Volume 41 B, number 5 PHYSICS LETTERS 30 October 1972

I M P O R T A N C E O F M O M E N T U M D E P E N D E N T T E R M S

IN R A D I A T I V E P I O N C A P T U R E S

J.D. VERGADOS and H.W. BAER Lawrence Berkeley Laboratory, Umverstty of Cahforma, Berkeley, Cahf 94720, USA

Received 24 August 1972

The contribution of the momentum dependent terms m radmtlve plon capture from bound atomic orbits is inves- tigated A shell model calculatmn for 6L1 shows that the momentum dependent terms represent ~ 2% of the 1 s cap- ture rate and up to 40% of 2p capture rates

Radiative plon capture has been found [ 1 ] to pro- vide an interesting new probe for obtaining nuclear structure information. However, before the detailed compartson between theory and experiment can be re- lied upon, an important point needs to be clarified. This concerns the role of the terms in the effective in- teraction [2] responsible for radiative plon capture which depend on the plon momentum. Estimates by Delorme and Ericson [3] based on a nuclear Fermi gas model indicated that such terms make only very small contribution to the capture rates from Is, 2p and 3d plonlc Bohr orbits. Thus nearly all subsequent authors have neglected such terms. Skupsky [4], how- ever, has suggested that for 2p capture In 12C the mo- mentum dependent terms make a sizable contribution. Unfortunately from his published results it is not clear exactly how big such contributions are. Since it is now known [6] that plon absorption even in light nuclei occurs predominantly from the 2p atomic orbit, the settlement of this question is of great importance for nuclear structure investigations. Furthermore, it is im- portant for understanding the relationship of radiative plon capture to other weak and electromagnetic rater- actions such as/3-decay, 3,-decay, electron scattering and muon capture. With the above motivation in mind we have performed a calculation for 6LI. The 6LI (g.s.) to 6He (g.s.) transition was chosen since the recent measurements of the Berkeley group [5] have Isolated this transition thereby permitting an unambiguous comparison with experiment. Our results support the conclusion that the momentum dependent terms are important.

:[:Work supported by the US Atomic Energy Commission.

560

The effective Hamiltonian responsible for plon capture was taken to be the nonrelatlvlStlC reduction of the plon photoproductlon amphtude known as CGLN (Chew-Goldberger-Low-Nambu) Interaction [2] which in the impulse approximation can be written as follows [3]:

A

Hef t = ~ exp ( - ik ' r l )H( / ) rb~o(r). 1=1

(2a)

qsf~(r) is the wave function for a plon bound in an atomic orbit and

H(j) = 27ri t+(/)[A oi "£x + B(o] "~)(q'k) +

+ C(ol.k)(q.ax) + iDq ' (k X ax)] 6 ( r - r I) (2b)

where 01 and r 1 are the Pauh spin and coordinate of the ]th nucleon and t+(1) Is the ordinary lsospln raising operator, which transforms a proton into a neutron. The photon polarization and propagation vectors are denoted by £x and k and the pion momentum and co- ordinate are denoted by q and r. The effective coupling constants A, B, C, D, which are functions of the four- momentum transfer, are related to the various multi- pole amplitudes in photoproductlon at threshold [2, 3]. Values of these constants have been given by Delorme and Ericson [3], Skupsky [2], Kawaguchl et al. [7], and others, with no two sets being alike. Since these constants have not been uniquely determined, we performed our calculations employing the coeffi- cients of Delorme and Ericson [3] (I) and of Kawa- guchl et al. [7] (II):

Volume 41B, number 5 PHYSICS LETTERS 30 October 1972

I. A = ( -0 .034 + 0.003)M~r2'

C = -0.017/If~-4;

/ l = c = 1),

II. A = - 0 . 0 3 2 M - 2 ' 7r

C = -0 .037 M~- 4 ,

B = 0.019M~4;

D = - 0 . 0 1 0 M ~ 4 (in umts of

B = 0.0075 M~4;

D -- - 0 . 0 1 4 M -4 . 7r

The pion wave function as written q~,o(r) = R~o(r) × Yt m 0"), where Y/m (r) as the spherical harmonic of rank/. In the present calculations pure hydrogenic wave functions are used. It is well known that the ra- &al part R~o(r) is distorted due to strong interactions and the finite nuclear size. However, as discussed later, this distortion can be taken into account by a multiplicative constant applied to the computed cap- ture rates. But even the Krell-Encson wave function [8] obtained by solving the Klein-Gordon equation in an optical potential can be expanded an the basis of hydrogenic wave functions and incorporated into our formalism, af necessary.

The nuclear wave functmns were obtained by dlag- onahzing the nuclear Hamdtonlan within the p-shell. The recent realistac Kuo-Lee [9] energy dependent matrix elements were used, whale the single particle energies were taken from experiment (epa/2 = 0, epa/2 = 6.24 MeV). The wave functions thus obtained are:

This model space ls ldentacal with the one employed by Cohen and Kurath [10]. It as anterestmg to note that the Kuo-Lee interaction [9] maxes the spatially symmetric and antlsymmetnc states apprecaably.

Perhaps the above model space is rather restricted in the sense that It ignores core excitations. The latter might be amportant m determining precisely how the capture strength ls dastnbuted among the different states. But for the purposes of the present investiga- tion, which is to study the effects of the momentum dependent terms on the pion capture rate, the p-shell model space should be adequate; e.g., for the/3-decay (6He ~ 6Li), the M1 transxtion 10+T=I) ~ I I+T=O), and the g.s. magnetic moment of 6La one obtains fo t = 1030 sec, B(M1) = 16.9 (eh/2mc) 2,/l(1 +) = 0.788 eh/2mc. These compare very well with the ex- perimental values [ 11 ] of 802 sec, 17.2 (eh/2mc) 2, and 0.822 e~t/2mc, respectwely.

The effective Hamdtoman given by eqs. (2a) and (2b) is decomposed anto xrreducable spherical tensors m the usual way. First the retardation factor exp (-tk.r) is expanded into spherical harmonacs (rank denoted by/a') . Thas spherical harmonic as cou- pled with the polanzataon vector to yaeld a vector spherical harmonic of rank/~. The momentum opera- tor (i.e., - I V ) operates on the pxon wave functmn and yaelds a spherical harmonac of rank l'. Then the fol lowmgcoupling scheme was used:

6He (g.s.)

IJ+r=0+, T=I) = 0.97841 [p~/2) + 0.20670 Lp~/2)

6La (g.s.): lj~r=l +, T=-0) = 0.79955 }p2/2)

- 0.02580 Ip2/2 ) -0 .60041 IP3/2Pl/2 ).

l ^

3

It as instructive to write these wave functions in the L-S coupling scheme. One gets.

[ fr=0+, T=-I) = 0.91821 [L=0, S=0)

-0 .39612 [L=I, S=I)

IJ'r=l +, T=0) = 0.93347 [L=0, $=1)

Term A

Term D

Terms B, C

+ 0.01840 IL=2, S=I) + 0.30093 IL=I, S=0). Fig. 1

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Volume 41B, number 5 PHYSICS LETTERS 30 October 1972

Once the operators are written in the above form the matrix element between the nuclear states can be cal- culated applying the Racah techniques. The transition probability (capture rate) m units of h = c = 1 is given by

2k 2 1 At(n/ ; 1 ~ f) - (4a) 7r (2,/1 + 1)(2l + 1) S l ~ f

S l , f = f~l(JfMflHefflJiM1)l 2. (4b) m ' s 71"

In the above summation care must be taken so that the polarization and propagation vector for the pho- ton are perpendicular. So the formulas are first de- rived in a frame in which the Z-axis coincides with the direction of the photon momentum (then ~ = +- 1). The results are generalized in an arbitrary frame by a mere rotation of the coordinate system.

The photon energy is E. r = 135 MeV and its mo- mentum In natural umts is k = 0.9677. Since this mo- mentum is fairly high, the multipole expansion of the retardation term converges rather slowly. So all/a-mul- tlpoles were retained. Also the long wavelength ap- proximation is no longer apphcable and the spherical Besset functions must be used.

The Importance of the momentum dependent terms comes partly from the nuclear structure and the geometry. In addition it depends crucially on the radial integrals involved. The radial integrals are of two types:

(i) (NLIfu(kr)R~o(r)IN'L'> (in term A)

(iQ (Ntl/u(kr)d~,~(r)lN'L') (in terms B, C, D)

where INL), IN'L') are the harmonic oscillator single particle states, RTo(r) is the radial part of the plon wave function and d~,~ is given by

, ' = , + 1

d~,~(r) =

< r ] R~°(r) = l - 1

no+l=n, = 0 , 1 , 2 . . . . , n - l ; n is theprxncipal quantum number specifying the energy of the Bohr orbit. One easily obtains 562

2 Z

X [ [no+l -1 +a(l ' ) ] R~O(u)/u - ½R~O(u)

where Z is the atomic number, a = h2/me 2, u = 2Z(r/a)/(no +l) and

/ - l , l' = l + 1 a(l') l+ l, l ' = l - 1 .

The important terms are those which contain the radial integral dwtded by u. We note that for spheri- cal orbits (n o = 1) the last term vanishes. The first term Is also absent when n o = 1 and l' = l + 1 (which is always the case for 1 s-plons). This is the reason why for 1 s absorption the momentum dependent terms are negligible.

To see why the momentum dependent terms are important for 2p absorption, two points must be con- sidered: First the contribution to every term of the CGLN Hamfltoman of each Individual operator char- actertzed by the set of quantum numbers {t/, ta, l, l', L, o, J}; second, the Interference between such contri- butions. E.g., for 2p absorption the most important contnbutton to the [A[ 2 comes from/a' = I, ta = 1, l = 1, /_ = 0, J = 1, while the most important contribu- tion to the [C[ 2 arises from/a' = 1,/a = 0, l' = 0, l = 1, /_ = 0, J = 1. (The /_ = 0 term dominates in both cases because the nuclear wave functions are characterized by L = 0.) Hence, both terms are characterized by the same "angular" nuclear matrix element. The ratio of the radial integrals involved satisfies [Rc/R A ]2 = 5.49 m 2, while the geometrical factors, resulting after the m-sums in eq. (4b) have been performed, yield

~2C/~2A = 3. So the two terms, apart from the values of the coupling constants, contribute equally. Smce the above contributions are the dominant ones in the IAI 2 and ICl 2 terms, we get (C2~2c/A2g2A)Rc/RA 12

1 (m agreement with the exact result II of table 1). In some cases, however, interference becomes important. The small value of the term A*C of table 1 is the result of destructwe interference, while the large value of the A*B term is the result of constructive in- terference. We have not been able to understand the physical reasons for these interference results.

Volume 41B, number 5 PHYSICS LETTERS

Table 1

Calculated radmtlve capture rates for the n- + 6Li ~ "r + 6He (g.s). (Columns I as defined in the text.)

30 October 1972

and II correspond to the sets of coefficients I and II

I. II. Delorme and Ericson Kawaguchl et al

Term a) A3,(I s ) AT(2p) AT(ls) A,y(2p)

1015 sec -1 1010 sec -1 1015 sec -1 1010 sec -1

A 2 2.788336 2.012230 2.469908 1 782433 B 2 0.000523 0.138319 0.000081 0.021550 C 2 0 0.293704 0 1.391276 D 2 . . . . . . . . . . . . A*B -0.037461 -0.526845 -0.013917 -0.195723 A*C 0.000978 0.040071 0.002003 0.082081 A*D . . . . . . . . . . . . B*C -0.000013 0.000071 -0.000011 0.000061 B*D . . . . . . . . . . . . C*D . . . . . . . . . . . .

Totfl 2.71587 1.47085 2.434139 2.968097

a) Since A*B = AB*, etc., terms such as AB*, etc., are not listed. Not all digits appearing in this table are significant. Many figures were included to gwe an idea of the order of magnitude of each term.

Since the pion wave-function is more or less urn- form inside the nucleus, the following approximation is usually made [3] :

( N t l / ~ ( k r ) R x o ( r ) l N t ) 2 = ( (R~o(r ) )2 ) (NLI /u (kr ) lNL) 2.

(5b)

With hydrogenic wave functions, this approximation is good to within 10%. The above approximation is the eqmvalent of the Konopmski-Uhlenbeck approxi- mation used m 3-decay and allows one to take care of the distortions of the pion wave-function in an aver- age way [3, 4]. This approximation is expected to hold for all n o, i.e., even for the Krell-Ericson [8] wave functions. It might be questionable, however, when off diagonal nuclear radial integrals are encoun- tered.

Our results are presented in table 1, columns I and II corresponding to the two sets of coefficients I and II described above. If one defines the fraction f a s the ratio of the momentum dependent to the momentum independent terms, i.e., t

t i n our notat ion IAI 2 means the contribution to eqs. (4a) and (4b) resulting from the first term of the CGLN Hamil- tonian, etc.

f ( n l ) = {IBI 2 + ICI 2 + IDI 2 + A * B + B * A + A ' C +

+ C * A + A * D + D * A + B * C + C * B + B * D +

+ D*B + C*D + D*C}/IAI 2

one gets

fi(1 s) = - 0 . 0 2 6 f l (2P) = - 0 . 2 6 9

f i i ( l s ) = - 0 . 0 1 5 f i i (2P) = 0.666.

Thus we see that the contr ibut ion of the momentum dependent terms for ls absorption is small (in agree- ment with ref. [3]) but important for 2p absorption. This depends crucially on the choice of A, B, C, D co- efficients (there is even a sign change in f ( 2 p ) between I and II).

In the radiative capture experiments performed to date, the capture rates from isolated pionic orbits are not measured. Rather, the measured quant i ty R 7 (in % of l r - atoms formed) represents the summed contri- bution of photons from all capture orbits. Thus R7 for any final nuclear state or group of states is related to the computed transition rates by

X. ~ A ( n / ) . ~, R 7 = Z . . / ~ , ~ w l , n o (7)

nl '~a (nt)

563

Volume 41B, number 5 PHYSICS LETTERS 30 October 1972

where A~(n/) and Aa(n/) are the radiative and total absorption rates, respectively, for the nl orblt The co(n/) are the probablhtles that the pxon is absorbed into the nucleus from orbit nl. The capture schedules for hght nuclei [6], which are deduced from the plon- lC X-ray intensity data, show that only l = 0 and 1 or- bits make appreciable contributions to the above sum. However, several values of n for each I contrib- ute. We make the assumption that Av(nl)/Aa(n/) is independent of n. This reduces eq. (7) to

AT(ls) + A (_7_~2P)

R-r = Aa(lS) cos Aa(2P) cop, (8)

where co s = 2nco(ns) and cop = 2;nco(np). The values obtained by Sapp et al. [6] for 6LI are cos = 0.40 +

0.09 and cop = 0.60 ± 0.09. The measured total level width are for the ls state [12] 0.15 ± 0.05 keV (2.3 × 1017 sec -1) and for the 2p state [6] 0.015 + 0.004 eV (2.3 × 1013 sec-1). Inserting these values

into eq. (8) together with the calculated values, we obtain

R ( I ) = 0 . 5 1 5 % and R r ( I I ) = 0 . 5 0 5 % .

The Berkeley experiments [5], which will be reported in detail elsewhere, yield R.y = 0.39 ± 0.10%. Deutsch

et al. [13] reported a value R, r = 1.0 ± 0.1%. Although the effective Hamlltonlan given by eqs.

(2a) and (2b) xs most commonly used, for more pre- cise calculations up to order m~/m N [14], eq. (4a) must be multiplied by 1 + m J m N. Furthermore, if realistic pion wave functions are used, this will effec- tively reduce the contr ibut ion of the ls state and in- crease the contr ibut ion of the 2p state. If we take this into account by multiplicative factors of 0.65 (1 s) and 1.10 (2p) and include the factor of 1 + m J m N, we obtain R~(I) = 0.407% and R,y(II) = 0.419%. Both of these compare very well with the results of the Berke- ley group [5] but are In &sagreement wxth the results of Deutsch et al. [13].

One addit ional point must be mentioned. Light nu- clei are characterized by a diffuse surface, consequent- ly the energy of the harmonic oscillator, which pro- vides the single particle wave functions, is not well de- fined. Energy calculations are consistent with hw = 15 MeV, while rms radms of 2.54 fm obtained by elas- tic electron scattering on 6Li [I 5] is best f i t ted with a size parameter a = 1.94 fm for the p-shell harmonic oscillator, i.e.,/~co = 11 MeV. Our results gwen above

564

were obtained using the first of the above energies which correspond to a size parameter of 1.66 fro. Hence the whole calculation, xncludlng the nuclear eigenvectors, was repeated with hw = 11 MeV. The results obtained are:

f l s ( I ) = - 0 . 0 3 2

f2p(I) = - 0 . 2 3 7

R y(1) = 0.307%

f l s ( I I ) = - o . o 1 2

f2p(II) = 0.281

R r( l l ) = 0.317%.

These values of R.r, which include both corrections mentioned above, are in as good an agreement with the Berkeley experiment as the earlier ones. The quanti ty R. r does not seem to be as sensitive to the variations considered here as the individual transition probabilities. But this may be a special feature of 6L1.

In conclusion, we can say that the momentum de- pendent terms contribute significantly to the radlatwe plon capture rate from the 2p orbit, the dominant capture orbit for light nuctel. Also a precise determi- nation of the values of the A, B, C, D coefficients is in order if detailed agreement with experiment is to

be sought.

The authors wish to thank Professor Kenneth M. Crowe for enhghtenmg &scussions.

References [ 1 ] J.A. Blstxrllch, K.M Crowe, A S.L Parsons, P. Sharek

P. Truol, Phys. Rev C5 (1972) 1867. [2] G F Chew, M.L Goldberger, F.E. Low and Y Nambu,

Phys. Rev 106 (1957) 1345. [3] J. Delorme and T.E.O Ericson, Phys Lett 21 (1966)

98 [4] S Skupsky, Nucl Phys. A178 (1971) 289,Phys. Lett

36B (1971) 271. [5] H W. Baer, J A. Blstlrhch, K M. Clowe, J.A Helland

and P TruiJl, Bull. Am. Phys. Soc. 17 (1972) 585, H. Baer and K M Crowe, private communication.

[6] W W Sapp Jr., M. Eckhause, G H Mllter and R.E Welsh, Phys Rev. C5 (1972) 690.

[7] M. Kawaguchl et al , Progr. Theoret. Phys. (Kyoto) Suppl. extra number (1969) 28.

[8] M Krell and T.E.O. Ericson, Nucl. Phys. B l l (1969) 521

[9] T T S Kuo and G E Brown, Nucl. Phys. 85 (1966) 40, T T.S Kuo and S.Y Lee, private commumcatlon.

[10] S. Cohen and D. Kurath, Nucl Phys. 73 (1965) 1. [ 11 ] T. Laurltsen and F Ajzenberg-Selove, Nucl. Phys 78

(1966) 1, R M Mutcheson et al., Nucl. Phys. A107 (1968) 266

[12] R.J. Harris Jr , W.B Shuler, M. Eckhause, R.T Siegel and R.E. Welsh, Phys. Rev. Lett. 20 (1968) 505.

[13] J.P Deutsch, L. Grenacs, P. Igo-Kemenes, P Llpmk and P C Macq, Phys. Lett. 26B (1968) 315