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[[File:Brendel-Bormann Oscillator Model.png|thumb|Brendel-Bormann oscillator model. The real (blue dashed line) and imaginary (orange solid line) components of relative permittivity are plotted for a single oscillator model with parameters <math>\omega_{0}</math> = 500 cm<math>^{-1}</math>, <math>s</math> = 0.25 cm<math>^{-2}</math>, <math>\Gamma</math> = 0.05 cm<math>^{-1}</math>, and <math>\sigma</math> = 0.25 cm<math>^{-1}</math>.]]
The '''Brendel–Bormann oscillator model''' is a mathematical formula for the frequency dependence of the complex-valued [[relative permittivity]], sometimes referred to as the dielectric function. The model has been used to fit to the complex [[refractive index]] of materials with absorption lineshapes exhibiting [[Spectral_line_shape#Lorentzian|non-Lorentzian]] broadening, such as metals<ref name = "Rakić1998"/> and amorphous insulators,<ref name = "Brendel1992"/><ref name = "Naiman1984"/><ref name = "Kučírková1994"/><ref name = "Hobert1996"/> across broad spectral ranges, typically near-[[ultraviolet]], [[visible spectrum|visible]], and [[infrared]] frequencies. The [[dispersion relation]] bears the names of R. Brendel and D. Bormann, who derived the model in 1992,<ref name = "Brendel1992"/> despite first being applied to optical constants in the literature by Andrei M. Efimov and E. G. Makarova in 1983.<ref name = "Efimov1983"/><ref name = "Efimov1985"/><ref name = "Efimov1996"/> Around that time, several other researchers also independently discovered the model.<ref name = "Naiman1984"/><ref name = "Kučírková1994"/><ref name = "Hobert1996"/> The Brendel-Bormann oscillator model is aphysical because it does not satisfy the [[Kramers–Kronig relations]]. The model is non-causal, due to a singularity at zero frequency, and [[Hermitian function|non-Hermitian]]. These drawbacks inspired J. Orosco and C. F. M. Coimbra to develop a similar, causal oscillator model.<ref name = "Orosco2018"/><ref name = "Orosco2018a"/>

== Mathematical formulation ==
The general form of an oscillator model is given by<ref name = "Brendel1992"/>

:<math>\varepsilon(\omega) = \varepsilon_{\infty} + \sum_{j} \chi_{j}</math>
where
* <math>\varepsilon</math> is the relative permittivity,
* <math>\varepsilon_{\infty}</math> is the value of the relative permittivity at infinite frequency,
* <math>\omega</math> is the [[angular frequency]],
* <math>\chi_{j}</math> is the contribution from the <math>j</math>th absorption mechanism oscillator.

The Brendel-Bormann oscillator is related to the [[Lorentz oscillator model|Lorentzian oscillator]] <math>\left(\chi^{L}\right)</math> and Gaussian oscillator <math>\left(\chi^{G}\right)</math>, given by

:<math>\chi_{j}^{L}(\omega; \omega_{0,j}) = \frac{s_{j}}{\omega_{0,j}^{2} - \omega^{2} - i \Gamma_{j} \omega} </math>
:<math>\chi_{j}^{G}(\omega) = \frac{1}{\sqrt{2 \pi} \sigma_{j}} \exp{\left[ -\left( \frac{\omega}{\sqrt{2} \sigma_{j}} \right)^{2} \right]}</math>
where
* <math>s_{j}</math> is the Lorentzian strength of the <math>j</math>th oscillator,
* <math>\omega_{0,j}</math> is the Lorentzian resonant frequency of the <math>j</math>th oscillator,
* <math>\Gamma_{j}</math> is the Lorentzian broadening of the <math>j</math>th oscillator,
* <math>\sigma_{j}</math> is the Gaussian broadening of the <math>j</math>th oscillator.

The Brendel-Bormann oscillator <math>\left(\chi^{BB}\right)</math> is obtained from the [[convolution]] of the two aforementioned oscillators in the manner of
:<math>\chi_{j}^{BB}(\omega) = \int_{-\infty}^{\infty} \chi_{j}^{G}(x-\omega_{0,j}) \chi_{j}^{L}(\omega; x) dx</math>,

which yields
:<math>\chi_{j}^{BB}(\omega) = \frac{i \sqrt{\pi} s_{j}}{2 \sqrt{2} \sigma_{j} a_{j}(\omega)} \left[ w\left( \frac{a_{j}(\omega) - \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) + w\left( \frac{a_{j}(\omega) + \omega_{0,j}}{\sqrt{2}\sigma_{j}} \right) \right]</math>
where
* <math>w(z)</math> is the [[Faddeeva function]],
* <math>a_{j} = \sqrt{\omega^{2}+i \Gamma_{j} \omega}</math>.

The square root in the definition of <math>a_{j}</math> must be taken such that its imaginary component is positive. This is achieved by:
: <math>\Re\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}+1}{2}}</math>
: <math>\Im\left( a_{j} \right) = \omega \sqrt{\frac{\sqrt{1+\left( \Gamma_{j}/\omega \right)^{2}}-1}{2}}</math>

== References ==
{{reflist | refs =
<ref name = "Brendel1992">{{cite journal
| last1       = Brendel
| first1      = R.
| last2       = Bormann
| first2      = D.
| title       = An infrared dielectric function model for amorphous solids
| journal     = Journal of Applied Physics
| volume      = 71
| issue       = 1
| pages       = 1–6
| year        = 1992
| url         = https://doi.org/10.1063/1.350737
| doi         = 10.1063/1.350737
| bibcode = 1992JAP....71....1B
| access-date = 2021-10-13
}}</ref>
<ref name = "Efimov1983">{{cite conference 
| last1 = Efimov
| first1 = Andrei M.
| last2 = Makarova
| first2 = E. G.
| title = [Vitreous state and the dispersion theory]
| year = 1983 
| book-title = Proc. Seventh All-Union Conf. on Vitreous State
| pages = 165–71
| language = ru
}}</ref>
<ref name = "Efimov1985">{{cite journal
| last1       = Efimov
| first1      = Andrei M.
| last2       = Makarova
| first2      = E. G.
| year        = 1985
| title       = [Dispersion equation for the complex equation constant of vitreous solids and dispersion analysis of their reflection spectra]
| url         = 
| language    = ru
| journal     = Fiz. Khim. Stekla
| trans-journal = The Soviet Journal of Glass Physics and Chemistry
| volume      = 11
| issue       = 4
| pages       = 385–401
}}</ref>
<ref name = "Naiman1984">{{cite journal
| last1       = Naiman
| first1      = M. L.
| last2       = Kirk
| first2      = C. T.
| last3       = Aucoin
| first3      = R. J.
| last4       = Terry
| first4      = F. L.
| last5       = Wyatt
| first5      = P. W.
| last6       = Senturia
| first6      = S. D.
| title       = Effect of Nitridation of Silicon Dioxide on Its Infrared Spectrum
| journal     = Journal of the Electrochemical Society
| volume      = 131
| issue       = 3
| pages       = 637–640
| year        = 1984
| url         = https://doi.org/10.1149/1.2115648
| doi         = 10.1149/1.2115648
| bibcode = 1984JElS..131..637N
| access-date = 2021-10-20
}}</ref>
<ref name = "Kučírková1994">{{cite journal
| last1       = Kučírková
| first1      = A.
| last2       = Navrátil
| first2      = K.
| title       = Interpretation of Infrared Transmittance Spectra of SiO2 Thin Films
| journal     = Applied Spectroscopy
| volume      = 48
| issue       = 1
| pages       = 113–120
| year        = 1994
| doi = 10.1366/0003702944027534
| bibcode = 1994ApSpe..48..113K
| s2cid = 98613649
| url         = http://www.osapublishing.org/as/abstract.cfm?URI=as-48-1-113
| access-date = 2021-10-20
}}</ref>
<ref name = "Hobert1996">{{cite journal
| last1       = Hobert
| first1      = H.
| last2       = Dunken
| first2      = H. H.
| title       = Modelling of dielectric functions of glasses by convolution
| journal     = Journal of Non-Crystalline Solids
| volume      = 195
| issue       = 1–2
| pages       = 64–71
| year        = 1996
| url         = https://doi.org/10.1016/0022-3093(95)00517-X
| doi         = 10.1016/0022-3093(95)00517-X
| bibcode = 1996JNCS..195...64H
| access-date = 2021-10-20
}}</ref>
<ref name = "Rakić1998">{{cite journal
| last1       = Rakić
| first1      = Aleksandar D.
| last2       = Djurišić
| first2      = Aleksandra B.
| last3       = Elazar
| first3      = Jovan M.
| last4       = Majewski
| first4      = Marian L.
| title       = Optical properties of metallic films for vertical-cavity optoelectronic devices
| journal     = Applied Optics
| volume      = 37
| issue       = 22
| pages       = 5271–5283
| year        = 1998
| url         = https://doi.org/10.1364/AO.37.005271
| doi         = 10.1364/AO.37.005271
| pmid = 18286006
| bibcode = 1998ApOpt..37.5271R
| access-date = 2021-10-13
}}</ref>
<ref name = "Efimov1996">{{cite journal
| last1       = Efimov
| first1      = A. M.
| title       = Quantitative IR spectroscopy: Applications to studying glass structure and properties
| journal     = Journal of Non-Crystalline Solids
| volume      = 203
| pages       = 1–11
| year        = 1996
| url         = https://doi.org/10.1016/0022-3093(96)00327-4
| doi         = 10.1016/0022-3093(96)00327-4
| bibcode = 1996JNCS..203....1E
| access-date = 2021-10-13
}}</ref>
<ref name = "Orosco2018">{{cite journal
| last1       = Orosco
| first1      = J.
| last2       = Coimbra
| first2      = C. F. M.
| title       = On a causal dispersion model for the optical properties of metals
| journal     = Applied Optics
| volume      = 57
| issue       = 19
| pages       = 5333–5347
| year        = 2018
| url         = https://doi.org/10.1364/AO.57.005333
| doi         = 10.1364/AO.57.005333
| pmid = 30117825
| bibcode = 2018ApOpt..57.5333O
| s2cid = 51760671
| access-date = 2021-10-14
}}</ref>
<ref name = "Orosco2018a">{{cite journal
| last1       = Orosco
| first1      = J.
| last2       = Coimbra
| first2      = C. F. M.
| title       = Optical response of thin amorphous films to infrared radiation
| journal     = Physical Review B
| volume      = 97
| issue       = 9
| pages       = 094301
| year        = 2018
| url         = https://doi.org/10.1103/PhysRevB.97.094301
| doi         = 10.1103/PhysRevB.97.094301
| bibcode = 2018PhRvB..97i4301O
| access-date = 2021-10-14
}}</ref>
}}

== See also ==
* [[Cauchy's equation|Cauchy equation]]
* [[Sellmeier equation]]
* [[Forouhi–Bloomer model]]
* [[Tauc–Lorentz model]]
* [[Lorentz oscillator model#Dielectric function|Lorentz oscillator model]]

{{DEFAULTSORT:Brendel-Bormann oscillator model}}
[[Category:Condensed matter physics]]
[[Category:Electric and magnetic fields in matter]]
[[Category:Optics]]