Paramagnetic properties of a vacuum embedded in a ...

Paramagnetic properties of a vacuum embedded in a strong magnetic field W. J. MIELNICZUK Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, Warsaw 02-668, Poland

D. R. LAMM Millimeter Wave Technolog))Division, Engineering Experiment Station, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. AND

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S. R. VALLURI' Department of Applied Mathematics, Universiv of Western Ontario, London, Ont., Canada N6A 5B9 Received January 22, 1988 It is shown that a vacuum embedded in a very strong magnetic field, B , behaves like an isotropic, paramagnetic medium. It is further demonstrated that the magnetic permeability k(B) of the vacuum differs from 1 by not more than 1.1 x for magnetic fields weaker than 4.4 x 1013G (typical paramagnetic materials at room temperature have corresponding values in the range of loe5- lo-'). On montre qu'un vide soumis a un champ magnktique trks intense, B , se comporte comme un milieu paramagnktique isotrope. On demontre de plus que la permeabilitk magnetique k ( B )du vide difftre de 1 par pas plus que 1 , l x pour des champs magnktiques plus faibles que 4,4 X 1013G (les valeurs correspondantes pour des materiaux parametiques typiques, h temperature ambiante, se situent dans I'intervalle 10-~-10-'). [Traduit par la revue] Can. J . Phys. 66. 692 (1988)

1. Introduction The hypothesis that very strong magnetic fields exist in the vicinity of neutron stars was formulated by Gold ( I ) and Pacini (2). This hypothesis was confirmed by astrophysical observations of magnetic fields in the order of 5 X 10I2G (3-5). Pulsars as neutron stars have the common characteristics of being strong emitters of electromagnetic radiation or sources of intense magnetic fields (6, 7). This paper is devoted to a very simple physical problem related to the existence of these strong magnetic fields. Consider a given volume of a vacuum embedded in a strong magnetic field characterized by the magnetic induction, B . Such a vacuum is called, following Stoneham (8), a magnetized one. Because of the virtual electron-pair creation in a background magnetic field (9-12), the magnetic field strength H differs from the magnetic induction vector B ,

Let the two vectors be parallel 11.21

HllB

Therefore, it is reasonable to introduce a measure of the difference between these two vectors, namely, the magnetic permeability k(B) determined by the constitutive relation (13) For weak magnetic fields, the magnetic permeability k(B) is given in an implicit form by Euler and Kockel (9). A natural question arises: What is the order of magnitude of the magnetic permeability k(B) for much stronger magnetic fields, for example, those observable near the surfaces of neutron stars? Another interesting question can also be asked: Is the magnetized vacuum a paramagnetic (k(B) > l), or diamagnetic (p,(B) < 1) one? These two questions are addressed in this paper. 'Author to whom all correspondence should be addressed. Printed in Canada I Imprime au Canada

2. Magnetic permeability of a vacuum It is shown in Appendix A that for all magnetic fields weaker than 4.4 x 1013G, the magnetic permeability k(B) is given by

where a is the fine-structure constant. The magnetic field B,, is defined by

where rn and e denote the mass and charge of the electron, respectively; c is the velocity of light; and h is Planck's constant divided by 2 r . The magnetic permeability k(B) has the two following properties. (i) k(B) fulfills the inequality

(ii) p(B) is always greater than 1 for a nonvanishing magnetic field, i.e., the vacuum behaves like an isotropic paramagnetic medium. Property (ii) in some sense justifies the name magnetized vacuum proposed by Stoneham (8) for a vacuum embedded in a strong magnetic field. Properties (i) and (ii) are proven in Appendix B. Inequality [2.3] shows that the magnetic permeability p(B) differs from 1 by less than 1.1 X lop4 for magnetic fields weaker than 4.4 X 1013G. However, in the case of ultra strong poloidal fields of 1015G (14), the correction is higher and could be -lop2. The magnetization density M is the difference between B and H and can be calculated for various field strengths. For comparison, typical values of p(B) -- 1 for paramag-

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netic substances are in the range of at room temperature, but they decrease at higher temperatures because of the randomizing effects of thermal excitations (13, 15).

(compare with ref. 16). The symbol tiik denotes the Kronecker delta function. The real part of the Lagrangian density, 2(Y, 9 ) , is given by

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3. Conclusions It has been shown that the magnetic permeability of the vacuum differs from 1 by a very small amount for fields weaker than the critical magnetic field. It is still of interest since the nonlinear effect is a purely quantum effect of the vacuum polarization. There is a net angular momentum associated with a paramagnetic substance that renders it paramagnetic. Is there also a net angular momentum associated with the vacuum that makes it paramagnetic? This question merits further investigation.

where 2::;denotes the effective Heisenberg-Euler Lagrangian density (10-12). The field invariants Y and 9 are given by 1 Y = -2 ( E ~ - B ~ ) and

8=E-B

where E denotes the fixed but arbitrary electric field. In the case when the electric field E vanishes, [A21 takes the form

Acknowledgements One of us (W.J.M.) wishes to thank Professor BialynickaBirula and Dr. J. Mostowski for illuminating discussions and Dr. R. Impey for assistance in the preparation of this paper. Another of us (D.R.L.) would like to thank the Millimeter Wave Technology Division of the Georgia Institute of Technology, Engineering Experiment Station, Atlanta, GA , for aid in the preparation of this paper. S.R.V. would like to thank Dr. R. Mendel at the Department of Applied Mathematics and Dr. J. Landstreet of the Department of Astronomy, University of Western Ontario, London, Ont., for informative discussions.

One can show that the real part of the Lagrangian density has the following behavior in the neighborhood of the point 9 = 0:

and that the partial derivative term of [A41 is finite. From [A31 and [A41 one gets

1. T. GOLD.Nature (London), 215, 731 (1968). 2. F. PACINI. Nature (London), 216, 567 (1968). 3. J. TRUMPER, W. PIETSCH, C. REPPIN, W. VOGES,R. STAUBERT, and and E. KENDZIORRA. Astrophys. J. 219, L105 (1978). Int. Astron. Union Circ. No. 3298, 1978. 4. W. WHEATON. 5. J. MATTESON, D. GRUBER, and J. HOFFMAN. NASA Tech. Rep. 79619, 1978. Thus, the permeability tensor p i k is proportional to the identity 6. J. P. OSTRIKER and J. E. GUNN.Astrophys. J. 157,1395 (1969). matrix. We can rewrite [A61 in the form and W. H. JULIAN. Astrophys. J. 157,869 (1969). 7. P. GOLDREICH 8. R. J. STONEHAM. J. Phys. A: Math. Gen. 12,2187 (1979). 9. H. EULER and B. KOCKEL. Naturwissenschaften, 23,216 (1935). and H. EULER.Z. Phys 98,714 (1936). 10. W. HEISENBERG 11. V. WEISSKOPF. Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 14, 6 (1936). where the magnetic permeability p(B) is given by 12. J. SCHWNGER. Phys. Rev. 82, 664 (1951). 13. J. D. JACKSON. Classical electrodynamics. Wiley Interscience, New York, NY. 1975. 14. Yu. VANDAKUROV. Astrophys. Lett. 5, 267 (1970). 15. D. E. GRAY(Editor). American institute of physics handbook. Equation [A81 conveys that the magnetized vacuum is iso3rd ed. McGraw-Hill Book Company, New York, NY. 1972. tropic from the point of view of magnetic properties. From the Sects. 5d, 5f. explicit form of the Heisenberg-Euler Lagrangian (lo), the 16. I. BIALYNICKI-BIRULA and Z. BIALYNICKA-BIRULA. Quantum following expression is obtained: electrodynamics. Pergarnon Press, Oxford, Great Britain. 1975. pp. 89-90. 17. S. R. VALLURI, D. R. LAMM,and W. J. MIELNICZUK. Phys. Rev. D, 25, 2729 (1982). 18. I. S. GRADSHTEYN and I. M. RHYZHIK. Tables of integrals, series and products. Academic Press, London, Great Britain. 1980.

Appendix A The purpose of this appendix is to derive [2.1]. The magnetic permeability tensor p i k is defined by the equation

1

Bi =

Because the calculations are perturbative in the fine-structure constant a,the final result for the magnetic permeability p(B) is

pikHk

The reciprocal tensor p,T1can be calculated from the equation Equation [A101 is identical to 12.11.

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Appendix B The proof of property (i) in Sect. 2 is based on the inequality

and is valid for every real number u. Inequality [Bl] follows immediately from [2.4] and [2.6] in a previous paper by Valluri et al. (17). Combining [A101 and [Bl] yields

Inequality (i) is a trivial consequence of combining [B2] with P31. To demonstrate property (ii), both sides of [A101 are differentiated with respect to B: [B4]

dp(B) a Bcr -dB

2rB2

Ix d;

( 2,)

- exp - u -

u

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By virtue of [B 11 and [B4] it is found that

Explicit integration (1 8) produces

Thus, the magnetic permeability p(B) is a monotonically increasing function of the magnetic induction B. This statement implies that p(B) is greater than 1 for B f 1. This proves property (ii) .

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