The **Stark effect** is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the pressure broadening (Stark broadening) of spectral lines by charged particles in plasmas. For most spectral lines, the Stark effect is either linear (proportional to the applied electric field) or quadratic with a high accuracy.

- History
- Mechanism
- Overview
- Multipole expansion
- Perturbation theory
- Applications
- See also
- References
- Further reading

The Stark effect can be observed both for emission and absorption lines. The latter is sometimes called the **inverse Stark effect**, but this term is no longer used in the modern literature.

The effect is named after the German physicist 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 and Stark was rewarded with the Nobel Prize in Physics in the year 1919.

Inspired by the magnetic Zeeman effect, and especially by Hendrik Lorentz's explanation of it, Woldemar Voigt ^{ [2] } 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 undertook measurements^{ [3] } on excited states of the hydrogen atom and succeeded in observing splittings.

By the use of the Bohr–Sommerfeld ("old") quantum theory, Paul Epstein ^{ [4] } and Karl Schwarzschild ^{ [5] } were independently able to derive equations for the linear and quadratic Stark effect in hydrogen. Four years later, Hendrik Kramers ^{ [6] } derived formulas for intensities of spectral transitions. Kramers also included the effect of fine structure, with corrections for relativistic kinetic energy and coupling between electron spin and orbital motion. The first quantum mechanical treatment (in the framework of Werner Heisenberg's matrix mechanics) was by Wolfgang Pauli.^{ [7] } Erwin Schrödinger discussed at length the Stark effect in his third paper^{ [8] } 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. Finally, Epstein reconsidered^{ [9] } 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's results obtained by the old quantum theory.

While the first-order-perturbation (linear) Stark effect in hydrogen is in agreement with both the old Bohr–Sommerfeld model and the quantum-mechanical theory of the atom, higher-order corrections are not.^{ [9] } Measurements of the Stark effect under high field strengths confirmed the correctness of the new quantum theory.

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.

Other things being equal, the effect of the electric field is greater for outer electron shells, because the electron is more distant from the nucleus, so it travels farther left and farther right.

The Stark effect can lead to splitting of degenerate energy levels. For example, in the Bohr model, an electron has the same energy whether it is in the 2s state or any of the 2p states. However, in an electric field, there will be hybrid orbitals (also called quantum superpositions) 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.

The Stark effect originates from the interaction between a charge distribution (atom or molecule) and an external electric field. The interaction energy of a continuous charge distribution , confined within a finite volume , with an external electrostatic potential is

- .

This expression is valid classically and quantum-mechanically alike. If the potential varies weakly over the charge distribution, the multipole expansion converges fast, so only a few first terms give an accurate approximation. Namely, keeping only the zero- and first-order terms,

- ,

where we introduced the electric field and assumed the origin **0** to be somewhere within . Therefore, the interaction becomes

- ,

where and are, respectively, the total charge (zero moment) and the dipole moment of the charge distribution.

Classical macroscopic objects are usually neutral or quasi-neutral (), so the first, monopole, term in the expression above is identically zero. This is also the case for a neutral atom or molecule. However, for an ion this is no longer true. Nevertheless, it is often justified to omit it in this case, too. Indeed, the Stark effect is observed in spectral lines, which are emitted when an electron "jumps" between two bound states. Since such a transition only alters the internal degrees of freedom of the radiator but not its charge, the effects of the monopole interaction on the initial and final states exactly cancel each other.

Turning now to quantum mechanics an atom or a molecule can be thought of 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

This operator is used as a perturbation in first- and second-order perturbation theory to account for the first- and second-order Stark effect.

Let the unperturbed atom or molecule be in a *g*-fold degenerate state with orthonormal zeroth-order state functions . (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

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 ,

Because the electric dipole moment is a vector (tensor of the first rank), the diagonal elements of the perturbation matrix **V**_{int} vanish between states with a certain parity. Atoms and molecules possessing inversion symmetry do not have a (permanent) dipole moment and hence do not show a linear Stark effect.

In order to obtain a non-zero matrix **V**_{int} for systems with an inversion center it is necessary that some of the unperturbed functions 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 or Rydberg states. Neglecting fine-structure effects, such a state with the principal quantum number *n* is *n*^{2}-fold degenerate and

where is the azimuthal (angular momentum) quantum number. For instance, the excited *n* = 4 state contains the following states,

The one-electron states with even are even under parity, while those with odd are odd under parity. Hence hydrogen-like atoms with *n*>1 show first-order Stark effect.

The first-order Stark effect occurs in rotational transitions of 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

with 2(2*J*+1)-fold degenerate energy for |K| > 0 and (2*J*+1)-fold degenerate energy for K=0. Here *D*^{J}_{MK} 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 in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent electric dipole moment of the symmetric top molecule.

As stated, the quadratic Stark effect is described by second-order perturbation theory. The zeroth-order eigenproblem

is assumed to be solved. The perturbation theory gives

with the components of the polarizability tensor α defined by

The energy *E*^{(2)} gives the quadratic Stark effect.

Neglecting the hyperfine structure (which is often justified — unless extremely weak electric fields are considered), the polarizability tensor of atoms is isotropic,

For some molecules this expression is a reasonable approximation, too.

It is important to note that for the ground state is *always* positive, i.e., the quadratic Stark shift is always negative.

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) resonances of finite width. 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).

The Stark effect is at the basis of the spectral shift measured for voltage-sensitive dyes used for imaging of the firing activity of neurons.^{ [10] }

In quantum mechanics, the **Hamiltonian** of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's *energy spectrum* or its set of *energy eigenvalues*, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

In theoretical physics, **quantum field theory** (**QFT**) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics, but *not* general relativity's description of gravity. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles.

In particle physics, **quantum electrodynamics** (**QED**) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

The **Zeeman effect**, named after Dutch physicist Pieter Zeeman, is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. Also similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden, as governed by the selection rules.

The **Klein–Gordon equation** is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

**Density-functional theory** (**DFT**) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

In quantum mechanics, **perturbation theory** is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system can be expressed as "corrections" to those of the simple system. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

A **rotational transition** is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

In quantum physics, **Fermi's golden rule** is a formula that describes the transition rate from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

In atomic physics, the **electron magnetic moment**, or more specifically the **electron magnetic dipole moment**, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10^{−24} J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10^{13}.

In quantum mechanics, the **Hellmann–Feynman theorem** relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, the **momentum operator** is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

In quantum mechanics, an energy level is **degenerate** if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

In quantum chemistry, ** n-electron valence state perturbation theory (NEVPT)** is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known second-order Møller–Plesset perturbation theory to multireference Complete Active Space cases. The theory is directly integrated into many quantum chemistry packages such as MOLCAS, Molpro, DALTON and ORCA.

In spectroscopy, the **Autler–Townes effect**, is a type of dynamical Stark effects corresponding to the case when an oscillating electric field is tuned in resonance to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line. The AC stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes.

A normalized **1s Slater-type function** is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. It is particularly important as the accurate quantum theory description of the smallest free atom, hydrogen. It has the form

The **quantum-confined Stark effect** (**QCSE**) describes the effect of an external electric field upon the light absorption spectrum or emission spectrum of a quantum well (QW). In the absence of an external electric field, electrons and holes within the quantum well may only occupy states within a discrete set of energy subbands. Only a discrete set of frequencies of light may be absorbed or emitted by the system. When an external electric field is applied, the electron states shift to lower energies, while the hole states shift to higher energies. This reduces the permitted light absorption or emission frequencies. Additionally, the external electric field shifts electrons and holes to opposite sides of the well, decreasing the overlap integral, which in turn reduces the recombination efficiency of the system. The spatial separation between the electrons and holes is limited by the presence of the potential barriers around the quantum well, meaning that excitons are able to exist in the system even under the influence of an electric field. The quantum-confined Stark effect is used in QCSE optical modulators, which allow optical communications signals to be switched on and off rapidly.

In quantum mechanics, **orbital magnetization**, **M**_{orb}, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, **M**_{spin}, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

In condensed matter and atomic physics, **Van Vleck paramagnetism** refers to a positive and temperature-independent contribution to the magnetic susceptibility of a material, derived from second order corrections to the Zeeman interaction. The quantum mechanical theory was developed by John Hasbrouck Van Vleck between the 1920s and the 1930s to explain the magnetic response of gaseous nitric oxide and of rare-earth salts. Alongside other magnetic effects like Paul Langevin's formulas for paramagnetism and diamagnetism, Van Vleck discovered an additional paramagnetic contribution of the same order as Langevin's diamagnetism. Van Vleck contribution is usually important for systems with one electron short of being half filled and this contribution vanishes for elements with closed shells.

- ↑ Courtney, Michael; Neal Spellmeyer; Hong Jiao; Daniel Kleppner (1995). "Classical, semiclassical, and quantum dynamics of lithium in an electric field".
*Physical Review A*.**51**(5): 3604–3620. Bibcode:1995PhRvA..51.3604C. doi:10.1103/PhysRevA.51.3604. PMID 9912027. - ↑ W. Voigt,
*Ueber das Elektrische Analogon des Zeemaneffectes*(On the electric analogue of the Zeeman effect), Annalen der Physik, vol.**309**, pp. 197–208 (1901). - ↑ 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. - ↑ P. S. Epstein,
*Zur Theorie des Starkeffektes*, Annalen der Physik, vol.**50**, pp. 489–520 (1916) - ↑ K. Schwarzschild, Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss. April 1916, p. 548
- ↑ 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) - ↑ 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) - ↑ E. Schrödinger,
*Quantisierung als Eigenwertproblem*, Annalen der Physik, vol.**385**Issue 13, 437–490 (1926) - 1 2 P. S. Epstein,
*The Stark Effect from the Point of View of Schroedinger's Quantum Theory*, Physical Review, vol**28**, pp. 695–710 (1926) - ↑ Sirbu, Dumitru; Butcher, John B.; Waddell, Paul G.; Andras, Peter; Benniston, Andrew C. (2017-09-18). "Locally Excited State-Charge Transfer State Coupled Dyes as Optically Responsive Neuron Firing Probes" (PDF).
*Chemistry - A European Journal*.**23**(58): 14639–14649. doi:10.1002/chem.201703366. ISSN 0947-6539. PMID 28833695.

- Edmond Taylor Whittaker (1987).
*A History of the Theories of Aether and Electricity. II. The Modern Theories (1800-1950)*. American Institute of Physics. ISBN 978-0-88318-523-0.*(Early history of the Stark effect)* - E. U. Condon & G. H. Shortley (1935).
*The Theory of Atomic Spectra*. Cambridge University Press. ISBN 978-0-521-09209-8.*(Chapter 17 provides a comprehensive treatment, as of 1935.)* - H. Friedrich (1990).
*Theoretical Atomic Physics*. Springer-Verlag, Berlin. ISBN 978-0-387-54179-2.*(Stark effect for atoms)* - H. W. Kroto (1992).
*Molecular Rotation Spectra*. Dover, New York. ISBN 978-0-486-67259-5.*(Stark effect for rotating molecules)*

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