Mössbauer tests of time reversal invariance

] 7.A.2

1

NucIe~r Physics A177 (1971) 493-512;

@ ~or~h-~~~~~n~ Putjlislting Co., Amsterdam

Not to be reproducedby photoprintor microfilm without writteenpermissionfrom the publisher

M&?&BAUER TESTS OF TiME REVERSAL INVARIANCE

Abstract: A detailed analysis is given of the MBssbaner absorption experiments designed to test T-invariance. We discuss in particular how the effects of a T-vioIation can be separated from the initial- and final-state atomic radiative interactions which simulate T-non-invariance.

1. Introduction

The origin of the small CP violation observed in the decay of the neutral K-meson [ref. “)J is still uncertain. Various theories have been proposed, suggestjn~ that the source of the violation is in superweak interactions ‘*3>, electromagnetic interactions 4- ‘), milliweak interactions *), or the weak interactions ‘- ’ ’ ). These theories are consistent with the Y’CP theorem, and consequently predict weak T-violations. The strongest effects on nuclear physics are given by the eIectromagnetic theories, which give a T-odd part in the nuclear interaction on the order of 10v3 times the T-even amplitude. Experiments to test T-invariance are difficult, and at present no T-violation has been observed tt. Furthermore, the experiments are not yet sufficiently accurate to eliminate any of the contending theories. However, with only a small improvement, a decision could be made on the theory that the source of the CP violation is in the electromagnetic interaction of the hadrons [ref.‘)]. The most sensitive tests of this theory have been experiments utilizing the MGssbauer effect 13-15). The purpose of this paper is to discuss how to improve the Miissbauer experiments. In particular we discuss how the effects of a T-violation can be distinguished from the final- and inititial-state atomic radiative interactions which simulate a T-violation ’ 6), and which give a major limiting factor to the accuracy of the experiments. The proposal of Bernstein, Feinberg and Lee “) is based on the observation that at present there is no substantial experimental evidence that the electromagnetic interactions of the strongly interacting particles are invariant under C and T. If strong ? Work supported in part by the Ofiice of Naval Research under Contract Number NooO14-68-A0503. t? Comprehensive reviews of the current theoretical and experimental situation of parity and timereversal invariance in nuclear physics are given by Henley lz). Recent experimental reviews are given by Hamilton 12) and Richter I*). 493

494

J. P. HANNON

violations should occur, then, through virtual electromagnetic processes, all strong reactions can violate C and T to order x = h. In particular the nuclear y-transition matrix elements would then contain a fractional admixture of ~-non-invariant amplitude on the order of 10e3 to lo-‘. Such a T-non-invariant amplitude would give an observable effect in nuclear transitions in which a y-ray of mixed multipolarity (e.g., Ml-E2) is emitted or absorbed. If T-invariance holds, then the reduced matrix elements of the current multipole moments which give the amplitudes of the respective radiations (to lowest order in e) are in phase or 180” out ofphase 17). On the other hand, if there is a small T-noninvariant admixture in the interactions, then the reduced matrix elements are no longer relatively real. For a mixed Ml-E2 transition ~fII~2lii~/

= ISI exp @

where the deviation of v] = 1~~- qM from 0 or x is proportional to the relative admixture of T-non-invariant interaction. Bernstein, Feinberg and Lee have estimated that under favorable conditions one might obtain an r7deviation of the order 10T3 - IO-*. Jacobsohn and Henley 18) and Stichel 19) have shown that this will result in the addition of a term proportional to sin ~(2. iFx 3X& *3)(t?-3) =

4 sin 9 sin 2a sin’ /? cos p

(2)

for the probability of emission or absorption of a y-ray in the direction k with polarization e^where3is the spin of the initial nuclear state and on the right-hand side z is the axial angle between e”and the projection of3 on the (6, e”x f;) plane, and j3 is the polar angle between fi and3. Kistner i3) has attempted to measure the effect of the term (2) in the resonant absorption of the 90 keV Mossbauer y-ray in 99Ru and he concludes rl = 7r--(1.0+1.7)x

10-3,

(3)

whereas Atac, Chrisman, Rebrunner and Frauenfelder r4) and Zech, Wagner, Korner and Kienle “) have carried out (emission) experiments utilizing the 73 keV Mijssbauer y-ray in ig31r and obtained ‘I = (l.lrt:3.8)x

10-3,

1 = (1.6&2.4)x 10-3,

(4)

respectively. Both transitions involved are of the Ml-E2 type. However, as we have discussed in a previous paper 16), the relative multipole phase difference does not arise solely from a possible phase difference of the nuclear currents as given by eq. (l), and thus is not a direct measure of T-invariance. In particular, the “conversion” currents induced in the inner electronic shells by a mixed Miissbauer r-transition give an E(L+ 1)- M(L) phase difference < = tE --tM, where tE, i.\, are given by cq. (15) of ref. 16), or more explicitly, by eqs. (16) and (17)

TIME REVERSAL

INVARIANCE

495

of ref. ” )- We have calculated the relative phase differences for the Ru and Ir MSssbauer transitions, and obtained values of a

= -6.3 x 10W3,

(i)

@r) = +0.9x 10W3.

(ii)

(5)

Since these phase shifts areof the same order of magnitude as the maximum expected deviation of 1 from 0 or 71due to I-non-invariance, these effects must be taken into account t. For emission experiments, we have shown 16) that the effect of the conversion current phase shift is to replace q by q’ = q+{ in expression (2), and in order to determine if one must correct the measurements by calculating 5 ++.Thus tf should be replaced by q’ in the left-hand side of eq. (4). Using the calculated value of r(Ir) given in eq. (5), gives sin q = (i-0.2+3.8) x 10m3 for the experiment of Atac et al. For the total cross section, however, we have shown that the term arising from the interference between photoelectric absorption and internal conversion following nuclear excitation will cancel out the <-dependent term which simulates T-noninvariance. Thus, for very thin films, an absorption experiment will give a direct measurement of T-non-invariance. However, even for thin films, multiple scattering effects are important. If the incident radiation is polarized, then only in the limit that the complex indexes of refraction are equal (i.e. the isotropic scattering limit) will the abso~tion be simply proportional to the total cross section. As first pointed out by Blume and Kistner 22), the transmitted intensity will contain a Faraday effect term in addition to the total cross-section term. Furthermore, as we noted in ref. 16) and show explicitly below, the multiple scattering gives a c-dependent term which has exactly the same symmetry properties as the T-non-invariant sin q term of the total cross section. 2. Theory The sin q term given by eq. (2) has two unique symmetry properties which distinguishit from allother contributions to the total cross section”+: (i) the sign of the term changes when the direction of the magnetic field at the nucleus is reversed, or more generally, fi + IL-& a + afnz; and (ii) the sign changes under an axial angle rotation a -* 3++(2n+ 1)~ or 3! + -a+mr. The sign of the term also changes with a change in sign of fn, (i.e. mO + -m. and + There are also nuclear radiative corrections considered by Henley and Jacobsohn ‘I). For the Mtissbauer experiments they are about three orders of magnitude smaller and can be neglected. +t We note that these statements also apply to non-Mtissbauer T-invariance experiments, such as /?y(M l-E2)-y(E2) correlation experiments, which involve a mixed multipole y-emission. +t? It should be noted that an implicit assumption in eq. (2) and in many of the results derived here is that Jz is conserved for the nucleus. This would not be the case for example if there were an EFG tensor non-coaxial with the direction of the magnetic field at the nucleus, or if there were fluctuating fiefds. See for example ref. ‘*), appendices A and C; Biume 23) and Tumolillo “).

J. P. HANNON

496

m, + -m,), or equivalently, the sin n term is an odd function of the Doppler shift measured about the center of symmetry of the absorption spectrum. However, there are also dispersion terms in the total cross section proportional to C&and CMwhich exhibit this asymmetry ‘*’ ‘- “). SOURCE

-

I ‘“5 2

“k

FILM

A

FILM

6

z-k

1 DETECTOR

1

Fig. I. Schematic of two film MBssbauer absorption experiments for measuring

T-violation effects.

It is also useful to note that the sin n term vanishes if the incident photon is unpolarized, while for a linearly polarized photon, the term maximizes for a = 45” and /? w 54$“, and vanishes for cx = &a or j3 = @IL The experiments measure the change of the intensity transmitted through a magnetically ordered film caused by field reversal, and axial rotation a -+ -a, or a Doppler shift u + -a. The necessary polarization of the incident photon can be achieved by using a magnetically ordered Zeeman split source, or by using a monochromatic source and passing the unpolarized y-ray through a magnetically ordered Mossbauer film which acts as a polarizing filter. If the first procedure is used the magnetic field in the source should be aligned perpendicular or parallel to the direction of observation in order to eliminate an undesired sin(q + r) contribution to the intensity, and to eliminate Faraday effects within the source. However, the appearance of a number of distinct frequency components greatly complicates the analysis and decreases the intensity in t De-Shalit also considered El conversion photoelectric interference and a description of a preliminary calculation by L. Tassie has been given by Lipkin [H. J. Lipkin, M6ssbauer Effect. edited by H. Frauenfelder and H. Lustig, US Airforce report TN 60-698, pp. 25-261.

497

TIME REVERSAL INVARIANCE

each component. For this reason it is preferable to use a monochromatic source and a polarizing filter, and we shall limit our consideration to this type of experiment (see fig, 1). 2.1. TRANSMISSION

THROUGH

SEVERAL MAGNETICALLY

ORDERED

FILMS

For an initial photon a,(&, w) = [a(& ~)~~~)+~~~, w)j@)j z f;) incident upon several magnetically ordered films A, B, . . *, N, the transmitted polarization and amplitude is given by ‘O*‘“)

where the 2 x 2 matrix UL(L = A, B, . . ., N) is given by

where (for q = I, 2) fq = 3 Trff(

- 1)‘“i”[a(TrSf2-detf]f

= WXX-UY&-(-- W+ “Eb&-fyu)2 elf) = [exp (if, Qrtexp (if2

+_&j?fyJf*

91t

6)

(91

andf,(a, b = x or y) is related to the atomic coherent forward scattering amplitude f(k, &,; k, $; o) for scattering a I&,&) photon into a ]k, $) photon by f brr= JO nf(k, &,; k, ;a; m),

00)

where n is the density and 2 = 2zJk. The f(k) and P(k) can be any orthogonal basis pe~endicular to k. The explicit expressions forf,, fjl~I~~ and& are given in appendix A for a linear basis #, 9 for a mixed Ml-E2 transition, Finally, the subscript L = A, B, . . ., N in eqs, (6) and (7) indicates that the matrix eiements are evaluated with the concentration, thickness and magnetic field orientation parameters of the Lth film +. The transmitted intensity is then given by JAB...N = (/dodf =

f

j Ls.,.N

i’)

dodRTrfd,...25,b,p’,a~a~...a~l,

(If)

t The thickness t of the Lth film, which enters through the eCt, factors given by eq. (9), is the thickness in the direction of the photon k.Thus if /J(i) is the minimum angle between 2 and the film. f = r(i) = r,jsin @(kj, where ti is the perpendicular thickness of the film.

498

J. P. HANNON

where the brackets ( ) indicate an average over initial polarization states, and in the second line we have introduced the 2 x 2 density matrix for the incident beam: P&9 0) = Y which is normalized to the incident intensity: Tr PO = ZO(&,0). For the experiments considered here we have two films, A and B as shown in fig. 1. The transmitted intensity is given by eq. (11) when the scattering in each film is analyzed in terms of the sume basis 9(k), j(k). It is convenient, however, to use different basis vectors 9,, jA and 9,, $a to analyze the scattering in the two films. In particular, in A it is convenient to use the linear basis 2, and j,+ where i, and j* form equal angles with the direction of the magnetic field HA in A; and in B, the linear basis 9, and QBwhich form equal angles with HB as shown in fig. 1. The angle of rotation from 9, to & counterclockwise about + k is designated as 4, and we note that 4 also designates the rotation angle from the (HA, k) plane to the (Ha, k) plane. With this choice of basis, we have SL*= CQ,= 45” in eqs. (A.l)-(A.3) of appendix A, and wesee that (fXX+fY,)A,B, (fxy)A,B, and (fyx)A,B are independent of sin V, while (“Lx-f,,)A,B is directly proportional to sin 9, which is the quantity of interest. Isolating the sin q term in this manner greatly simplifies the further analysis. The transmitted intensity is now given by Z A6 =

s

dodL Tr [&,&A dA& aLfliA al],

(12)

where 0 is given by eq. (7), BeA is the rotation matrix: cos 4 &A

and for an unpolarized

=

-sin

4

sin 4 cos C#J’

(13)

source, PO = jZ,(ff, 0; o,)L

(14)

where 1 is the 2 x 2 unity matrix, and Z,,(ri, o; G,) is the intensity of the (fi, w) component of the source. radiation (recoilless and non-recoilless). Our interest is to obtain the terms proportional to sin q, Te and CM,i.e. to make an expansion of ZABin terms of sin q, rE and &,, and to exhibit the symmetry properties of these terms. Since sin II, tE and thi are small (5 10 -‘), it is sufficient to keep only linear terms. The development of such an expansion is given in appendix B, and the resulting expression for the transmitted intensity is zAB(“s;

dbPA

9

fA 9

PII hl> 9

= (* sin 24[F’“ed’(3) + J&)(3)(& - &,J + Jpdd’(3){, + Jr”“)(3)&, + 3 cos 2+[G’““‘(2) + K:““d’(2)5E+ K~d)(2)~, + U@“)(3) sin ~1 + [H’“‘)(O)+ +L$Od’(l)tE + jtihd’( l)t, + + Pad)(3) sin ~1).

+ T@“)(2)sin rjj (15)

TIME REVERSAL INVARIANCE

499

The explicit expressions for the coefficient FCPed)(3),J&‘(3), etc. are given by eqs. (B. 3 i-xi) in appendix B. The superscripts of the coefficients in (15) describe the following properties of the functions: The first designation, i.e. s or a, indicates that the functions are symmetric or antisymmetric under the simultaneous reversal of the fields H,, and Hn (i.e. H, + -H, and H, --f -HB). The second designation, e or o, indicates that the functions are even or odd functions of the source velocity U,measured from the center of the Zeeman spectrum (assuming, of course, that the spectrum is perfectly symmetric in the limit rM, tE --f 0). The third designation, d, when it occurs, indicates that these are dispersive type terms - i.e. near a well isolated transition these functions are proportional to x/(x2+ l), where x is the deviation from resonance. At exact resonance, the dispersive terms contain no contribution from the resonant transition, but they are finite because of the contribution of neighboring transitions. Since these contributions are of the form y(y2 + I) where y is the Zeeman splitting, the dispersive terms are generally smaller than the non-dispersive terms if the Zeeman splitting is large. The designation 0, I, 2, or 3 of the coefficients gives the first order of the concentration-thickness dependence of the function, i.e. this designates the order of the dependence of the leading term when a series expansion is made of the exponentials involved (see appendix B). In general, the higher the order of this dependence, the smaller the coefficient is. The #-dependence of Isll is contained entirely in the factors sin 24, cos 24, and 1, so that the symmetry properties of the terms with respect to changes in C$are explicit in eq. (15). We note two additional properties which follow from the discussion and expressions given in appendix B: The coefficients of sin tl and (tE- tM) are proportional to (T,(ML)T,(Ef.+ 1)) and thus will only be finite for a multipole mixture as we would expect. The remaining coefficients will be finite for a pure multipole transition. Also, far off resonance, all terms are zero except Hoe’(O). In this limit H@‘) gives the usual nonresonant transmission, H’““(O) = ZOe-na(r*+‘g). Our interest is in measuring sin q. The dominant sin q term is [&in 247$$ sin ~1. The remaining sin q terms are generally much smaller, being dispersive and of higher order in the concentration thickness. The coefficient of sin 4, sin 24 T”“‘(2), is quadratic in the concentration thickness, and changes sign for simultaneous field reversal (/I* --f n-/IA, /Ia --f II -Be, 4 + 4 +nn), for the &rotations 4 + $+$(2n+ 1)x or 4 + 4+nn, and for a Doppler shift v, + -v,. We see however that there are several additional contributions to the transmitted intensity which exhibit some or all of the asymmetries of sin 24 T$;)). Of primary interest in what follows are the remaining terms proportional to sin 24. The leading sin 241 contribution, fsin 24 F (aed)( 3) , gives the Faraday effect considered by Blume and Kistner “). The F (acd)( 3 )’IS a dispersive-type function which is

500

J. P. HANNON

of third order in the concentration thickness, and is antisymmetric under field reversal; but in contrast to T’““‘(2), FCacd)(3)is even with respect to u, + -u,. The second term in (15), )sin 24 &‘(3)(&-&,J, gives the dominant effect of the multipole currents induced in the electron cloud. This term has exactly the same asymmetries of sin 24 T’““‘(2), and hence simulates T-non-invariance. However J@(3) is of third order in the concentration thickness, as opposed to the secondorder dependence of fi“‘)(2), and this fact enables an experimental separation of any T-violation effects. The remaining two terms also give effects of the induced multipole currents, and again they simulate a T-violation. These terms are also of third order in the concentration thickness, but they are doubly dispersive, and generally negligible with respect to &‘(3) if there is strong Zeeman splitting. The remaining terms in eq. (15) are either #-independent or have a cos 24 dependence, and are all symmetric under field reversal. However, the terms proportional to &, C& and sin q are all odd with respect to u, + -us, and hence will be picked up in an experiment measuring the intensity change for v, + -0,. Intensity change. For a simultaneous field reversal /IA -+ PA = rt--./I,,, pB -+ /& = + $ = ++f(2n+ l)rt or 4 + $ = ~--_a, 4-B =4+ nlc, or for a +-rotation C#J -4 +nn, the intensity change is given by

= sin 2~[F’Bed’(3)+J~~)(3Xr,-

5M)+J~Odd)(3)5E+J~Odd)(3)rM+ T’““‘(2) sin ~1. (16)

Here AZ gives the relative intensity change, e.g. AZ,,,,,, = (I,,,-Z&/(Z,,,+Z;i,), and the right-hand side is evaluated with the initial parameters 4, j?,, and &,. Here up is kept constant during the measurement of AI. We see that AIii,BA and AZ, contain a Faraday contribution and contributions due to the induced multipole currents in addition to the T-non-invariance contribution. If measurements are made for +u,, the intensity change is given by

= sin 24LM(3x~,

- &)+ J8”“)(3)& +J$“dd)(3)5, + T (“92) sin 11]

+ cos 24[KLA’(2)5e+

K$ti’(2)&4 + lYSod)(3)sin ~1

+ [~~_Od’(l)~~+~~“‘(l)~,+

P93)

sin q].

(17)

The right-hand side of (17) is evaluated with v, = + o,; 4, PA and j3a are held constant during the measurement of AI,,, _“,. In contrast to the measurements (16), AZ,., _“a contains no Faraday contribution, but there are six additional contributions proportional to tE, CM,or sin tl.

501

TIME REVERSAL INVARIANCE

3. Determination of sin 1 Sin q can be determined by utilizing the different dependence of the F, J and T coefficients upon the concentration thickness, the polar angles /IA and j&, and ZJ~. There is in fact a particular geometry for which the F and J coefficients vanish, so that in this geometry dlAB,i~ or AI,, i; gives a direct measurement of sin q. However 35 XldL

0

FIG.

2

0.S

1.0

ni2

2.0

2.5

f% Fig. 2. Calculated curve-sfor the contributions to the relative intensity change AZ,,. r;rias a function of the polar angle &, with & 3 n/2. Both films are 2.3 mm thick Ru-Fe fihns containii 4.8 at. % 99Ru. The M = f 1 curves are for the source in resonance with the F fi + r 4 transitions of 99Ru. Po4)(3) gives the Faraday effect, f, (“’ ( 3) gives the coe5cient of the induced multipole current effect, and !P)(2) gives the coefficient of the T-violation effect.

J. P. HANNON

502

we first discuss a method of indirect determination of sin q in order to compare our calculations with experimental values: (i) An indirect de?er~inut~~n of sin q can be obtained by treating
t(A LB; G&L

s P,) + AIMS; itit - ~33A))

= F’“+(3 ; 0sPd.

(ii) (18)

Here, the coefficients on the right-hand side are understood to be divided by (I‘4,(% Btl) f &ii(a., /&I)). The values of sin ? and r can then be determined by using the different dependence 7 The behavior in ref. 22).

of F(*C“)(3) and T(‘*)(Z) with respect to 0. and /?a is discussed

by Blume and Kistner

TIME REVERSAL

503

INVARIANCE

of their coefficients on the concentration, film thickness, resonance denominator, or angular changes /$, to obtain two independent equations from eq. (ISi). For example, if measurements are made in the Kistner experiment for Z& = 54.75” and pi, = ‘70” with fu, in resonance with the r$ -+ ~3 99Ru transitions, the relative changes of the sin q and r coefficients are P’(2;

v,, 70°)/T’“‘(2; v,, 59) = 0.7,

.ZF$(3 ; 0,) 70°)/~~~(3 ; v, ,55O) = 1.4.

(19)

Kistner’s data are not sufficient to determine both sin PZand 5, but if we assume T-invariance in (I%), i.e. sin q = 0, then Kistner’s measurements give (0

= (-4.355.0)x

(F”‘Cd’(3; 1.8 mm/set, 54$‘)) = (1.4+0.1)x

10-3, 10m4,

(20)

which agree well both in sign and magnitude with the calculated values: 5 = -6.5x F’aed’(3; 1.8 mmlsec, 543’) =

10-3,

1.6 x 10e4.

(21) (ii) Direct ~eter~i~ution of sin q. It has been pointed out by Chrisman 28) that the Faraday effect &sin 24 F (acd)( 3 ) in (I 5) will vanish for two identical films which are oriented such that /IB = x-PA. As we show below, this particular orientation will also eliminate afl & and rM contributions to LIZ,,, nA (or AZ++),so that a direct determination of sin q can be obtained. To see this, we make use of the development given in appendix B. If the two films are of identical composition (but not necessarily of identical thickness) and subject to identical fields, and if fiB = x-_B~, then the scattering amplitudes in B are related to the scattering amplitudes in A by the relations (BAi-iv) of appendix B. Substituting these relations into (B.2iii-vi) we find that T$“’ = -Tp), Ftcdf = -Fpd), JW = -Jf&, Lr) = Lp), and &?$ = L$,,d . We note that these relations will BEM also hold if the films contain different concentrations of Mossbauer atoms (if the Zeeman splitting is identical), since these coefficients are independent of the concentration thickness. The explicit expressions for the coefficients F, J, T of eq. (16) are then given by PQ(2)

= / do dk 41, ABLER)+

higher order,

(9

j=‘“ed’(3)= 1 dw dk 2ZoFa”““‘{[B’“e’(l),4’Se’(2)- /$““‘(I)z+““(~)]@ - Z!~)[BtJOd’(l)A’““(2) - A’“Od)( 1)Boe)(2)]),

(ii)

do dli 2Zo~~~~~~~)~~(*‘)(l)z4’“‘(2) - A(“)( 1)8’““(2)] - Z$~‘[ZP+( 1)A°C’(2)- ~(‘~)(1)~(**)(2)]~. We omit the expressions for @“rd)(3) and .@“rd’(3).

(iii)

(22)

J. P. HANNON

504

In the limit that the concentration thickness of B becomes identical with that of A, then the concentration thickness dependent parameters of the two films become equal - i.e. fP’( 1) -P Atset( I), 8’““(2) -+ A’““‘(2), etc. Then F(Ped)(3) = .@(3) = 0, while T”“(2) is finite. It is also easily verified that Jgodd) and JcDdd) are zero. Thus a measurement of Ali,, BA or of .4& gives a direct measurement of sin q in the Chrisman geometry. We note that this is not true for a measurement of N_,,_,. The cos 2# and &independent terms proportionat to tE and CMin (17) do not vanish for the Chrisman geometry, and hence df_,_ OSdoes not give a direct measure of sin n. Of course in this case measurements of LIZ-,,, ,,, at i-# can be used to determine sin q. (iii) Maximizing the sin q contribution. Since in the Chrisman geometry only a single orientation is required to determine sin q, the orientation and the concentration thickness of the two identical films can be chosen to maximize T’*“‘(2), and hence the intensity change: (I,, +I&Ai,B;

~6 = 2+)(2j

sin q = (jd,dr41,[a’“‘(l)lz~~)~~)]

sin q.

(23)

Here we have taken 4 = 45”. We note again that I,(& w, u,) is the intensity of the (R, o) component of the source radiation; A’““(1) is given explicitly by (B. 2ii) and contains all the ~on~ntration thickness dependence; and the factors Ly), rr’ are given by (B, 2vi and viii) and depend only on fi*, the Zeeman transition, and the multipole mixture of the transition. As a function of B* = n-j3u, the dominant angular dependence of T’““‘(2) is contained in the factor (L~‘T~“‘) which is independent of the film thickness. Thus to first approximation, the optimum angle is determined from (d~d~)(~~)~~)) = 0. The optimum angle will depend in genera1 on the multipolarity of the transition, the mixing ratio f,(EL-t I)/T,(ML), the particular Zeeman transition involved, and the splitting. Once the optimum angle has been determined, the concentration thickness can be varied to maximize the intensity change, T’““‘(2)sin ‘1, or aiternativefy, to maximize the relaGue intensity change (1, +1&-i ~(ao)(2)sin 11.The first procedure, which we consider below, offers the advantage of requiring shorter counting times to measure theeffect, but gives a smaller intensity change relative to the total transmitted intensity. To gain a qualitative picture of the dependence of T”‘)(2) on the concentration thickness, we consider the limit of a delta function source. In this limit, the film thickness f T which maximizes the absohtte intensity change (for a fixed con~ntration P) is given by Ptf = sin P log (U1/UJ/C2KfI” while the optimum concentration is given by

-_f?)l~

(9

(24

P* of Miissbauer atoms (for a fixed thickness 1,)

(24)

TIMERJYERSALINVARIANCE

505

In obtaining eq. (24) we have expressed f,,q = 1, 2, as fq = (1 - P)fc”’ + Pfm where of Miissbauer atoms, fz” is the contribution to f, from the non-Miissbauer atoms andf,” is the contribution from the Miissbauer atoms. It is clear from eqs. (24i, ii) that in general there does not exist a simultaneous solution P*, t*, i.e. there is no optimum concentration thickness. Instead, as we might expect, T”“(2) increases monotonically as P increases monotonically from 0 -i 1, and it is advantageous to use as high a concentration of Miissbauer atoms as possible. Explicitly, with t* determined by eq. (24i), the concentration-thickness dependent coefficient A( 1) is given by P is the concentration

A( 1) = 3(y - l)y[Y’(’- y)‘,

(25)

where y = IfJZfl and we recall from the defining relation [eq. (8)] that If1 > Ifz. The value of y is between 0 and 1, and as y decreases from 1 + 0, IA(l)1 increases from 0 + 0.5. In the limit of strong Zeeman splitting, fi is approximately equal to the electronic scattering amplitude, and y is given approximately by y z ((l-P)o:“+Pa;) x (P~~e-“““)“K(~)[(2jo+

1x1 +cr)(x2+1)4rr]-’

+(l -P)c$“+

Porn)-‘,

(26)

where P is the concentration of Mijssbauer atoms, ai” and 0: are the cross sections for electronic absorption processes for the non-Mvlbssbauer and MSssbauer atoms, c1is the internal conversion coefficient, x = 2(w-dE)/P, and K(b) is a factor of the order of unity which contains the angular functions of fl, the Clebsch-Gordan coefficients, and the ratios of P,(ML) and Py(E2) to P, (for example, K(B) = 0.8 for the 99Ru +- + +$ transitions and /I = 55”. It is easily verified from eq. (26) that y decreases monotonically as P increases from 0 + 1, and hence IA(l)1 and T”“‘(2) increase monotonically. From eqs. (23)-(26) we can obtain an estimate of the maximum T-violation effect which can be expected for a given experiment. We denote the incident intensity as ~f~I,d(o-AE), where f: is the MSssbauer factor for the source, I, is the incident intensity (recoilless and non-recoilless), AE is a resonance energy, and the factor 4 is a scaling factor which is introduced to bring the b-function calculation of T’““‘(2) into quantitative agreement with the value which would be obtained using the exact source distribution. Then with ti = 1: determined by eq. (24.i) and A(1) by eq. (25), the maximum intensity change due to T-non-invariance is T”“‘(2) sin rj = 3fA I0 I_!r)Tp’[O.5(y - l)~(~‘(’ -y))]2 sin q.

(27)

As noted previously, the factor (Lp)Ty)) dep en d s on PA, the Zeeman transition, and the multipole mixture, and will have a maximum value on the order of unity for a strong Ml-E2 mixture. As an example, we consider “Ru-Fe films containing a fraction P of 99R~, and we assume the incident radiation is in resonance with the -3 -_) -+ transition and

506

1. P. HANNON

that T = 4.2” (fi = 0.12). The maximum value of Ly’T:““’ = 0.9 occurs near fiA = 55”. Taking p = 55” and P = 0.048 as in the Kistner experiment, we then find tz = t,* = 1.2 mm, y = 0.5, A(1) = 0.13, and the maximum intensity change for P = 0.048 is T’““‘(f)sin q z (2.8 x 10e3 sin q)Z,. If the film thicknesses are taken as 2.3 mm as in the Kistner experiment, then the intensity change is z (1.8 x 10e3 sin q)Z,, so that about 14 x the counting time would be required. If the Kistner geometry were also used, then 6 x the counting time would be required since four measurements are then required to isolate the effect. The maximum possible effect which can be obtained with the -3 + -3 9gRu transition occurs for P + 1, which is, of course, purely hypotheticat, and gives y = 0.146, t* = 0.11 mm, and ~(ao)(2)sin q x (2.3 x lo-’ sin q)Z,, which is up by a factor of 8.3 over the P = 0.048 case. 4. Summary We have given a detailed analysis to show how the effects of the initial- and finalstate atomic radiative corrections can be distinguished from any T-violation effects in MSssbauer T-invariance experiments. Such a separation is necessary in order to improve the accuracy of the experiments sufficiently to compare with the predictions based on the theory of a T-violating hadron current. The dominant atomic radiative correction exactly simulates a T-violation under field reversal, axial field rotations, and Doppler shifts about the center of the Zeeman pattern. However the term has a different dependence on the concentration thickness of the films, and on the polar angle of the field, so that a separation is possible. A field reversal experiment carried out in the Chrisman geometry gives a direct measure of T-violation. With this choice of experiment, we can vary the thickness, concentration, and the orientation angle j? to maximize any T-violation effects. Of course the maximum possible effect depends on the choice of Miissbauer transition. In general, the strongest possible effects will occur for those MSssbauer transitions which exhibit (i) a strong multipole mixture (so that T’,B” z unity in eq. 27), (ii) a small recoil (fga: unity and y < 1 in 27), and (iii) for which the condition Zj& > I&, is satisfied (y < I in 27). This latter condition is best satisfied when there is a high concentration of Miissbauer atoms, strong Zeeman splitting, a small recoil, and when the total cross section for resonance absorption is much greater than the photoelectric absorption cross section. Part of this work was carried out while the author was a guest at the Technische Hochschule Mtlnchen, Teilinstitut Mossbauer. I would particularly like to thank Dr. R. L. Mijssbauer and Dr. P. Kienle for their hospitality during my visit. I also wish to thank Dr. B. Chrisman and Dr. P. Debrunner of the University of Illinois for useful discussions. Finally, the author is indebted to Dr. G. T. Trammel1 of Rice University for innumerable contributions.

507

TIME REVERSAL INVARIANCE

Appendix A We give the scattering amplitudes introduced in eqs. (7) and (8) for a fraction P of Mijssbauer atoms in a uniformly magnetized medium. The MSssbauer transition is taken as an Ml-E2 mixture, and it is assumed that .i, is a good quantum number. For simplicity, it is also assumed that the y-ray energy is far off resonance for any electronic transitions, so that there is no anomalous electronic scattering. The scattering ampfitudes f, tf,,f,, -& J& and& are given in terms of a Iinear basis R, j, where (9, 9, g) form a right-hand coordinate system. The a and fl are the azimuthal and polar angles in the ($9, f) coordinate system defining the direction of the magnetic field at the nucleus. For (& +_&) we then have,

W)

The notation for the Cfebsch-Gordan coefficients C(L, Mm,) = C(j&&, m,Mm,) is that of Rose 29), ISI = [r,(E2)/r,(Ml)]* is the E2/Ml mixing ratio, and the constant K(m,) is given by K(m,) = ~~PnP(mo)r,(M1)(8nr)“exp-((k * r)2>, where n is the density, and P(Q) is the probability that the initial internal state of the nucleus is IJ,, me). The frequency variables x(M, me) are defined by x(M, me) = me)-ke]fr. Finally, Z,, Z,,e and a:” are the charges 2K(J,, m,+M)-%&I, and the total absorption cross sections for electronic absorption processes for the Mtissbauer and non-M~ssbauer atoms, respectively. For (AX-fry ):

+M &*1(cz(2; Mm,)~6~2[5(cos2 2/I--cos2 8) cos 2cc]e2’ca + C’(1; MmO)[3 sin’ B cos 2a]e2’rM-i-C(2; Mm&(1

; Mmo)fSj

x [2JiX(h4 sin’ j? cos 2a cos q - 2 cos fl sin2 j? sin 2a sin rjjei(e~+CM))(x(h4mo) - i>- * + (C2(2; 0m,)16]2[~

sin’ 218cos 2a]ezrcB-- C2(1 ; Om(t)fS sin’ fr cos 2a]e2@=

+C(2; Om&(l ; ~~~)~~1~2~~ sin 28 sin fl sin 2a sin ~]ei~~~~~~~X~(O~~~-i)-‘1. (A-2)

J, P. HANNON

For & : fyx = c G%){M m

g*2C”(2;

x +[(a. sin2 2/?-sin’ +M;*1(C2(2;

~f%W

/?) sin 2a+fiM

~m)l~12W

sin 2/I sin /YJe2ieE(x(Mm0)- i)-’

cos2 2/3-cos2 /I) sin 2~ + iM cos p cos 2j3]e2ite

+ C2(1 ; Mme)3[+ sin2 B sin 2a+ iM cos b]e2icM +C(2;

Mmo)C(l; M~~)l~l~~[M

sin’ /? sin 2a cos q + 2 cos j? sin2 /I?cos 2a sin 9

- i2 cos3 ,9 cos ~]e~(~~‘~“‘)(~(M~~) - i)- ’ +(C2(2;

Om0)1612+$[sin’ 28 sin 2a]e2*CE- C2(1 ; 0m,)3[sin2 B sin 2a]e2’rM

- C(2; Om,)C(f ; OmrJSIJS x [sin p sin 2/?(cos 2a sin q + i cos q)]e’(~E+~M))(x(On2,)- i)- ‘}.

(A.31

The expression of& is given by eq. (A.3) with the replacement of all the quantities in square brackets, [---I, by their complex conjugates. Appendix B In order to make the expansion of the transmitted intensity IAB in terms proportional to sin I, &, and &,, and in order to exhibit the symmet~ properties of these terms, we first note the functional form of the expansion of five basic function combinations: fI J2

yCL,+f,.J = CW4 + (se)&- 5d) + i((se> + @odXt~ - hdlei2% (i> (fyx-.fx7)= C(@o> f (aed>& - td) + i((aed) -k(ao)&- 5ho)lei2cM, (ii> (iii) (B. 1) (k-.f,d = CW)+ iW1 sin‘1%

where here x, y refers to the Iinear basis with a = 45”. The first designation of the functional form, i.e. s or a, indicates that the functions are symmetric or antisymmetric under the interchange x ++ y. The second designation, e or o, indicates that the functions are even or odd functions of the frequency w measured from the center of the Zeeman spectrum (assuming, of course, that the spectrum is perfectly symmetric in the limit CM,& -+ 0). The third designation, d, when it occurs, indicates that these are dispersive type terms - i.e. near a well isolated transition these functions are proportional to x/(x’+ I), where x is the deviation from resonance. The expressions for the remaining functions of interest, e.g. fi -f2,~‘.,fyx, can be obtained immediately from eqs. (B.1). The expansions [eqs. (B. I)] can be immediately verified from the explicit expressions given in appendix A, with a = 45”. We note that eq. (B-Ii) gives the functional form of frlf2 and K-,.+&h and d oes not imply equality, We also note that the functional form

509

TIME REVERSAL INVARIANCE

given for(Axz4f,,) and(L -fpy1is on/y valid

for our particular choice of basis, i.e. r = 45”. This basis is chosen because it greatly facilitates making the expansion in terms of sin 17,c&.and tM, but of course the final expression, eq. (15) is independent of our choice of basis. The exact expression for IAa, given by substituting eqs. (7) (13) and (14) into eq. (12), is too long to justify writing out explicitly. However, when the operations are carried out, one finds that the expression contains the products of ten functions relating to the film A, and the same ten functions with the parameters of the film B. Three of these functions depend on the concentration-thickness parameter of the film, while the remaining seven are independent of the con~ntration thickness. For film A, the three factors ~e~e~~e~~ on the con~ntration thickness are [&te,+,l”], = a’S”(0)+aZ”d’(l)e,+a~~)(l)Shil, [fef+)ec_JA

(9

= ~~(e-ZU”-e-2rf2*)+ie-r(fi+n~rsin(R(f,-f,)t)],

= (A’S”(1)+A~~‘(l)~E+A~pd~(l)~M)fi(A~sOd~(l)+A~~(l)~E +Ar’(l)f&), C!&,-,i23a = A’“e’(2)+A~Dd)(2)~E+A~~)(2)5h(,

(iii)

(ii) (B.2)

where e(,, is given by eq. (9). The right-hand side of eqs. (B.Zi-iii) gives the Taylor series expansion to the first order in
(iv) (9

510

J. P. HANNON

1

fxx-fYY = T+) sinq+i~('ed) --._ sin rj, A

[ f,-fi A

(viii)

We note that all the functions here are purely nuclear in origin in addition to being independent of the concentration-thickness, and that they are functions of #?_.,, CO,the Zeeman splitting, and the multipole mixture. We also note that the coefficients of sin 4 and (tp.-
Jgodd) ( 3 )’ is given by eq. (B.3iii) with the interchange of all E and M subscripts.

TIME REVERSAL

INVARIANCE

511

For the cos 24 and &independent terms, we only write out the expressions for the leading terms, G’““(2) and H’““(O): G’““‘(2) = 21,,{[,4’“‘( l)br’ - ,@“d’(l)@r)][B’““( l)fig’ - B’=+( l)kp’] _ A’““(2)B’““(2>F:““‘F’,“‘d’),

(v)

H’“‘(0) = 10{[a(S’)(0)+ A’Se’(2)G~e~][~‘se’(0) + B’““(2)G~)] + [A’“‘( l)M’,““’+ k”d’( l)M~ed’][P)(

1)MF’ + Pd’( l)M~d’]}.

(vi)

(B.3)

Finally we note that the symmetry designation s or o, which was introduced in eqs. (B. 1) as designating the symmetry of the functions under the interchange x t, y, gives the symmetry of the functions defined in eqs. (B.3) under the simultaneous reversal of the fields in A and B (j?* + n-j?*, and /Ia + IL-/&). This is easily seen by noting that if fl = n-/3, and a c 45” in eqs. (A.l)-(A.3), then (II * 216 = VI, 2189 o-y, -LIB

= (fxx -_fJs 9

(9

(ii)

(fX,)P = (fYX)S3

(iii)

&)a

(iv) (B.4)

= (fX,)S *

Thus field reversal in a medium is equivalent to the interchange x~y choice of basis.

with our

References 1) J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13 (1964) 138 2) 3) 4) 5) 6) 7i 8)

L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562 T. D. Lee and L. Wolfenstein, Phys. Rev. 1Xl (1965) B1490 J. Bernstein, G. Feinberg and T. D. Lee, Phys. Rev. 139 (1965) B1650 S. Barshay, Phys. Lett. 17 (1965) 78 J. Prentki and M. Veltman. Phys. Lett. 15 (1965) 88 T. D. Lee, Phys. Rev. 140.(1965) B959, B967 B. A. Arzubov, A. T. Filippov, ZhETF Pisma 8 (1968) 493 [Transl. JETP Lett. (Sov. Phys.) 8 (1968) 3021 9) K. Nishijima and L. J. Swank, Nucl. Phys. B3 (1967) 553, 565 JO) T. Das, Phys. Rev. Lett. 21 (1968) 409 11) S. Okubo, Ann. of Phys. 49 (1968) 219 12) E. M. Henley, Ann. Rev. Nucl. Sci. 19 (1969) 367; F. Bdhm, Hyperfine structure and nuclear radiations (North-Holland. Amsterdam, 1968) p. 279; D. Hamilton, Proc. Third Conf. on high-energy physics and nuclear structure, ed. S. Devon (Plenum Press, New York, London, 1970) p. 696; A. Richter, Talk at the summer school on intermediate-energy physics, Banff, Canada, August 1970 13) 0. C. Kistner, Phys. Rev. Lett. 19 (1967) 872; and Hypertine structure and nuclear radiations (North-Holland, Amsterdam, 1968) p. 295 14) M. Atac, B. Chrisman, P. Debrunner and H. Frauenfelder, Phys. Rev. Lett. 20 (1968) 691 15) E. Zech, F. Wagner, H. J. Korner and P. Kienle, in Hypertine structure and nuclear radiations (North-Holland, Amsterdam, 1968) p. 314

sit

16) i7) 18) 19) 20) 21) 22) 23) 24) 25)

26) 27) 28) 29)

J. P. HANNON J. P. Hannon and G. T. Trammelt, Phys. Rev. Lett. 21 (1968) 726 S. P. Lloyd, Phys. Rev. 83 (1951) 716 B. A. Jacobsohn and E. M. Henley, Phys. Rev. 113 (1959) 234 P. Stichel, 2. Phys. 150 (1958) 264 J. P. Harmon and G. T. Trammeli, Phys. Rev. 186 (1969) 306 E. M. Henley and B. A. Jacobsohn, Phys. Rev. Lett. 16 (1966) 706 M. Blume and 0. C. Kistner, Phys. Rev. 171 (1968) 417 M. Blume, Phys. Rev. A4 (1968) 351 T. A. Tumolillo, Nucl. Phys. Al43 (1970) 78 G. T. Trammel1 and J. P. Hannon, Phys. Rev. 180 (1969) 351; Yu. M. Kagan, A. M. Afanasev and V. K. Voitovetskii, ZhETF Pisma 9 (1969) I55 [English tramdation: JETP Lett. (Sov. Phys.) 9 (1969) 911 M. E. Rose, internal conversion coefficients (North-HolIand, Amsterdam, 1958) 0. C. Kistner, Phys. Rev. 144 (1966) 1022 B. Chrisman, private communication M. E. Rose, Elementary theory of angular momentum (Wiley, New York, 1957) pp. 32-48