Editing Photoelectric effect (section)

==Emission mechanism==
The photons of a light beam have a characteristic energy, called [[photon energy]], which is proportional to the frequency of the light. In the photoemission process, when an electron within some material absorbs the energy of a photon and acquires more energy than its [[electron binding energy|binding energy]], it is likely to be ejected. If the photon energy is too low, the electron is unable to escape the material. Since an increase in the intensity of low-frequency light will only increase the number of low-energy photons, this change in intensity will not create any single photon with enough energy to dislodge an electron. Moreover, the energy of the emitted electrons will not depend on the intensity of the incoming light of a given frequency, but only on the energy of the individual photons.<ref>{{Cite web |title=The Photoelectric Effect {{!}} Physics |url=https://courses.lumenlearning.com/suny-physics/chapter/29-2-the-photoelectric-effect/ |access-date=2024-07-08 |website=courses.lumenlearning.com}}</ref>

While free electrons can absorb any energy when [[irradiated]] as long as this is followed by an immediate re-emission, like in the [[Compton scattering|Compton effect]], in quantum systems all of the energy from one photon is absorbed—if the process is allowed by [[Introduction to quantum mechanics|quantum mechanics]]—or none at all. Part of the acquired energy is used to liberate the electron from its atomic binding, and the rest contributes to the electron's [[kinetic energy]] as a free particle.<ref name="Ref_Lenard">{{cite journal
 |last1=Lenard |first1=P.
 |year=1902
 |title=Ueber die lichtelektrische Wirkung
 |journal=[[Annalen der Physik]]
 |volume=313 |issue=5 |pages=149–198
 |bibcode=1902AnP...313..149L
 |doi=10.1002/andp.19023130510
|url=https://zenodo.org/record/1424009
 }}</ref><ref name="Ref_Millikan">{{cite journal
 |last1=Millikan |first1=R.
 |year=1914
 |title=A Direct Determination of "''h''."
 |journal=[[Physical Review]]
 |volume=4 |issue=1 |pages=73–75
 |bibcode=1914PhRv....4R..73M
 |doi=10.1103/PhysRev.4.73.2
|url=https://authors.library.caltech.edu/42736/
 }}</ref><ref>{{cite journal
 |last1=Millikan
 |first1=R.
 |year=1916
 |title=A Direct Photoelectric Determination of Planck's "''h''"
 |journal=[[Physical Review]]
 |volume=7
 |issue=3
 |pages=355–388
 |bibcode=1916PhRv....7..355M
 |doi=10.1103/PhysRev.7.355
 |doi-access=free
 }}</ref> Because electrons in a material occupy many different quantum states with different binding energies, and because they can sustain energy losses on their way out of the material, the emitted electrons will have a range of kinetic energies. The electrons from the highest occupied states will have the highest kinetic energy. In metals, those electrons will be emitted from the [[Fermi level]].

When the photoelectron is emitted into a solid rather than into a vacuum, the term ''internal photoemission'' is often used, and emission into a vacuum is distinguished as ''external photoemission''.

===Experimental observation of photoelectric emission===
Even though photoemission can occur from any material, it is most readily observed from metals and other conductors. This is because the process produces a charge imbalance which, if not neutralized by current flow, results in the increasing potential barrier until the emission completely ceases. The energy barrier to photoemission is usually increased by nonconductive oxide layers on metal surfaces, so most practical experiments and devices based on the photoelectric effect use clean metal surfaces in evacuated tubes. Vacuum also helps observing the electrons since it prevents gases from impeding their flow between the electrodes.{{citation needed|date=November 2023}}

Sunlight is an inconsistent and variable source of ultraviolet light. Cloud cover, ozone concentration, altitude, and surface reflection all alter the amount of UV. Laboratory sources of UV are based on [[xenon arc lamps]] or, for more uniform but weaker light, [[fluorescent lamps]].<ref>{{Cite journal |last=Diffey |first=Brian L. |date=2002-09-01 |title=Sources and measurement of ultraviolet radiation |url=https://linkinghub.elsevier.com/retrieve/pii/S1046202302002049 |journal=Methods |volume=28 |issue=1 |pages=4–13 |doi=10.1016/S1046-2023(02)00204-9 |pmid=12231182 |issn=1046-2023}}</ref>
More specialized sources include [[Laser|ultraviolet lasers]]<ref>{{Cite journal |last=Savage |first=Neil |date=February 2007 |title=Ultraviolet lasers |url=https://www.nature.com/articles/nphoton.2006.95 |journal=Nature Photonics |language=en |volume=1 |issue=2 |pages=83–85 |doi=10.1038/nphoton.2006.95 |bibcode=2007NaPho...1...83S |issn=1749-4893|url-access=subscription }}</ref> and [[synchrotron radiation]].<ref>{{Cite journal |last1=Ederer |first1=D.L. |last2=Saloman |first2=E.B. |last3=Ebner |first3=S.C. |last4=Madden |first4=R.P. |date=November 1975 |title=The use of synchrotron radiation as an absolute source of VUV radiation |url=https://nvlpubs.nist.gov/nistpubs/jres/79A/jresv79An6p761_A1b.pdf |journal=Journal of Research of the National Bureau of Standards Section A: Physics and Chemistry |language=en |volume=79A |issue=6 |pages=761–774 |doi=10.6028/jres.079A.032 |issn=0022-4332 |pmc=6589412 |pmid=32184529}}</ref>

[[File: Photoelectric effect measurement apparatus - microscopic picture.svg|thumb|Schematic of the experiment to demonstrate the photoelectric effect. Filtered, monochromatic light of a certain wavelength strikes the emitting electrode (E) inside a vacuum tube. The collector electrode (C) is biased to a voltage V<sub>C</sub> that can be set to attract the emitted electrons, when positive, or prevent any of them from reaching the collector when negative.|alt=]]
The classical setup to observe the photoelectric effect includes a light source, a set of filters to [[Monochromator|monochromatize]] the light, a [[vacuum tube]] transparent to ultraviolet light, an emitting electrode (E) exposed to the light, and a collector (C) whose voltage ''V''<sub>C</sub> can be externally controlled.{{citation needed|date=November 2023}}

A positive external voltage is used to direct the photoemitted electrons onto the collector. If the frequency and the intensity of the incident radiation are fixed, the photoelectric current ''I'' increases with an increase in the positive voltage, as more and more electrons are directed onto the electrode. When no additional photoelectrons can be collected, the photoelectric current attains a saturation value. This current can only increase with the increase of the intensity of light.{{citation needed|date=November 2023}}

An increasing negative voltage prevents all but the highest-energy electrons from reaching the collector. When no current is observed through the tube, the negative voltage has reached the value that is high enough to slow down and stop the most energetic photoelectrons of kinetic energy ''K''<sub>max</sub>. This value of the retarding voltage is called the ''stopping potential'' or ''cut off'' potential ''V''<sub>o</sub>.<ref>{{cite book|last1=Gautreau|first1=R.|title=Schaum's Outline of Modern Physics|last2=Savin|first2=W.|publisher=[[McGraw-Hill]]|year=1999|isbn=0-07-024830-3|edition=2nd|pages=60–61}}</ref> Since the work done by the retarding potential in stopping the electron of charge ''e'' is ''eV''<sub>o</sub>, the following must hold ''eV''<sub>o</sub>&nbsp;=&nbsp;''K''<sub>max.</sub>

The current-voltage curve is sigmoidal, but its exact shape depends on the experimental geometry and the electrode material properties.

For a given metal surface, there exists a certain minimum frequency of incident [[radiation]] below which no photoelectrons are emitted. This frequency is called the ''threshold frequency''. Increasing the frequency of the incident beam increases the maximum kinetic energy of the emitted photoelectrons, and the stopping voltage has to increase. The number of emitted electrons may also change because the [[probability]] that each photon results in an emitted electron is a function of photon energy{{Citation needed|reason=give a more precise reason for this probabilist effect, other than frequency|date=May 2024}}.

An increase in the intensity of the same monochromatic light (so long as the intensity is not too high<ref name=" Zhang1996">{{cite journal|last1=Zhang|first1=Q.|year=1996|title=Intensity dependence of the photoelectric effect induced by a circularly polarized laser beam|journal=[[Physics Letters A]]|volume=216|issue=1–5|page=125|bibcode=1996PhLA..216..125Z|doi=10.1016/0375-9601(96)00259-9}}</ref>), which is proportional to the number of photons impinging on the surface in a given time, increases the rate at which electrons are ejected—the photoelectric current ''I—''but the kinetic energy of the photoelectrons and the stopping voltage remain the same. For a given metal and frequency of incident radiation, the rate at which photoelectrons are ejected is directly proportional to the intensity of the incident light.

The time lag between the incidence of radiation and the emission of a photoelectron is very small, less than 10<sup>−9</sup> second. Angular distribution of the photoelectrons is highly dependent on [[polarization (waves)|polarization]] (the direction of the electric field) of the incident light, as well as the emitting material's quantum properties such as atomic and molecular orbital symmetries and the [[electronic band structure]] of crystalline solids. In materials without macroscopic order, the distribution of electrons tends to peak in the direction of polarization of linearly polarized light.<ref name="Bupp">{{cite journal
 |last1=Bubb |first1=F.
 |year=1924
 |title=Direction of Ejection of Photo-Electrons by Polarized X-rays
 |journal=[[Physical Review]]
 |volume=23|issue=2|pages=137–143
 |bibcode=1924PhRv...23..137B
 |doi=10.1103/PhysRev.23.137
}}</ref> The experimental technique that can measure these distributions to infer the material's properties is [[angle-resolved photoemission spectroscopy]].

===Theoretical explanation===
 [[File:Photoelectric effect - stopping voltage diagram for zinc - English.svg|thumb|Diagram of the maximum kinetic energy as a function of the frequency of light on zinc|alt=]]
In 1905, [[Albert Einstein|Einstein]] proposed a theory of the photoelectric effect using a concept that light consists of tiny packets of energy known as [[photons]] or light quanta. Each packet carries energy <math>h\nu</math> that is proportional to the frequency <math>\nu</math> of the corresponding electromagnetic wave. The proportionality constant <math>h</math> has become known as the [[Planck constant]]. In the range of kinetic energies of the electrons that are removed from their varying atomic bindings by the absorption of a photon of energy <math>h\nu</math>, the highest kinetic energy <math>K_\max</math> is
<math display="block">K_\max = h\,\nu - W.</math>
Here, <math>W</math> is the minimum energy required to remove an electron from the surface of the material. It is called the [[work function]] of the surface and is sometimes denoted <math>\Phi</math> or <math>\varphi</math>.<ref>{{cite book
 |last1=Mee |first1=C.
 |last2=Crundell |first2=M.
 |last3=Arnold |first3=B.
 |last4=Brown |first4=W.
 |year=2011
 |title=International A/AS Level Physics
 |page=241
 |publisher=[[Hodder Education]]
 |isbn=978-0-340-94564-3
}}</ref> If the work function is written as
<math display="block">W = h\,\nu_o,</math>
the formula for the maximum kinetic energy of the ejected electrons becomes
<math display="block">K_\max = h \left(\nu - \nu_o\right).</math>

Kinetic energy is positive, and <math>\nu > \nu_o</math> is required for the photoelectric effect to occur.<ref name="Fromhold1991">{{cite book
 |last=Fromhold |first=A. T.
 |year=1991
 |title=Quantum Mechanics for Applied Physics and Engineering
 |pages=5–6
 |publisher=[[Courier Dover Publications]]
 |isbn=978-0-486-66741-6
}}</ref> The frequency <math>\nu_o</math> is the threshold frequency for the given material. Above that frequency, the maximum kinetic energy of the photoelectrons as well as the stopping voltage in the experiment <math display="inline">V_o = \frac{h}{e} \left(\nu - \nu_o\right)</math> rise linearly with the frequency, and have no dependence on the number of photons and the intensity of the impinging monochromatic light. Einstein's formula, however simple, explained all the phenomenology of the photoelectric effect, and had far-reaching consequences in the [[History of quantum mechanics|development of quantum mechanics]].

=== Photoemission from atoms, molecules and solids ===
Electrons that are bound in atoms, molecules and solids each occupy distinct states of well-defined [[Binding energy|binding energies]]. When light quanta deliver more than this amount of energy to an individual electron, the electron may be emitted into free space with excess (kinetic) energy that is <math>h\nu</math> higher than the electron's binding energy. The distribution of kinetic energies thus reflects the distribution of the binding energies of the electrons in the atomic, molecular or crystalline system: an electron emitted from the state at binding energy <math>E_B</math> is found at kinetic energy <math>E_k=h\nu-E_B</math>. This distribution is one of the main characteristics of the quantum system, and can be used for further studies in quantum chemistry and quantum physics.{{citation needed|date=November 2023}}

====Models of photoemission from solids====
The electronic properties of ordered, crystalline solids are determined by the distribution of the electronic states with respect to energy and momentum—the electronic band structure of the solid. Theoretical models of photoemission from solids show that this distribution is, for the most part, preserved in the photoelectric effect. The phenomenological ''three-step model''<ref>{{Cite journal|last1=Berglund|first1=C. N.|last2=Spicer|first2=W. E.|date=1964-11-16|title=Photoemission Studies of Copper and Silver: Theory|journal=Physical Review|volume=136|issue=4A|pages=A1030–A1044|doi=10.1103/PhysRev.136.A1030|bibcode=1964PhRv..136.1030B}}</ref> for ultraviolet and soft X-ray excitation decomposes the effect into these steps:<ref name="Stefan2003">{{cite book|last=Hüfner|first=S.|title=Photoelectron Spectroscopy: Principles and Applications|publisher=[[Springer (publisher)|Springer]]|year=2003|isbn=3-540-41802-4}}</ref><ref>{{Cite journal|last1=Damascelli|first1=Andrea|last2=Shen|first2=Zhi-Xun|last3=Hussain|first3=Zahid|date=2003-04-17|title=Angle-resolved photoemission spectroscopy of the cuprate superconductors|arxiv=cond-mat/0208504|journal=Reviews of Modern Physics|volume=75|issue=2|pages=473–541|doi=10.1103/RevModPhys.75.473|s2cid=118433150|issn=0034-6861}}</ref><ref name=":0">{{cite journal|last1=Sobota|first1=Jonathan A.|last2=He|first2=Yu|last3=Shen|first3=Zhi-Xun|title=Angle-resolved photoemission studies of quantum materials|journal=Reviews of Modern Physics|year=2021|volume=93|issue=2|page=025006|doi=10.1103/RevModPhys.93.025006|arxiv=2008.02378|bibcode=2021RvMP...93b5006S|s2cid=221006368}}</ref>

# Inner photoelectric effect in the bulk of the material that is a direct optical transition between an occupied and an unoccupied electronic state. This effect is subject to quantum-mechanical [[selection rule]]s for dipole transitions. The hole left behind the electron can give rise to secondary electron emission, or the so-called [[Auger effect]], which may be visible even when the primary photoelectron does not leave the material. In molecular solids [[phonon]]s are excited in this step and may be visible as satellite lines in the final electron energy.
# Electron propagation to the surface in which some electrons may be scattered because of interactions with other constituents of the solid. Electrons that originate deeper in the solid are much more likely to suffer collisions and emerge with altered energy and momentum. Their mean-free path is a [[Inelastic mean free path|universal curve]] dependent on electron's energy.
# Electron escape through the surface barrier into free-electron-like states of the vacuum. In this step the electron loses energy in the amount of the ''work function of the surface'', and suffers from the momentum loss in the direction perpendicular to the surface. Because the binding energy of electrons in solids is conveniently expressed with respect to the highest occupied state at the Fermi energy <math>E_F</math>, and the difference to the free-space (vacuum) energy is the work function of the surface, the kinetic energy of the electrons emitted from solids is usually written as <math>E_k = h\nu -W - E_B</math>.

There are cases where the three-step model fails to explain peculiarities of the photoelectron intensity distributions. The more elaborate ''one-step model''<ref>{{Cite journal|last=Mahan|first=G. D.|date=1970-12-01|title=Theory of Photoemission in Simple Metals|journal=Physical Review B|volume=2|issue=11|pages=4334–4350|doi=10.1103/PhysRevB.2.4334|bibcode=1970PhRvB...2.4334M}}</ref> treats the effect as a coherent process of photoexcitation into the final state of a finite crystal for which the wave function is free-electron-like outside of the crystal, but has a decaying envelope inside.<ref name=":0" />