- Email: [email protected]
Study of the (π, nn) reaction on 12C
B. IL JAiN Nuclear Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay-85, India Received 29 October 1971 (Revised 22 February 1972) Abstract: S~~~~d expressions for the tradition strength of the (FF~ nn) reaction OReyes-ma$s mzcfeiand for the back-to-back emission of neutrons are presented. These expressions are then used to study the effect of various factors on the transition strength and to analyse the recent available data on the 12C(1c-, nn) reaction. A good agreement with the experimental data is found.
It is known that, due to kinematical considerations, negative pions in nuclei are preferentially absorbed by a correlated pair of nucleons. That is why the reactions like (z-, NN) are normally used to study the short-range correlations in nuclei I). The information about co~ela~i~~s obtained this way, ~ow~~e~, is subject to un~~r~ai~ties due to the following reasons: (i) The outgoing channel in this reaction consists of two nucleons each moving with an energy of about 70 MeV. Therefore, due to their mutual scattering the correlations are also introduced in the final state. The experimental data, therefore, depend on the correlations in the initial as well as in the final state. At present, there does not seem to be any reliable way of separating the one from the other. (ii) In addition to the two n&eons, the final state also consists of a recoiling nucleus. Hence it is a three-body system and its solution cannot be found exactly. In correlation studies, only the mutual interaction of nucleons is taken into account. Their interaction with the residual nucleus is neglected. There is, however, another aspect of this process. The emission of two ~ucI~ons following the abso~~on of a bound pion creates a two-hole state in the residual nucleus. This is similar to the creation of one-hole states in the (p, 2p) reaction. From the successful use of the (p, 2p) reaction in the study of the one-hole states in nuclei 2, it seems logical to use (z-, NN) reactions to investigate two-hole states in nuclei, This potenti~jty of pion absorption studies was pointed out by Ericson 3, and has been emphasized recently by ~~~~i~so~ “). motivated by this Jain and Banerjee ‘)? developed a general formalism for the study of two-hole states in nuclei with the t In our NUOVOCim. 69 (1970) 419 paper there is a mistake in eq. (17). There should also be an inference term between M,, and MI.
513
514
B. K. JAIN
(n-, NN) reaction. In this formalism, pion capture in a nucleus is assumed to proceed through the elementary reaction n+NN + N+N. (1) The interaction Hamiltonian for this process has been constructed by Eckstein “). The parameters of this interaction are obtained by analysing the data on the elementary reaction. Since this interaction Hamiltonian contains within it the effect of the correlations in the initial and the final states, the semi-phenomenological approach avoids both the explicit introduction of a correlation function in the initial wave function and the treatment of the N-N scattering in the final state. This approach then also makes it easier to include the interaction of the outgoing nucleons with the residual nucleus. In our earlier paper ‘), using this approach, we have presented the expressions for the transition probability in the general case. These expressions are very complicated, and hence the calculations are very involved. In this paper we observe that in the special cases where the isospin of either the initial nucleus or that of the final nucleus is zero and the observations of the outgoing nucleons are made at back-to-back angles, the expressions for the transition probability gets very much simplified. Using these simplified expressions, we have studied in this paper the various aspects of the (n-, nn) reaction on the “C nucleus. We have also analysed the recently available data of Cheshire and Sabottka 7, on “C. In their paper, Cheshire and Sabottka have also compared their results with the theoretical predictions of Guy et al. *). However, the calculations of Guy et al. do not include the scattering of the outgoing nucleons with the residual nucleus. Hence the conclusions in ref. ‘) cannot be taken with confidence. Besides, their approach for the analysis of the (z-, NN) reaction is different from ours.
2. Formalism The absorption
cross section for the (z-, dw =
nn) reaction
can be written as (c = rZ = 1)
(2)
where p is the density of states in the final system, Z, and m, are respectively the orbital and magnetic quantum numbers of the pionic orbit, J and M are the angular momentum and its projection quantum number for the absorbing pair of nucleons, M9 is the spin projection quantum number of the neutron pair in the final state, and A is the mass number of the absorbing nucleus and M, the mass of pion. Matrix element M,i is defined as
(3) where Ti, Vi and T,, vf are the isospin and isospin projection quantum numbers for the initial and the final nucleus respectively. The Go and G, correspond respectively
(n-, nn) ON ‘*C
515
to the contributions from the isospin singlet (T = 0) and isospin triplet (T = 1) states of absorbing nucleons in nuclei; G,, and G, are defined as G, =
Ms,) ~~i(LS
c 2”k,,,LMLM.s
= 1J T = O)(LlM,M,IJM) x(llMs,M,-M,IIM,)F~~~,
G, = c (- l)MS’k_M,, Xfi(LS = 0 J T = l)Fk;;, Mr.
(4a) (4b)
where k, is the spherical component of the vector k; k itself is given by k = +(k, -k,) where k, and k, are the momentum vectors of the outgoing neutrons; L and ML and S, MS are the orbital angular momentum and its projection quantum numbers for the absorbing nucleons respectively; .Yfi(LSJT) is the spectroscopic amplitude for removing a pair of nucleons with the quantum numbers (LSJT) from the initial nuclear state i and leaving the final nucleus in the state f; (abc$lcy) is the Clebsch-Gordon coefficient. The integral F$‘; is given by F$;
=
s x-
dR *(k, Rh- *(k, W,M,@, Wr~m,@), 9
2
(5)
where the x are the distorted waves for the outgoing neutrons, 4LMr. is the space part of the two-nucleon wave function in nucleus, and 4,,,,% is the pion wave function. In eq. (3) go and g1 are the interaction parameters of Eckstein’s Hamiltonian. They 33P, transitions of two nucleons. respectively determine the 13S1 -+ 33P1 and 31S,, + The modulii of these parameters can be obtained by analysing the reactions .n+ +d + p+p, P+P
+ nO+p+p,
(7)
The values of 1gol and (gll recently obtained by Figureau and Ericson ing the latest experimental data of Rose ’ “) on the above reactions are )go12 = 0.64
‘) by analys-
+0.05 fm’,
lg1j2 = 0.155kO.08
fm*.
(8)
In the analysis of reaction (7) one does not need to know anything about the relative phase of go and gi. However, for the calculation of the transition amplitude for the (n-, nn) reaction, we need the magnitudes as well as the relative phase of go and gi. This happens, as we see from eqs. (2) and (3) because Go and G, contribute coherently. In certain many cases where either the isospin of the initial or that of the final state is zero, this difficulty can be avoided. In these cases either T = 0 or T = 1 contribute to Mri* Therefore, for these transitions, the matrix element Mri is
IMfil’ = lg0121Go12, for
T = 0,
for IMfil’ = l~~12(T,~~~‘flTi~~)21~~12~ These equations
are applicable
to many even-mass
(94
T=l.
(9b)
nuclei (e.g. 6Li, 12C, 160, etc.).
B. K. JAIN
516
Furthermore, if we consider the emission of nucleons at 180” to each other only, the expressions for G, and G, are considerably simplified. As we shall now see in this case the sums over M, and M, disappear in eqs. (4) for G, and G,. This can be seen very easily. Let us consider the partial-wave expansion of the distorted waves x in the integral Fk$‘;. If we take the Z-axis parallel to the vector k, and since k, is antiparallel to k,, only the zero projection of the angular momentum would contribute to the partial-wave expansions. Consequantly, due to the angular integration of vector R in integral (5), Fk$‘; would have non-vanishing contributions only when M,+m,
= 0,
or M, = -m,.
(10)
Secondly, since the vector k = +(k, -k2) is parallel to k, MS = Ms.,
for
kws-Ms+ = k, = 0,
otherwise, M,, = 0,
for
k-MS, = k, = 0,
(lla)
otherwise.
(lib)
These two conditions [i.e. (10) and (1 l)] thus remove the summations over ML and MS in eqs. (4). In addition, the summation over MS in eq. (2) is also restricted because
(1 lM,, M,-M&M,)
= (1 lM,O]lM,)
= 2-+Ms,
= +2-3,
for
MS, = 21,
= 0,
for
Ms. = 0.
(12)
Substituting eqs. (lo), (11) and (12) in eqs. (4) and (2) we get for T = 0
do
= p A+=+2Z,+l)-1]g,]zk2 (2~)~ 2M, X
IC
Yfi(LS
C Jm,Ms*
(25+1)-r = f 1
= 1 J T = 0)(Ll-
m, M,.IJM)F$E=
-m,i2y
L
(134
and for T = 1 P dw = m
A@-- -1’)(2[,+l)-1]g1]2k2(T,lv10]Tivi)2 2M, c (2J+l)-‘]&(LS
= 0 JT = 1)F$;12.
(13b)
m,L=.J
In the case of absorption from the 1s pionic orbit the summation over m, also disappears.
(n-, nn) ON l*C
517
Finally, it may be mentioned that the simplified expressions (13) for the transition probability are quite useful for studying the two-hole states in nuclei. This is due to the fact that in the 180” geometry of outgoing nucleons we can scan the whole momentum distribution of nucleons pair in nuclei by just recording the events corresponding to the various energies of one nucleon. 3. Results and discussions Recently Cheshire and Sobottka ‘) have measured the cross section for the (x, nn) reaction on ‘*C in which two outgoing neutrons are detected at 180” to each other. This experiment had also been done earlier by Calligaries et al. ‘l). But the resolution and the statistics of Cheshire and Sobottka are much better. These people in their summed energy spectrum have observed three peaks at the excitation energies of “B at about 5 MeV, 22 MeV and 37 MeV. From the ‘*C(p, 2p) reaction’*) we know that the binding energies of proton in the lp and 1s shell in “C are about 16 MeV and 34 MeV respectively. Therefore, considering ‘*C in the (1~)~ (lp)* configuration, the observed three peaks can be attributed to the capture of pion on (lp)*, (lslp) and (1s)’ pairs respectively. Since the formalism presented in sect. 2 is applicable to nucleons in equivalent orbits we will be restricting ourselves to the absorption of pions on (IS)* and (lp)* pairs only. Chesire and Sobottka have also measured the one-neutron energy spectrum corresponding to the three peaks in the summed energy spectrum. For the theoretical calculations, since “C in its ground state has isospin equal to zero, expressions (13) for transition probabilities are directly applicable. For the absorption by (lp)* nucleons, as the peak in the excitation energy curve appears at about 5 MeV, we have included the contribution from the state of “B up to 6.0 MeV excitation. The strength of the transitions to various excited states of “B is governed by the spectroscopic amplitude Yri. The values of 4ri for various transitions are calculated by using the intermediate coupling wave functions of Boyarkina 13) for r*C and “B. These wave functions are obtained by the energy level fitting of the lp shell nuclei using the L-S coupled basis states. It may be mentioned here that even in the calculation of the shape of the one-neutron energy spectrum for the absorption on (1~)’ nucleons it is necessary to calculate the contribution from each state of ’ 'B. This is required due to the fact that firstly the contribution to FLMr. comes from L = 0 and 2 and secondly these contributions are coherent for T = 1. Therefore, even the shape of the one-neutron energy spectrum is different for different states of “B. The radial part of the bound state wave functions for the neutron and the proton are taken as the solution of the Woods-Saxon potential. The parameters of this potential are taken from the work of Elton and Swift I’) on the analysis of the elastic scattering of the 420 MeV electrons on the “C nucleus. These wave functions of Elton and Swift have already been found to give a good description of the (p, 2p) [ref. ’ “)I and (p, d) [ref. ’ “)I reactions on ‘*C. For the distorted waves we have used the partial-wave expansion. The parameters
B. K. JAIN
518
of the optical potential, which is required in the neutron energy region of 10-70 MeV, are assumed to vary linearly with energy. Determining the parameters of these linear relations by using the sparsely available optical parameters in literature ’ 7), we have U,(MeV)
= -48 +0.3&
W,(MeV)
= -(2.63
W,(MeV)
= -(1.86+0.114
10 MeV c E S 70 MeV, E < 40 MeV,
+ O.O935B),
(14)
E 2 40 MeV.
E),
Here V0 and W0 are the depths of the real and imaginary parts of the optical potential. The shapes of U and W are taken to be Woods-Saxon, with the radius (rc,) and diffuseness parameters fixed at 1.25 fm and 0.65 fm respectively. For pions in the 1s orbit we have used the wave function of Seki and Cromer ‘*). These wave functions are obtained, in an approximate way, by solving a set of coupled equations in which the effect of the strong pion-nucleon interaction is simulated by a complex square-well potential and that of a finite charge distribution of nucleons by a uniformly charged sphere. The exact calculations of Fulcher et al. lg) show that the approximate wave functions of Seki and Cromer are in reasonably good agreement with their wave functions, and therefore are sufficiently accurate for most purposes. The Seki-Cromer wave function is written as $rs = c(e+YR -e-yR)e+8R4/R,
for
RSR,,
As = ce -““%(R/~o)h,,
for
R>R,,
(15)
where R, is the rms radius of nucleus and is fixed by the electron scattering experiment on the relevant nucleus, and a, = h2/(pZe2), with p as the reduced mass of pion and nucleus. The parameters y, 6 and C in eq. (15) are determined by fitting the energy shift of the 1S level and the spread of the 2P + 1S transition energies. These parameters are listed in the paper of Seki and Cromer; cp is a smoothly varying modulating function. For the 2P orbit we have used the hydrogenic wave function. Since the measured recoil momentum distribution in the (n-, nn) reaction does not discriminate amongst the contributions from the different pionic orbits, the experimentally observed yield for the 2n branch is equal to the weighted-average yield for this branch from various orbits, i.e. Lp(2n)
= Y2,(2n) + (P(lS)/P(2P))Y1s(2n).
(16)
Here we have assumed that the absorption from orbits other than 1S and 2P is negligible; P(n1) is the population of pions in the nl orbit. If we assume that the 1S orbit is primarly populated by the 2P -+ 1S transition, then W)/WP)
=
Kp(W/WipW
+
W,P
(a>>,
(17)
where W,,(X) is the X-ray transition rate for the 2P-1S transition; Wnl(a) is the total absorption rate of pions in the nl orbit. As pointed out by Ericson ““) W&a)
nn) ON ‘“C
(n-,
519
I
I
I
I
2
::
2
0
( SllNn
WV)
l-
q
8’SW
I 8 ‘SW P 3P/mP
1 hi .t? a
B. K. JAIN
520
can be deduced from the intensity attenuation of the 2P-1S X-ray line. The latest value of W,,(a) for 12C, due to Anderson et al. 2’) is 1.25 eV or equivalently 1.8 x 10” set-I. Taking W,,(X) equal to 2.14 x 10 l4 set-I, P(lS)/P(2P)
w 0.1.
(18)
Hence Y,,,(2n) The branching
= Y2r(2n) + 0.1 Y,s(2n).
(19)
ratios Y,,(2n) are defined as Y,,(2n)
= W2,(2n)/( I%(a) + I%(X)),
Yrs(2n) = Ws(2n)W1s(a). From eq. (19) it is clear that in “C 90 ‘A of the pions are absorbed in the 2P orbit and only the remaining 10 % are absorbed in the 1s orbit. In figs. 1 and 2 we have plotted the calculated one-neutron energy spectrum for the absorption of pions on (1~)~ and (1~)~ nucleons along with the experimental data of Cheshire and Sobottka. The solid (---) and dot-dash (-.-)lines in thesefigurescorrespond respectively to the contribution of the only 1S pion and the combined (1s +2P)pion as given by eq. (19). The normalization for the (1~)’ curves are chosen such that the area under them is the same. The area under the dot-dash curve for (1~)~ is about 15 % more than under the solid-line curve. This is done to bring out the best possible agreement between the computed and the experimental results. As we see from figs. 1 and 2, the combined (1s +2P) curves predict the maximum yield at the one-neutron energies less than the energies at which the experimental peak is observed. In other words, experimental maximum yield for the absorption on (1~)~ and (1 s)~ nucleons are observed at about Q, the recoil momentum, equal to zero while the combined (1 S + 2P) distribution predicts them at Q > 0. On the other hand, the distributionfromthe absorption of only 1S pions on (1~)’ nucleons is in quite good agreement with the corresponding experimental results. For the (1 s)~ nucleons, however, the 1S theoretical curve falls off rather too rapidly in comparison to the experimental one. The slow fall off of the experimental (1~)~ data can partly be attributed to the insufficient experimental resolution. In the present experiment, as we see from their excitation energy spectrum, although the peak corresponding to (lp) 2 is well resolved, those corresponding to (Is)’ and (lslp) are not quite resolved from each other. Therefore, it may be possible that the measured events for (1s)’ contain some events from the absorption on (Islp) and vice versa. And, since in the absorption of the 1S pion on the (1~1~) nucleon-pair the orbital angular momentum (L) which can be transferred is 1, the neutron energy spectrum for (1 s lp) would peak at some value of the recoil momentum greater than zero. Consequently the contribution of the peak around 22 MeV (due to (lslp) nucleons) to that around 37 MeV (due to (Is)’ nucleons) in the excitation energy spectrum would increase the cross section at high recoil momentum (or equivalently low neutron energy) for the absorption on the (IS)’ pair. The mixing of (Is)’ events
(n-, nn) ON l*C
521
into the (lslp) events would also distort the one-neutron energy spectrum for the absorption on (lslp) nucleons. The orbital angular momentum transfer in the (Is)? absorption is zero, and hence the neutron energy distribution for the (1~)~ peaks at zero recoil momentum (i.e. around 35 MeV neutron energy). Therefore any contribution from the absorption on (1 s)~ to that on (Is1 p) would increase the cross section
----.-._-K--X-
0.3663
IFL.
o12 (HWw*nic
0.4050
1 F,
. %I2 IHWwnicl
1 x 10
IF,,zi21Seki-Cromer)
5
4
IFL12
3
(1o-6 frli3)
2
1
I
I
I
I
I
I
I
10
20
30
40
50
60
70
E (MeV 1 Fig. 3. Variation of the integral IFLIz with the one-neutron energy (I?) for the absorption pions on (1~)~ nucleons using the hydrogenic and Seki-Cromer pion wave functions.
of 1s
for (lslp) around zero recoil momentum (or equivalently around 35 MeV neutron energy). This fact is born out by the discrepancy at low neutron energies between the experimental and the calculated (Islp) neutron energy spectrum presented in the paper of Cheshire and Sobottka. From these considerations we feel that the discrepancy between the 1s calculated and observed one-neutron energy spectra for the absorption on the (IS)’ and (1 sip) nucleons may partly be due to the insufficient resolution of the experiment.
522
B. K. JAIN
In order to see the effect of the pionic wave function on the transition probability, in fig. 3 we have plotted the lFL12 for (1~)’ nucleons using the hydrogenic and SekiCromer form for the 1S pion wave function. As we see, the use of the Seki-Cromer wave function which includes the effect of the strong pion-nucleus interaction and finite nuclear charge distribution, reduced the value of lFL12 for L = 0 and L = 2 by the factors of 2.73 and 2.46 respectively. The shape of the distribution is also changed. lF,.,l’ ----_
0.5372
a-+
IFL=
_._.-._
0.3160
I Distorted lFL_l*
Wave) IPlant
t~2Klistortrd
Wavcl
Wave)
~FL,r~21Pl.nc
Wove1
5
4
IV2
(lo-%‘; 2
1
10
20
30
10
50
I
I
60
70
E (MeV) Fig. 4. Variation of the integral IFLIz with the one-neutron energy (E) for the absorption pions on (1~)~ nucleons using plane and distorted waves for the outgoing neutrons.
of 1S
As we mentioned earlier, most of the calculations ‘) on pion absorption are done without including the interaction of the outgoing nucleons with the residual nucleus. In order to see the effect of the distortion of the outgoing neutrons by the residual nucleus, in fig. 4 we have plotted the IFLIz for (1~)~ nucleons using the Seki-Cromer wave function for the 1S pion, for plane and distorted waves for the outgoing neutrons. We see that the distortion reduces the peak values of lFL12 for L = 0 and L = 2 by the factors of 1.86 and 3.16 respectively. The shape of the distribution is also changed significantly. The effect of distortion on the transition probability for the 1s pion is shown in figs. 1 and 2 where the solid line corresponds to the distorted-wave results and
(n-,nn)
ON
i2C
523
the dashed line to the plane-wave results. The distortion reduces the transition strengths for the (1~)~ and (1~)~ nucleons by factors of 1.75 and 1.52 respectively. The shape of the neutron energy spectrum is also broadened and thus becomes in better agreement with the experimental results. It may also be noted that the effect of distortion on the (1~)~ neutron energy spectrum is mainly that of its effect on [FL12 for L = 0.This happens because, as we find from the calculations of the spectroscopic amplitude (Y& the main contribution to the pion absorption on the (1~)~ nucleons comes from L = 0 only.
4. Conclusions From the analysis presented in this paper the following conclusions may be drawn: (i) A good agreement between the calculated and measured one-neutron energy spectrum for the absorption of pion on the (1~)~ nucleons is obtained using only 1s pions provided the proper bound state wave functions for the nucleons are used and the scattering of outgoing nucleons with the residual nucleus is taken into account. (ii) The discrepancy between the calculated and observed spectra for the (1s)’ and (1 slp) nucleons seems partly due to the insufficient resolution of the experiment. (iii) The inclusion of the proper pion wave function (i.e. those including the effect of the strong pion-nucleons interaction and the finite charge distribution of nucleus) and the distortion of the outgoing nucleons by the residual nucleus is essential. Each of these factors reduces the magnitude of the pion absorption probability and changes the shape of the one-neutron energy spectrum significantly. (iv) The calculated weighted-average one-neutron energy spectrum from the 1s and 2P pion orbits, where the weightage is taken from the data on the 2P-1s X-ray yield, is in clear disagreement with the experimental results. The computed distribution peaks at Q (the recoil momentum) > 0 while the experimental data peaks around Q z 0. The author wishes to express his thanks to Dr. R. Ramanna for his encouragement and interest in this work, to Prof. D. F. Jackson for providing the Is nucleon radial wave functions, and to Prof. T. E. 0. Ericson for some useful comments.
References 1) E. H. S. Burhcp, High energy physics, vol. 3 (New York, 1969); G. Backenstoss, Ann. Rev. Nucl. Sci. 20 (1970) 467 2) D. H. Wilkinson, Comm. in Nucl. and Particle Phys. 2 (1968) 48 3) T. E. 0. Ericson, Phys. Lett. 2 (1962) 278 4) D. H. Wilkinson, Proc. Int. Conf. on nuclear structure, Tokyo, 1967, p. 469; Cornm. and Particle Phys. 2 (1958) 83 5) B. K. Jain and B. Banerjee, Nuovo Cim. 69 (1970) 419; B. K. Jain, Bhabha Atomic Research Centre, Bombay, 1970, report-512 6) S. G. Eckstein, Phys. Rev. 129 (1963) 413
in Nucl.
524
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20) 21)
B. K. JAIN D. L. Cheshire and S. E. Sobottka, Phys. Lett. 3OB (1969) 244 R. Guy, J. M. Eisenberg and J. Le Tourneux, Nucl. Phys All2 (1968) 689 A. Figureau and M. Ericson, Nucl. Phys. BlO (1969) 349 C. M. Rose, Phys. Rev. 154 (1967) 1305 F. Calligaris, C. Cernigoi, I. Gabrielli and F. Pellegrini, Nucl. Phys. Al26 (1969) 225 H. Tyren, S. Kullander, 0. Sundberg, R. Ramachandran, P. Isacason and T. Berggren, Nucl. Phys. 79 (1966) 321 A. Boyarkina, Bull. Acad. Sci. USSR (whys. ser.) 28 (1964) 255 L. R. B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52 B. K. Jain and D. F. Jackson, Nucl. Phys. A99 (1967) 113; R. Shanta and B. K. Jain, Nucl. Phys. 175 (1971) 417 I. S. Towner, Nucl. Phys. A93 (1967) 145 P. H. Bowen, J. F. Scanlon, G. H. Stafford, J. J. Thresher and P. E. Hodgson, Nucl. Phys. 22 (1961) 640; P. E. Hodgson, Proc. Conf. on direct interactions and nuclear reaction mechanisms, Padua, 1962, p. 103; D. R. Winner and R. M. Drisko, Tech. Rep. Dept. of Phys. Sarah Mellon Scaife Radiation Lab., June, 1965, unpublished R. Seki and A. H. Cromer, Phys. Rev. 156 (1967) 93 L. Fulcher, J. M. Eisenberg and J. Le Tourneux, Can. J. Phys. 45 (1967) 3313 M. Ericson, Comp. Rend. 258 (1964) 1471 D. K. Anderson et al., Phys. Rev. Lett. 24 (1970) 71
It is known that, due to kinematical considerations, negative pions in nuclei are preferentially absorbed by a correlated pair of nucleons. That is why the reactions like (z-, NN) are normally used to study the short-range correlations in nuclei I). The information about co~ela~i~~s obtained this way, ~ow~~e~, is subject to un~~r~ai~ties due to the following reasons: (i) The outgoing channel in this reaction consists of two nucleons each moving with an energy of about 70 MeV. Therefore, due to their mutual scattering the correlations are also introduced in the final state. The experimental data, therefore, depend on the correlations in the initial as well as in the final state. At present, there does not seem to be any reliable way of separating the one from the other. (ii) In addition to the two n&eons, the final state also consists of a recoiling nucleus. Hence it is a three-body system and its solution cannot be found exactly. In correlation studies, only the mutual interaction of nucleons is taken into account. Their interaction with the residual nucleus is neglected. There is, however, another aspect of this process. The emission of two ~ucI~ons following the abso~~on of a bound pion creates a two-hole state in the residual nucleus. This is similar to the creation of one-hole states in the (p, 2p) reaction. From the successful use of the (p, 2p) reaction in the study of the one-hole states in nuclei 2, it seems logical to use (z-, NN) reactions to investigate two-hole states in nuclei, This potenti~jty of pion absorption studies was pointed out by Ericson 3, and has been emphasized recently by ~~~~i~so~ “). motivated by this Jain and Banerjee ‘)? developed a general formalism for the study of two-hole states in nuclei with the t In our NUOVOCim. 69 (1970) 419 paper there is a mistake in eq. (17). There should also be an inference term between M,, and MI.
513
514
B. K. JAIN
(n-, NN) reaction. In this formalism, pion capture in a nucleus is assumed to proceed through the elementary reaction n+NN + N+N. (1) The interaction Hamiltonian for this process has been constructed by Eckstein “). The parameters of this interaction are obtained by analysing the data on the elementary reaction. Since this interaction Hamiltonian contains within it the effect of the correlations in the initial and the final states, the semi-phenomenological approach avoids both the explicit introduction of a correlation function in the initial wave function and the treatment of the N-N scattering in the final state. This approach then also makes it easier to include the interaction of the outgoing nucleons with the residual nucleus. In our earlier paper ‘), using this approach, we have presented the expressions for the transition probability in the general case. These expressions are very complicated, and hence the calculations are very involved. In this paper we observe that in the special cases where the isospin of either the initial nucleus or that of the final nucleus is zero and the observations of the outgoing nucleons are made at back-to-back angles, the expressions for the transition probability gets very much simplified. Using these simplified expressions, we have studied in this paper the various aspects of the (n-, nn) reaction on the “C nucleus. We have also analysed the recently available data of Cheshire and Sabottka 7, on “C. In their paper, Cheshire and Sabottka have also compared their results with the theoretical predictions of Guy et al. *). However, the calculations of Guy et al. do not include the scattering of the outgoing nucleons with the residual nucleus. Hence the conclusions in ref. ‘) cannot be taken with confidence. Besides, their approach for the analysis of the (z-, NN) reaction is different from ours.
2. Formalism The absorption
cross section for the (z-, dw =
nn) reaction
can be written as (c = rZ = 1)
(2)
where p is the density of states in the final system, Z, and m, are respectively the orbital and magnetic quantum numbers of the pionic orbit, J and M are the angular momentum and its projection quantum number for the absorbing pair of nucleons, M9 is the spin projection quantum number of the neutron pair in the final state, and A is the mass number of the absorbing nucleus and M, the mass of pion. Matrix element M,i is defined as
(3) where Ti, Vi and T,, vf are the isospin and isospin projection quantum numbers for the initial and the final nucleus respectively. The Go and G, correspond respectively
(n-, nn) ON ‘*C
515
to the contributions from the isospin singlet (T = 0) and isospin triplet (T = 1) states of absorbing nucleons in nuclei; G,, and G, are defined as G, =
Ms,) ~~i(LS
c 2”k,,,LMLM.s
= 1J T = O)(LlM,M,IJM) x(llMs,M,-M,IIM,)F~~~,
G, = c (- l)MS’k_M,, Xfi(LS = 0 J T = l)Fk;;, Mr.
(4a) (4b)
where k, is the spherical component of the vector k; k itself is given by k = +(k, -k,) where k, and k, are the momentum vectors of the outgoing neutrons; L and ML and S, MS are the orbital angular momentum and its projection quantum numbers for the absorbing nucleons respectively; .Yfi(LSJT) is the spectroscopic amplitude for removing a pair of nucleons with the quantum numbers (LSJT) from the initial nuclear state i and leaving the final nucleus in the state f; (abc$lcy) is the Clebsch-Gordon coefficient. The integral F$‘; is given by F$;
=
s x-
dR *(k, Rh- *(k, W,M,@, Wr~m,@), 9
2
(5)
where the x are the distorted waves for the outgoing neutrons, 4LMr. is the space part of the two-nucleon wave function in nucleus, and 4,,,,% is the pion wave function. In eq. (3) go and g1 are the interaction parameters of Eckstein’s Hamiltonian. They 33P, transitions of two nucleons. respectively determine the 13S1 -+ 33P1 and 31S,, + The modulii of these parameters can be obtained by analysing the reactions .n+ +d + p+p, P+P
+ nO+p+p,
(7)
The values of 1gol and (gll recently obtained by Figureau and Ericson ing the latest experimental data of Rose ’ “) on the above reactions are )go12 = 0.64
‘) by analys-
+0.05 fm’,
lg1j2 = 0.155kO.08
fm*.
(8)
In the analysis of reaction (7) one does not need to know anything about the relative phase of go and gi. However, for the calculation of the transition amplitude for the (n-, nn) reaction, we need the magnitudes as well as the relative phase of go and gi. This happens, as we see from eqs. (2) and (3) because Go and G, contribute coherently. In certain many cases where either the isospin of the initial or that of the final state is zero, this difficulty can be avoided. In these cases either T = 0 or T = 1 contribute to Mri* Therefore, for these transitions, the matrix element Mri is
IMfil’ = lg0121Go12, for
T = 0,
for IMfil’ = l~~12(T,~~~‘flTi~~)21~~12~ These equations
are applicable
to many even-mass
(94
T=l.
(9b)
nuclei (e.g. 6Li, 12C, 160, etc.).
B. K. JAIN
516
Furthermore, if we consider the emission of nucleons at 180” to each other only, the expressions for G, and G, are considerably simplified. As we shall now see in this case the sums over M, and M, disappear in eqs. (4) for G, and G,. This can be seen very easily. Let us consider the partial-wave expansion of the distorted waves x in the integral Fk$‘;. If we take the Z-axis parallel to the vector k, and since k, is antiparallel to k,, only the zero projection of the angular momentum would contribute to the partial-wave expansions. Consequantly, due to the angular integration of vector R in integral (5), Fk$‘; would have non-vanishing contributions only when M,+m,
= 0,
or M, = -m,.
(10)
Secondly, since the vector k = +(k, -k2) is parallel to k, MS = Ms.,
for
kws-Ms+ = k, = 0,
otherwise, M,, = 0,
for
k-MS, = k, = 0,
(lla)
otherwise.
(lib)
These two conditions [i.e. (10) and (1 l)] thus remove the summations over ML and MS in eqs. (4). In addition, the summation over MS in eq. (2) is also restricted because
(1 lM,, M,-M&M,)
= (1 lM,O]lM,)
= 2-+Ms,
= +2-3,
for
MS, = 21,
= 0,
for
Ms. = 0.
(12)
Substituting eqs. (lo), (11) and (12) in eqs. (4) and (2) we get for T = 0
do
= p A+=+2Z,+l)-1]g,]zk2 (2~)~ 2M, X
IC
Yfi(LS
C Jm,Ms*
(25+1)-r = f 1
= 1 J T = 0)(Ll-
m, M,.IJM)F$E=
-m,i2y
L
(134
and for T = 1 P dw = m
A@-- -1’)(2[,+l)-1]g1]2k2(T,lv10]Tivi)2 2M, c (2J+l)-‘]&(LS
= 0 JT = 1)F$;12.
(13b)
m,L=.J
In the case of absorption from the 1s pionic orbit the summation over m, also disappears.
(n-, nn) ON l*C
517
Finally, it may be mentioned that the simplified expressions (13) for the transition probability are quite useful for studying the two-hole states in nuclei. This is due to the fact that in the 180” geometry of outgoing nucleons we can scan the whole momentum distribution of nucleons pair in nuclei by just recording the events corresponding to the various energies of one nucleon. 3. Results and discussions Recently Cheshire and Sobottka ‘) have measured the cross section for the (x, nn) reaction on ‘*C in which two outgoing neutrons are detected at 180” to each other. This experiment had also been done earlier by Calligaries et al. ‘l). But the resolution and the statistics of Cheshire and Sobottka are much better. These people in their summed energy spectrum have observed three peaks at the excitation energies of “B at about 5 MeV, 22 MeV and 37 MeV. From the ‘*C(p, 2p) reaction’*) we know that the binding energies of proton in the lp and 1s shell in “C are about 16 MeV and 34 MeV respectively. Therefore, considering ‘*C in the (1~)~ (lp)* configuration, the observed three peaks can be attributed to the capture of pion on (lp)*, (lslp) and (1s)’ pairs respectively. Since the formalism presented in sect. 2 is applicable to nucleons in equivalent orbits we will be restricting ourselves to the absorption of pions on (IS)* and (lp)* pairs only. Chesire and Sobottka have also measured the one-neutron energy spectrum corresponding to the three peaks in the summed energy spectrum. For the theoretical calculations, since “C in its ground state has isospin equal to zero, expressions (13) for transition probabilities are directly applicable. For the absorption by (lp)* nucleons, as the peak in the excitation energy curve appears at about 5 MeV, we have included the contribution from the state of “B up to 6.0 MeV excitation. The strength of the transitions to various excited states of “B is governed by the spectroscopic amplitude Yri. The values of 4ri for various transitions are calculated by using the intermediate coupling wave functions of Boyarkina 13) for r*C and “B. These wave functions are obtained by the energy level fitting of the lp shell nuclei using the L-S coupled basis states. It may be mentioned here that even in the calculation of the shape of the one-neutron energy spectrum for the absorption on (1~)’ nucleons it is necessary to calculate the contribution from each state of ’ 'B. This is required due to the fact that firstly the contribution to FLMr. comes from L = 0 and 2 and secondly these contributions are coherent for T = 1. Therefore, even the shape of the one-neutron energy spectrum is different for different states of “B. The radial part of the bound state wave functions for the neutron and the proton are taken as the solution of the Woods-Saxon potential. The parameters of this potential are taken from the work of Elton and Swift I’) on the analysis of the elastic scattering of the 420 MeV electrons on the “C nucleus. These wave functions of Elton and Swift have already been found to give a good description of the (p, 2p) [ref. ’ “)I and (p, d) [ref. ’ “)I reactions on ‘*C. For the distorted waves we have used the partial-wave expansion. The parameters
B. K. JAIN
518
of the optical potential, which is required in the neutron energy region of 10-70 MeV, are assumed to vary linearly with energy. Determining the parameters of these linear relations by using the sparsely available optical parameters in literature ’ 7), we have U,(MeV)
= -48 +0.3&
W,(MeV)
= -(2.63
W,(MeV)
= -(1.86+0.114
10 MeV c E S 70 MeV, E < 40 MeV,
+ O.O935B),
(14)
E 2 40 MeV.
E),
Here V0 and W0 are the depths of the real and imaginary parts of the optical potential. The shapes of U and W are taken to be Woods-Saxon, with the radius (rc,) and diffuseness parameters fixed at 1.25 fm and 0.65 fm respectively. For pions in the 1s orbit we have used the wave function of Seki and Cromer ‘*). These wave functions are obtained, in an approximate way, by solving a set of coupled equations in which the effect of the strong pion-nucleon interaction is simulated by a complex square-well potential and that of a finite charge distribution of nucleons by a uniformly charged sphere. The exact calculations of Fulcher et al. lg) show that the approximate wave functions of Seki and Cromer are in reasonably good agreement with their wave functions, and therefore are sufficiently accurate for most purposes. The Seki-Cromer wave function is written as $rs = c(e+YR -e-yR)e+8R4/R,
for
RSR,,
As = ce -““%(R/~o)h,,
for
R>R,,
(15)
where R, is the rms radius of nucleus and is fixed by the electron scattering experiment on the relevant nucleus, and a, = h2/(pZe2), with p as the reduced mass of pion and nucleus. The parameters y, 6 and C in eq. (15) are determined by fitting the energy shift of the 1S level and the spread of the 2P + 1S transition energies. These parameters are listed in the paper of Seki and Cromer; cp is a smoothly varying modulating function. For the 2P orbit we have used the hydrogenic wave function. Since the measured recoil momentum distribution in the (n-, nn) reaction does not discriminate amongst the contributions from the different pionic orbits, the experimentally observed yield for the 2n branch is equal to the weighted-average yield for this branch from various orbits, i.e. Lp(2n)
= Y2,(2n) + (P(lS)/P(2P))Y1s(2n).
(16)
Here we have assumed that the absorption from orbits other than 1S and 2P is negligible; P(n1) is the population of pions in the nl orbit. If we assume that the 1S orbit is primarly populated by the 2P -+ 1S transition, then W)/WP)
=
Kp(W/WipW
+
W,P
(a>>,
(17)
where W,,(X) is the X-ray transition rate for the 2P-1S transition; Wnl(a) is the total absorption rate of pions in the nl orbit. As pointed out by Ericson ““) W&a)
nn) ON ‘“C
(n-,
519
I
I
I
I
2
::
2
0
( SllNn
WV)
l-
q
8’SW
I 8 ‘SW P 3P/mP
1 hi .t? a
B. K. JAIN
520
can be deduced from the intensity attenuation of the 2P-1S X-ray line. The latest value of W,,(a) for 12C, due to Anderson et al. 2’) is 1.25 eV or equivalently 1.8 x 10” set-I. Taking W,,(X) equal to 2.14 x 10 l4 set-I, P(lS)/P(2P)
w 0.1.
(18)
Hence Y,,,(2n) The branching
= Y2r(2n) + 0.1 Y,s(2n).
(19)
ratios Y,,(2n) are defined as Y,,(2n)
= W2,(2n)/( I%(a) + I%(X)),
Yrs(2n) = Ws(2n)W1s(a). From eq. (19) it is clear that in “C 90 ‘A of the pions are absorbed in the 2P orbit and only the remaining 10 % are absorbed in the 1s orbit. In figs. 1 and 2 we have plotted the calculated one-neutron energy spectrum for the absorption of pions on (1~)~ and (1~)~ nucleons along with the experimental data of Cheshire and Sobottka. The solid (---) and dot-dash (-.-)lines in thesefigurescorrespond respectively to the contribution of the only 1S pion and the combined (1s +2P)pion as given by eq. (19). The normalization for the (1~)’ curves are chosen such that the area under them is the same. The area under the dot-dash curve for (1~)~ is about 15 % more than under the solid-line curve. This is done to bring out the best possible agreement between the computed and the experimental results. As we see from figs. 1 and 2, the combined (1s +2P) curves predict the maximum yield at the one-neutron energies less than the energies at which the experimental peak is observed. In other words, experimental maximum yield for the absorption on (1~)~ and (1 s)~ nucleons are observed at about Q, the recoil momentum, equal to zero while the combined (1 S + 2P) distribution predicts them at Q > 0. On the other hand, the distributionfromthe absorption of only 1S pions on (1~)’ nucleons is in quite good agreement with the corresponding experimental results. For the (1 s)~ nucleons, however, the 1S theoretical curve falls off rather too rapidly in comparison to the experimental one. The slow fall off of the experimental (1~)~ data can partly be attributed to the insufficient experimental resolution. In the present experiment, as we see from their excitation energy spectrum, although the peak corresponding to (lp) 2 is well resolved, those corresponding to (Is)’ and (lslp) are not quite resolved from each other. Therefore, it may be possible that the measured events for (1s)’ contain some events from the absorption on (Islp) and vice versa. And, since in the absorption of the 1S pion on the (1~1~) nucleon-pair the orbital angular momentum (L) which can be transferred is 1, the neutron energy spectrum for (1 s lp) would peak at some value of the recoil momentum greater than zero. Consequently the contribution of the peak around 22 MeV (due to (lslp) nucleons) to that around 37 MeV (due to (Is)’ nucleons) in the excitation energy spectrum would increase the cross section at high recoil momentum (or equivalently low neutron energy) for the absorption on the (IS)’ pair. The mixing of (Is)’ events
(n-, nn) ON l*C
521
into the (lslp) events would also distort the one-neutron energy spectrum for the absorption on (lslp) nucleons. The orbital angular momentum transfer in the (Is)? absorption is zero, and hence the neutron energy distribution for the (1~)~ peaks at zero recoil momentum (i.e. around 35 MeV neutron energy). Therefore any contribution from the absorption on (1 s)~ to that on (Is1 p) would increase the cross section
----.-._-K--X-
0.3663
IFL.
o12 (HWw*nic
0.4050
1 F,
. %I2 IHWwnicl
1 x 10
IF,,zi21Seki-Cromer)
5
4
IFL12
3
(1o-6 frli3)
2
1
I
I
I
I
I
I
I
10
20
30
40
50
60
70
E (MeV 1 Fig. 3. Variation of the integral IFLIz with the one-neutron energy (I?) for the absorption pions on (1~)~ nucleons using the hydrogenic and Seki-Cromer pion wave functions.
of 1s
for (lslp) around zero recoil momentum (or equivalently around 35 MeV neutron energy). This fact is born out by the discrepancy at low neutron energies between the experimental and the calculated (Islp) neutron energy spectrum presented in the paper of Cheshire and Sobottka. From these considerations we feel that the discrepancy between the 1s calculated and observed one-neutron energy spectra for the absorption on the (IS)’ and (1 sip) nucleons may partly be due to the insufficient resolution of the experiment.
522
B. K. JAIN
In order to see the effect of the pionic wave function on the transition probability, in fig. 3 we have plotted the lFL12 for (1~)’ nucleons using the hydrogenic and SekiCromer form for the 1S pion wave function. As we see, the use of the Seki-Cromer wave function which includes the effect of the strong pion-nucleus interaction and finite nuclear charge distribution, reduced the value of lFL12 for L = 0 and L = 2 by the factors of 2.73 and 2.46 respectively. The shape of the distribution is also changed. lF,.,l’ ----_
0.5372
a-+
IFL=
_._.-._
0.3160
I Distorted lFL_l*
Wave) IPlant
t~2Klistortrd
Wavcl
Wave)
~FL,r~21Pl.nc
Wove1
5
4
IV2
(lo-%‘; 2
1
10
20
30
10
50
I
I
60
70
E (MeV) Fig. 4. Variation of the integral IFLIz with the one-neutron energy (E) for the absorption pions on (1~)~ nucleons using plane and distorted waves for the outgoing neutrons.
of 1S
As we mentioned earlier, most of the calculations ‘) on pion absorption are done without including the interaction of the outgoing nucleons with the residual nucleus. In order to see the effect of the distortion of the outgoing neutrons by the residual nucleus, in fig. 4 we have plotted the IFLIz for (1~)~ nucleons using the Seki-Cromer wave function for the 1S pion, for plane and distorted waves for the outgoing neutrons. We see that the distortion reduces the peak values of lFL12 for L = 0 and L = 2 by the factors of 1.86 and 3.16 respectively. The shape of the distribution is also changed significantly. The effect of distortion on the transition probability for the 1s pion is shown in figs. 1 and 2 where the solid line corresponds to the distorted-wave results and
(n-,nn)
ON
i2C
523
the dashed line to the plane-wave results. The distortion reduces the transition strengths for the (1~)~ and (1~)~ nucleons by factors of 1.75 and 1.52 respectively. The shape of the neutron energy spectrum is also broadened and thus becomes in better agreement with the experimental results. It may also be noted that the effect of distortion on the (1~)~ neutron energy spectrum is mainly that of its effect on [FL12 for L = 0.This happens because, as we find from the calculations of the spectroscopic amplitude (Y& the main contribution to the pion absorption on the (1~)~ nucleons comes from L = 0 only.
4. Conclusions From the analysis presented in this paper the following conclusions may be drawn: (i) A good agreement between the calculated and measured one-neutron energy spectrum for the absorption of pion on the (1~)~ nucleons is obtained using only 1s pions provided the proper bound state wave functions for the nucleons are used and the scattering of outgoing nucleons with the residual nucleus is taken into account. (ii) The discrepancy between the calculated and observed spectra for the (1s)’ and (1 slp) nucleons seems partly due to the insufficient resolution of the experiment. (iii) The inclusion of the proper pion wave function (i.e. those including the effect of the strong pion-nucleons interaction and the finite charge distribution of nucleus) and the distortion of the outgoing nucleons by the residual nucleus is essential. Each of these factors reduces the magnitude of the pion absorption probability and changes the shape of the one-neutron energy spectrum significantly. (iv) The calculated weighted-average one-neutron energy spectrum from the 1s and 2P pion orbits, where the weightage is taken from the data on the 2P-1s X-ray yield, is in clear disagreement with the experimental results. The computed distribution peaks at Q (the recoil momentum) > 0 while the experimental data peaks around Q z 0. The author wishes to express his thanks to Dr. R. Ramanna for his encouragement and interest in this work, to Prof. D. F. Jackson for providing the Is nucleon radial wave functions, and to Prof. T. E. 0. Ericson for some useful comments.
References 1) E. H. S. Burhcp, High energy physics, vol. 3 (New York, 1969); G. Backenstoss, Ann. Rev. Nucl. Sci. 20 (1970) 467 2) D. H. Wilkinson, Comm. in Nucl. and Particle Phys. 2 (1968) 48 3) T. E. 0. Ericson, Phys. Lett. 2 (1962) 278 4) D. H. Wilkinson, Proc. Int. Conf. on nuclear structure, Tokyo, 1967, p. 469; Cornm. and Particle Phys. 2 (1958) 83 5) B. K. Jain and B. Banerjee, Nuovo Cim. 69 (1970) 419; B. K. Jain, Bhabha Atomic Research Centre, Bombay, 1970, report-512 6) S. G. Eckstein, Phys. Rev. 129 (1963) 413
in Nucl.
524
7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)
18) 19) 20) 21)
B. K. JAIN D. L. Cheshire and S. E. Sobottka, Phys. Lett. 3OB (1969) 244 R. Guy, J. M. Eisenberg and J. Le Tourneux, Nucl. Phys All2 (1968) 689 A. Figureau and M. Ericson, Nucl. Phys. BlO (1969) 349 C. M. Rose, Phys. Rev. 154 (1967) 1305 F. Calligaris, C. Cernigoi, I. Gabrielli and F. Pellegrini, Nucl. Phys. Al26 (1969) 225 H. Tyren, S. Kullander, 0. Sundberg, R. Ramachandran, P. Isacason and T. Berggren, Nucl. Phys. 79 (1966) 321 A. Boyarkina, Bull. Acad. Sci. USSR (whys. ser.) 28 (1964) 255 L. R. B. Elton and A. Swift, Nucl. Phys. A94 (1967) 52 B. K. Jain and D. F. Jackson, Nucl. Phys. A99 (1967) 113; R. Shanta and B. K. Jain, Nucl. Phys. 175 (1971) 417 I. S. Towner, Nucl. Phys. A93 (1967) 145 P. H. Bowen, J. F. Scanlon, G. H. Stafford, J. J. Thresher and P. E. Hodgson, Nucl. Phys. 22 (1961) 640; P. E. Hodgson, Proc. Conf. on direct interactions and nuclear reaction mechanisms, Padua, 1962, p. 103; D. R. Winner and R. M. Drisko, Tech. Rep. Dept. of Phys. Sarah Mellon Scaife Radiation Lab., June, 1965, unpublished R. Seki and A. H. Cromer, Phys. Rev. 156 (1967) 93 L. Fulcher, J. M. Eisenberg and J. Le Tourneux, Can. J. Phys. 45 (1967) 3313 M. Ericson, Comp. Rend. 258 (1964) 1471 D. K. Anderson et al., Phys. Rev. Lett. 24 (1970) 71