Polarization effects in photonuclear reactions

Naclear Physics 85 (1966) 327--336; ~

North-rtolland Publishiny Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

POLARIZATION EFFECTS IN PHOTONUCLEAR REACTIONS R. RAPHAEL and H. OBERALL

Physics Department, The Catholic University of America, Washington, D.C. Received 8 March 1966 Abstract: We derive explicit formulae for angular asymmetries and longitudinal and transverse polarizations of nucleons emitted from nuclei in the (?', N) reaction, corresponding to linearly or circularly polarized as well as unpolarized incident photons. The reaction is assumed to proceed in two stages, namely, by photo-excitation of a single nuclear level and its subsequent decay via nucleon emission. Our expressions are given in terms of the nuclear transverse multipole form factors for the excitation and of the reduced matrix elements corresponding to the decay. The results are illustrated by obtaining nucleon angular distributions and transverse polarizations following photo-excitation of the giant dipole resonance in a60, using Gillet's model and R-matrix theory. The comparison of these results with experiment is satisfactory.

The nuclear photoeffect 1,2) has long been an object of experimental investigation, and one has now, besides performing measurements of angular distributions 3, 4), proceeded to a study of the polarization effects involved, concerning either polarization of the photons 5) initiating the reaction or an observation of polarizations of the emitted nucleons 6). In connection with this experimental interest, we have ,t in the present paper derived explicit formulae for the angular asymmetries in (7, N) reactions with all polarization effects except those of the nucleus taken into account, viz. (i) incident photon linearly on circularly polarized and (ii) angular distribution and longitudinal or transverse polarization of the emitted nucleon (or generally: spin-½ particle) considered observed. It is assumed that the process proceeds in two steps, first by photo-excitation and formation of a compound nucleus and subsequently by its decay with nucleon emission. (We also take the excited state to correspond to a single isolated level of the compound nucleus; in our formalism, this can easily be generalized.) Accordingly, our formulae depend on two types of reduced matrix elements, the transverse electromagnetic multipole form factors of the excitation and the matrix elements of the decay of the compound state. General expressions for angular and polarization correlations have been given in Goldfarb's review articles (refs. 8, 9)), also on the basis of the compound nucleus hypothesis and using the channel formalism. It appears unwise 10), however, to treat the photons, whose electromagnetic interaction can be handled e.g. by perturbation theory with appreciable accuracy, on the same basis when employing the channel formalism as one treats * Work supported in part by a grant of the National Science Foundation. tt An analogous calculation of angular distributions, but not polarizations, has been carried out by us for the case of electro-excitation of nuclear levels and their subsequent decay 7). 327

328

R. RAPHAEL

AND

H.

UBERALL

the nucleons, whose strong interactions can be calculated with no such accuracy. Therefore, we have explicitly introduced all the electromagnetic multipole matrix elements of the excitation into our calculation, and thus at least the first step of our process is expressed by electromagnetic nuclear form factors which are well-known (or can be accurately tested against nuclear models); only the decay part of the reaction is given in terms of corresponding reduced matrix elements which are less accurately known, but which can still be evaluated in terms of nuclear models (by the use of Wigner R-matrix theory 11), e.g.). Our results moreover show the explicit dependence on the azimuth tp of the emitted particle in the form of cos 2q~ or sin 2tp. A relation derived by Agodi 12) between the nucleon azimuthal asymmetry when using linearly polarized photons and the coefficients in the polar angle distribution when the photon is unpolarized, is generalized. By way of illustration of our results, we have evaluated angular distributions and transverse polarizations of nucleons emitted following photo-excitation of the 22.5 MeV giant dipole level in 160, using the model of Gillet 13,14) and R-matrix theory; the comparison with experimental results is satisfactory. For the electromagnetic excitation part of the photonuclear reaction to be treated in perturbation theory, we use the Hamiltonian H = - e ( 2 k ) - ~ f d a r ( j • A +t*" V x A),

(1)

w h e r e e 2 --- --A-4/z and j , / , are the nuclear charge current and magnetization density, 137 ' respectively. The vector potential is taken as a plane wave propagating along the z-axis, and expanded in multipoles as)

A~ = - 22-½(g,x + i2~y)etk~ = (2n) ~ ~ iLL{-- k IV x [JL(kr)XLZ(r)] -- ljL(kr)XLa(r)};

(2)

L=I

here k is the photon momentum, L = (2L+1) ~, ? = r/r and 2 = 1 ( - 1 ) for right (left) circularly polarized photons; XLz is a vector spherical harmonic is). This gives 16)

H (~) = e(n/k) ~ ~

iL£(~'(L~)--1-2 J - ( M ) ) ,

(3)

L=I

using the transverse electric and magnetic nuclear form factors J'[~)(k) = k -1 f dar{j • V x [jL(kr)XLa(P) + kZl~ • jL(kr)XLa(P)},

(4)

ftL~)(k ) = f dar{p • V x [jL(kr)XLz(P)] +j" jL(kr)XLa(P)}.

(5)

For photons polarized linearly in the (x, z) plane, one has

n('° = -e(n/2k) ½i iL~ E 2(J'Lz (E)+ 2,~'La Oa,). L=I

A = :i: 1

(6)

PHOTONUCLEAR REACTIONS

329

We introduce reduced matrix elements by the Wigner-Eckart theorem: (JoMoI,°~(LE~)'(M)[JiMi) = j o l ( J i M i ,

L 2 ] J o M o ) ~ ' ( L E)'(M),

(7a)

where (7b)

~-(E), (M) ----- (J0[I~---(E), (M)llJi),

and where the quantum numbers of the initial nuclear state were called Ji Mi, of the compound state Jo Mo ; we also call TL~ = j-~E)+ 2j-~m.

(7C)

Then the density matrix for the photo-excitation of a compound level (integrated over its width) by a linearly polarized photon is found to be adk)MoM,o = k-'(zce/ZJo) 2 Z

Z iL'-LLL'~'2'

Mt2).' LL"

x(JiMi,L,~lJoMo)(JiMi, 1_22 ' , IJoMo)T~z , * TL,z,.

(8)

If the photon were circularly polarized, one would leave out ~a~, in eq. (8), set 2' = 2 in the remainder and multiply by 2. For unpolarized photons, the latter result would still be averaged over the two values 2 = _ 1. In order to obtain the differential cross section combining excitation and decay, we have to multiply eq. (8) with the density matrix of nucleon decay PMoU'o to the final state JMj, whose magnetic quantum number we sum over, and obtain

dQp

-

Z

MoM'o \ d

adk

Z PMo~ro,

(9)

/MoM'o Mj

where dg2p refers to the direction (0, ~o) of the momentum p of the outgoing nucleon with respect to k as the z-axis. The density matrix for the decay is given by t ---

.

-j~

Z Pmo~ro = Z ( - l)-M'°(JoMo, J o - M o l J M ) f s o s ( J ms ~

),

(10a)

F~o~(J~) = Jo~Z E ~ ( j l , j'l 3 ( - L)~÷~- J W(Jo Jo JJ', J J) ~, jlj'l"

× ~i~,(g)(jlljlllJo)*(j[Ij'I'liJo);

(10b)

here, ~ is the rotation matrix s, 9) corresponding to a rotation R that brings the z-axis into the direction of the outgoing nucleons, which carry total and orbital angular m o m e n t u m j and l, respectively. The quantities (J] [jl]IJo) are the corresponding reduced nuclear matrix elements for the decay. There appear three types of radiation parameters c)~t,, for spin ½ emitted particles 9, 17): one corresponding to unobserved polarization of the particles (but not yet summed over this polarization), c~O) 2~, = (167r)- a))'(j½, j' - ½ l J 0 ) ( - 1)J+J-&[1 + ( - 1)s-+t+v]6~,o

(1 la)

330

R. RAPHAEL AND H. UBERALL

one corresponding to their longitudinal polarization, s being a unit vector along their spin direction, and p = p/p: c-a) ~,

=

1 1 -((16n)-ls • Pj)'(-1)J-~(j½,j ' -~lJO)[

1 )Y + l + r

]O~,o,

(llb)

and finally one corresponding to their transverse polarization (t/ = +_ 1):

c(n) ) ~ , = _~lJ(16n)-l(sx_itlsy)3j,(j½,J,½1J1)(_l)J-J'+'6~,n.

(11c)

Inserting eqs. (8) and (10) into eq. (9), one finds

d~o~ - ½ ~.,_CYLL, ~ 22'T*zTL,z,(L2, E LL'J~

dQp

dal x)

dOp d a ~ x)

-,t'lJ~)~jt~(O)e-'~7%

- -½s. p E CYLL"~_2)"T~aTLw( L2, L'-2'IJM')~J~(O) e-'~'' LL'Jo -

--

dOp

~'~

(12a)

(12b)

),A'M _

¼(sx--itlsy)Z ~lJCJLL"Z )')"TL*zTL'z" LL'J

AA'M

x (L)~, L' - A'iJ2~)[(2J + 1)/4n]*~,(t#, O, - ¢p). (12c) The summation indices Je(Jo) mean that only even (odd) values of J should be summed over. (The assumption of an isolated intermediate state Jo implied 9) that l+l' had to be even.) We used the notation xs) Ylm( O, q) ) = "-~,m( O)e imp,

(13)

and have introduced the symbols

C~L, = (e2n/akJ2)[(2J+ 1)/4k]-~( - 1 ) - J ° J 2 ( - 1)J+~ .L,_L,~',.~,pz × , 1,1., [,-- 1 ) L W ( J J o L J i , and

Joff,)(Sj/F),

(14a)

CLJL, , similar to this, except for the replacement

Sj ---}Sj.

(14b)

Sj = ~, jj'W(JoJojj'; JJ)(j½,j' -½[JO)(J[]jl][Jo)*(Jllj'l'llJo),

(15a)

We must still define jj'll"

Sf = ~.~3)'W(JoJojj'; JJ)(j½,J'½lYl)(- 1)J+t-J'+*(JlljlllJo)*(Jllj'l'llJo). (15b) jj'll"

One may note that 7,/7)

So = 7o 1(_ 1)Zo-s+~Fsoj '

(16)

PHOTONUCLEAR REACTIONS

331

where the partial decay width from Jo to J for nucleon emission is given by

Fsos = ~ (JlljlllJo)*(Jlljl'llJo) ; ill"

(17)

finally, F is the total decay width of the compound nucleus state Jo. Next, in the expressions da (x) ofeqs. (12), the summations over 2, 2' will be carried out. Further, the corresponding expressions da (~) t'or circulary polarized photons are worked out, and both are written in a unified form which turns out differently depending on whether ( - 1 ) L+L'-j is 1 or - 1 . Separating out again the electric and magnetic matrix elements in eq. (7c), we may distinguish two cases. (i) For the terms in eqs. (12) containing multipoles corresponding to L and L' of same type (both electric, or both magnetic), L+L' must be even, which means ( - 1 ) L+L'-J is + 1 if J = Je, - 1 if J = Jo. (ii) For the multipoles L, L ' of different type (i.e. electric-magnetic cross terms), the opposite is the case. Using this observation, the equations may be simplified, and we finally obtain the following results for the angular distributions of the emitted nucleons summed over their polarizations (da = 2dao), for linear (x) or circular (2) polarization of the incident photon: da(x) -

dQp

C~L,(L1, E - 1 J0]/~-"(E)] I, L "{- ~"(M)~*/~-(E)L ] \ L' LL'So --[ ~ C{L,(L1, L'IIJZ)(~--(LE)+ J-(M)a*tyCE)L J ~ L'

2{ ~

"[- ~ " ( M ) ~1" _ L , JO\{0~]

--Y(M)L@-L' J" S2tt0alJJCOS 2~b}, (18a)

LL'Je

da(~-) dQp

-

2 ~ _ C)LL,(L1,L' -- 1,J 0 ,,v ~ --(E) L + oq-(M)~*t.y-(E) L , , L' + Y~L'M))~-@)O(0) •

(18b)

LL'Je

Since the cross section in eq. (18b) is independent of 2, it is equal to the differential cross section for unpolarized incident photons: da

da (a)

(18c)

- -

df2p

df2p "

Next, the longitudinal polarizations of the emitted particles are obtained, given by p[x) = [da~X)(s = p)_da}X)(s = _p)]/da(X).

(19)

We write them in the form p[x) = 2dwJ*)/da (*);

(20a)

similarly, the transverse polarizations of the emitted particles P(xx) along X = p × (p × k), and P(rx) along Y = k × p are written as n(xx) =

2dw(x*)/da(x),

n(rx) =

2dw(r*)/da(X);

(20b)

all this refers to incident photons linearly polarized in the x, z plane where z//k. The corresponding expressions for circularly polarized photons, P}a)x,r, are easily

332

R. RAPHAEL AND H. UBERALL

obtained also, where 2 = 1 ( - 1 ) refers to right (left) circular photons, and so are the expressions P~, x, Y for unpolarized photons. We finally have i ~ __ CfzL,(L1, E1 J2"ffc~"(E)+~--(M)~*/~ -'(E) j-(M)~_ [0 ~ sin 2~, ]\ L L ] k L' -L' J d2k ]

dw}X) dQp

(21a)

LL'Jo

dw~xX)- (2i)-'{ Z_ CJLL.(L1, L'I J2V~--(E)+ - ' L E )J'(M)~*tJ L : tJ I, L-Y-~'M))E(23+1)/4rc]~df-(O) dOp LL'Je + 2 CJLL'(LI' /JlIJ2)(~--'(LE)+~--(M)'~*/J'(E)L ] k L' --~'(M)~I-r2J±L' :ct ~ lW4rD~dJ:O ~ : j ~ +, :~ sin 2cp, LL'Jo (21b) d w T _ ( 2 0 - ' { 2 y. _ Ci.:(L1,

df2p

L' - I lJ0)(~'--~E) + ~~1 ML ) , ]:1 ~ "~ -L"~ ) ~T ~ "¢ ~ L" M )]~~ J_l \ ~u ]

LL'Je

+ [ ~_ C{.(L1, L'ltJ2)(S~ ~) + : ~ ) ) * ( J - ' ? , ) - 9-~,~))[(2J + 1)/4~]~d{ (0) LL'J~

+ ~_ C~,:(L1, L'~132)(~-k~)+ Sr~))*(~-~,)_ ~-~,~))[(2~ + 1)/4~]~d~_(0)] cos 2~}, (21c)

LL'Jo

dw~~)

_

df2p dwtx~)

:~ Z C{.(L1, L' - 11~0)(~-IE~+ :IM~).(~-i~ + ~-i,~)~jo(0), (22a) LL'Jo

- )° 2 CiL'( LI' E -- I IJ0)(~--(E) + ~--(M))*(,ST(LE)+ ~--~LM))~,~j~(0),

df2p

dw~rZ) dl2~

(22b)

LL'Jo

i 2 - CJLL'(LI' 12 --IIJ0)(~-~LE)-F ~"°7"(M)'~*/af-fE)'a°7"(M)'~(O~ (22C) L ] K~L" ~ L' ] ~ d l k V ) ~ ~c'so

as well as for unpolarized photons:

dwr d~-2p

i ~ _ C2LL,(L1, E

--

dwt _ O, df2p

(23a)

dwx _ O, df2p

(23b)

llJ0)(J~LE)±J-(m~*¢JtE)+Jfta)~ - :0 J""~ aL J k L' L' ] dlk

(23C)

LL'Je

We have introduced the angular functions

d~(o) = d2,(O)+_d_2,(O), s J

(24a)

with the d~.m(O) determined from 18) ~@~,m(~fl~) = e-~"'d~,m(fl)e-'mL

(24b)

PHOTONUCLEAR REACTIONS

333

The following observations can be made: (i) For linearly polarized photons, the azimuthal asymmetry of the emitted nucleons eq. (18a), is of the form

(25)

da(X)/dl2p ,-~ 1 + ~(0) cos 2q);

also, the nucleon angular distribution is independent of any circular photon polarization. (ii) The longitudinal nucleon polarization is proportional to sin 2~0 for linearly polarized photons; for circularly polarized photons, it changes sign when the photon changes handedness. Both cases lead to zero longitudinal polarization if averaged over photon polarization, owing to parity conservation. (iii) The same is true for the transverse component of nucleon polarization in the direction X. (iv) The transverse polarization in direction Yis independent of any circular photon polarization. The corresponding Y polarization for unpolarized photons which violates time reversal, is non-vanishing due to the nucleon-nucleus interaction. The result of eq. (25) has already been obtained by Agodi 1~). His derivation is actually more general than ours, not assuming any compound state of definite spin and parity as we do, but only definite spin and parity of initial and final nucleus. On the other hand, Agodi's results do not show the explicit dependence on the electromagnetic form factors. If the transition is such that only a single multipole L contributes, we obtain

Z CJLL(LI' LIIJ2)~y2(O) a(O) = (--1)" ao

(26)

Z C2LL(L1, L-llYO)~yo(O) ' J~

where a = 1 for an electric, 0 for a magnetic multipole. If the corresponding nucleon angular distribution for unpolarized photons is written d~

d~p

- ~.bLf~yo(O),

(27)

so

where the coefficients bL) may be determined from an analysis of the experimental angular distribution, then owing to the appearance of C L)L only in eq. (26), the azimuthal asymmetry is uniquely determined by these coefficients: ~(0) = (-- 1)'[ ~. ,/e

(L1, LIIJ2) L - llJO)

bLj,.~o(O)]-i ~_ bL) (El, ./~

~12(0),

(28)

the sign being determined by the type of multipole. Using eq. (28), we can reproduce Agodi's result that for angular distributions containing Je = 0 and 2 only, written in

334

a. RAPHAEL AND H. 0BERALL

the form da - - -----a o + a 2 COS2 0, dO e

(29)

the azimuthal asymmetry at 0 = ½n is simply ct(½n) = ( - 1 ) ~ ( L I . L l ! 2 2 ) 6_ ~ a 2 " " (L1, L-1120) ao'

(30a)

which for a dipole becomes ~(½n) = ( - 1 ; a_2.

(30b)

ao

This formula has been used by Shoda 5) to determine experimentally the E1 character of the 17.3 MeV level in 160. The validity of eq. (28) seems to be based on our assumption of the compound nucleus hypothesis. However, upon studying Agodi's arguments, it becomes clear that it is also valid under his less stringent assumptions. F o r the purposes of illustration, we shall now apply some of our formulae to the excitation and nucleon decay of the 22.5 MeV 1- giant electric dipole state in 12C and especially in 160. For a Jo = 1 state (and Ji = 0; J = ~ for 12C, J = ½ for x60 if only ground state transitions are considered) reached by an electric dipole transition, we find from eq. (18c) for the angular distribution: da _ ne 2 1~--(1E)[2 dO v 2k

..]_(__ 1)j+~_(~)~_ ?

/

This is the same angular distribution as the one obtained previously 7) for the (e, e'N ) reaction with e' emerging in the forward direction; we may thus take over the angular distributions from ref. 7) and list them in table 1, column 2. In that calculation, the nuclear decay matrix elements ( J I[ j l [ l J o ) were obtained using R-matrix theory 4, 11); the procedure is described in some earlier papers by one of us iT, 19). In ref. 19), partial and total neutron and proton widths of the giant resonance states of 12C and 160 have been evaluated which were used in ref. 7). Numerical values are based on Gillet's 13,14) particle-hole wave functions. It may be noted again that for 12C, the total widths are assumed to come from just the ground state transitions to 11B and 11C, as required by the single particle-hole model. In 160, we also considered only ground state transitions to 150 and 15N; the values of the matrix elements then depend only on small components of Gillet's wave functions (since the ~50 or ~5N ground state is a (lp~)- 1 hole, whereas the model assumes the dipole states to contain a (lp~) -1 hole), and must be considered less reliable than in the case of 12C. The experimental values, column 3 of table 1, were quoted 20)t in ref. 4) or obtained more recently 21). Column 4 then presents predictions of nucleon azimuthal t Dodge and Barber s0) give the high value for 160(7, p)lSN.

PHOTONUCLEAR REACTIONS

335

asymmetries for linearly polarized photons via eq. (30b), based on the theoretical or experimental values for the angular distributions. It may be noted here that for the ~60(7, n)150 reaction, the transition to the 6.2 MeV 3 - excited state of ~50 has recently been determined 22) to amount to 18 of the total transition (with very little transition to any but this and the ground state). F r o m our earlier calculation 19), we had the corresponding partial neutron widths F ~ ) ~ 0.15 MeV, F ~ ) ~ 0.60 MeV, which gives us for the corresponding ratio 20 ~ , in good agreement. TABLE 1 Theoretical and experimental photonucleon angular distributions, azimuthal asymmetries and polarization for the 22.5 MeV E1 states of 1~C and xeO

Reaction

(da/d-Q)th

(dcr/d.Q)exp

(dtr(X)/dO)°p~ed~ --Y,/90=4"5°th PO=45°Y, exp

12C(y, n)11C

1-0.51cos20 1--0.41 P~

1-0.5 P~ a)

(1 +0.51cos 2tp)th (1 +0.6 cos 2q0)exp

12C(y, p)llB

1-- 0.57cos20 1--0.47 P~

1--0.6 P~ a)

(1 +0.57cos 2~0)th (1 +0.7 cos 2tp)exp

160(y, n)150

1--0.42cos 20 1--0.32 P~

160(7, p)lSN

1--0.36cos 20 1--0.27 P2

a) Ref. 4).

(1 +0.42cos 2~p)th

1-- (0.4-0.7)cos~0 d) (1 +0.36cos 2q0)th [1 + (0.4-0.7)cos 2cp]exp

b) Ref. s).

c) Ref. 23).

0.39

0.264-0.06b) 0.38±0.10 e)

0.35

a) Refs. so,21).

Finally, we calculated the transverse nucleon polarization P r for a transition from Ji = 0 to an E1 state using eq. (23c), and find pr(0 ) =

i(-1)'+~2~$2 sin 0 cos 0 Faoa + ( - 1)J + ~(~)42-S2 P2(cos 0)"

(32)

This shows that Pr(90 °) = 0, in the absence of interferences. Applying this to the 160(7, N) ground state reaction via the 22.5 MeV 1- giant dipole state, and again using R-matrix theory, we find the values shown in column 5 of table 1. The experimental values of the last column for 160(7, n)150 are from ref. 6), or from a more recent result of the same experimentalists, as quoted by Weiss 23). The agreement with the latter result is excellent, and is actually much better than the check of proton angular distributions in 160; note that here, as mentioned before, only small components of the particle-hole wave function (which may not be so reliable) enter in the ground state transition.

336

R. RAPHAELAND H. UBERALL

F o r the cases considered a b o v e as a n illustration o f o u r general f o r m u l a e for a n g u l a r d i s t r i b u t i o n s a n d polarizations, namely, the g i a n t d i p o l e transitions in 12C a n d 160, some m o r e detailed theoretical analysis has been m a d e 4, 23). Such analysis was however always restricted to the specific case in question, whereas o u r results, eqs. (18), (21), (22) a n d (23), are o f r a t h e r general validity. I n conclusion, we m a y state t h a t we have derived general f o r m u l a e for all a n g u l a r a n d p o l a r i z a t i o n correlations o f p h o t o n s a n d nucleons in p h o t o n u c l e a r reactions, given explicitly in terms o f the nuclear e l e c t r o m a g n e t i c f o r m factors. O u r theory, a l t h o u g h it does n o t c o n t a i n a n y effects s t e m m i n g f r o m o v e r l a p p i n g levels 4) (since it is based on the a s s u m p t i o n o f p r o c e e d i n g t h r o u g h a single isolated level o f the c o m p o u n d nucleus), nevertheless contains all interferences between various multipoles. I f need be, a g e n e r a l i z a t i o n to several c o m p o u n d nucleus levels w o u l d be s t r a i g h t f o r w a r d . W e have generalized a relation o f A g o d i between a z i m u t h a l asymmetry coefficients with linearly p o l a r i z e d p h o t o n s a n d a n g u l a r d i s t r i b u t i o n coefficients with u n p o l a r i z e d p h o t o n s (going b e y o n d the c o m p o u n d nucleus hypothesis). Finally, o u r results were illustrated b y o b t a i n i n g nucleon a n g u l a r distributions a n d p o l a r i z a tions f r o m the p h o t o - e x c i t a t i o n o f the giant d i p o l e resonance in 12C a n d 160, which c o m p a r e f a v o r a b l y with experiment.

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