# Physics/Essays/Fedosin/Magnetic monopole

< Physics‎ | Essays‎ | Fedosin

The magnetic monopole is a hypothetical particle in physics that is a magnet with only one pole. In more technical terms, it would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unification theory and superstring theory, which predict their existence. [1] [2]

The magnetic monopole was first hypothesized by Pierre Curie in 1894, [3] but the quantum theory of magnetic charge started with a 1931 paper by Paul Dirac. [4] In this paper, Dirac showed that the existence of magnetic monopoles was consistent with Maxwell's equations only if electric charges are quantized, which is observed. Since then, several systematic monopole searches have been performed. Experiments in 1975 (Price et all) [5] and 1982 (Blas Cabrera) [6] produced candidate events that were initially interpreted as monopoles, but are now regarded as inconclusive.

Monopole detection is an open problem in experimental physics. Within theoretical physics, some modern approaches assume their existence. Joseph Polchinski, a prominent string-theorist, described the existence of monopoles as "one of the safest bets that one can make about physics not yet seen." [7] These theories are not necessarily inconsistent with the experimental evidence: in some models magnetic monopoles are unlikely to be observed, because they are too massive to be created in particle accelerators, and too rare in the universe to enter a particle detector.

Some condensed matter systems propose a superficially similar structure, known as a flux tube. The ends of a flux tube form a magnetic dipole, but since they move independently, they can be treated for many purposes as independent magnetic monopole quasiparticles. [8]

In late 2009 a large number of popular publications incorrectly reported this phenomenon as the long-awaited discovery of magnetic monopoles, [9] but the two phenomena are not related. [10]

## Main properties of magnetic monopoles

### Magnetic monopole charge

Magnetic monopole charge in the CGS units has the same units as the electric charge:

${\displaystyle q_{m}={\frac {c\hbar }{2e}}={\frac {e}{2\alpha }}=\approx 137e/2,\ }$

where ${\displaystyle c}$ is the speed of light in vacuum, ${\displaystyle \hbar }$ is the reduced Planck constant and ${\displaystyle e}$ is the elementary charge.

In the SI units magnetic charge has other units:

${\displaystyle q_{m}={\frac {h}{e}},\ }$

where ${\displaystyle h}$ is the Planck constant.

### Coupling constant for monopole

It is known that electric charges have a small value of the coupling constat (s.c. fine structure constant). It has the following value in the CGS units:

${\displaystyle \alpha ={\frac {e^{2}}{c\hbar }}\approx 1/137.\ }$

In the SI units its expression is more complicated:

${\displaystyle \alpha ={\frac {e^{2}}{2hc\varepsilon _{0}}}\approx 1/137,\ }$

where ${\displaystyle \varepsilon _{0}}$ is the electric constant.

By analogically, we could define the magnetic coupling constant in the CGS units (Giacomelli et al, 2003): [11]

${\displaystyle \beta ={\frac {q_{m}^{2}}{c\hbar }}={\frac {1}{4\alpha }}=34.25.\ }$

In the SI units this coupling constant is (Yakymakha O.L.,1989): [12]

${\displaystyle \beta ={\frac {q_{m}^{2}}{2hc\mu _{0}}}={\frac {1}{4\alpha }}=34.25,\ }$

where ${\displaystyle \mu _{0}}$ is the vacuum permeability. It is worth noting, that the magnetic coupling constant is more higher then 1, and thus the perturbative methods cannot be used (in the quantum electrodynamics).

### Magnetic monopole mass

Dirac theory doesn't predict the monopole mass. Therefore, there is no real prediction for the mass of the classical Dirac monopole (experiments pointed the lower border limit of mass only). Furthermore, the electron mass is defined from experiments only too, and it isn't predicted by any theory.

#### Low border limit for monopole mass

Low border limit for monopole mass could be obtained using the classical electron radius (SI units):

${\displaystyle r_{0}={\frac {e^{2}}{4\pi \varepsilon _{0}m_{e}c^{2}}}={\frac {\alpha \lambda _{e}}{2\pi }},\ }$

where ${\displaystyle \lambda _{e}}$ is the Compton wave-length for electron, and ${\displaystyle m_{e}}$ is the rest mass of electron.

By analogy, we could define a value for the classical monopole radius (SI units):

${\displaystyle r_{D0}={\frac {q_{m}^{2}}{4\pi \mu _{0}m_{D}c^{2}}},\ }$

where ${\displaystyle m_{D}}$ is the monopole rest mass. Thus, equating the classical radiuses for electron and monopole, we could obtain the low border limit for the monopole mass:

${\displaystyle m_{D}=({\frac {q_{m}}{e}})^{2}{\frac {\varepsilon _{0}}{\mu _{0}}}m_{e}={\frac {1}{4\alpha ^{2}}}m_{e}\approx 4692m_{e}.\ }$

#### High border limit for monopole mass

In the general case, magnetic monopoles are results of the dynamic interaction of the particles with electric charges, and therefore it shouldn't has the rest mass (or static mass). For example, the gravitational interaction of the two static electron masses will be defined by the Newton law:

${\displaystyle F_{g}={\frac {1}{4\pi \varepsilon _{g}}}{\frac {m_{e}^{2}}{r^{2}}}=\alpha _{g}{\frac {c\hbar }{r^{2}}},\ }$

where ${\displaystyle \alpha _{g}={\frac {m_{e}^{2}}{2ch\varepsilon _{g}}}\approx 1.7518\cdot 10^{-45}\ }$ is the gravitational coupling constant, ${\displaystyle ~\varepsilon _{g}={\frac {1}{4\pi G}}}$ is the gravitoelectric gravitational constant, ${\displaystyle ~G}$ is the gravitational constant.

It seems that magnetic monopole should to have the "dynamic torsion mass" ${\displaystyle h/m_{e}}$, since it has the "dynamic magnetic charge" ${\displaystyle h/e}$. Then, the interaction of two dynamic torsion masses will be defined as:

${\displaystyle F_{\Omega }={\frac {1}{4\pi \mu _{g}}}{\frac {(h/m_{e})^{2}}{r^{2}}}=\beta _{g}{\frac {c\hbar }{r^{2}}},\ }$

where ${\displaystyle \beta _{g}={\frac {(h/m_{e})^{2}}{2ch\mu _{g}}}\approx 1,428\cdot 10^{44}\ }$ is the dynamic torsion coupling constant, ${\displaystyle ~\mu _{g}={\frac {4\pi G}{c^{2}}}}$ is the gravitomagnetic gravitational constant.

On the other hand, magnetic monopole could have the upper border limit for the rest mass, which gravitational force is:

${\displaystyle F_{gM}={\frac {1}{4\pi \varepsilon _{g}}}{\frac {m_{D}^{2}}{r^{2}}}.\ }$

Equating the static and dynamic forces

${\displaystyle F_{gM}=F_{\Omega },\ }$

we could find out the upper limit for the monopole mass:

${\displaystyle m_{D}={\frac {1}{2\alpha _{g}}}m_{e}\approx 2.86\cdot 10^{44}m_{e}.\ }$

It is evident that this value is so great, that it belongs to the cosmological scale. Therefore, any attempts of building such experimental devices to discover even one monopole are out of the scope of any civilization.

It is worth noting, that the supermassive monopoles, predicted by Grand Unified Theory theories, and connected with Early Universe, have the monopole mass values considerably lesser then predicted above.

## "Monopoles" in condensed-matter systems

While a magnetic monopole particle has never been conclusively observed, there are a number of phenomena in condensed-matter physics where a substance, due to the collective behaviour of its electrons and ions, can show emergent phenomena that resemble magnetic monopoles in some respect. [13] [14] [15]

These should not be confused with actual monopole particles; in particular, the divergence of the microscopic magnetic B-field is zero everywhere in these systems, unlike in the presence of a true magnetic monopole particle. The behaviour of these quasiparticles would only become indistinguishable from true magnetic monopoles — and they would truly deserve the name — if the so-called magnetic flux tubes connecting these quasiparticles became unobservable which also means that these flux tubes would have to be infinitely thin, obey the Dirac quantization rule, and deserve to be called Dirac strings.

In a paper published in Science in September 2009, researchers Jonathan Morris and Alan Tennant from the “Helmholtz-Zentrum Berlin fur Materialien und Energie” (HZB) along with Santiago Grigera from “Instituto de Fisica de Liquidos y Sistemas Biologicos” (IFLYSIB, CONICET) and other colleagues from Dresden University of Technology, University of St. Andrews and Oxford University described the observation of quasiparticles resembling monopoles. [16] A single crystal of dysprosium titanate in a highly frustrated pyrochlore lattice (F d -3 m) was cooled to 0.6 to 2 K. Using neutron scattering, the magnetic moments were shown to align in the spin ice into interwoven tube-like bundles resembling Dirac strings. The ends of each tube are at the crystallographic defects, where the magnetic field looks like that of a monopole. Using an applied magnetic field to break the symmetry of the system, the researchers were able to control the density and orientation of these strings. A contribution to the heat capacity of the system from an effective gas of these quasiparticles is also described.

## References

1. Wen, Xiao-Gang; Witten, Edward, Electric and magnetic charges in superstring models,Nuclear Physics B, Volume 261, p. 651-677.
2. S. Coleman, The Magnetic Monopole 50 years Later, reprinted in Aspects of Symmetry.
3. Pierre Curie, Sur la possibilite d'existence de la conductibilite magnetique et du magnetisme libre (On the possible existence of magnetic conductivity and free magnetism), Seances de la Societe Francaise de Physique (Paris), p76 (1894). (fr)Free access online copy
4. Paul Dirac, "Quantised Singularities in the Electromagnetic Field". Proc. Roy. Soc. (London) A 133, 60 (1931). Free web link
5. P. B. Price, E. K. Shirk; W. Z. Osborne; L. S. Pinsky “Evidence for Detection of a Moving Magnetic Monopole”. Physical Review Letters, vol. 35, No 8, p.487 (1975).
6. Blas Cabrera. “First Results from a Superconductive Detector for Moving Magnetic Monopoles”, Physical Review Letters, vol. 48, No 20, p.1378 (1982).
7. C. Castelnovo, R. Moessner and S. L. Sondhi, Nature 451, 42-45 (3 January 2008) Magnetic monopoles in spin ice.
8. G. Giacomelli and L. Patrizii. Magnetic Monopole Searches. arXiv:hep-ex/0302011v2.
9. Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN 5-11-002309-3.
10. Zhong, Fang; Naoto Nagosa, Mei S. Takahashi, Atsushi Asamitsu, Roland Mathieu, Takeshi Ogasawara, Hiroyuki Yamada, Masashi Kawasaki, Yoshinori Tokura, Kiyoyuki Terakura (October 3, 2003). "The Anomalous Hall Effect and Magnetic Monopoles in Momentum Space". Science 302 (5642): 92-95. doi:10.1126/science.1089408. ISSN 1095-9203. Retrieved on 2 August 2007.
11. Xiao-Liang Qi, Rundong Li, Jiadong Zang, Shou-Cheng Zhang. Science Express magazine, 29 January 2009 Inducing a Magnetic Monopole with Topological Surface States.
12. Steven Bramwell et al. Measurement of the charge and current of magnetic monopoles in spin ice. Nature 461, 956-959 (15 October 2009); doi:10.1038/nature08500].
13. D.J.P. Morris, D.A. Tennant, S.A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Czter-nasty, M. Meissner, K.C. Rule, J.-U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky, and R.S. Perry. Dirac Strings and Magnetic Monopoles in Spin Ice Dy2Ti2O7. Science, 2009, Vol. 326 no. 5951 pp. 411-414. [1]