Casimir effect

Illustration of the calculation of the Casimir force on two parallel plates assuming hypothetical vacuum fluctuations.

The Casimir effect is a quantum-theoretically interpretable effect of microphysics, which causes that a force acts on two parallel, conductive plates in a vacuum , which presses both together. The effect was predicted in 1948 by Hendrik Casimir and also named after him. In 1956 the experimental confirmation was carried out by Boris Derjaguin , II Abrikosowa and Jewgeni M. Lifschitz in the Soviet Union and in 1958 by Marcus Sparnaay from the Philips research laboratories in Eindhoven.

Scientists are investigating the possibilities of utilizing the Casimir effect in the field of nanotechnology for microsystems .

history

Joseph Cugnon reports in his article how Casimir came up with his simplified calculation. The calculation of van der Waals forces between bodies is very complex. As Casimir now z. B. had found an unexpectedly simple formula for the van der Waals force between an atom and a conductive plate, he doubted whether it could be correct. He then followed advice from Niels Bohr: “Why not calculate the effect by determining the difference in the zero point energies of the electromagnetic field?” He then calculated the forces between two atoms and between an atom and a conductive plate. Eventually he realized that the calculation is even easier for two conductive plates, and he finally published this result.

Predicting the theory

Casimir's result can be derived from two fundamentally different theoretical approaches, a quantum field theoretical approach, which includes the assumption of vacuum fluctuations (equivalent to zero point energy of the electromagnetic field ) and an approach from classical electrodynamics . According to Cugnon, there is a “duality of physical interpretation”.

Interpretation as the effect of vacuum fluctuations

The Casimir force between conductive plates can be calculated assuming the vacuum is a space full of virtual particles called vacuum fluctuation. A de Broglie wavelength can be assigned to such particles. Outside the plates there are all possible wavelengths, there is a continuum of virtual particles with any momentum

${\displaystyle p=\hbar k}$

assume (i.e. have a continuous spectrum) with

• ${\displaystyle \hbar ={\frac {h}{2\pi }}}$: reduced Planck's quantum of action
• ${\displaystyle k}$: Circular wavenumber .

The excitations of the discrete pulse spectrum can be understood as standing waves between the two plates. The distance between the two plates must be a multiple of half the wavelength of the virtual particles. All other states of virtual particles are “forbidden” there, since they do not meet the boundary conditions of the wave field . More (“allowed”) virtual particles collide from outside than in the space between the plates, and a pressure difference is created . The particles are virtual, but the pressure difference is real. This Casimir “pressure” acts as a force on the panels of the respective surface and presses them together. It is specified as the "negative pressure" (minus sign) between the plates and for perfectly conductive plates in a vacuum is: ${\displaystyle p_{c}}$ ${\displaystyle F_{c}}$ ${\displaystyle A}$

${\displaystyle p_{c}=\mathrm {-} {F_{c} \over A}=\mathrm {-} {\frac {\pi ^{2}\hbar c}{240}}\cdot {\frac {1}{d^{4}}}}$

with the sizes

• ${\displaystyle \pi }$: Circle number
• ${\displaystyle \hbar }$: reduced Planck's quantum of action
• ${\displaystyle c}$: Vacuum speed of light
• ${\displaystyle d}$: Distance between both plates.

According to this formula, the distance of 190 nm results in a negative pressure of 1 Pa, at 11 nm one reaches 100 kPa (1  bar ).

The very exact agreement of the measurements with this result does not prove the existence of vacuum fluctuations, although some physicists claim it.

Interpretation as van der Waals force between macroscopic objects

The Casimir force can also be interpreted as the sum of the forces of attraction between atoms across a gap (the effect of these forces is not linked to the existence of a vacuum).

Interactions between atoms / molecules ( van der Waals forces ) had already been considered by Fritz London . With the help of retarded van der Waals potentials, Casimir and Polder calculated the force that a perfectly conductive plate exerts on a single atom and came up with a simple formula using certain approximations. The result published a little later by Casimir, which was obtained by assuming vacuum fluctuations between two metal plates, agreed very well with this.

In 1956 EM Lifschitz calculated in general the force between two opposing blocks of dielectric material with plane-parallel surfaces.

The specialization in two perfectly conductive parallel metal plates produced exactly the result of Casimir

So the Casimir effect can be explained without reference to vacuum fluctuations.

Temperature dependence

Brevik, Ellingsen and Milton calculated the Casimir effect for any absolute temperature with the help of the temperature-dependent Green function . They found an analytical expression for the force of attraction between the plates per area (the "Casimir pressure") , from which they derived the limit values ​​for very high and very low temperatures: ${\displaystyle T}$ ${\displaystyle P(T)}$

${\displaystyle P(T)\to \mathrm {-} {\frac {k_{B}T}{4\pi \hbar cd^{3}}}\zeta (3)\mathrm {-} {\frac {k_{B}T}{2\pi \hbar cd^{3}}}\left(1+t+{\frac {t^{2}}{2}}\right)\mathrm {e} ^{-t}\qquad \left({\frac {1}{k_{B}T}}\ll {\frac {4\pi }{\hbar c}}\cdot d\right)}$

${\displaystyle P(T)\to \mathrm {-} {\frac {\pi ^{2}\hbar c}{240d^{4}}}\left[1+{\frac {16}{3}}\cdot {\left({\frac {k_{B}Td}{\hbar c}}\right)}^{4}\mathrm {-} {\frac {240}{\pi }}\cdot {\frac {k_{B}Td}{\hbar c}}\mathrm {e} ^{-\left({\frac {\pi \hbar c}{k_{B}Td}}\right)}\right]\qquad \left({\frac {1}{k_{B}T}}\gg {\frac {4\pi }{\hbar c}}\cdot d\right)}$

Here , ${\displaystyle t={\frac {4\pi k_{B}}{\hbar c}}\cdot d\cdot T}$

• ${\displaystyle k_{B}}$: the Boltzmann constant ,
• ${\displaystyle T}$the absolute temperature ,
• ${\displaystyle d}$ the plate spacing,
• ${\displaystyle \zeta (3)=1{,}2020569...}$a numeric constant ( zeta function for ). ${\displaystyle \zeta (z)}$ ${\displaystyle z=3}$

For large , the force of attraction increases linearly with temperature, for one obtains the result of Casimir. ${\displaystyle T}$ ${\displaystyle T\to 0}$

For real metals, material properties (including finite conductivity) must be included and thermodynamic considerations must be made.

Measurement results

Quantitative measurements were taken by Steve Lamoreaux ( Seattle , 1997) and Umar Mohideen and Anushree Roy ( Riverside , 1998).

In 2009, Alexej Weber from the University of Heidelberg and Holger Gies from the University of Jena showed that the Casimir effect shows different properties when plates are tilted against each other; so it grows z. B. faster with increasing surface temperature than in the case of parallel plates.

Other effects

Reverse Casimir effect

There are special cases in which the Casimir effect can cause repulsive forces between (uncharged) objects. Yevgeny M. Lifschitz had already predicted this in 1956 . The repulsive forces should most easily occur in liquids. After suitable metamaterials were available, the effect was again predicted by Eyal Buks and Michael L. Roukes in 2002. In 2007, physicists led by Ulf Leonhardt from the University of St Andrews theoretically predicted that with the help of metamaterial with a negative refractive index it would be possible to reverse the Casimir effect, i.e. to achieve a repulsion of the plates. This is called the reverse or repulsive Casimir effect or also quantum levitation .

An experimental demonstration of the repulsion predicted by Lifschitz due to the reversal of the Casimir effect was provided by Munday et al. Carried out in 2009, which also referred to the effect as quantum levitation.

Other scientists have suggested the use of laser-active media to achieve a similar levitation effect , although this is controversial as these materials appear to violate basic causality and thermodynamic equilibrium requirements ( Kramers-Kronig relationships ). Casimir and Casimir-Polder repulsion can actually occur with sufficiently anisotropic electrical bodies. For a review of the problems associated with rejection, see Milton et al. For more on the controllable ( English tunable ) repulsive Casimir effect see Qing-Dong Jiang et al. (2019).

Dynamic Casimir effect

As early as 1970, physicist Gerald T. Moore derived from quantum field theory that virtual particles that are in a vacuum can become real when they are reflected by a mirror that moves almost at the speed of light . This effect was later also called the dynamic Casimir effect. The experimental physicist Per Delsing and colleagues from the University of Gothenburg were able to prove this in 2011.

Casimir torque

In addition to the Casimir force between parallel plates, there is also a Casimir torque. This was proven in 2018 by twisting liquid crystals . The acting torques were on the order of a few billionths of a newton meter.

Others

The Casimir effect was also researched in NASA's Breakthrough Propulsion Physics Project . Since 2008 DARPA has been running a research program, the Casimir Effect Enhancement program .

literature

Books

• William MR Simpson, et al .: Forces of the quantum vacuum - an introduction to Casimir physics. World Scientific, New Jersey 2015, ISBN 978-981-4632-90-4 .
• Michael Bordag, et al .: Advances in the Casimir Effect. Oxford Univ. Pr., Oxford 2009, ISBN 978-0-19-923874-3 .
• Kimball A. Milton : The Casimir Effect. World Scientific, Singapore 2001, ISBN 981-02-4397-9 .
• Vladimir M. Mostepanenko, et al .: The Casimir effect and its applications. Clarendon Press, Oxford 1997, ISBN 0-19-853998-3 .
• Frank S. Levin, David A. Micha: Long-range Casimir forces . Plenum Press, New York 1993. ISBN 0-306-44385-6 .

items

• Michael Bordag: The Casimir effect 50 years later . In: Proceedings of the 4th Workshop on Quantum Field Theory under the Influence of External Conditions . World Scientific, Singapore 1999, ISBN 981-02-3820-7
• G. Jordan Maclay, (et al.): Of some theoretical significance - implications of Casimir effects. European Journal of Physics, 22, pp. 463-469, 2001 Abstract pdf at arxiv
• Gerold Gründler: Zero Point Energy and the Casimir Effect. The main arguments for and against the assumption of a physically effective zero point energy. Astrophysical Institute Neunhof. Communication sd08011, February 2013. Online: [9]
• Christopher Hertlein (et al.): Direct measurement of critical Casimir forces. In: Nature. 451, No. 7175, 172-175 (2008) Abstract
• JN Munday, F. Capasso , VA Parsegian: Measured long-range repulsive Casimir-Lifshitz forces . In: Nature 457, Letter, pp. 170-173, January 8, 2009 online
• Steven K. Lamoreaux: The Casimir force: background, experiments, and applications. Rep. Prog. Phys., 68 (2005) 201-236, doi : 10.1088 / 0034-4885 / 68/1 / R04 , pdf .

Web links

Commons : Casimir effect  - collection of images, videos and audio files

• Rainer Scharf: Sometimes nothing also repels In: Frankfurter Allgemeine Zeitung No. 11, January 14, 2009, p. N1; on-line
• Forces from nowhere - press release from the Max Planck Institute for Metals Research (2008)
• A force from empty space: the Casimir Effect - Astronomy Picture of the Day from January 3, 2010.
• Physicists Confirm Power of Nothing, Measuring Force of Universal Flux The New York Times, January 21, 1997
• Casimir Introduction @ mit.edu, accessed August 12, 2011
• What is the Casimir effect? @ scientificamerican.com
• Astrid Lambrecht: The power from nothing . A generalized scattering theory based on quantum optics allows the Casimir effect to be calculated for any disjoint objects. In: Physics Journal . tape 15 , no. 09 , 2016, p. 51-55 .
• M [arkus] Fierz: For the attraction of conductive levels in a vacuum. Cern, Geneva 1959. Typo / manuscript (PDF, 5 pages). Online [10]

Video

• What is the Casimir Effect? from the alpha-Centauri television series(approx. 15 minutes). First broadcast on Sep 1 2004.
• The experimental proof of the Kasimir effect Video portrait about Marcus Sparnaay, with Wolfgang Werner, 2007, approx. 12 min., Accessed on March 28, 2011
• Peter Milonni: Casimir Effects Colloquium @Institute for Quantum Computing, University of Waterloo , February 2011, @youtube, accessed May 25, 2012

Individual evidence

1. ↑ Astrid Lambrecht: The vacuum gains strength: The Casimir effect . In: Physics in Our Time . 36, No. 2, March 2005, pp. 85-91. ISSN  1521-3943 . doi : 10.1002 / piuz.200501061 . Retrieved July 28, 2012.
2. ↑ "The Casimir effect, in its simplest form, is the attraction between two electrically neutral, infinitely large, parallel conducting planes placed in a vacuum" in: Michael Bordag, et al .: Advances in the Casimir effect. Oxford Univ. Pr., Oxford 2009, ISBN 978-0-19-923874-3 , p. 1, @ google books
3. ↑ Hendrik Casimir: On the attraction between two perfectly conducting plates. Proc. Con. Nederland. Akad. Wetensch. B51, 793 (1948) pdf reprint accessed online on August 12, 2011
4. ↑ Diego Dalvit, et al .: Casimir physics. Springer, Berlin 2011, ISBN 978-3-642-20287-2 ; P. 1ff .: How the Casimir Force was Discovered , @ google books & P. 394ff: The History of Casimir-Polder Experiments. , @ google books. Retrieved September 19, 2012
5. ^ EM Lifshitz: The Theory of Molecular Attractive Forces Between Solids. Soviet Physics 2, 73 (1956), pdf, accessed online August 12, 2011
6. ↑ Experiment with parallel plates by Sparnaay , pp. 513-514, Chapter 18 General Requirements for Casimir Force measurements, in: Michael Bordag, et al .: Advances in the Casimir effect. Oxford Univ. Pr., Oxford 2009, ISBN 978-0-19-923874-3 .
7. ^ MJ Sparnaay: Measurements of attractive forces between flat plates. In: Physica. 24, 1958, p. 751, doi : 10.1016 / S0031-8914 (58) 80090-7 .
8. ↑ Federico Capasso, et al .: Attractive and Repulsive Casimir – Lifshitz Forces, QED Torques, and Applications to Nanomachines. Pp. 249-286, in: Diego Dalvit, et al .: Casimir physics. Springer, Berlin 2011, ISBN 978-3-642-20287-2 .
9. ↑ Federico Capasso, et al .: Casimir forces and quantum electrodynamical torques: physics and nanomechanics. In: IEEE Journal of Selected Topics in Quantum Electronics , Vol.13, issue 2, 2007, pp. 400-414, doi: 10.1109 / JSTQE.2007.893082 .
10. ^ Metamaterials Could Reduce Friction in Nanomachines sciencedaily.com, accessed November 22, 2012
11. ^ Joseph Cugnon: The Casimir Effect and the Vacuum Energy: Duality in the Physical Interpretation. In: Few-Body Systems. 53.1-2 (2012): p. 185. Online: [1]
12. ↑ HBG Casimir, D. Polder The Influence of Retardation on the London-van der Waals Forces Phys. Rev. 73, 360 - Published February 15, 1948, doi: 10.1103 / PhysRev.73.360
13. ^ Joseph Cugnon: The Casimir Effect and the Vacuum Energy: Duality in the Physical Interpretation. In: Few-Body Systems. 53.1-2 (2012) (title, p. 187), ( [2] )
14. ↑ Gerold Gründler : Zero point energy and Casimir effect: The essential arguments for and against the assumption of a physically effective zero point energy. Astrophysical Institute Neunhof. Communication sd08011, February 2013 (PDF, 48 pages). Online: [3] Well explanatory article at textbook level. Lots of scientific history details.
15. ^ Joseph Cugnon: The Casimir Effect and the Vacuum Energy: Duality in the Physical Interpretation. In: Few-Body Systems. 53.1-2 (2012): p. 187. [4]
16. ^ RL Jaffe: The Casimir Effect and the Quatum Vacuum. Online: [5]
17. ^ HBG Casimir and D. Polder: The Influence of Retardation on the London-van der Waals forces . Phys. Rev. 73, 360 (February 1948)
18. ^ EM Lifschitz: The Theory of Molecular Attractive Forces between Solids. JETP Vol. 2, No. 1, p. 73 (Russian original: ZhETF Vol. 29, No. 1, p. 94), online [6]
19. ↑ Formula (4.3) on page 80
20. ^ Joseph Cugnon: The Casimir Effect and the Vacuum Energy: Duality in the Physical Interpretation. In: Few-Body Systems. 53.1-2 (2012): p. 187. [7]
21. ^ Iver Brevik, Simen A. Ellingsen, Kimball A. Milton: Thermal corrections to the Casimir effect . New Journal of Physics, Vol. 8, Oct. 2006, online [8]
22. ↑ The original work uses Planck units ( ) ${\displaystyle \hbar =c=k_{B}=1}$
23. ↑ Steve K. Lamoreaux: Demonstration of the Casimir Force in the 0.6 to 6 μm Range . In: Physical Review Lett. Volume 78, 5-8 (1997) Abstract pdf online, accessed on August 12, 2011
24. ↑ Quantum Effects in Nanostructures In: Spektrum der Wissenschaft, September 2009, p. 12; Interplay between geometry and temperature for inclined Casimir plates , bibcode : 2009PhRvD..80f5033W
25. ↑ Physicists Demonstrate that the Warmer a Surface is, the Stronger its Ability to Attract Nearby Atoms. AZoNano.com, February 8, 2007, accessed June 21, 2016 .  
26. ↑ IE Dzyaloshinskii, EM Lifshitz, LP Pitaevskii: The general theory of van der Waals forces . In: Advances in Physics . 10, No. 38, 1961, p. 165. bibcode : 1961AdPhy..10..165D . doi : 10.1080 / 00018736100101281 .
27. ↑ Eyal Buks, Michael L. Roukes: Quantum physics: Casimir force changes sign . In: Nature . 419, No. 6903, September 12, 2002, pp. 119-120. ISSN  0028-0836 . doi : 10.1038 / 419119a . Retrieved July 28, 2012.
28. ↑ Ulf Leonhardt, Thomas G Philbin: Quantum levitation by left-handed metamaterials . In: New Journal of Physics . 9, No. 8, August 10, 2007, ISSN  1367-2630 , pp. 254-254. doi : 10.1088 / 1367-2630 / 9/8/254 . Retrieved July 28, 2012.
29. ↑ Ulf Leonhardt: Quantum Levitation , University St. Andrews (via WayBack WebArchive of March 31, 2016)
30. ↑ JN Munday, F. Capasso, VA Parsegian: Measured long-range repulsive Casimir-Lifshitz forces . In: Nature . 457, No. 7226, 2009, pp. 170-3. bibcode : 2009Natur.457..170M . doi : 10.1038 / nature07610 . PMID 19129843 . PMC 4169270 (free full text).
31. ^ Roger Highfield: Physicists have 'solved' the mystery of levitation . In: The Daily Telegraph , August 6, 2007. Retrieved April 28, 2010. 
32. ^ KA Milton, EK Abalo, Prachi Parashar, Nima Pourtolami, Iver Brevik, Simen A. Ellingsen: Repulsive Casimir and Casimir-Polder Forces . In: J. Phys. A . 45, No. 37, 2012, p. 4006. arxiv : 1202.6415 . bibcode : 2012JPhA ... 45K4006M . doi : 10.1088 / 1751-8113 / 45/37/374006 .
33. ^ Qing-Dong Jiang, Frank Wilczek: Chiral Casimir forces: Repulsive, enhanced, tunable . In: Physical Review B . 99, No. 12, March 4, 2019, p. 125403. doi : 10.1103 / PhysRevB.99.125403 .
34. ↑ Robert Gast: Quantum Physics: The Casimir Effect can also be different , on: Spektrum.de from March 6, 2019
35. ↑ Rüdiger Vaas: Nothing comes from nothing . January 2012. Accessed January 2017.
36. ↑ Maike Pollmann: Light generated from vacuum . November 2016. Accessed January 2017.
37. ↑ Gerald T. Moore: Quantum Theory of the Electromagnetic Field in a Variable-Length One-Dimensional Cavity . September 1970, bibcode : 1970JMP .... 11.2679M .
38. ↑ Jan Oliver Löfken: Virtual photons twist liquid crystal . January 11, 2019. Retrieved January 11, 2019.
39. ↑ Marc G. Millis: Breakthrough Propulsion Physics Workshop Preliminary Results nasa.gov, (pdf; 821 kB); ASSESSING POTENTIAL PROPULSION BREAKTHROUGHS grc.nasa.gov, accessed August 30, 2012
40. ^ Research in a Vacuum - DARPA Tries to Tap Elusive Casimir Effect for Breakthrough Technology scientificamerican.com; DARPA seeks sticky-goldenballs Casimir forcefields theregister.co.uk, accessed March 28, 2011