\langle \psi^0_0 | V_\mathrm{int} | \psi^0_k \rangle \langle \psi^0_k | V_\mathrm{int} | \psi^0_0 \rangle}{E^{(0)}_0 - E^{(0)}_k}
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| =- \frac{1}{2} \sum_{i,j=1}^3 F_i \alpha_{ij} F_j
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| </math>
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| with the components of the [[polarizability]] tensor α defined by
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| :<math> | |
| \alpha_{ij}\equiv -2\sum_{k>0} \frac{\langle \psi^0_0 | \mu_i | \psi^0_k \rangle \langle \psi^0_k | \mu_j | \psi^0_0\rangle}{E^{(0)}_0 - E^{(0)}_k}.
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| </math>
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| The energy ''E''<sup>(2)</sup> gives the quadratic Stark effect.
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| Because of their spherical symmetry the polarizability tensor of atoms is isotropic,
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| :<math>
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| \alpha_{ij} = \alpha_0 \delta_{ij} \Longrightarrow E^{(2)} = -\frac{1}{2} \alpha_0 F^2,
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| </math>
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| which is the quadratic Stark shift for atoms. For many molecules this expression is not too bad an approximation, because molecular tensors are often reasonably isotropic.
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| ====Problems====
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| The perturbative treatment of the Stark effect has some problems. In the presence of an electric field, states of atoms and molecules that were previously bound ([[square-integrable]]), become formally (non-square-integrable) [[resonance]]s of finite width.
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| These resonances may decay in finite time via field ionization. For low lying states and not too strong fields the decay times are so long, however, that for all practical purposes the system can be regarded as bound. For highly excited states and/or very strong fields ionization may have to be accounted for. (See also the article on the [[Rydberg atom]]).
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| == Quantum-confined Stark effect ==
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| {{main|Quantum-confined Stark effect}}
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| In a semiconductor heterostructure, where a small bandgap material is sandwiched between two layers of a larger bandgap material, the Stark effect can be dramatically enhanced by bound [[excitons]]. This is because the [[electron]] and [[Electron hole|hole]] which form the exciton are pulled in opposite directions by the applied electric field, but they remain confined in the smaller bandgap material, so the exciton is not merely pulled apart by the field. The quantum-confined Stark effect is widely used for semiconductor-based optical modulators, particularly for [[optical fiber]] communications.
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| == See also ==
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| *[[Zeeman splitting]]
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| *[[Autler-Townes effect]]
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| *[[Stark spectroscopy]]
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| *[[Inglis-Teller Equation]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{cite book | author=E. U. Condon and G. H. Shortley|title=The Theory of Atomic Spectra | publisher=Cambridge University Press |year=1935 |isbn=0-521-09209-4 }} ''(Chapter 17 provides a comprehensive treatment, as of 1935.)''
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| * {{cite book | author=H. W. Kroto|title=Molecular Rotation Spectra| publisher=Dover, New York |year=1992 |isbn=0-486-67259-X }} ''(Stark effect for rotating molecules)''
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| * {{cite book | author=H. Friedrich|title=Theoretical Atomic Physics| publisher=Springer-Verlag, Berlin |year=1990 |isbn=0-387-54179-9 }} ''(Stark effect for atoms)''
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| ==Further reading==
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| * {{cite book |author= Edmond Taylor Whitaker |title=A History of the Theories of Aether and Electricity. II. The Modern Theories (1800-1950) |publisher=American Institute of Physics |year=1987 |isbn=978-0-88318-523-0}} ''(Early history of the Stark effect)''
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| {{DEFAULTSORT:Stark Effect}}
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| [[Category:Atomic physics]]
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| [[Category:Foundational quantum physics]]
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| [[Category:Physical phenomena]]
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| [[Category:Spectroscopy]]
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