1926 GLOECKNER, I A%SON, AND SERDUKE

' R. M. Freeman and A. Gallmann, Nucl. Phys.A156, 305 (1970).

' D. Kurath and R, D. Lawson, Phys. Rev. C 6, 901 (1972),P. W. M. Glaudemans, B. H. Wildenthal, and J. B.

McGrory, Phys. Lett. 21, 427 (1966)."G. A. P. Engelbertink and G. van Middelkop, Nucl. Phys.

A138, 588 (1969)."A. N. James, P. R. Aldeson, D. C. Bailey, P. E. Carr, J.

L. Durell, L. L. Green, M. W. Greene, and J. F.Sharpey-Schafer, Nucl. Phys. A168, 56 (1971).

"C, W. Towsley, D. Cline, and R. N. Horoshko, Phys. Rev.Lett. 28, 368 (1972).

W. Fitz, R. Jahr, and R. Santo, Nucl. Phys. A114, 392(1968).

"R. Bass and F. M. Saleh-Bass, Nucl. Phys. A95, 38(1967).

' S. A. Moszkowski, in Alpha-, Beta-, and Gamma-RaySpectroscopy, edited by K, Siegbahn (North-Holland,Amsterdam, 1965), Vol. 2, p. 863.

"D. Kurath and R. D. Lawson, Phys. Rev. 161, 915 (1967)."G, A. P. Engelbertink, E. K. Warburton, and J. W. Olness,

Phys. Rev, C 5, 128 (1972).' D. B. Fossan and A. R. Poletti, Phys. Rev. 152, 984(1966); Phys. Lett. 248, 38 (1967).' R. D. Lawson and M. H. Macfarlane, Nucl. Phys. 66, 80(1965).

"O. Skeppstedt, R. Hardell, and S. E. Arnell, Ark. Fys.35, 527 (1967).

"R. Hardell and C. Beer, Phys. Scr. 1, 85 (1970)."M, Bini, P. G. Bizzeti, A. M. Bizzeti-Sona, R. A. Ricci,

Phys. Rev. C 6, 784 (1972).' I. Talmi, Phys. Rev. 126, 1096 (1962).

PHYSICAL REVIE W C VOLUME 7, NU MBER 5 MAY 1973

Slow-Neutron Scattering by a Standing Electromagnetic Wave

C. StassisAmes I aboratory —U. S. Atomic Energy Commission and DePartrnent of Physics, Iona State University,

Ames, Iozoa 50020(Received 3 August 1972)

W'e examine the elastic scattering of slow neutrons by a standing electromagnetic wave.It is shown that the real part of the coherent scattering amplitude arises essentially fromthe interaction of the induced electric dipole moment in the neutron with the electric fieldof the wave. The imaginary part of the coherent scattering amplitude arises from the inter-ference between the magnetic dipole and spin-orbit interaction terms.

I. INTRODUCTION

The advent of lasers has motivated a consider-able amount of theoretical work on the interactionof the radiation field with photons and free elec-trons. ' The particular focus of interest has beenon the possib&e existence of nonlinear effects' infamiliar quantum processes such as Compton scat-tering. The photon densities required for the ex-perimental observation of these effects is of theorder of 10"photons/cm' at optical frequencies.In considering the scattering of particles by a pho-ton beam of such high density the polarization ofthe particle by the radiation electric field shouldbe taken into account. In the scattering of acharged particle the contribution to the scatteringcross section of the interaction of the induced elec-tric dipole moment in the particle with the electricfield is insignificant, compared to that arisingfrom the coupling of the charge of the particle tothe electric field. In the case of a neutral particle,on the other hand, the latter term is absent andthe contribution to the scattering cross section ofthe induced electric dipole interaction term maybe of importance. The main interest of such scat-

tering experiments would come from the possibil-ity of obtaining information about the polarizabilityof the particle. In the present paper we examine,in some detail, the scattering of long-wavelengthneutrons by the field of an electromagnetic wave.

The main physical aspects of the problem maybe qualitatively understood by examining the inter-action of a neutron with the electromagnetic fieldof the wave. The static electric dipole moment ofthe neutron has been experimentally found to bezero to high accuracy, ' as required by parity andtime-reversal invariance in electromagnetic inter-actions. 4 However, when a neutron encounters anelectromagnetic wave an electric dipole momentd =nE will be induced in the neutron by the electricfield F. of the wave, if its electric polarizability nhas a finite nonzero value. ' The interaction, ——,'nE',of this induced neutron electric dipole moment withthe electric field will contribute to the neutronelastic scattering cross section, in the first orderof perturbation theory. In addition the neutron pos-sesses a magnetic moment p, of -1.91'„whiCh cou-ples to the electromagnetic field through the well-known magnetic dipole' and spin-orbit interactionterm. ' Both these terms will contribute, in the

SLOW-NEUTRON S CAT TE RING . 1927

second order, to the neutron elastic scatteringcross section.

It is easily seen that the neutron scattering by aplane traveling electromagnetic wave is incoherent,and that the contribution of the neutron-polariza-tion term to the scattering amplitude is negligiblecompared to that of the magnetic dipole term. Infact, o/(p, '/&d), where &u is the wave frequency isof the order of 10 ' for optical frequencies and nof the order of 10 ' cm'.

In the case of a standing electromagnetic wave,in addition to incoherent scattering, coherent scat-tering of the neutrons must also occur. This canbe easily seen by considering the neutron-polariza-tion interaction term ——,'zE'. The time-indepen-dent part of this term is proportional to cos(2k. x),where +k are the wave vectors of the two Fouriercomponents of the standing wave and x the neutronposition vector. Thus the neutron senses a time-independent Periodic potential with a period of ~X,where X is the light wavelength. Therefore coher-ent scattering of the neutron will occur if theBragg condition for coherent elastic scatteringis satisfied, i.e. , for a neutron scattering vectorof +2k.

Both the magnetic dipole and the spin-orbit inter-action contribute, in the second order, to the co-herent neutron scattering amplitude. However, itwill be shown that the real part of the coherentscattering amplitude arises essentially from theinteraction of the induced electric dipole momentin the neutron with the electric field of the wave.The imaginary part of the coherent scattering am-plitude, on the other hand, arises from the inter-ference between the magnetic dipole and spin-orbitinteraction terms. The latter term is neutron spindependent and its magnitude evidently depends onthe angle between the direction of the incident neu-tron momentum and that of the electric field of thewave. For a standing wave polarized normally tothe scattering plane this interference term is thedominant contribution to the coherent scatteringamplitude, assuming a value of 10 "cm' for theneutron polarizability. On the other hand, if thestanding wave is polarized in the scattering planethe interference term vanishes and the cross sec-tion is essentially determined by the neutron polar-iz ability.

In Sec. II we calculate, to the second order ofperturbation theory, the coherent-neutron-scatter-ing cross section by a standing electromagnetic

wave, and in Sec. III we briefly discuss the mainphysical aspects of the scattering problem.

II. CALCULATION OF THE COHERENT-SCATTERING CROSS SECTION

In considering the scattering of slow neutronsby an electromagnetic wave the following interac-tion Hamiltonian may be adopted:

Exp&=-po ~ H —po ~ ——oE

where o is the Pauli matrix, p. the neutron mag-netic moment, nz the neutron mass, n the neutronpolarizability, and E and H the electric and mag-netic field of the wave, respectively. One recog-nizes in the first term the well-known magneticdipole interaction, ' in the second the spin-orbitinteraction, ' and in the third term the interactionof the induced electric dipole moment in the neu-tron with the electric field. The Foldy term' doesnot appear in Eq. (1) as a result of the transversecharacter of the electric field.

The elastic neutron scattering cross section bya standing electromagnetic wave can be easily cal-culated using the second quantization formalism.The initial state ~i) of the system consists of theincident neutron, of wave vector k, , and the stand-ing wave which is considered as the superpositionof two beams of photons, of equal intensity and op-posite polarizations, propagating in opposite direc-tions:

ik] .xIz) =10 .ni„0 n i, 0 0) v'V

In Eq. (2) the subscript k describes the photonstate of wave vector k and polarization e, the pho-ton occupation number n„and n „are assumedequal, and P is a normalization volume. We willcalculate the transition matrix element for theprocess in which the neutron is elastically scat-tered to a final state of wave vector k& as a resultof a photon undergoing a Compton transition k- k':

eiQ X

If)=I0 . .n, —1, 0 Ii;, 0 n „, 0. 0)

Using Eq. (1) the transition matrix element, tothe second order of perturbation theory, is given

x(o~(e„~ e,.)+p /id[(X. B+C D)+io" (AxB+CxD)])5(Q —(k-k')), (4)

1928 C. STASSIS

where is simply

(10)

is the neutron scattering vector and we defined

e' x [k' —(&u/m)(k; +k)]&v[1 —v/2m —(k; ~ k)/&um]

'

8=ex —k, —k

e x [(&o/m)(k; —k') —k](v[1 + &u/2m —(k,. ~ k')/(om ] '

D = e' x [k' —(~/m)k, ].From Eq. (4) it is easily seen that the scattering

cross section consists of two terms: the incoher-ent-scattering cross section proportional to thelight intensity and the coherent cross section pro-portional to the square of the light intensity. Inthe coherent scattering of the neutron a photon isscattered from the state (%, e} to the state (-k, -e)and conservation of linear momentum requiresQ =2%. Evidently the reverse process, in whicha photon undergoes the transition (-%, -e) —(k, e)is also possible and in this case Q =-2k. There-fore the condition for coherent scattering to occur

(4v)(2w)'2~a

x [-o.(u —p, '(td/m) ~4i(p, '/m)(k, h)(v e)]

x 5(Q v (k —k')}, (11)

where h is a unit vector defined by

ex%[exk[' (12)

Using Eq. (11}the differential coherent neutron

The transition matrix element for coherent scatter-ing with a momentum transfer of -2k is obtainedfrom that with a momentum transfer of 2k by sim-ply changing the sign of the wave vector %.

The transition matrix element, for the coherentscattering of neutrons, reduces to a very simpleform in all cases of practical interest. In fact forslow neutrons and an electromagnetic wave of op-tical frequency ~/m and k/k, are of the order of10 ' and 10 ', respectively. Under these condi-tions the transition matrix element for coherentneutron scattering reduces to

scattering cross section o „„(0)is

x p ((s&~ —n —(dtj.'/m) +4i(p, '/ma&)(k; h)(o ~ e)~s;)]'5(/+2k).

Q, Sy ys j

III. DISCUSSION

The neutron diffraction pattern by a standingelectromagnetic wave consists of two coherentelastic peaks at Q =+2k, superimposed on the in-coherent background. The condition for coherentscattering to occur Eq. (10) is equivalent to Bragg'slaw for coherent elastic scattering by a lattice withspacing —,'X:

2(—,X) sin8=X„, (14)

where 20 is the scattering angle and X, X„ the lightand neutron wavelength, respectively. Taking aneutron wavelength of 6 A and X=3400 A (nitrogenlaser), one obtains a scattering angle of twelve

In Eq. (13) n is the photon density, s& and s, denotethe final and initial neutron spin states, respective-ly, Bnd the bar over the summation sign indicatesaveraging over the initial neutron spin states.

minutes of arc.The spin-independent part of the coherent-scat-

tering amplitude arises essentially from the inter-action of the induced electric dipole moment in theneutron with the electric field of the wave. In factn/(p'/m) is of the order of one hundred for n= 10 "cm'. The spin-dependent part of the co-herent-scattering amplitude ari. ses from the inter-ference between the magnetic dipole and spin-orbitinteraction. For a standing wave polarized normal-ly to the scattering plane this interference term isthe dominant contribution to the coherent-scatter-ing amplitude, assuming a value of 10 "cm' forthe neutron polarizability. It is seen that for astanding wave polarized in the scattering plane thelatter term vanishes and the neutron-polarizationcontribution is the dominant term in the coherent-scattering amplitude.

For slow neutrons and typical laser frequenciesthe coherent amplitude per photon is of the orderof 10 "cm for a standing wave polarized normally

SLOW-NEUTRON SCATTERING ~ ~ 1929

to the scattering plane and of the order of 10 "cmif the wave is polarized in the scattering plane.Thus, with presently available neutron intensities,the experimental observation of the coherent neu-tron scattering from a standing electromagnetic

wave requires photon densities of the order of 1027

photons/cms. The photon densities required forsuch experiment are very high but not beyond thepossibilities of pulsed-laser technology.

~See for instance Z. Fried and W. M. Frank, NuovoCimento 2V, 218 (1963); H. R. Reise, J. Math. Phys, 3,59 (1962); A. I. Nikishov and V. I. Ritus, Zh. Eksperim.i Teor. Fiz. 46, 776 (1963) ftransl. : Soviet Phys. —JETP19, 529 (1964)].

L. S. Brown and T. W B. Kibble, Phys. Rev. 133,A705 (1964); I. I. Goldmann, Phys. Letters 8, 103 (1964);Z. Fried and J. H. Eberly, Phys. Rev. 136, B8V1 (1964);T. W B. Kibble, ibid. 138, B740 (1965); J. H. Eberly,Phys. Rev. Letters 15, 91 (1965).

3J. H. Smith, E. M. Purcell, and ¹ F. Ramsey, Phys.Rev. 108, 120 (1957); P. D. Miller, W. B. Dress, J. K.Baird, and ¹ F. Ramsey, Phys, Rev. Letters 19, 381(1967); C. G. Shull and R. Nathans, ibid. 19, 384 (196V);W. B. Dress, J. K. Baird, P. D. Miller, and N. F. Ram-sey, Phys. Rev. 170, 1200 (1968); V. W. Cohen, R. Na-thans, H. B. Silsbee, E. Lipworth, and N. F. Ramsey,aid. 17V, 1942 (1969); J. K. Baird, P. D. Miller, W. B.Dress, and N, F. Ramsey, ibid. 179, 1285 (1969).

4L. Landau, Zh. Eksperim. i Teor. Fiz. 32, 405 (1957)[transl. : Soviet phys. —JETP 5, 336 (1957)]; T. D. Leeand C. N. Yang, Brookhaven National Laboratory ReportNo. BNL-433 (T-91), 1957 (unpublished).

Estimates of the neutron polarizability, based on cur-rent meson theories or on photomeson production data

suggest a value of the order of 10 42 cm3. Currentlyavailable experiment on the scattering of fast neutronsby heavy elements only place an upper limit of about10 4~ cm3 on the neutron polarizability; see the follow-ing: Y. A. Aleksandrov, Zh. Eksperim. i Teor. Fiz. 33,294 (1957) ttransl. : Soviet Phys. —JETP 6, 228 (1958);R. M. Thaler, Phys. Rev. 114, 827 (1959); M. Walt andD. B. Fossan, ibid. 137, B629 (1965); A. J. Elwyn, J. E.Monahan, R. O. Lane, A. Langsdorf, Jr. , and F. P.Mooring, ibid. 142, 758 (1966); Y. A. Aleksandrov, G. S.Samosvat, Z. Sereeter, and Tsoi Gen Sov, Zh. Eksperim.i Teor. Fiz. -Redakt. 4, 196 (1966) ttransl. : JETP Let-ters 4, 134 (1966)].

~F. Bloch, Phys. Rev. 50, 259 (1936); O. Halpern andM. H. Johnson, ibid. 55, 898 (1939).

VJ. Schwinger, Phys. Rev. 73, 407 (1948); S. B. Gera-simov, A. I. Lebedev, and V. A, Putrunkin, Zh.Eksperim. i Teor. Fiz. 43, 1872 (1962) ttransl. : SovietPhys. —JETP 16, 1321 (1963)]; C. G. Shull, Phys. Rev.Letters 10, 297 (1963); M. Blume, Phys. Rev. 133,A1366 (1964); G. Obermair, Z. Physik 204, 215 (1967);S. W. Lovesey, J. Phys. 2, 981 (1969).

SL. L Fo1dy, Phys. Rev. 83, 633 (1951); L. L. Fo1dy,Rev. Mod. Phys. 30, 471 (1958).