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Type of mathematical relation
The Kramers–Kronig relations, sometimes abbreviated as KK relations, are bidirectional mathematical relations, connecting the real and imaginary parts
Kramers–Kronig_relations
Topics referred to by the same term
Look up kronig in Wiktionary, the free dictionary. Kronig or Krönig may refer to: People: Alfred Kronig (1928–2020), Swiss cross country skier who competed
Kronig
Kronig isthmus is a band of resonance representing the apex of lung, also it's described as the narrow strap-like portion of the resonant field that extends
Kronig_isthmus
German physicist (1904–1995)
include the Kronig–Penney model, the Coster–Kronig transition and the Kramers–Kronig relations. Ralph Kronig (later Ralph de Laer Kronig) was born on
Ralph_Kronig
Model in Quantum Physics
\left(n=0,\pm 1,\dots ,\pm {\frac {N}{2}}\right).} The Kronig–Penney model (named after Ralph Kronig and William Penney) is a simple, idealized quantum-mechanical
Particle in a one-dimensional lattice
Particle_in_a_one-dimensional_lattice
Swiss football player (born 2000)
Jan Kronig (born 24 June 2000) is a Swiss footballer who plays for Sion as a defender. On 12 June 2018, Kronig signed his first professional contract with
Jan_Kronig
German chemist and physicist (1822–1879)
August Karl Krönig (German: [ˈkʁøːnɪç]; 20 September 1822 – 5 June 1879) was a German chemist and physicist who published an account of the kinetic theory
August_Krönig
The Coster–Kronig transition is a special case of the Auger process in which the vacancy is filled by an electron from a higher subshell of the same shell
Coster–Kronig_transition
Swiss cross-country skier (1928–2020)
Alfred Kronig (8 November 1928 – 28 April 2020) was a Swiss cross-country skier who competed in the 1950s. He finished 30th in the 18 km event at the 1952
Alfred_Kronig
Swiss cross-country skier
Victor Kronig (born 18 December 1933) is a Swiss cross-country skier. He competed in the men's 15 kilometre event at the 1956 Winter Olympics. Evans, Hilary;
Victor_Kronig
Equation of the state of a hypothetical ideal gas
the microscopic kinetic theory, as was achieved (independently) by August Krönig in 1856 and Rudolf Clausius in 1857. The state of an amount of gas is determined
Ideal_gas_law
Amnesic and analgesic state induced by morphine and scopolamine
"Freiburg method" when performed according to Gauss and Kronig's specific technique. Gauss and Kronig's research showed that the use of scopolamine during
Twilight_sleep
Dutch physicist (1889–1950)
Bibcode:1923Natur.111...79C. doi:10.1038/111079a0. Coster, D; de Laer Kronig, R (1935). "A new type of Auger effect and its influence on the x-ray spectrum"
Dirk_Coster
Empirical relationship between refractive index and wavelength
resonances as dirac delta functions, followed by the application of the Kramers-Kronig relations. This results in real and imaginary parts of the refractive index
Sellmeier_equation
American physicist (1898–1988)
experiment, which convinced him of the validity of quantum mechanics. With Ralph Kronig, Francis Bitter, Mark Zemansky and others, he set out to extend the Schrödinger
Isidor_Rabi
Effect of a material on light
formula such as the Cauchy or Sellmeier equations. Because of the Kramers–Kronig relations, the wavelength dependence of the real part of the refractive
Dispersion_(optics)
Property in optics
refractive index are not independent, and are connected through the Kramers–Kronig relations, i.e. the complex refractive index is a linear response function
Refractive_index
Austrian theoretical physicist (1880–1933)
Igor Tamm, Oskar Klein, J. Robert Oppenheimer, Walter Elsasser, Ralph Kronig, Werner Heisenberg, Paul Dirac, and David Dennison. Ehrenfest had ambivalent
Paul_Ehrenfest
English mathematician and physicist (1909–1991)
Groningen in the Netherlands, where he worked with Ralph Kronig. Together, they developed the Kronig–Penney model, which described the motions of electrons
William_Penney,_Baron_Penney
Relationship between volume and temperature of a gas at constant pressure
Regnault's Observations on Steam", Philosophical Magazine, 4. Extract. Krönig, A. (1856), "Grundzüge einer Theorie der Gase", Annalen der Physik, 001
Charles's_law
Relation of wavelength/wavenumber as a function of a wave's frequency
geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency-dependence of wave propagation and attenuation
Dispersion_relation
Yellow food coloring
by modulating refractive indices at longer wavelengths via the Kramers–Kronig relations. This property has been independently verified by multiple laboratories
Tartrazine
or how much energy waves lose) and are designed to satisfy the Kramers-Kronig relations, ensuring physical consistency between attenuation and dispersion
Azimi_Q_models
Measure of the electric polarizability of a dielectric, compared with that of a vacuum
Electret Ferroelectricity Green–Kubo relations High-κ dielectric Kramers–Kronig relations Linear response function Low-κ dielectric Dielectric loss Permittivity
Relative_permittivity
Dutch physicist (1894–1952)
was nonrelativistic. He is also credited for the Kramers–Kronig relations with Ralph Kronig which are mathematical equations relating real and imaginary
Hans_Kramers
Type of caesarean section
cases where Pfannenstiel incision cannot be done (such as large baby), Kronig incision (low vertical incision), classical (midline), J or T-shaped incisions
Lower segment Caesarean section
Lower_segment_Caesarean_section
Auxiliary input/output device for a computer
A.; Furrer, Simeon; Petermann, Martin; Rothuizen, Hugo; Brach, Stella; Kronig, Luzius; Iliadis, Ilias; Weiss, Beat; Childers, Ed R.; Pease, David (2025-01-08)
Peripheral
Association football team in Switzerland
Ruberto 13 FW SUI Winsley Boteli 14 DF SUI Numa Lavanchy 17 DF SUI Jan Kronig 19 FW SUI Franck Surdez No. Pos. Nation Player 20 DF SUI Nias Hefti 21
FC_Sion
Opposition to the passage of an electric current
frequency (see Drude model, deep-level traps, resonant frequency, Kramers–Kronig relations, etc.) Resistors (and other elements with resistance) oppose the
Electrical resistance and conductance
Electrical_resistance_and_conductance
Validation of Impedance Spectra
the Kramers–Kronig relations, where the real part can be computed from the imaginary part (or vice versa). In contrast to the Kramers–Kronig relations,
Z-HIT
Surgical procedure to deliver a baby
incision Extraperitoneal CS and then moving to low transverse incision (Krönig, 1912)[clarification needed] Adherence to principles of asepsis Anesthesia
Caesarean_section
Physical phenomenon
Charge carrier generation and recombination Characteristic X-ray Coster–Kronig transition Electron capture Radiative Auger effect IUPAC, Compendium of
Auger_effect
Semiconductor material
direct. The refractive index is related with the bandgap via the Kramers–Kronig relations and varies between 2.9 (x = 1) and 3.5 (x = 0). This allows the
Aluminium_gallium_arsenide
Physics award
1964 Samuel Goudsmit and George Uhlenbeck 1963 Rudolf Peierls 1962 Ralph Kronig 1961 Eugene Wigner 1960 Lev Landau 1959 Oskar Klein 1958 Wolfgang Pauli
Max_Planck_Medal
Changes in optical properties from applied electric fields
refractive index for wavelengths near the absorption edge, due to the Kramers–Kronig relation. Using a less strict definition of the electro-optic effect allowing
Electro–optic_effect
Series of TV adaptations of Shakespeare's plays
Italian renaissance theatre. Bennett and his costume designer, Barbara Kronig, gave things a mid-sixteenth century feel, taking inspiration from Durer
BBC_Television_Shakespeare
Speed of electromagnetic waves
refraction and to the attenuation coefficient, are linked by the Kramers–Kronig relations. In practical terms, this means that in a material with refractive
Speed_of_light
Measure of the electric polarizability of a dielectric material
effectively χ(Δt) = 0 for Δt < 0), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ(0). As opposed to the response of a
Permittivity
Dutch theoretical physicist
Netherlands) is a Dutch theoretical physicist in the line of Dirk Polder, Ralph Kronig, and Nico van Kampen. His scientific work inter alia involves laser theory
Hans_Dekker
Solid-state physics model
Electronic band structure Electronic band structure Nearly free electron model Kronig–Penney model Band properties Band gap Effective mass Density of states Fermi
K·p_perturbation_theory
Weisstor 11 September 1858 John Llewelyn Davies, Johann Zumtaugwald, Johann Kronig, Hieronymous Brantschen From Domhütte via Northern flank to the summit (PD)
List of mountains of the Alps over 4000 metres
List_of_mountains_of_the_Alps_over_4000_metres
Swiss-Austrian actor
policeman 1956 Von der Liebe besiegt [de] (Schicksal am Matterhorn) Beni Kronig – mountain guide 1957 The Big Chance Chaplain Sommer 1958 Resurrection Simonson
Robert_Freitag
Mathematical formula
of χ T L ( E ) {\displaystyle \chi ^{TL}(E)} is obtained via the Kramers-Kronig transform of its imaginary component. Mathematically, they are given by
Tauc–Lorentz_model
Force resulting from the quantisation of a field
causality constraints and the requirement of thermodynamic equilibrium (Kramers–Kronig relations). Casimir and Casimir–Polder repulsion can in fact occur for sufficiently
Casimir_effect
Mirror-like wave reflection
therefore be indirectly determined from the reflection spectrum by a Kramers-Kronig transform. The polarization of the reflected light depends on the symmetry
Specular_reflection
Experimental technique to determine the distribution of electrons in solids
bare band can be found in a self-consistent way by enforcing the Kramers-Kronig relation between the two components of the complex function Σ ( E ) {\displaystyle
Angle-resolved photoemission spectroscopy
Angle-resolved_photoemission_spectroscopy
Austrian scientist (1821–1895)
independently of Krönig, developed a similar, but much more sophisticated version of the theory which included translational and contrary to Krönig also rotational
Johann_Josef_Loschmidt
German obstetrician and gynecologist
Geburtshülfe, Leipzig 1897. Operative Gynäkologie, Leipzig 1905; (with Bernhard Krönig). Handbuch der Geburtshilfe, 4 volumes, Wiesbaden 1915–1921. Döderlein,
Albert_Döderlein
Scientific background leading to the discovery of subatomic particles
atom. However problems with the hypothesis soon became apparent. Ralph Kronig pointed out in 1926 that the observed hyperfine structure of atomic spectra
Discovery_of_the_neutron
BSC Young Boys 2026–27 football season
Sion Bern 14:00 Blum 8', 22' Sanches 40' Essende 66', 73' Kabacalman 43' Kronig 45+2' Boteli 56' Sow 74' Stadium: Stadion Wankdorf Referee: Lukas Fähndrich
2026–27_BSC_Young_Boys_season
Classic textbook in by Charles Kittel
electron model 7 Energy Bands nearly free electron model, Bloch's theorem, Kronig-Penney model, crystal momentum 8 Semiconductor Crystals band gap, electron
Introduction to Solid State Physics
Introduction_to_Solid_State_Physics
Austrian physicist (1900–1958)
new two-valued degree of freedom. The idea of spin originated with Ralph Kronig. A year later, George Uhlenbeck and Samuel Goudsmit identified Pauli's new
Wolfgang_Pauli
Complex analysis theorem
Sokhotski–Plemelj theorem for the unit circle and a closed Jordan curve) Kramers–Kronig relations Hilbert transform Kress, Rainer (2012). Linear Integral Equations
Sokhotski–Plemelj_theorem
Chemical compound – an oxide of copper with formula Cu2O
coefficient to be deduced. It can be shown using Cu2O that the Kramers–Kronig relations do not apply to polaritons. In December 2021, Toshiba disclosed
Copper(I)_oxide
Property of a material
polished samples under normal incidence (in combination with a Kramers–Kronig analysis) or using variable incidence angles. For samples that can be prepared
Optical_conductivity
In control theory, when an LTI system and its inverse are causal and stable
mixed phase. All-pass filter – A special non-minimum-phase case. Kramers–Kronig relation – Minimum phase system in physics Hassibi, Babak; Kailath, Thomas;
Minimum_phase
Spectroscopic techniques that measure the absorption of radiation
extinction and index coefficients are quantitatively related through the Kramers–Kronig relations. Therefore, the absorption spectrum can be derived from a scattering
Absorption_spectroscopy
Fundamental theorem in condensed matter physics
periodic potential. Specific periodic one-dimensional equations include the Kronig–Penney model and Mathieu's equation. Mathematically, various theorems similar
Bloch's_theorem
Vector field related to displacement current and flux density
of the applied field. The constraint of causality leads to the Kramers–Kronig relations, which place limitations upon the form of the frequency dependence
Electric_displacement_field
Specific structure observed in X-ray absorption spectroscopy
of electron volts past the edges was called the “Kronig structure” after the scientist, Ralph Kronig, who assigned this structure in the high energy range
X-ray absorption fine structure
X-ray_absorption_fine_structure
Kreimer Kurt Kremer Erich Kretschmann Herbert Kroemer August Krönig Bernd J. Kröger Ralph Kronig Eckhard Krotscheck Rudolf Kühnhold Helmuth Kulenkampff August
List_of_German_physicists
Understanding of gas properties in terms of molecular motion
the development of mechanical explanations of gravitation. In 1856 August Krönig created a simple gas-kinetic model, which only considered the translational
Kinetic_theory_of_gases
Conclusion of the human pregnancy
was first used in Germany and popularised by German physicians Bernard Kronig and Karl Gauss. This concoction offered minor pain relief but mostly allowed
Childbirth
Modelling the appearance of paint coatings
and macroscopic levels. However, using an analysis based on the Kramers–Kronig relations, the coefficients were shown to be dependent quantities related
Kubelka–Munk_theory
Attempt by E. Whymper, Reginald J. S. MacDonald, Johann zum Taugwald, Johann Krönig, Luc Meynet (3657 m) July 9–10: Attempt by E. Whymper, R. J. S. MacDonald
Timeline of climbing the Matterhorn
Timeline_of_climbing_the_Matterhorn
1925 physics article by Werner Heisenberg
limit, as the primary driver of Heisenberg's argument. His letter to Ralph Kronig one month before the publishing of the Umdeutung paper closely follows the
Umdeutung_paper
Ratio of the number of particles in a sample to Avogadro's constant
of a current is constant for a constant quantity of electricity". 1856: Krönig derives the ideal gas law from kinetic theory. Clausius publishes an independent
Amount_of_substance
Intrinsic quantum property of particles
interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Alfred Landé's assistants, suggested in early 1925 that it was produced
Spin_(physics)
Distortion of the periodic lattice of a one-dimensional crystal
zone k a = ± π {\displaystyle ka=\pm \pi } (similar to the result of the Kronig–Penney model, which helps to explain the origin of band gaps in semiconductors)
Peierls_transition
FC Aarau 2022–23 football season
Gashi (captain) 11 MF KOS Milot Avdyli 13 MF POR Ivo Candé 14 DF SUI Jan Kronig 17 MF SUI Valon Fazliu No. Pos. Nation Player 19 MF SUI Silvan Schwegler
2022–23_FC_Aarau_season
Political party in Poland
call became controversial within DSAP, and some members (such as Arthur Kronig, Otto Heike, Ludwig Kuk and Gustav Ewald) left the party as a result. In
German Socialist Labour Party of Poland
German_Socialist_Labour_Party_of_Poland
Dutch university
iodide process Jan Hendrik de Boer, discoverer of the particle spin Ralph Kronig, discoverer of the Einstein–de Haas effect Wander Johannes de Haas and discoverer
Delft University of Technology
Delft_University_of_Technology
Mountain in the Pennine Alps
southern face. He came to Zermatt with Melchior Anderegg, and engaged Johann Kronig as second guide. They slept at the huts of Schallenbergalp and started at
Weisshorn
Analytical technique used specifically in the study of surfaces
of the type LMM and KLL along with faster Coster–Kronig transitions such as LLM. While Coster–Kronig transitions are faster, they are also less energetic
Auger_electron_spectroscopy
FC Zürich 1993–94 football season
Exhibition game Luzern 1–1 Zürich Unterallmend, Root, Switzerland Koch 54' (1:0) FCZ summary 76' (1:1) Şahin Attendance: 1,000 Referee: Walter Kronig
1993–94_FC_Zürich_season
of an integral transform in the frequency domain that are of the Kramers-Krönig type. This effect is described in the article ‘Dispersive body waves’ by
Dispersive_body_waves
theory had two significant flaws: two values computed by Kronig were off by a factor of two. Kronig's senior colleagues discouraged his work and it was never
History_of_quantum_mechanics
Technique of measuring electromagnetic spectra
optical constant are related by the Kramers–Kronig relations. There is a difficulty in applying the Kramers-Kronig relations as written, because information
Terahertz time-domain spectroscopy
Terahertz_time-domain_spectroscopy
Degree of polarization
{\displaystyle \Delta t<0} ), a consequence of causality, imposes Kramers–Kronig constraints on the susceptibility χ e ( 0 ) {\displaystyle \chi _{\text{e}}(0)}
Electric_susceptibility
meat-packing industries of Geelong, Victoria, Australia. 1856 – August Krönig simplistic foundation of kinetic theory of gases. 1857 – Rudolf Clausius
Low-temperature technology timeline
Low-temperature_technology_timeline
Grasshopper Club Zurich 2025–26 football season
Football League. 6 March 2026. Retrieved 6 March 2026. "Eine Spielsperre gegen Kronig und Frey" (in Swiss High German). Swiss Football League. 9 March 2026. Retrieved
2025–26 Grasshopper Club Zurich season
2025–26_Grasshopper_Club_Zurich_season
City in North Rhine-Westphalia, Germany
physicist, physical chemistry August Junkermann (1832–1915), actor August Krönig (1822–1879), chemist and physicist Wilhelmine Lohmann (1872-?), German teacher
Bielefeld
How one process influences another
effect). This constraint has mathematical implications such as the Kramers-Kronig relations. Causality is one of the most fundamental and essential notions
Causality
Swiss Postal Telegraph and Telephone agency
Geschichte der Schweizerischen Post. PTT General Directorate, Bern 1949. Karl Kronig (ed.): Ab die Post. 150 Jahre schweizerische Post. Museum für Kommunikation
PTT_(Switzerland)
Optical imaging technique
Kramers-Kronig (KK) relations. This is because the imaginary part of the refractive index can be tied to the absorption spectra using Kramer-Kronig relations
Spectroscopic optical coherence tomography
Spectroscopic_optical_coherence_tomography
seismic geophysical applications. Because these models satisfies the Krämers–Krönig relations they should be preferable to the Kolsky model in seismic inverse
Kjartansson_constant_Q_model
American physicist (1873–1937)
substances (1897) Doctoral advisor Arthur Gordon Webster Doctoral students Isidor Isaac Rabi Francis Bitter Ralph De Laer Kronig Shirley Leon Quimby
Albert_Potter_Wills
Relation between static and dynamic quantities
one can use dispersion relations or equivalently for GDH, the Kramers–Kronig relations), unitary and Lorentz and gauge invariant. These three assumptions
Sum rules (quantum field theory)
Sum_rules_(quantum_field_theory)
Ledenbestand (Digitaal Wetenschapshistorisch Centrum). Retrieved 3 May 2020. "R. Kronig", KNAW Historisch Ledenbestand (Digitaal Wetenschapshistorisch Centrum)
Members of the Royal Netherlands Academy of Arts and Sciences (K)
Members_of_the_Royal_Netherlands_Academy_of_Arts_and_Sciences_(K)
Popular optical dispersion relation
of the refractive index are related to one another through the Kramers-Kronig relations. Forouhi and Bloomer derived a formula for κ ( E ) {\displaystyle
Forouhi–Bloomer_model
at 1. FC Nürnberg) 30 MF SUI Sandro Lauper (from Thun) 51 DF SUI Jan Kronig (from Young Boys U21) Out: No. Pos. Nation Player 4 DF SUI Marco Bürki
List of Swiss football transfers summer 2018
List_of_Swiss_football_transfers_summer_2018
Electrically insulating substance able to be polarised by an applied electric field
{\displaystyle \Delta t<0} ), a consequence of causality, imposes Kramers–Kronig constraints on the real and imaginary parts of the susceptibility χ e (
Dielectric
Surname list
Kramers' opacity law, Kramers theorem, Kramers–Heisenberg formula, Kramers–Kronig relations, and Kramers–Wannier duality. Cramer (surname) Cremer surname
Kramer_(surname)
Graduate textbook by J.D. Jackson
some new topics were added, including the Stokes parameters, the Kramers–Kronig dispersion relations, and the Sommerfeld–Brillouin problem. The two chapters
Classical Electrodynamics (book)
Classical_Electrodynamics_(book)
Mountain in the Pennine Alps, Switzerland
(north-west ridge) by John Llewelyn Davies with guides Johann Zumtaugwald, Johann Krönig and Hieronymous Brantschen. Davies published an account of his ascent in
Dom_(mountain)
Index of lists with the same name
metencephalon Isthmus tubae uterinae, which links the fallopian tube to the uterus Kronig isthmus, a band of resonance representing the apex of the lung Thyroid isthmus
List_of_anatomical_isthmi
=1/T} is the inverse temperature. This relation is an example of Kramers-Kronig relation. The spectral function can also be related to the imaginary-time
Numerical analytic continuation
Numerical_analytic_continuation
Intereference effect in quantum optics
one of many diverse mechanisms which can produce slow light. The Kramers–Kronig relations dictate that a change in absorption (or gain) over a narrow spectral
Electromagnetically induced transparency
Electromagnetically_induced_transparency
National under-21 association football team representing Switzerland
Müller (2000-09-18)18 September 2000 (aged 22) 6 0 Schaffhausen 17 2DF Jan Kronig (2000-06-24)24 June 2000 (aged 22) 13 1 Aarau 20 2DF Bećir Omeragić (2002-01-20)20
Switzerland national under-21 football team
Switzerland_national_under-21_football_team
Mountain in Switzerland
Switzerland Location Valais, Switzerland Parent range Pennine Alps Climbing First ascent 31 July 1867 by George Edward Foster, Hans Baumann and Johann Kronig
Mont_Collon
KRONIG
KRONIG
KRONIG
KRONIG
KRONIG
KRONIG
KRONIG
KRONIG
KRONIG