Behavior of neutrons in electromagnetic fields

PHYSICS LETTERS A

PhysicsLettersA 181 (1993) 109—113 North-Holland

Behavior of neutrons in electromagnetic fields R. Simonovits and H. Mitter Instilutefor Theoretical Physics, University of Graz, Universilalsplatz 5, A-8010 Graz, Austria Received 15 July 1993; accepted for publication 10 August 1993 Communicated by J.P. Vigier

The behavior ofneutral particles with spin and anomalousmagnetic respectively electric moment in external electromagnetic fields is considered up to second order effects in v/c. The resulting two-component wave equation is briefly discussed. Its exact solution for constant fields is used for an analysis of spin interference.

1. Introduction Since a long time the structure of the neutron is of interest for hadron physics and has been investigated both in theory and experiment. Whereas the magnetic structure is quite well-known, it is still an open problem whether the particle has a small (intrinsic) electric dipole moment. Such a moment is predicted by particle physics theories as a consequence of CP violating effects [1]. Due to the smallnessof the predicted value, the moment has so far escaped experimental observation [2,3]. Considerable progress in the accuracy obtained in neutron interferometry [4,5] leaves some hope that the situation could be improved in an appropriate experiment. At present, neutron interferometers allow for a control of energy differences of about lO~~ eV; in order to improve the present bound [2,3] on the electric moment, one would need still rather strong electric fields (>500 kY/cm), but the issue should be discussed quantatively. A final answer meets with a difficulty: there is a controversy in the literature [6,7] on the magnitude of the acceleration of the particle in a combined electric and magnetic field; this acceleration is of relevance for the measurement in the used semiclassical description of the spin. The controversy could only be settled by a theory in which the spin is treated as an intrinsically quantum variable, Another problem of interest in this context is the observation of quantum mechanical interferences caused by the magnetic moment. Although neutron

interferometers require the use of very slow neutrons, the high accuracy allows for a control of relativistic effects. To some extent this might provide for a control ofinterference patterns predicted by relativistic quantum mechanics. For both purposes one needs an adaption of relativistic quantum mechanics to the situation, i.e. a wave equation for slow neutrons in presence of elecisomagnetic fields, which contains relativistic corrections. We shall develop such a theory and follow its consequences in the context of neutron spin interferometry.

2. Wave equation As a starting point we shall use Dirac’s equation, treating the neutron as a relativistic particle moving in a prescribed (external) electromagnetic field. The particle will be coupled to the field via a Pauli term involving the magnetic (i’) and electric (d) dipole moment. The equation reads / 1 mc\ o,~(pF’~’—dF~’) u=0. (1) —

Here m is the neutron mass, ~ the electromagnetic field tensor and P~the dual tensor. For the Dirac matrices (y,~, a,~)we use the notation of ref. [8]. The sign of the term involving d has been chosen in such a way, that the entire coupling term is invariant under exchange of electricity and magnetism.

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It should be observed, that the structure of (1) emerges from any reasonable quantum field theory for sufficiently low momentum transfer, since the static properties of the particle are fixed by Lorentz invariance and low energy theorems (cf. ref. [8], i’• 160). Structural corrections (e.g. a form factor for ~t) can be neglected at energies small in comparison with the rest energy of the pion. For the electric moment these corrections involve even much higher masses [1]. Quantum field theory corrections to the description of F~~’ as an external field can safely be neglected for laboratory field strengths, in particular for slowly varying fields. Of course the equation (with ~ interpreted as a one-particle wave function) would not provide for a consequent description of pair production phenomena, which can arise at very high energies and/or field strengths. A nonrelativistic reduction can be obtained with the familiar transformation We shall useFoldy—Wouthuysen an algebraic approach [10], which [9]. can be outlined as follows. In terms of the operators

ih

Qg

+y

2gYO,

0~F,

F=uB+dE

YoQg =‘ 1

1 1 Q =——y-IhV+-y mc

y0Q~=

—

\

0GI

c

\.

An expansion in powers of the Compton wavelength h/mc of the neutron reads ~

—

=y0mc2(1 + ~Q~)_Qg

+ ~ [Q~, [Q~,

QglJ

+....

Here we have neglected terms ofthird and higher order. The r.h.s. of (2) does not contribute to the order considered here. The “large” components are contained in i,u 4.. For the two-component spinor 2

~~=~±exp(—imc i/h) we obtain a Pauli-type equation (3)

H

—

ôt

—

with

2Q2+ ~[Q~, [Q

,Q

]]

.

(4)

H= —~~F+ ~mc

In order to identify individual terms in the Hamiltonian and their meaning, the expressions on the r.h.s. must be evaluated. The computation of the Q~-term is straightforward. The term contains the rest energy and first order relativistic corrections. The second .

j’

G ~iE dB

2u Yo,

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2 (1 + iQ~) yt. Foreq. (1)wetakes Q5çv=y0mc mally havethe (1 form + iQ~) = ~ K~with

order corrections are contained in the double commutator, whose evaluation is a bit mOre tedious. It is convenient to use Maxwell’s equations for E and B (in absence ofsources, since the neutron shall move outside of the source region of the external field). The result can be written as a sum H=HP+HR. The

K=exp (~iarctanQ~) y ,

0K=K~y0.

first term takes the form

The spinor ~=K~ fulfils the equation

1

KQaK1~yomc2,Jl+Q~

We split the operator on the l.h.s. into an even and an odd part KQgK~V++V,

(yomc2.Ji~~u — 17+ )~,±

110

—

~

kyomc 2V

+a’(—F+K”~+Kt2~) ,

(5)

with K~’~ ~(P~ihV)xG,

By straightforward elimination we obtain two decoupled equations, which read

V

H~= ~-~P

2V±KQgK~±K’QgK

and consider the projections

— —

1 2+ ~-~G2

U —

~+1~_1v —

‘

‘2’/

2m2c

2

(P—~ihV)F’P.

(6) (7)

Here P denotes the momentum operator of the partide (i.e. P~o=—ihV~,in the coordinate representation), V applies to the fields. The whereas remainder is givenexclusively by

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HR

=

—

2mc 2mc

~‘

(—~-~

\mc

PHYSICS LETTERS A

G(F- G)

[V(FG)+ (F.v)G1).

(8)

j~=I~(~><~,.+ _-~-_a(F.P))co. mc \ 2mc

(9)

For d=0 and Kt2~=0(i.e. up to first order corrections) Hamiltonian (5) agrees with forms, which have been used in the literature, e.g. by Anandan (see eq. (8) in ref. [6]). The second term in (5) can be interpreted as a polarization contribution. Neglecting contributions from the electric dipole moment we have G2=~a(~)E2,

—

sults “automatically” for a charged particle). The electric polarizability predicted from dynamical hadron theories [11] is about 20 times larger 3) and has been measured recently (-~ l0~ (fm)accuracy [12]. Higher is needed, if one wants to decide, whether a~ can be separated or not.

This term is extremely small for any imaginable external field situation. Taking the term into account would also be inconsequent. The first expression in (8) should become important, if the field energy is comparable with the reset energy of the neutron. In the second expression the variation of the fields over a Compton wavelength of the neutron must be large enough in order that the term contributes. Such fields would produce neutron—antineutron pairs and the context of the one-particle Dirac theory (from which we started) had to be abandoned. We shall therefore drop this term, The Pauli equation (3) with Hamiltonian (5) describes the behavior of neutrons in arbitrary electromagnetic fields up to second order relativistic corrections in one-particle quantum mechanics. it can be shown, that the probability density p= ~, and the currentf =j. +1. fulfil a continuity equation. Hereby the convection currentj~has the familiar Schrödinger form and the spin current ,j. is

a(~)=

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-~--~

mc

—6.0x105 (fm)3,

where a~ has the meaning of a (static) electric polarizability. Thus Dirac’s theory for a neutral particle with anomalous magnetic moment ~uprovides “automatically” (i.e. as a consequence of relativity) for a corresponding electric polarizabiity (in analogy to the way, in which the normal magnetic moment re-

3. Solution for constant fields, interference pattern In order to discuss spin interference experiments we considerconstant fields. In this case the quantum problem is stationary and hamiltonian (5) cornmutes with P. The corresponding momentum operator in the Heisenberg picture is time-independent and the particle is not accelerated. In any experiment with constant fields produced by laboratory equipments accelerations of the particle can only be due to edge regions, where the fields change. For d= 0 and K~2~ = 0 both facts have already been stated and discussed in ref. [71.They are a direct consequence of quantum mechanics for neutral particles, for which the fields are coupled only to the spin (cf. (1)). It has to be noted, however, that the spin operator in the Heisenberg picture is (of course) not constant. In the momentum representation ~,= ~ t) the Hamiltonian assumes the form H~=

with 12

2m

±..~

G2,

2mc

~=—F+ -~--pxG+ 2 (F~p)p. mc 2(mc)

(10)

In terms of a constant Pauli spinor w, a solution of eq. (3) reads ç9(p, t)=exp(~or.zt)wexp(_ ~ Since the eigenvalues of a’ ~ are ±a ±I I, the rotation angle ofthe spin vector a is at/h. It is easy to show, that rLt can be obtained by a Lorentz boost (up to terms (v/c)2) from F‘t’, where the primed quantities refer to the rest system of the particle. The interference pattern is obtained in standard —

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fashion [131. If one of the paths passes the field region, the intensity is given by 1= ~I~[1 +cos(at/h)] . Here I~is the incoming

(11) intensity and t is the time, during which the spin is exposed to the field. If fields with opposite directions are applied in the two paths, the intensity takes the form I=~I0[l+cos(2at/h)].

(12)

The relevant quantity a (which does not depend on the direction of spin quantization) is determined by (10) in terms of the neutron speed /3= v/c, the angles ~= L (E, B), y= L (fi, B), 5= L (fi, E) and the field strengths. Up to terms $2 we have 2=B2[u2(l _/32 cos2y)+/32d2 sin2y] a +2EB[~u2+d2)flf(a,y,5)+jtd(l—fl2)cosij] +E2[d2(l —/J2cos2ô)+fl2ji2 sin2ô] The function f is defined by reads

f(~,y,

.

fi- (BxE) =J3BEf and

5)= (1 —cos2i—cos2y—cos2ö

Form (13) allows for an identification ofthe various contributions. Since, however, d is much smaller than ~uand /3<< 1 for reactor neutrons, we can in practice use the approximation

+

jt

2{B2 (I

—

/32 cos2y)

+ 2$BE [f+ (d/$~)COS?)]

(/JE)2[sin2ô+ (d//4t)2] }

.

(14)

This form shows the difficulties encountered in an experimental observation of the electric dipole moment. With d< 1025 e cm, ~i—~2x1014 e cm and /3,.~l06 we have d//3u<5x106. Contributions from the electric moment d show up in the second and third term in (14). In both terms they are, however, dominated by much larger contributions from the magnetic moment (-...fresp. —. sin2ô), unless the angles assume particular values, so that fresp. sin2ô vanishes. These values must then, however, be adjusted and controlled with extremely high precision. In the setups used in neutron interferometry these requirements can certainly not be met. Therefore the present experimental bound [2] on the dipole moment cannot be reached even with a further im112

provement of interferometric accuracy. Contributions from the magnetic moment can be controlled within the present accuracy ofneutron interferometry. In addition to measurements of the nonrelativistic phase shifts [14] also first order corrections in /3 have already been investigated for special field configurations. The influence of an electric field on neutrons (as caused by (6) resp. by the second term in (14)), referred to as Aharonov—Casher effect [151 for neutrons, has been observed [161. In order to extend the control of quantum interference patterns to relativistic quantum mechanics one should, however, measure the angular dependence contained in (14). The second order term should in principle be accessible in a pure electric field; even very weak remnant magnetic fields would, however, dominate.

(13)

+2 cos ,j cos y cos

a2

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4. Summary

The Pauli equation (3) with (5) describes the behavior of neutrons in external magnetic fields up to second order relativistic corrections and allows for a physical interpretation in the standard framework of one-particle quantum mechanics. For constant fields the equation can be solved exactly. We have concentrated on consequences for neutron spin interferometry. In this case the effects of a (small) intrinsic electric moment are dominated by those of the magnetic moment and cannot be observed. The role of relativistic corrections has been discussed.

References [1] J. Ellis, Nuci. Instrum. Methods A 284 (1989) 33. [2] N. Ramsey, Annu. Rev. Nucl. Part. Sci. 40 (1990) 1. [31IS. Altarev et al., Phys. Lett. B 276 (1992) 242. [4] J. Summhammer and A. Zeilinger, Physica B 174 (1991) 396. [5]G. Badurek, H. Rauch and D. Tuppinger, Phys. Rev. A 34 (1986) 2600. [61J. Anandan, Phys. Lett. A 138 (1989) 347. [7] R. Casella and S. Werner, Phys. Rev. Lctt. 69 (1992) 1625. [8] C. Itzykson and J.1980). Zuber, Quantum field theory (McGrawHill, New York, [9] L. Foldy and S. Wouthuysen, Phys. Rev. 78 (1950) 29. [l0]H.NeuerandP.Urban,ActaPhys.Austr. 15(1962)380.

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[11] J.L. Friar, in: Workshop on Electron—nucleus scattering, eds. A. Fabriocini, S. Fantoni, S. Rosati and M. Viviani (World Scientific, Singapore, 1989) p. 3 and references therein. [12] J. Schmiedmayer, P. Riehs, J.A. Harvey and NW. Hill, Phys. Rev. Lett. 66(1991) 1015. [131 G. Badurek, H. Rauch, A. Zeilinger, W. Bauspiess and U. Bonse,Phys.Rev.D 14(1976)1177. [14] H. Rauch, A. Zeilinger, G. Badurek, A. Wilfing, W. Bauspiess and U. Bonse, Phys. Lett. A 54 (1975) 425; S.A. Werner, R. Colella, A.W. Overhauser and C.F. Eagen, Phys. Rev. 1..ett. 35 (1975) 1053;

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H. Rauch, A. Wilfing, W. Bauspiess and U. Bonse, Z. Phys. B 29 (1978) 281. [151 Y. Aharonov and A. Casher, Phys. Rev. Lett. 53 (1984) 319. [l6]A. Cimmino, G.I. Opal, A.G. Klein, H.A. Kaiser, S.A. Werner, M. Arifand R. Clothier, Phys. Rev. Lett. 63 (1989) 380; R.C. Casella, Phys. Rev. Lett. 65(1990) 2217.

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