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The 16O(d, 3He) excitation spectrum
I
2.G J
Nuclear Physics A203 (1973) 513--531; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE 160(d, 3He) EXCITATION SPECTRUM A M A N D FAESSLER and S. K U S U N O Institut flir Kernphysik der Kernforschungsanlage Jiilich GmbH, 517 Jiilich, West Germany
and G. L. STROBEL
University of Georgia, Georgia, USA Received 19 June 1972 (Revised 4 December 1972) Abstract: The t60(d, 3He) excitation spectrum has been calculated in the framework of the Gross-
Lipperheide model which explicitly takes into account the effect of residual interactions and the decay of the residual nucleus by particle emission. The reaction mechanism is described by the DWBA for a direct pick-up reaction. The theory is extended to include the higher-order effects of the residual interaction on the self-energy operator, in terms of which the spectral function is expressed. The real part of the self-energy operator is calculated up to the fourth order (2p-3h configurations) in the strength of the residual interaction. The calculation of the selfenergy operator up to the second order yields the leading term only for separation energy below 38 MeV. Higher-order effects have already become important for separation energies as low as 10 MeV. The Lemmer-Shakin potential is used for the residual interaction. The calculated spectrum exhibits a shell structure of groups of narrow peaks, which depends very much upon the choice of the absorption potential for the 3He optical parameters.
1. Introduction Nucleon hole states in nuclei have been studied through pick-up reactions such as (p, d), (n, d), (p, 2p) and (d, 3He) reactions. Because of the higher beam resolution possible, the last reaction is the most suitable for studying the highly excited proton hole states in nuclei. One can interpret peaks in the excitation spectrum as corresponding to a single-particle level structure in the nucleus. The identification of the angular momentum of such low excited states has been possible through a study of the reaction angular distribution using the distorted-wave Born approximation (DWBA) to describe the reaction mechanism. For higher excitations nucleonnucleon correlations in the target and final nuclei can destroy this simple picture. Inclusion of such correlations between nucleons in the nucleus via a residual twonucleon interaction results in a simple hole state being coupled to np-(n + 1)h states. As a result the expected excitation spectrum is more difficult to unravel experimentally or to calculate theoretically. We calculated an excitation spectrum for the (d, 3He) reaction. Particle decay of the residual nucleus into the continuum is taken into account. We apply the spectral function theory of Gross and Lipperheide t) which is formulated only up to the 513
514-
A. FAESSLER et
al.
second order of the residual interaction (lp-2h states in the final nucleus). It was applied by Wille, Gross and Lipperheide for the 12C(p, 2p) spectrum 2,3), and by Wille and Lipperheide for 160(p, 2p) spectrum 3). In this theory the one-nucleon pick-up cross section is expressed in terms of the spectral function of the target nucleus. We extend this theory in order to include 2p-3h excitations and describe the (d, 3He) reaction mechanism as a direct reaction using the DWBA. The spectroscopic factor, which is usually set to 2j+ 1 for the pick-up of nucleons out of a level with low excitation energy of angular momentum j, is calculated here through the use of the spectral function. The spectral function has a discrete and continuous part. The continuous part occurs for excitation energies where the residual nucleus can undergo subsequent particle decay. The residue of the discrete contribution to the spectral function is the "usual spectroscopic factor" associated with the bound states of the final nucleus 3). We consider the target to be a x60 nucleus described by the shell-model as having full ls~, lp~ and lp~ shells, and all other levels empty. We take as a residual interaction the Lemmer-Shakin potential *), which may be appropriate for this nucleus. With no residual interactions present, the (d, 3He) reaction on 160 would produce ls~, lp~, l p , hole states in the final 15N nucleus. The (d, 3He) spectral function would consist of three peaks corresponding to proton pick-up from one of the three filled shells of the target nucleus. The position of these peaks would permit a determination of the proton single-particle binding energies in the target nucleus. With a residual interaction included, other configurations are coupled to single-hole states. The effect of the residual interaction appears through the self-energy operator of the nucleus by which the spectral function is expressed. The self-energy operator can be calculated by the technique of the partial summation of many-body diagrams 5). We include all possible second- and third-order diagrams and a typical fourth order one in the calculation of the self-energy operator. This means that all possible lp-2h and several types of 2p-3h configurations that can couple to the single-hole states are included. In order to see the effect of the residual interaction, let us consider only the lp-2h configurations coupled to single-hole states for simplicity, For weak enough residual interaction the energies of the lp-2h configurations will not change appreciably when coupled to single-hole and other lp-2h configurations. We then expect to observe aH at final-state energies corresponding to an energy transfer to the final nucleus equal to the excitation energy of one of the single-hole and lp-2h configurations. Now instead of one peak resulting from the pick-up of a 1s~ proton, twenty peaks can be expected in the spectral function. This is so because we include the nineteen different lp-2h configurations that can couple to ls~ single-hole configurations. The spectrum resulting from lp~_ or lp½ proton pick-up will be complicated by a similar amount due to lp-2h configurations coupled to the corresponding single-hole configuration. The calculated spectral function has peaks at energies near the lp-2h
l~O(d, 3He) SPECTRUM
515
configuration energies, but is a continuous function of energy, where the residual nucleus can decay via a particle emission into the continuum. If we allow the higher-order excitations further, the excitation spectrum is expected to become so complicated that the sharp peaks corresponding to lp-2h excitations may be smoothed out. We have calculated the real part of the self-energy operator up to the fourth order in the residual interaction. The perturbation expansion of the self-energy operator shows only fair convergence for separation energies below 20 MeV. The imaginary part and the cross section for the 160(d, 3He ) reaction was calculated with a self-energy operator obtained up to the second order only. The ground-state correlations are neglected. In sect. 2 we present the theoretical details of the calculation of the excitation spectrum, and in sect. 3 we discuss the results.
2. Theoretical details of the (d, 3He) excitation spectra We follow the theory of Gross, Lipperheide and Wille (GLW) t) developed for (p, 2p) and (d, 3He) reactions. The momentum-energy spectral function of G L W is assumed expanded in shell-model states as
S(q, q'; W) = E qgv(q)~Pv'(q')Svv'(W),
(1)
where v denotes the quantum numbers of a shell-model hole state, and ~Pv(q) is the momentum-space representation of the bound-state wave function of such a hole state. Eq. (1) covers the case where total angular momentum and parity do not totally distinguish a bound state, such as for nuclei where the 2s state is occupied at least partially, in addition to the more tightly bound Is state. We assume total angular momentum and parity do completely specify all bound states for an t 6 0 target, in which case eq. (1) becomes
S(q, q'; W) = ~ ~pv(q)q~v(q')Sv(W).
(2)
v
Such a description ignores any nucleons occupying the 2s~ shell-model state in the target nucleus 160. Such an event could result from the correlations in the target nucleus but are ignored here. We take the ls½, lp~ and lp~. levels as completely full, and all other target levels as completely empty. Here I4/ is the energy removed from the nucleus, so S = - 141 is the energy given to the final nucleus. We shall call S the "separation energy' for later discussions. The quantity IV is related to various c.m. kinetic energies (E) or positive binding energies (B) by
W = E,--Ed+Bd-B~,
(3)
where the d or z subscript refers to the initial deuteron or final 3He projectiles. Eq. (2) assumes that the momentum distribution for a given state is independent of the energy transfer. The DWBA cross section for detecting 3He particles with an
A. FAESSLER ¢t al.
516 energy E is given by d17
E IL(w)I2S,(W),
-
dt2dE
(4)
where fv(W) is the usual D W B A amplitude for a deuteron pick-up of a proton from the state v, with the final aHe leaving with an energy related to W via eq. (3). The spectral function S~(W) is the probability that when the proton, in the state v, is picked up, an energy transfer of S to the final nucleus will result. The spectral function S,(IV) is related to the single-particle Green function t) by
S,¢(W) = lim 1 _ {G~¢(W-bl)-Gv¢(W + bl)). ,~o 2hi
(5)
The Green function G is related to the self-energy operator 2: via the Dyson equation 5) by Z ((z- e,)5,~,-- Z,,.~,(z)}G~,¢(z)= 6,~,, (6) /t
or with
Z~,(z) = 6v,~,(_~),
(7)
one can solve for the Green function to obtain
G.~(O = 6v,(z-
~,-
~v(z))-'
(8)
Here ev is the energy of the single-hole state v. The self-energy operator Z' is composed of all possible orders of the (irreducible) self-energy diagrams. The first-order
k
~¢k 1
....
'1
(Q]
(b)
2
(c)
(d) ~k
1
2
2
4
!}iiii ..... _ y (e)
If)
[g)
Fig. 1. The second-, third- and. one typical fourth-order self-energy diagrams. Each graph a-g corresponds to eqs. (A.9), (A.16), (A.17), (A.18), (A.19), (A.20) and (A.21), respectively.
160(d, aHe) SPECTRUM
517
contribution (with respect to the strength of the residual interaction) corresponds to the Hartree-Fock (bubble and open oyster) diagram, which is reduced to the HartreeFock single-particle energy in the Dyson equation, as long as the single-particle potential is chosen self-consistently 5). This condition is assumed to be fulfilled in the following calculation 3): The second- and higher-order contributions to the self-energy operator can be evaluated according to the method proposed by Villars 6). The results are shown in the appendix (fig. 1). Now the self-energy operator is the sum of contributions of each order
a,(z) = cr~2)(z) + a~3)(z) + . . . .
(9)
It should be noted that the sum over particle states in the expression for the selfenergy operator in the appendix symbolizes automatically integrations over the continuum states also. For example the expression for a(~2)(W) is
I(vklVAIlm)12 --½ lim ~
a~2)(W) = ½ ~, l, ra
ek--Etm
F
f ~ risk I(vklVAllm)l 2 , Etm+iq-e k
(10)
r/--*0 l,m0
where
E~m = *l+~m-- W.
(11)
The summations in eq. (10) are over all the quantum numbers indicated by /, m such that the energy of the hole state is indeed less than the Fermi level. The particle states k in the first sum are over the bound states that lie above the Fermi level. In the second sum the particle states k are summed over all continuum states. The angular momentum selection rule restricts the angular momenta of the continuum states to { or less. For the residual interaction V, we utilize a delta-function residual interaction first used by Lemmer and Shakin 4) V(r x - -
r2)
(a + bal" a2)f(f21 - •2)6(r 1
=
-
-
r2)/r
(12)
2 .
The value used for a is - 5 0 3 . 7 MeV- fm a and for bis - 7 8 . 6 MeV- fm s. The matrix element of this interaction is also given in ref. 4). The principal value integral ofeq. (10) is done by the following substitution 7) u kr ( r )eu kc( ,)
Qo
lim |
n-.o j o
dek
a
B
,
- gtk+)(r, r', Etm) - ~u~(r)u~(r').
Elm + i~l - ek
.
(13)
Elm -- ekn
Here u~(u~) denotes the continuum (bound) state radial wave function with quantum number k which is a solution of the Schr6dinger equation d2u
t 2M E + i
V (
l(l+ 1)}
O.
(14)
" =
The sum over n denotes the various bound states for a given parity and total angular momentum specified by k. In general there can be more than one such state. For a 6 0 target, there are either one or zero such bound states depending on the parity
518
A. FAESSLER et al.
and total angular momentum. Now the Green function 0~+) (r, r', Etm) is given by gtg+)(r' ,.,, Elm) _ 2 M uk(r<)6k(r> ) h~
W(uk,6~)
(15) '
where r< (r>) is the smaller (larger) o f r and r'. The quantity W ( u , 6) is the Wronskian between u and C. The wave function u(r) is the regular solution (at the origin), and C(r) is a solution that satisfies outgoing boundary conditions, of eq. (14). We note that 9kt+)(r, r', E~m) is real if the energy E~m is negative, and that E~m is not in general the eigenvalue for the state k. If one substitutes eqs. (A.6) and (13)into the last term of eq. (10), and interchanges the order of integration over energy with coordinates, one encounters, with a finite-range residual interaction, a four-dimensional radial integral. The alternative to this four-dimensional integral is to do the principal value integral in eqs. (10). Instead of this, we assume a delta-function residual interaction 4) whereupon we obtain a two-dimensional radial integral of the form IiV,k = J J t" d, • ,., dr ,uvQ .)ut(r)um(r)ut(t ., )um(t ., )uv(t J )gk(+) (r, r ,, Etm).
(16)
This integral is done numerically using a Simpson quadrature. The subscript k on yg(+)"it', r ' , Etm ) is to emphasize that it depends on the quantum numbers of the particle k. The potential Vws of eq. (14) used to generate the Green function and the boundstate wave functions is of the form Vws = V c ( , . ) _ V o f ( r o , a ) + 2
(+)2
V~.o.(l d f(rs .... as.o.), r • s) drr
(17)
where f ( r , a) = (1 +eX) - ',
(18)
x = ( r - r o ( A - 1)~)/a.
(19)
Here A is the mass number of the target nucleus. The single-particle energies used are listed in table 1. These energies and Woods-Saxon potential parameters were taken from Buck and Hill 8). The same set of parameters has also been used for 160 in ref. 3). The ls~ binding energy was obtained through use of the same set of WoodsSaxon potential parameters that fit the 2s~ binding energy. A Woods-Saxon radius and diffuseness of r~.o. = r o = 1.25 and a .... = a = 0.53 fm were used throughout. The proton bound states included a Coulomb potential Vc(r ) due to a uniformly charged sphere of radius equal to the Woods-Saxon radius. The potential depth Vo was adjusted for the d~ state to produce a resonance at the resonance energy (see table 1). The potential parameters for the If states were taken to be the average of those states with lower angular momenta. Wille and Lipperheide 3) calculated the Hartree-Fock single-particle energies from the two-body interaction (12) and
160(d, aHe) S P E C T R U M
519
TABLE 1 ~60 single-particle energies a n d W o o d s - S a x o n potential parameters
nlj
charge (e)
I s½
1
lp½ lp. t_ ld~ l d 1. lfk lf.I. 2s~ Is½ lp~ lP½ ld½ ld~ I f_~ lf~ 2s.t_
1 1 l l 1 1 1 0 0 0 0 0 0 0 0
rv (MeV)
Iio (MeV)
Vs.°. (MeV)
27.63 --18.45 -- 12.13 -- 0.60 + 4.27
47.09 52.52 52.52 47.44 46.80 49.0 49.0 47.09 56.97 57.42 57.45 54.99 54.92 56.0 56.0 56.97
0 9.88 9.88 5.24 5.24 7.56 7.56 0 0 9.66 9.66 5.24 5.24 7.45 7.45 0
--
-- 0.10 --35.91 --21.83 --15.63 -- 4.14 -- 0.94
-- 3.27
compared them with the Woods-Saxon single-particle energies of table 1. The selfconsistency condition was found rather well fulfilled. The final expression obtained for the self-energy operator a~v2) is <2,(w)
-
1 32rc 2
+ + + 6~+},
E A{(C+D)(F+~+,+F+~+)+2(C-D)F+
(20)
klm¢
where z denotes the isospin z-component of the particle and the sum over l, m (k) denotes the sum over parities and angular momentum for the hole (particle) states. The quantities C and D are related to the parameters of the residual potential by C = (a-3b)2No,
(21)
D = (3a+3b)2N~,
(22)
where N~ is a sum of the products of Clebsch-Gordan coefficients and 9-j symbols,
=
:
½
'k L)t,.
,.,
(23)
The quantity F is a factor resulting from the radial integrals of eqs. (10) and (A.5). The subscripts of F denote the charge state for nucleons vklm (in this order) in the expression F~.kU. =
t, v ( r ) u k ( r ) u l ( r ) u m ( r ) r 2 d r
Etm -- ek) - 1 _ i , m~.k.
(24)
The sum over T in eq. (20) is carried over the terms r = + (proton) and z = -
520
A. FAESSLER
et aL
TABLE 2 160 lp-2h configurations for s½(P) hole pick-up No.
l
m
k
S (MeV)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Pf(P)
P~r(P)
p~(p) p~(p) p½(p) Pt-(P) p,t.(p) p½(p) p~(p) pt_(p) Pt-(P) p~(p) p~(p) pt.(p) p.l_(p) p/_(p)
p~r(n) P½(n) p~.(n) Pg-(P) p~_(n) p~(n) P½(n) p~(p) p~(n) p~(n) p~}(p) P'I-(P) Pz2(P) p~(n)
2st-CP) 2sg(n) d.t.(n) ¢ d½(n) dl-(P) d.t.(n) 2s.t.(n) 2s~(n) d~(p) ¢ d?~(n): dt.(n) ~ d~(n) d~(p) 2s~(p) 2s½(n)
24.16 24.53 28.74 29.82 29.98 29.98 30.69 30.85 34.88 34.90 35.06 36.14 36.30 36.80 37.01
16
p~(p)
p~(p)
d~(p)~
41.20
17 18 19
P-t-(P) s~iP) s~(p)
p.}(n) s~_(p) s.t_(n)
d~(n) c 2s~(p) 2s÷(n)
41.22 55.16 60.27
TABLE 3 160 lp-2h configurations for p proton pick-up No.
I
m
k
S (MeV)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
s~_(p) s½(p) s~(p) s½(p) p~(p ~ p~(p ) s½(P) st_(p) s~(p) pt.(p) s x(p) st~(p) s½(p) P-t-(P) p.~(p) s~(p) s½(P) p{_(p)
p,~(Pl pt_(n) p~(p) p~(n) s~(n ) s.t_(n ) Pcz(P) pk(n ) p.t_(n) s~(n) P-I-(P) p~_(p) p~(n) s~(n) s½(n) p.l.lp) pg_(n) s½(n)
d~r(P) dl.(n) 2s½(P) 2s~(n) di.(n) 2s.i_(n) d~(P) c d~(n) dl_(n) d½(n) ¢ d~(p) 2st_(p) 2s½(n) d.t.(n) 2s½(n) d.l.(p )¢ d.~(n) c dt(n)C
39.16 39.16 39.66 40.03 43.90 44.77 44.03 44.24 45.32 48.98 45.48 45.98 46.19 50.22 51.09 50.35 50.40 55.30
Configurations with the sign ; are excluded for the p~(p) hole case by the angular momentum selection rule.
leO(d, 3He) SPECTRUM
521
A = 1,1~2~1,, 3k 3~ 3,,.
(25)
(neutron). The quantity A is The lp(k)-2h(/, m) configurations possible and the energy transfer needed to excite such a configuration are listed in tables 2 and 3. The optical-model potential used to generate the scattering wave function is (h
~2
VDw = Vc(r)--Vof(ro,a)+ ~c~c) V~'°'(l'S)r
.
,
,
_~xf r~.o.,a~.o.)+,Wof(ro,ao) +4iWt, ~d, f(r'o, a').
(26)
The DWBA cross section was calculated in a nearly standard manner using the code D W U C K 9). The only difference was that the binding energy of the picked-up proton, an input parameter for DWUCK, was not related to the reaction Q-value as the final projectile energy is specified independently. The DWBA parameters for d -160 scattering are taken from ref. xo), where Ea = 52 MeV. Those for 3He-~ 5N scattering are not so well-known over a wide range of 3He energy. Those values corresponding to E, = 39.8 MeV [ref. ~1)] are used. They are listed in table 4. TABLE 4 Optical parameters for the reaction 160(d, 3He)t 5N used in this calculation Vo (MeV) d+160
Wo (MeV)
88.9
4 Wo Vs.o. a (MeV)(MeV)(fm) 10.5
a" (fro)
as... (fm)
ro (fm)
r'o (fro)
rs... (fm)
rc (fm)
Ref.
11.275 0.792 0.792 0.792
1.05
1.36
1.05
1.3
lo)
0.595 0.858
1.23
1.80
1.3
11)
--12.44 3HeWlSN
160.0
(-40.0)
One gets both continuous and discrete contributions to the spectral function from eq. (5). The continuous contribution to the spectral function in terms of self-energy operator trv is
S~(W) =
-1 Im trv(W) rc ( W - e ~ - R e a~(W))2+(Im a02"
The discrete contribution occurs at those energies
W,,,-,~-a,,(W.,)
W~, where
(27)
W satisfies
= 0.
(28)
The discrete contribution to the spectral function at these energies is
s~i .... t°(w) = Z ,~(w-w,,,) / ( 1+ da~(W)]dW 'w=w~,
(29)
The rate of change of the self-energy operator a~ with W is always negative, so the discrete contribution to the spectral function is always less than one.
522
A. FAESSLER
e t al.
3. Discussion Let us consider first the calculation up to second order in eq. (9). Fig. 2 (3) shows the real and imaginary part of the self-energy operator t~ 2) for a proton pick-up from a ls½ (lp~) state v e r s u s the separation energy S. The peaks in I m a~,2) at energy transfer around 29, 35 and 41 MeV (44, 49, 50 and 55 MeV) for s(p) wave pick-up, are related to the lp-2h configurations in which the particle is in the continuum state ld~. The structure of the self-energy operator for proton pick-up from the lp~ state (not shown) is similar to that of the lp~ mass operator, except for the absence of three poles in the real part of o-c2),,. The missing poles are absent due to the angular momentum selection rule which prevents three configurations, labeled x in table 3, from contributing to the mass operator of the lP½ state. These configurations do contribute to the lp~ self-energy operator however. The various poles of Re ~(2) o~ occur near energies which can also be correlated to the lp-2h configuration energies. Intersections of the straight line in figs. 2 and 3 with 200 O~'{d, He 3) N Is v = S 7/~
t=
/
~c
~o
• 91 2
• 6/-
:g
t~
3O
StN~ Fig. 2. The real and imaginary part of the second-order self-energy operator for the v = s~_proton hole m the t60 nucleus c e r s u s the separation energy S. The line corresponding to eq. (28) is also shown.
l~O(d, aHe) SPECTRUM
523
016(d. He 3 ) N '5 v =P3/~ 106
"~5c
0
i
20
30
40
50
60
~7o
S
30
tD
5 - 2G
E tO
i 2O
I
30
L
40
i
h
50
i
60
70
S (M~Vl
Fig. 3. The same as fig. 2 for the v = P-t-proton hole in the 160 nucleus. Re tr(~2) determine the energies where the m a x i m a of the spectral function occur. The m a x i m a o f the spectral function, as seen in eq. (27), are S v = (zr Im a(~2))-1
(30)
I f Re ov-(2) were independent o f W = - S, the width of the peaks would be given by I m or-(2). Then we would expect peak widths of one MeV or more for peaks at S greater than a b o u t 30 MeV. However, in fact, Re O_(2) varies very rapidly with W v wherever the spectral function has a m a x i m u m . In fact the rate o f change of Re a(~2) with W near most m a x i m a is so large that this consideration determines the width. I f we write Re t r t 2 ) ( W ) = Re trio2)+ r d-dT Re a(2)l Low d
AW,
(31)
Wo
where W o is chosen so that W o - e = a(o2), we can see why the calculated peaks have such a n a r r o w width. Let the subscript 0 denote evaluation at an energy
524
A. F A E S S L E R et aL
corresponding to a peak in the spectral function. Then, near such an energy eq. (27) can now be rewritten as 1 Im a~o2) Sv = n ([d Re a(2'/dW]wo A W) 2 + ( I m a(2)) 2"
(32)
The half-width at half maximum of the spectral function will then be Im a(o2) A W~ = [d Re a(2)/dW]wo "
(33)
This says that the half-width is independent of the residual interaction strength chosen. This is much less than other estimates 12) and much less than what is expected by crudely setting the half-width equal to I m a ~z). As the energies of lp-2h configurations exhibit a shell structure and Re a C2) can be correlated to these energies, we can see in the spectral function, figs. 2 and 3, this same shell structure. Namely, at energies where you have many lp-2h configurations, we calculate many corresponding peaks in the spectral function. Fig. 4 shows the excitation spectrum for the 160(d, 3He ) reaction with E a = 52 MeV which is calculated by using DWBA and the second-order self-energies o_(2) r . Where the discrete contributions to the spectral function occur depends somewhat on the residual interaction strength, but the position of the continuum peak contribution is nearly independent of the residual interaction strength. Although the optical parameters for the final channel depend upon the energy of the 3He particle, we have fixed those values derived from the E~ = 39.8 MeV analysis it), because there are very few data available. In order to see the effect of the optical-potential parameters we have also calculated the excitation functions only by increasing the absolute value of the volume absorption potential W o to - 4 0 MeV. The result is shown in fig. 5. The other parameters remain exactly the same as in fig. 4. As a matter of fact, the excitation is reduced on account of the rather strong volume absorption. The experimental data which can give information about highly excited lp-2h states are not so numerous. Figs. 4 and 5 contain the experimental data taken from ref. 13). The experimental data (E d = 52 MeV) are only available up to S ~ 24 MeV at present. We could not get peaks corresponding to (b), (d) and (e) in figs. 4 and 5. This is because we have neglected the ground-state correlation (the second term in eq. (A.8)). This term becomes effective for small values of separation energy S. We have also calculated the excitation spectrum corresponding to Ea = 80 MeV. It is shown in fig. 6. There is only one experiment at E d = 82 MeV [ref. 14)]. But only three peaks which correspond to (a), (b) and (c) of fig. 4 are reported. Therefore we do not try to compare our results at Ed = 80 MeV with experiment. Fig. 6 is different from fig. 4 in the appearance of highly excited peaks at S > 50 MeV. We have used the same optical parameters in fig. 6 as in fig. 4, because such sets of parameters are not so well known apart from those reported in ref. 14).
160(d, 3He) o
0
SPECTRUM
2p
19
525
Ex(MeV)
3p
P#2
160(d, 3He}15N
(c)
Ed=52 HeY E)Lob=13.0" W,=-12. LJ-, MeV
.-x40
x40
P
(b) I o,_
10
x40
15
20
P 30
25
t.o
35
t~5
50
S(MeV)
Fig. 4. The t60(d, aHe) excitation spectrum (solid line) for Ea = 52 MoV. The second-order selfenergy operator is used in this calculation. The depth o f the volume absorption potential for 3He-15 N scattering is --12.44 MeV (table 4). The quantities below S = 25 MeV should be enlarged 40 times. The experimental data (dotted line) are taken from ref. 13) and are normalized to the theoretical p÷ peak at S = 14 MeV. The experimental spectrum is shifted also by 4.45 MeV to the left so that the p~. separation energy agrees with the theory. The upper energy scale gives the excitation energy in t SN.
0
o
p~
1.0
2,0
160( d, 3He)15N Ed =52 MeV OLob=l 3.0" W.='-/.O.O MeV
P~ (c)
(o)
Ex(MeV}
2p
x20
o
I
x2°jil 0
1"0
!J 15
^ 2"0
25
30
p~ p" s
35
p
\
~0
z.5
50
S(MeV)
Fig. 5. The same as fig. 4 for the volume absorption potential We = --40 MeV. The quantities below S = 25 MeV should be enlarged 20 times. T h e experimental spectrum is shifted as in fig. 4.
526
A. FAESSLER
e t al.
160(d, 3He)lSN Ed--8OMeV OLob=13.0" W.=-12.44 MeV
o
i1 5
10
$
15
20
25
30
P
35
40
45
50
S(M,eV)
55
60
F i g . 6. T h e z 6 0 ( d , 314e) e x c i t a t i o n s p e c t r u m f o r Ea ~ 80 M e V . T h e s e c o n d - o r d e r s e l f - e n e r g y o p e r a t o r is u s e d in t h i s c a l c u l a t i o n . T h e s a m e o p t i c a l - p o t e n t i a l p a r a m e t e r s as f o r t h e E a = 52 M e V case are used also.
The main effect of using distorted waves instead of plane waves for the scattering wave functions is to reduce the excitation spectra for large values of S. Contributions to the excitation spectra resulting from proton pick-up from the ls~ shell is inhibited in comparison to pick-up from the lp~ or l p , shells. This is true for all scattering angles between 10° and 90 ° and is not restricted to just the angle 13° for which the spectrum is calculated (figs. 4-6). The most time-consuming part of this work is to calculate the single-particle Green function #Ok+)(,", r', Ezra), and subsequently F (see eqs. (15), (16) and (24)). This calculation must be repeated for each value of If', and hence each Ezra. TABLE 5
V a l u e s o f t h e s e l f - e n e r g y o p e r a t o r s v e r s u s the s e p a r a t i o n e n e r g y S = - - 1 4 / f o r t h e k = s~ p r o t o n s t a t e in t h e ~ 6 0 n u c l e u s S
a
b
10 16 22 28 34 40 46 50
1.252 1.430 1.666 1.997 2.494 3.328 5.040 7.854
0.015 0.017 0.020 0.024 0.031 0.044 0.072 0.128
c 0.039 0.052 0.073 0.108 0.178 0.348 0.952 3.000
d 0.586 0.670 0.782 0.940 1.179 1.583 2.431 3.872
e ---------
0.449 0.587 0.800 1.155 1.813 3.267 7.694 19.580
f
g
0.552 0.722 0.983 1.419 2.229 4.016 9.458 24.068
0.025 0.033 0.046 0.066 0.105 0.191 0.455 1.191
T h e letter, a - g c o r r e s p o n d t o t h e r e s p e c t i v e d i a g r a m s s h o w n in fig. 1. T h e y a r e i n M e V .
160(d, aHe) SPECTRUM
Values of the self-energy operators
527
TABLE 6 the separation energy S = -- W for the k = pk proton state in the 160 nucleus
versus
S
a
c
d
10 16 22 28 34 40 46 50
1.478 1.770 2.206 2.935 4.440 -- 4.339 --70.319 44.097
0.028 0.040 0.064 0.117 0.288 3.536 4897.9 45.054
0.100 0.118 0.146 0.189 0.272 0.499 --16.982 4.537
e 0.019 0.028 0.046 0.088 0.216 --6370.7 --2693.8 237.27
f
g
0.493 0.709 1.111 1.997 4.749 4537.1 2129.l 1517.2
0.034 0.050 0.079 0.145 0.355 1313.7 2016.4 191.31
The letters a-g correspond to the respective diagrams shown in fig. 1. They are in MeV. Diagram b is not shown in the table because it vanishes on account of the parity selection rule. Values of the self-energy operators S
a
10 16 22 28 34 40 46 50
1.926 2.305 2.873 3.82l 5.7643 --75.275 --63.986 74.025
versus
TABLE 7 the separation energy S = -- W for the k = p~ proton state in the ~ O nucleus
c
d
e
0.037 0.054 0.083 0.156 0.385 14.690 1056.9 36.115
0.025 0.313 0.041 0.060 0.113 --16.317 -- 0.315 -- 0.110
0.006 0.091 0.015 0.026 0.050 --5460.9 2970.9 --164.24
f 0.605 0.872 1.365 2.455 5.863 79794.0 3416.1 2557.5
g 0.041 0.059 0.093 0.169 0.400 4565.2 796.88 302.28
See table 6 for further explanation. I n o r d e r to see t h e effect o f the h i g h e r - o r d e r e x c i t a t i o n s we h a v e c a l c u l a t e d the self-energy o p e r a t o r o f all p o s s i b l e t h i r d - a n d o n e t y p i c a l f o u r t h - o r d e r g r a p h s (fig. 1). I n this case t h e h a r m o n i c o s c i l l a t o r w a v e f u n c t i o n s are u s e d to a v o i d the c o n t i n u u m e n e r g y integral. T h e s a m e set o f s i n g l e - p a r t i c l e e n e r g i e s s h o w n in t a b l e 1 is u s e d as in t h e case o f W o o d s - S a x o n w a v e f u n c t i o n . T h e I f o r b i t s a r e e x c l u d e d in this calc u l a t i o n . T h e results are s h o w n in tables 5 - 7 f o r v = s~r, p~ a n d p~ h o l e states, respectively. T h e c o n t r i b u t i o n o f t h e h i g h e r - o r d e r self-energy o p e r a t o r a l r e a d y b e c o m e s i m p o r t a n t for S = 20 M e V . T h e c a l c u l a t i o n o f the self-energy o p e r a t o r u p to s e c o n d o r d e r yields o n l y t h e l e a d i n g t e r m f o r S < 38 M e V .
4. Conclusions T h i s w o r k s h o w s t h a t the c a l c u l a t i o n o f the s p e c t r a l f u n c t i o n by e x p a n d i n g t h e self-energy o p e r a t o r i n t o a p e r t u r b a t i o n series u p to s e c o n d o r d e r is n o t reliable e v e n at s m a l l s e p a r a t i o n energies. T h i s a p p r o a c h has f o r e x a m p l e b e e n u s e d by G r o s s , L i p p e r h e i d e a n d W i l l e 1,2,3) a n d by E n g e l b r e c h t a n d W e i d e n m i J l l e r 15). I n o u r
528
A. FAESSLER et al.
paper the formulation of the self-energy operator has been extended up to thirdand even up to one fourth-order graph (see fig. 1). Tables 5-7 contain the real part of the mass operator for separation energies S = 10-50 MeV for ls~, lp~ and l p , proton pick-up, respectively. An inspection of these results suggests that the perturbation series is only converging for the separation energies below 20 MeV. The second-order contribution is already corrected by at least 40 % by the third-order contribution at 10 MeV separation energy. These results therefore demand a reformulation of the method for calculation of the spectral function. An approach not based on perturbation theory is needed. The authors appreciate discussions with Dr. P. Turek about the experimental data.
Appendix CALCULATION
OF THE
SELF-ENERGY
OPERATOR
The Hamiltonian of the total system is divided into two parts H = no+V,
(A.1)
where H o is the Hartree-Fock single-particle Hamiltonian n o = E g, :a+ al :,
(A.2)
1
and V is the residual two-body interaction V = ½ ~ (121V134) ".al+a2+a4a3 "
(A.3)
1234.
The two-body matrix elements are not antisymmetrized in eq. (A.3). Here the notation : : is the normal product in the Hartree-Fock representation. The operator a-~(a,) creates (annihilates) a particle in state 1 with respect to the pure vacuum (not in the sense of Hartree-Fock representation). Let us define operators J ~ and Jk by J~ = [V, a [ ] = ½ ~ : a+ a~(12lVAIka)a,:, (A.4) 124-
Jk = --[I,, ak] = ½ ~ : af(k21VA134)a4a3:,
(A.5)
234-
where the antisymmetrized matrix element is defined by (1211,~,134) = (121 V134)- (121 V143).
(A.6)
The self-energy operator of a given order can be calculated according to Villars' formula 6).
160(d, aHe) SPECTRUM
529
(i) The second order. The formula shows that Z~,2k)(W) = (0[J~-
1
W+Ho-iq
Jk,[O) + (0lJk,
I
W-Ho+i q
10),
(A.7)
where 10) is the Hartree-Fock ground state and q is the infinitesimal positive value. The first term ofeq. (A.7) is represented by fig. la. By using eqs. (A.2-5), eq. (A.7) is expressed in an algebraic form
E~2,~(W) = ½~ (12IVAIk3)(k'3[VA[12) +½ ~ (k'l]VAI23)(23[VAIkl) 123 WW83--sl--e2--ir/
(A.8)
1~,3 W[-el--82--e3"]-iq
Here 1 (1) means that the sum only goes over particle (hole) states in the HartreeFock representation. The sum over particle states also symbolises automatically the integral over the continuum single-particle energy with the corresponding quantum numbers from the threshold to infinity. The second term in eq. (A.8) shows the 2p-lh intermediate excitations, which are due to the ground-state correlations in the target nucleus. We shall neglect the second term, as is discussed in ref. 3). Then eq. (A.8) becomes
Z'(kZ')k(W)= ½6,~- ~s W +ea-eiJ[V(12k3j)12-e2-iq,
(A.9)
with the notation J = 2J+l, (A.10)
(~ = J ; l~jkJk' ~mkmk'"
We have used the expansion of the antisymmetrized matrix element in terms of those with definite angular momentum J (121 I,),134) = ~ (Jl rn, J2 m21JM)(j3 m3 J4 m,IJM)V(1234J). J
(A.I1)
(ii) The third order. Neglecting the ground-state correlation, the third-order selfenergy operator is divided into three parts si 2(w) = 13; )(w) +
)(w) +
)(w),
(A.12)
1 1 J~.[0), 2"~,3~)(W) = - ( 0 I V Ho J~ W + Ho-irl
,Y,(k3,~)(W)= --(0[J~
=
-
(olJ;
1
V
1
Jk' 1
W+Ho-iq W + Ho--iq
1
W+Ho-iq Ho VIO).
(A.13)
jk,[0) '
(A.14)
(A.15)
530
A. F A E S S L E R
et al.
Eq. (A.13) is the sum of two graphs, figs. lb and c, which are expressed respectively as
~,~a;b)(W) = _¼3 Z
~,'TV(1234J)V(k512J)V(34k5J)
t 2~g s --
X (133 +/3,--/3 t --/32)-1(W-4-/35--8 l --/32)-1,
Z(k3,~c)(W) = --6 Z
(A.16)
~" V(1245J)V(35k2J')V(k431J")
1234-5
JJ'J'"
X(/34-[-EE,--/31--/3,~)-1(W-.[-/34--133--1-~l)-l,~j'j"(--)Jt+Ja+J+J'+J"{j I
J5 • (A.17) Jk J4
Here { } denotes a 9-j symbol. Eq. (A.14) is the sum of three graphs given by figs. ld-f. Their explicit forms are 27(k3kZd)(w) = 6 y'
Z V(12k4J)V(3415J')V(k532J") 1237~ ss's"
x(W+/3-~-/3,-/3~)-~(w+.~-/3.--~)-~JJT"(-) j'÷j~'~
(A.18)
~, JV(12k5J)V(k534J)V(3412J)
~ 3 ~ , = _¼3 ~
1234-5- J ---
~(k3~f'= --6 Z
....
J~ ~'-, j~, ~,, ja J~
X (W"[-/35 --/3 l --/31)-1(1/1/'-~-/35 --/33 - - / 3 4 ) - t
~ V(12k4J)V(3425J')V(k513J")
t Z3~,5
/
J,l'J"
J~ x(14: +/34-/3~-/32)-~(w + / 3 5 - ~ t - / 3 3 ) - ' J J ' ] " ( - ) j~-i~÷J+J'' J3 J"
'}
J2 J' J4 . Js J~
(A.19)
(A.20)
We do not show the explicit form of eq. (A.15), because it is given by graphs similar to figs. lb and c. (iii) The fourth order. We shall take only a typical fourth-order graph in fig. lg which comes from
1
w+m-i,
(v
1
,2
w+Iio-i )
J ,lo>.
It is expressed by
~(kS~)(W) = ½0 Z y' V(12k6J)V(k652J)V(3417J') 12345~,~ ss" × I / ( 5 7 3 4 J ' ) ( W +/36 -/31 - / 3 2 ) - 1( W --[-/36- e2 - E5) - [ X ( W -~-/36 -{-/37 --/32 --/33 --/34)- 1J..~'[3 t .
(A.21)
160(d, aHe) S P E C T R U M
531
References 1) D. H. E. Gross and R. Lipperheide, Nucl. Phys. A150 (1970) 449 2) U. Wille, D. H. E. Gross and R. Lipperheide, Phys. Rev. C4 (1971) 1070 3) U. Wille and R. Lipperheide, Nucl. Phys. A189 (1972) 113; U. Wille, dissertation F U Berlin, 1971 4) R. H. Lemmer and C. M. Shakin, Ann. of Phys. 27 (1964) 13 5) R. D. Mattuck, A guide to Feynman diagrams in the many-body problem (McGraw-Hill, London, 1967) 6) F. Villars, Fundamentals in nuclear theory (IAEA, 1967) ch. 5 7) C. Mahaux and H. A. Weidenmiiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) p. 12 8) B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271 9) P. D. Kunz, University of Colorado, unpublished 10) B. Duelli, F. Hinterberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. Wagner, Phys. Lett. 23 (1966) 485 11) G. C. Ball and J. Cerny, Phys. Rev. 177 (1968) 1466 12) M. Sakai and K. Kubo, Nucl. Phys. A185 (1972) 219 13) D. Hartwig, dissertation, Univ. of Karlsruhe, 1971 14) H. Doubre et aL, Phys. Lett. 29B (1969) 355 15) C. A. Englebrecht and H. A. Weidenmiiller, Nucl. Phys. AI84 (1972) 385
2.G J
Nuclear Physics A203 (1973) 513--531; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE 160(d, 3He) EXCITATION SPECTRUM A M A N D FAESSLER and S. K U S U N O Institut flir Kernphysik der Kernforschungsanlage Jiilich GmbH, 517 Jiilich, West Germany
and G. L. STROBEL
University of Georgia, Georgia, USA Received 19 June 1972 (Revised 4 December 1972) Abstract: The t60(d, 3He) excitation spectrum has been calculated in the framework of the Gross-
Lipperheide model which explicitly takes into account the effect of residual interactions and the decay of the residual nucleus by particle emission. The reaction mechanism is described by the DWBA for a direct pick-up reaction. The theory is extended to include the higher-order effects of the residual interaction on the self-energy operator, in terms of which the spectral function is expressed. The real part of the self-energy operator is calculated up to the fourth order (2p-3h configurations) in the strength of the residual interaction. The calculation of the selfenergy operator up to the second order yields the leading term only for separation energy below 38 MeV. Higher-order effects have already become important for separation energies as low as 10 MeV. The Lemmer-Shakin potential is used for the residual interaction. The calculated spectrum exhibits a shell structure of groups of narrow peaks, which depends very much upon the choice of the absorption potential for the 3He optical parameters.
1. Introduction Nucleon hole states in nuclei have been studied through pick-up reactions such as (p, d), (n, d), (p, 2p) and (d, 3He) reactions. Because of the higher beam resolution possible, the last reaction is the most suitable for studying the highly excited proton hole states in nuclei. One can interpret peaks in the excitation spectrum as corresponding to a single-particle level structure in the nucleus. The identification of the angular momentum of such low excited states has been possible through a study of the reaction angular distribution using the distorted-wave Born approximation (DWBA) to describe the reaction mechanism. For higher excitations nucleonnucleon correlations in the target and final nuclei can destroy this simple picture. Inclusion of such correlations between nucleons in the nucleus via a residual twonucleon interaction results in a simple hole state being coupled to np-(n + 1)h states. As a result the expected excitation spectrum is more difficult to unravel experimentally or to calculate theoretically. We calculated an excitation spectrum for the (d, 3He) reaction. Particle decay of the residual nucleus into the continuum is taken into account. We apply the spectral function theory of Gross and Lipperheide t) which is formulated only up to the 513
514-
A. FAESSLER et
al.
second order of the residual interaction (lp-2h states in the final nucleus). It was applied by Wille, Gross and Lipperheide for the 12C(p, 2p) spectrum 2,3), and by Wille and Lipperheide for 160(p, 2p) spectrum 3). In this theory the one-nucleon pick-up cross section is expressed in terms of the spectral function of the target nucleus. We extend this theory in order to include 2p-3h excitations and describe the (d, 3He) reaction mechanism as a direct reaction using the DWBA. The spectroscopic factor, which is usually set to 2j+ 1 for the pick-up of nucleons out of a level with low excitation energy of angular momentum j, is calculated here through the use of the spectral function. The spectral function has a discrete and continuous part. The continuous part occurs for excitation energies where the residual nucleus can undergo subsequent particle decay. The residue of the discrete contribution to the spectral function is the "usual spectroscopic factor" associated with the bound states of the final nucleus 3). We consider the target to be a x60 nucleus described by the shell-model as having full ls~, lp~ and lp~ shells, and all other levels empty. We take as a residual interaction the Lemmer-Shakin potential *), which may be appropriate for this nucleus. With no residual interactions present, the (d, 3He) reaction on 160 would produce ls~, lp~, l p , hole states in the final 15N nucleus. The (d, 3He) spectral function would consist of three peaks corresponding to proton pick-up from one of the three filled shells of the target nucleus. The position of these peaks would permit a determination of the proton single-particle binding energies in the target nucleus. With a residual interaction included, other configurations are coupled to single-hole states. The effect of the residual interaction appears through the self-energy operator of the nucleus by which the spectral function is expressed. The self-energy operator can be calculated by the technique of the partial summation of many-body diagrams 5). We include all possible second- and third-order diagrams and a typical fourth order one in the calculation of the self-energy operator. This means that all possible lp-2h and several types of 2p-3h configurations that can couple to the single-hole states are included. In order to see the effect of the residual interaction, let us consider only the lp-2h configurations coupled to single-hole states for simplicity, For weak enough residual interaction the energies of the lp-2h configurations will not change appreciably when coupled to single-hole and other lp-2h configurations. We then expect to observe aH at final-state energies corresponding to an energy transfer to the final nucleus equal to the excitation energy of one of the single-hole and lp-2h configurations. Now instead of one peak resulting from the pick-up of a 1s~ proton, twenty peaks can be expected in the spectral function. This is so because we include the nineteen different lp-2h configurations that can couple to ls~ single-hole configurations. The spectrum resulting from lp~_ or lp½ proton pick-up will be complicated by a similar amount due to lp-2h configurations coupled to the corresponding single-hole configuration. The calculated spectral function has peaks at energies near the lp-2h
l~O(d, 3He) SPECTRUM
515
configuration energies, but is a continuous function of energy, where the residual nucleus can decay via a particle emission into the continuum. If we allow the higher-order excitations further, the excitation spectrum is expected to become so complicated that the sharp peaks corresponding to lp-2h excitations may be smoothed out. We have calculated the real part of the self-energy operator up to the fourth order in the residual interaction. The perturbation expansion of the self-energy operator shows only fair convergence for separation energies below 20 MeV. The imaginary part and the cross section for the 160(d, 3He ) reaction was calculated with a self-energy operator obtained up to the second order only. The ground-state correlations are neglected. In sect. 2 we present the theoretical details of the calculation of the excitation spectrum, and in sect. 3 we discuss the results.
2. Theoretical details of the (d, 3He) excitation spectra We follow the theory of Gross, Lipperheide and Wille (GLW) t) developed for (p, 2p) and (d, 3He) reactions. The momentum-energy spectral function of G L W is assumed expanded in shell-model states as
S(q, q'; W) = E qgv(q)~Pv'(q')Svv'(W),
(1)
where v denotes the quantum numbers of a shell-model hole state, and ~Pv(q) is the momentum-space representation of the bound-state wave function of such a hole state. Eq. (1) covers the case where total angular momentum and parity do not totally distinguish a bound state, such as for nuclei where the 2s state is occupied at least partially, in addition to the more tightly bound Is state. We assume total angular momentum and parity do completely specify all bound states for an t 6 0 target, in which case eq. (1) becomes
S(q, q'; W) = ~ ~pv(q)q~v(q')Sv(W).
(2)
v
Such a description ignores any nucleons occupying the 2s~ shell-model state in the target nucleus 160. Such an event could result from the correlations in the target nucleus but are ignored here. We take the ls½, lp~ and lp~. levels as completely full, and all other target levels as completely empty. Here I4/ is the energy removed from the nucleus, so S = - 141 is the energy given to the final nucleus. We shall call S the "separation energy' for later discussions. The quantity IV is related to various c.m. kinetic energies (E) or positive binding energies (B) by
W = E,--Ed+Bd-B~,
(3)
where the d or z subscript refers to the initial deuteron or final 3He projectiles. Eq. (2) assumes that the momentum distribution for a given state is independent of the energy transfer. The DWBA cross section for detecting 3He particles with an
A. FAESSLER ¢t al.
516 energy E is given by d17
E IL(w)I2S,(W),
-
dt2dE
(4)
where fv(W) is the usual D W B A amplitude for a deuteron pick-up of a proton from the state v, with the final aHe leaving with an energy related to W via eq. (3). The spectral function S~(W) is the probability that when the proton, in the state v, is picked up, an energy transfer of S to the final nucleus will result. The spectral function S,(IV) is related to the single-particle Green function t) by
S,¢(W) = lim 1 _ {G~¢(W-bl)-Gv¢(W + bl)). ,~o 2hi
(5)
The Green function G is related to the self-energy operator 2: via the Dyson equation 5) by Z ((z- e,)5,~,-- Z,,.~,(z)}G~,¢(z)= 6,~,, (6) /t
or with
Z~,(z) = 6v,~,(_~),
(7)
one can solve for the Green function to obtain
G.~(O = 6v,(z-
~,-
~v(z))-'
(8)
Here ev is the energy of the single-hole state v. The self-energy operator Z' is composed of all possible orders of the (irreducible) self-energy diagrams. The first-order
k
~¢k 1
....
'1
(Q]
(b)
2
(c)
(d) ~k
1
2
2
4
!}iiii ..... _ y (e)
If)
[g)
Fig. 1. The second-, third- and. one typical fourth-order self-energy diagrams. Each graph a-g corresponds to eqs. (A.9), (A.16), (A.17), (A.18), (A.19), (A.20) and (A.21), respectively.
160(d, aHe) SPECTRUM
517
contribution (with respect to the strength of the residual interaction) corresponds to the Hartree-Fock (bubble and open oyster) diagram, which is reduced to the HartreeFock single-particle energy in the Dyson equation, as long as the single-particle potential is chosen self-consistently 5). This condition is assumed to be fulfilled in the following calculation 3): The second- and higher-order contributions to the self-energy operator can be evaluated according to the method proposed by Villars 6). The results are shown in the appendix (fig. 1). Now the self-energy operator is the sum of contributions of each order
a,(z) = cr~2)(z) + a~3)(z) + . . . .
(9)
It should be noted that the sum over particle states in the expression for the selfenergy operator in the appendix symbolizes automatically integrations over the continuum states also. For example the expression for a(~2)(W) is
I(vklVAIlm)12 --½ lim ~
a~2)(W) = ½ ~, l, ra
ek--Etm
F
f ~ risk I(vklVAllm)l 2 , Etm+iq-e k
(10)
r/--*0 l,m
where
E~m = *l+~m-- W.
(11)
The summations in eq. (10) are over all the quantum numbers indicated by /, m such that the energy of the hole state is indeed less than the Fermi level. The particle states k in the first sum are over the bound states that lie above the Fermi level. In the second sum the particle states k are summed over all continuum states. The angular momentum selection rule restricts the angular momenta of the continuum states to { or less. For the residual interaction V, we utilize a delta-function residual interaction first used by Lemmer and Shakin 4) V(r x - -
r2)
(a + bal" a2)f(f21 - •2)6(r 1
=
-
-
r2)/r
(12)
2 .
The value used for a is - 5 0 3 . 7 MeV- fm a and for bis - 7 8 . 6 MeV- fm s. The matrix element of this interaction is also given in ref. 4). The principal value integral ofeq. (10) is done by the following substitution 7) u kr ( r )eu kc( ,)
Qo
lim |
n-.o j o
dek
a
B
,
- gtk+)(r, r', Etm) - ~u~(r)u~(r').
Elm + i~l - ek
.
(13)
Elm -- ekn
Here u~(u~) denotes the continuum (bound) state radial wave function with quantum number k which is a solution of the Schr6dinger equation d2u
t 2M E + i
V (
l(l+ 1)}
O.
(14)
" =
The sum over n denotes the various bound states for a given parity and total angular momentum specified by k. In general there can be more than one such state. For a 6 0 target, there are either one or zero such bound states depending on the parity
518
A. FAESSLER et al.
and total angular momentum. Now the Green function 0~+) (r, r', Etm) is given by gtg+)(r' ,.,, Elm) _ 2 M uk(r<)6k(r> ) h~
W(uk,6~)
(15) '
where r< (r>) is the smaller (larger) o f r and r'. The quantity W ( u , 6) is the Wronskian between u and C. The wave function u(r) is the regular solution (at the origin), and C(r) is a solution that satisfies outgoing boundary conditions, of eq. (14). We note that 9kt+)(r, r', E~m) is real if the energy E~m is negative, and that E~m is not in general the eigenvalue for the state k. If one substitutes eqs. (A.6) and (13)into the last term of eq. (10), and interchanges the order of integration over energy with coordinates, one encounters, with a finite-range residual interaction, a four-dimensional radial integral. The alternative to this four-dimensional integral is to do the principal value integral in eqs. (10). Instead of this, we assume a delta-function residual interaction 4) whereupon we obtain a two-dimensional radial integral of the form IiV,k = J J t" d, • ,., dr ,uvQ .)ut(r)um(r)ut(t ., )um(t ., )uv(t J )gk(+) (r, r ,, Etm).
(16)
This integral is done numerically using a Simpson quadrature. The subscript k on yg(+)"it', r ' , Etm ) is to emphasize that it depends on the quantum numbers of the particle k. The potential Vws of eq. (14) used to generate the Green function and the boundstate wave functions is of the form Vws = V c ( , . ) _ V o f ( r o , a ) + 2
(+)2
V~.o.(l d f(rs .... as.o.), r • s) drr
(17)
where f ( r , a) = (1 +eX) - ',
(18)
x = ( r - r o ( A - 1)~)/a.
(19)
Here A is the mass number of the target nucleus. The single-particle energies used are listed in table 1. These energies and Woods-Saxon potential parameters were taken from Buck and Hill 8). The same set of parameters has also been used for 160 in ref. 3). The ls~ binding energy was obtained through use of the same set of WoodsSaxon potential parameters that fit the 2s~ binding energy. A Woods-Saxon radius and diffuseness of r~.o. = r o = 1.25 and a .... = a = 0.53 fm were used throughout. The proton bound states included a Coulomb potential Vc(r ) due to a uniformly charged sphere of radius equal to the Woods-Saxon radius. The potential depth Vo was adjusted for the d~ state to produce a resonance at the resonance energy (see table 1). The potential parameters for the If states were taken to be the average of those states with lower angular momenta. Wille and Lipperheide 3) calculated the Hartree-Fock single-particle energies from the two-body interaction (12) and
160(d, aHe) S P E C T R U M
519
TABLE 1 ~60 single-particle energies a n d W o o d s - S a x o n potential parameters
nlj
charge (e)
I s½
1
lp½ lp. t_ ld~ l d 1. lfk lf.I. 2s~ Is½ lp~ lP½ ld½ ld~ I f_~ lf~ 2s.t_
1 1 l l 1 1 1 0 0 0 0 0 0 0 0
rv (MeV)
Iio (MeV)
Vs.°. (MeV)
27.63 --18.45 -- 12.13 -- 0.60 + 4.27
47.09 52.52 52.52 47.44 46.80 49.0 49.0 47.09 56.97 57.42 57.45 54.99 54.92 56.0 56.0 56.97
0 9.88 9.88 5.24 5.24 7.56 7.56 0 0 9.66 9.66 5.24 5.24 7.45 7.45 0
--
-- 0.10 --35.91 --21.83 --15.63 -- 4.14 -- 0.94
-- 3.27
compared them with the Woods-Saxon single-particle energies of table 1. The selfconsistency condition was found rather well fulfilled. The final expression obtained for the self-energy operator a~v2) is <2,(w)
-
1 32rc 2
+ + + 6~+},
E A{(C+D)(F+~+,+F+~+)+2(C-D)F+
(20)
klm¢
where z denotes the isospin z-component of the particle and the sum over l, m (k) denotes the sum over parities and angular momentum for the hole (particle) states. The quantities C and D are related to the parameters of the residual potential by C = (a-3b)2No,
(21)
D = (3a+3b)2N~,
(22)
where N~ is a sum of the products of Clebsch-Gordan coefficients and 9-j symbols,
=
:
½
'k L)t,.
,.,
(23)
The quantity F is a factor resulting from the radial integrals of eqs. (10) and (A.5). The subscripts of F denote the charge state for nucleons vklm (in this order) in the expression F~.kU. =
t, v ( r ) u k ( r ) u l ( r ) u m ( r ) r 2 d r
Etm -- ek) - 1 _ i , m~.k.
(24)
The sum over T in eq. (20) is carried over the terms r = + (proton) and z = -
520
A. FAESSLER
et aL
TABLE 2 160 lp-2h configurations for s½(P) hole pick-up No.
l
m
k
S (MeV)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Pf(P)
P~r(P)
p~(p) p~(p) p½(p) Pt-(P) p,t.(p) p½(p) p~(p) pt_(p) Pt-(P) p~(p) p~(p) pt.(p) p.l_(p) p/_(p)
p~r(n) P½(n) p~.(n) Pg-(P) p~_(n) p~(n) P½(n) p~(p) p~(n) p~(n) p~}(p) P'I-(P) Pz2(P) p~(n)
2st-CP) 2sg(n) d.t.(n) ¢ d½(n) dl-(P) d.t.(n) 2s.t.(n) 2s~(n) d~(p) ¢ d?~(n): dt.(n) ~ d~(n) d~(p) 2s~(p) 2s½(n)
24.16 24.53 28.74 29.82 29.98 29.98 30.69 30.85 34.88 34.90 35.06 36.14 36.30 36.80 37.01
16
p~(p)
p~(p)
d~(p)~
41.20
17 18 19
P-t-(P) s~iP) s~(p)
p.}(n) s~_(p) s.t_(n)
d~(n) c 2s~(p) 2s÷(n)
41.22 55.16 60.27
TABLE 3 160 lp-2h configurations for p proton pick-up No.
I
m
k
S (MeV)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
s~_(p) s½(p) s~(p) s½(p) p~(p ~ p~(p ) s½(P) st_(p) s~(p) pt.(p) s x(p) st~(p) s½(p) P-t-(P) p.~(p) s~(p) s½(P) p{_(p)
p,~(Pl pt_(n) p~(p) p~(n) s~(n ) s.t_(n ) Pcz(P) pk(n ) p.t_(n) s~(n) P-I-(P) p~_(p) p~(n) s~(n) s½(n) p.l.lp) pg_(n) s½(n)
d~r(P) dl.(n) 2s½(P) 2s~(n) di.(n) 2s.i_(n) d~(P) c d~(n) dl_(n) d½(n) ¢ d~(p) 2st_(p) 2s½(n) d.t.(n) 2s½(n) d.l.(p )¢ d.~(n) c dt(n)C
39.16 39.16 39.66 40.03 43.90 44.77 44.03 44.24 45.32 48.98 45.48 45.98 46.19 50.22 51.09 50.35 50.40 55.30
Configurations with the sign ; are excluded for the p~(p) hole case by the angular momentum selection rule.
leO(d, 3He) SPECTRUM
521
A = 1,1~2~1,, 3k 3~ 3,,.
(25)
(neutron). The quantity A is The lp(k)-2h(/, m) configurations possible and the energy transfer needed to excite such a configuration are listed in tables 2 and 3. The optical-model potential used to generate the scattering wave function is (h
~2
VDw = Vc(r)--Vof(ro,a)+ ~c~c) V~'°'(l'S)r
.
,
,
_~xf r~.o.,a~.o.)+,Wof(ro,ao) +4iWt, ~d, f(r'o, a').
(26)
The DWBA cross section was calculated in a nearly standard manner using the code D W U C K 9). The only difference was that the binding energy of the picked-up proton, an input parameter for DWUCK, was not related to the reaction Q-value as the final projectile energy is specified independently. The DWBA parameters for d -160 scattering are taken from ref. xo), where Ea = 52 MeV. Those for 3He-~ 5N scattering are not so well-known over a wide range of 3He energy. Those values corresponding to E, = 39.8 MeV [ref. ~1)] are used. They are listed in table 4. TABLE 4 Optical parameters for the reaction 160(d, 3He)t 5N used in this calculation Vo (MeV) d+160
Wo (MeV)
88.9
4 Wo Vs.o. a (MeV)(MeV)(fm) 10.5
a" (fro)
as... (fm)
ro (fm)
r'o (fro)
rs... (fm)
rc (fm)
Ref.
11.275 0.792 0.792 0.792
1.05
1.36
1.05
1.3
lo)
0.595 0.858
1.23
1.80
1.3
11)
--12.44 3HeWlSN
160.0
(-40.0)
One gets both continuous and discrete contributions to the spectral function from eq. (5). The continuous contribution to the spectral function in terms of self-energy operator trv is
S~(W) =
-1 Im trv(W) rc ( W - e ~ - R e a~(W))2+(Im a02"
The discrete contribution occurs at those energies
W,,,-,~-a,,(W.,)
W~, where
(27)
W satisfies
= 0.
(28)
The discrete contribution to the spectral function at these energies is
s~i .... t°(w) = Z ,~(w-w,,,) / ( 1+ da~(W)]dW 'w=w~,
(29)
The rate of change of the self-energy operator a~ with W is always negative, so the discrete contribution to the spectral function is always less than one.
522
A. FAESSLER
e t al.
3. Discussion Let us consider first the calculation up to second order in eq. (9). Fig. 2 (3) shows the real and imaginary part of the self-energy operator t~ 2) for a proton pick-up from a ls½ (lp~) state v e r s u s the separation energy S. The peaks in I m a~,2) at energy transfer around 29, 35 and 41 MeV (44, 49, 50 and 55 MeV) for s(p) wave pick-up, are related to the lp-2h configurations in which the particle is in the continuum state ld~. The structure of the self-energy operator for proton pick-up from the lp~ state (not shown) is similar to that of the lp~ mass operator, except for the absence of three poles in the real part of o-c2),,. The missing poles are absent due to the angular momentum selection rule which prevents three configurations, labeled x in table 3, from contributing to the mass operator of the lP½ state. These configurations do contribute to the lp~ self-energy operator however. The various poles of Re ~(2) o~ occur near energies which can also be correlated to the lp-2h configuration energies. Intersections of the straight line in figs. 2 and 3 with 200 O~'{d, He 3) N Is v = S 7/~
t=
/
~c
~o
• 91 2
• 6/-
:g
t~
3O
StN~ Fig. 2. The real and imaginary part of the second-order self-energy operator for the v = s~_proton hole m the t60 nucleus c e r s u s the separation energy S. The line corresponding to eq. (28) is also shown.
l~O(d, aHe) SPECTRUM
523
016(d. He 3 ) N '5 v =P3/~ 106
"~5c
0
i
20
30
40
50
60
~7o
S
30
tD
5 - 2G
E tO
i 2O
I
30
L
40
i
h
50
i
60
70
S (M~Vl
Fig. 3. The same as fig. 2 for the v = P-t-proton hole in the 160 nucleus. Re tr(~2) determine the energies where the m a x i m a of the spectral function occur. The m a x i m a o f the spectral function, as seen in eq. (27), are S v = (zr Im a(~2))-1
(30)
I f Re ov-(2) were independent o f W = - S, the width of the peaks would be given by I m or-(2). Then we would expect peak widths of one MeV or more for peaks at S greater than a b o u t 30 MeV. However, in fact, Re O_(2) varies very rapidly with W v wherever the spectral function has a m a x i m u m . In fact the rate o f change of Re a(~2) with W near most m a x i m a is so large that this consideration determines the width. I f we write Re t r t 2 ) ( W ) = Re trio2)+ r d-dT Re a(2)l Low d
AW,
(31)
Wo
where W o is chosen so that W o - e = a(o2), we can see why the calculated peaks have such a n a r r o w width. Let the subscript 0 denote evaluation at an energy
524
A. F A E S S L E R et aL
corresponding to a peak in the spectral function. Then, near such an energy eq. (27) can now be rewritten as 1 Im a~o2) Sv = n ([d Re a(2'/dW]wo A W) 2 + ( I m a(2)) 2"
(32)
The half-width at half maximum of the spectral function will then be Im a(o2) A W~ = [d Re a(2)/dW]wo "
(33)
This says that the half-width is independent of the residual interaction strength chosen. This is much less than other estimates 12) and much less than what is expected by crudely setting the half-width equal to I m a ~z). As the energies of lp-2h configurations exhibit a shell structure and Re a C2) can be correlated to these energies, we can see in the spectral function, figs. 2 and 3, this same shell structure. Namely, at energies where you have many lp-2h configurations, we calculate many corresponding peaks in the spectral function. Fig. 4 shows the excitation spectrum for the 160(d, 3He ) reaction with E a = 52 MeV which is calculated by using DWBA and the second-order self-energies o_(2) r . Where the discrete contributions to the spectral function occur depends somewhat on the residual interaction strength, but the position of the continuum peak contribution is nearly independent of the residual interaction strength. Although the optical parameters for the final channel depend upon the energy of the 3He particle, we have fixed those values derived from the E~ = 39.8 MeV analysis it), because there are very few data available. In order to see the effect of the optical-potential parameters we have also calculated the excitation functions only by increasing the absolute value of the volume absorption potential W o to - 4 0 MeV. The result is shown in fig. 5. The other parameters remain exactly the same as in fig. 4. As a matter of fact, the excitation is reduced on account of the rather strong volume absorption. The experimental data which can give information about highly excited lp-2h states are not so numerous. Figs. 4 and 5 contain the experimental data taken from ref. 13). The experimental data (E d = 52 MeV) are only available up to S ~ 24 MeV at present. We could not get peaks corresponding to (b), (d) and (e) in figs. 4 and 5. This is because we have neglected the ground-state correlation (the second term in eq. (A.8)). This term becomes effective for small values of separation energy S. We have also calculated the excitation spectrum corresponding to Ea = 80 MeV. It is shown in fig. 6. There is only one experiment at E d = 82 MeV [ref. 14)]. But only three peaks which correspond to (a), (b) and (c) of fig. 4 are reported. Therefore we do not try to compare our results at Ed = 80 MeV with experiment. Fig. 6 is different from fig. 4 in the appearance of highly excited peaks at S > 50 MeV. We have used the same optical parameters in fig. 6 as in fig. 4, because such sets of parameters are not so well known apart from those reported in ref. 14).
160(d, 3He) o
0
SPECTRUM
2p
19
525
Ex(MeV)
3p
P#2
160(d, 3He}15N
(c)
Ed=52 HeY E)Lob=13.0" W,=-12. LJ-, MeV
.-x40
x40
P
(b) I o,_
10
x40
15
20
P 30
25
t.o
35
t~5
50
S(MeV)
Fig. 4. The t60(d, aHe) excitation spectrum (solid line) for Ea = 52 MoV. The second-order selfenergy operator is used in this calculation. The depth o f the volume absorption potential for 3He-15 N scattering is --12.44 MeV (table 4). The quantities below S = 25 MeV should be enlarged 40 times. The experimental data (dotted line) are taken from ref. 13) and are normalized to the theoretical p÷ peak at S = 14 MeV. The experimental spectrum is shifted also by 4.45 MeV to the left so that the p~. separation energy agrees with the theory. The upper energy scale gives the excitation energy in t SN.
0
o
p~
1.0
2,0
160( d, 3He)15N Ed =52 MeV OLob=l 3.0" W.='-/.O.O MeV
P~ (c)
(o)
Ex(MeV}
2p
x20
o
I
x2°jil 0
1"0
!J 15
^ 2"0
25
30
p~ p" s
35
p
\
~0
z.5
50
S(MeV)
Fig. 5. The same as fig. 4 for the volume absorption potential We = --40 MeV. The quantities below S = 25 MeV should be enlarged 20 times. T h e experimental spectrum is shifted as in fig. 4.
526
A. FAESSLER
e t al.
160(d, 3He)lSN Ed--8OMeV OLob=13.0" W.=-12.44 MeV
o
i1 5
10
$
15
20
25
30
P
35
40
45
50
S(M,eV)
55
60
F i g . 6. T h e z 6 0 ( d , 314e) e x c i t a t i o n s p e c t r u m f o r Ea ~ 80 M e V . T h e s e c o n d - o r d e r s e l f - e n e r g y o p e r a t o r is u s e d in t h i s c a l c u l a t i o n . T h e s a m e o p t i c a l - p o t e n t i a l p a r a m e t e r s as f o r t h e E a = 52 M e V case are used also.
The main effect of using distorted waves instead of plane waves for the scattering wave functions is to reduce the excitation spectra for large values of S. Contributions to the excitation spectra resulting from proton pick-up from the ls~ shell is inhibited in comparison to pick-up from the lp~ or l p , shells. This is true for all scattering angles between 10° and 90 ° and is not restricted to just the angle 13° for which the spectrum is calculated (figs. 4-6). The most time-consuming part of this work is to calculate the single-particle Green function #Ok+)(,", r', Ezra), and subsequently F (see eqs. (15), (16) and (24)). This calculation must be repeated for each value of If', and hence each Ezra. TABLE 5
V a l u e s o f t h e s e l f - e n e r g y o p e r a t o r s v e r s u s the s e p a r a t i o n e n e r g y S = - - 1 4 / f o r t h e k = s~ p r o t o n s t a t e in t h e ~ 6 0 n u c l e u s S
a
b
10 16 22 28 34 40 46 50
1.252 1.430 1.666 1.997 2.494 3.328 5.040 7.854
0.015 0.017 0.020 0.024 0.031 0.044 0.072 0.128
c 0.039 0.052 0.073 0.108 0.178 0.348 0.952 3.000
d 0.586 0.670 0.782 0.940 1.179 1.583 2.431 3.872
e ---------
0.449 0.587 0.800 1.155 1.813 3.267 7.694 19.580
f
g
0.552 0.722 0.983 1.419 2.229 4.016 9.458 24.068
0.025 0.033 0.046 0.066 0.105 0.191 0.455 1.191
T h e letter, a - g c o r r e s p o n d t o t h e r e s p e c t i v e d i a g r a m s s h o w n in fig. 1. T h e y a r e i n M e V .
160(d, aHe) SPECTRUM
Values of the self-energy operators
527
TABLE 6 the separation energy S = -- W for the k = pk proton state in the 160 nucleus
versus
S
a
c
d
10 16 22 28 34 40 46 50
1.478 1.770 2.206 2.935 4.440 -- 4.339 --70.319 44.097
0.028 0.040 0.064 0.117 0.288 3.536 4897.9 45.054
0.100 0.118 0.146 0.189 0.272 0.499 --16.982 4.537
e 0.019 0.028 0.046 0.088 0.216 --6370.7 --2693.8 237.27
f
g
0.493 0.709 1.111 1.997 4.749 4537.1 2129.l 1517.2
0.034 0.050 0.079 0.145 0.355 1313.7 2016.4 191.31
The letters a-g correspond to the respective diagrams shown in fig. 1. They are in MeV. Diagram b is not shown in the table because it vanishes on account of the parity selection rule. Values of the self-energy operators S
a
10 16 22 28 34 40 46 50
1.926 2.305 2.873 3.82l 5.7643 --75.275 --63.986 74.025
versus
TABLE 7 the separation energy S = -- W for the k = p~ proton state in the ~ O nucleus
c
d
e
0.037 0.054 0.083 0.156 0.385 14.690 1056.9 36.115
0.025 0.313 0.041 0.060 0.113 --16.317 -- 0.315 -- 0.110
0.006 0.091 0.015 0.026 0.050 --5460.9 2970.9 --164.24
f 0.605 0.872 1.365 2.455 5.863 79794.0 3416.1 2557.5
g 0.041 0.059 0.093 0.169 0.400 4565.2 796.88 302.28
See table 6 for further explanation. I n o r d e r to see t h e effect o f the h i g h e r - o r d e r e x c i t a t i o n s we h a v e c a l c u l a t e d the self-energy o p e r a t o r o f all p o s s i b l e t h i r d - a n d o n e t y p i c a l f o u r t h - o r d e r g r a p h s (fig. 1). I n this case t h e h a r m o n i c o s c i l l a t o r w a v e f u n c t i o n s are u s e d to a v o i d the c o n t i n u u m e n e r g y integral. T h e s a m e set o f s i n g l e - p a r t i c l e e n e r g i e s s h o w n in t a b l e 1 is u s e d as in t h e case o f W o o d s - S a x o n w a v e f u n c t i o n . T h e I f o r b i t s a r e e x c l u d e d in this calc u l a t i o n . T h e results are s h o w n in tables 5 - 7 f o r v = s~r, p~ a n d p~ h o l e states, respectively. T h e c o n t r i b u t i o n o f t h e h i g h e r - o r d e r self-energy o p e r a t o r a l r e a d y b e c o m e s i m p o r t a n t for S = 20 M e V . T h e c a l c u l a t i o n o f the self-energy o p e r a t o r u p to s e c o n d o r d e r yields o n l y t h e l e a d i n g t e r m f o r S < 38 M e V .
4. Conclusions T h i s w o r k s h o w s t h a t the c a l c u l a t i o n o f the s p e c t r a l f u n c t i o n by e x p a n d i n g t h e self-energy o p e r a t o r i n t o a p e r t u r b a t i o n series u p to s e c o n d o r d e r is n o t reliable e v e n at s m a l l s e p a r a t i o n energies. T h i s a p p r o a c h has f o r e x a m p l e b e e n u s e d by G r o s s , L i p p e r h e i d e a n d W i l l e 1,2,3) a n d by E n g e l b r e c h t a n d W e i d e n m i J l l e r 15). I n o u r
528
A. FAESSLER et al.
paper the formulation of the self-energy operator has been extended up to thirdand even up to one fourth-order graph (see fig. 1). Tables 5-7 contain the real part of the mass operator for separation energies S = 10-50 MeV for ls~, lp~ and l p , proton pick-up, respectively. An inspection of these results suggests that the perturbation series is only converging for the separation energies below 20 MeV. The second-order contribution is already corrected by at least 40 % by the third-order contribution at 10 MeV separation energy. These results therefore demand a reformulation of the method for calculation of the spectral function. An approach not based on perturbation theory is needed. The authors appreciate discussions with Dr. P. Turek about the experimental data.
Appendix CALCULATION
OF THE
SELF-ENERGY
OPERATOR
The Hamiltonian of the total system is divided into two parts H = no+V,
(A.1)
where H o is the Hartree-Fock single-particle Hamiltonian n o = E g, :a+ al :,
(A.2)
1
and V is the residual two-body interaction V = ½ ~ (121V134) ".al+a2+a4a3 "
(A.3)
1234.
The two-body matrix elements are not antisymmetrized in eq. (A.3). Here the notation : : is the normal product in the Hartree-Fock representation. The operator a-~(a,) creates (annihilates) a particle in state 1 with respect to the pure vacuum (not in the sense of Hartree-Fock representation). Let us define operators J ~ and Jk by J~ = [V, a [ ] = ½ ~ : a+ a~(12lVAIka)a,:, (A.4) 124-
Jk = --[I,, ak] = ½ ~ : af(k21VA134)a4a3:,
(A.5)
234-
where the antisymmetrized matrix element is defined by (1211,~,134) = (121 V134)- (121 V143).
(A.6)
The self-energy operator of a given order can be calculated according to Villars' formula 6).
160(d, aHe) SPECTRUM
529
(i) The second order. The formula shows that Z~,2k)(W) = (0[J~-
1
W+Ho-iq
Jk,[O) + (0lJk,
I
W-Ho+i q
10),
(A.7)
where 10) is the Hartree-Fock ground state and q is the infinitesimal positive value. The first term ofeq. (A.7) is represented by fig. la. By using eqs. (A.2-5), eq. (A.7) is expressed in an algebraic form
E~2,~(W) = ½~ (12IVAIk3)(k'3[VA[12) +½ ~ (k'l]VAI23)(23[VAIkl) 123 WW83--sl--e2--ir/
(A.8)
1~,3 W[-el--82--e3"]-iq
Here 1 (1) means that the sum only goes over particle (hole) states in the HartreeFock representation. The sum over particle states also symbolises automatically the integral over the continuum single-particle energy with the corresponding quantum numbers from the threshold to infinity. The second term in eq. (A.8) shows the 2p-lh intermediate excitations, which are due to the ground-state correlations in the target nucleus. We shall neglect the second term, as is discussed in ref. 3). Then eq. (A.8) becomes
Z'(kZ')k(W)= ½6,~- ~s W +ea-eiJ[V(12k3j)12-e2-iq,
(A.9)
with the notation J = 2J+l, (A.10)
(~ = J ; l~jkJk' ~mkmk'"
We have used the expansion of the antisymmetrized matrix element in terms of those with definite angular momentum J (121 I,),134) = ~ (Jl rn, J2 m21JM)(j3 m3 J4 m,IJM)V(1234J). J
(A.I1)
(ii) The third order. Neglecting the ground-state correlation, the third-order selfenergy operator is divided into three parts si 2(w) = 13; )(w) +
)(w) +
)(w),
(A.12)
1 1 J~.[0), 2"~,3~)(W) = - ( 0 I V Ho J~ W + Ho-irl
,Y,(k3,~)(W)= --(0[J~
=
-
(olJ;
1
V
1
Jk' 1
W+Ho-iq W + Ho--iq
1
W+Ho-iq Ho VIO).
(A.13)
jk,[0) '
(A.14)
(A.15)
530
A. F A E S S L E R
et al.
Eq. (A.13) is the sum of two graphs, figs. lb and c, which are expressed respectively as
~,~a;b)(W) = _¼3 Z
~,'TV(1234J)V(k512J)V(34k5J)
t 2~g s --
X (133 +/3,--/3 t --/32)-1(W-4-/35--8 l --/32)-1,
Z(k3,~c)(W) = --6 Z
(A.16)
~" V(1245J)V(35k2J')V(k431J")
1234-5
JJ'J'"
X(/34-[-EE,--/31--/3,~)-1(W-.[-/34--133--1-~l)-l,~j'j"(--)Jt+Ja+J+J'+J"{j I
J5 • (A.17) Jk J4
Here { } denotes a 9-j symbol. Eq. (A.14) is the sum of three graphs given by figs. ld-f. Their explicit forms are 27(k3kZd)(w) = 6 y'
Z V(12k4J)V(3415J')V(k532J") 1237~ ss's"
x(W+/3-~-/3,-/3~)-~(w+.~-/3.--~)-~JJT"(-) j'÷j~'~
(A.18)
~, JV(12k5J)V(k534J)V(3412J)
~ 3 ~ , = _¼3 ~
1234-5- J ---
~(k3~f'= --6 Z
....
J~ ~'-, j~, ~,, ja J~
X (W"[-/35 --/3 l --/31)-1(1/1/'-~-/35 --/33 - - / 3 4 ) - t
~ V(12k4J)V(3425J')V(k513J")
t Z3~,5
/
J,l'J"
J~ x(14: +/34-/3~-/32)-~(w + / 3 5 - ~ t - / 3 3 ) - ' J J ' ] " ( - ) j~-i~÷J+J'' J3 J"
'}
J2 J' J4 . Js J~
(A.19)
(A.20)
We do not show the explicit form of eq. (A.15), because it is given by graphs similar to figs. lb and c. (iii) The fourth order. We shall take only a typical fourth-order graph in fig. lg which comes from
1
w+m-i,
(v
1
,2
w+Iio-i )
J ,lo>.
It is expressed by
~(kS~)(W) = ½0 Z y' V(12k6J)V(k652J)V(3417J') 12345~,~ ss" × I / ( 5 7 3 4 J ' ) ( W +/36 -/31 - / 3 2 ) - 1( W --[-/36- e2 - E5) - [ X ( W -~-/36 -{-/37 --/32 --/33 --/34)- 1J..~'[3 t .
(A.21)
160(d, aHe) S P E C T R U M
531
References 1) D. H. E. Gross and R. Lipperheide, Nucl. Phys. A150 (1970) 449 2) U. Wille, D. H. E. Gross and R. Lipperheide, Phys. Rev. C4 (1971) 1070 3) U. Wille and R. Lipperheide, Nucl. Phys. A189 (1972) 113; U. Wille, dissertation F U Berlin, 1971 4) R. H. Lemmer and C. M. Shakin, Ann. of Phys. 27 (1964) 13 5) R. D. Mattuck, A guide to Feynman diagrams in the many-body problem (McGraw-Hill, London, 1967) 6) F. Villars, Fundamentals in nuclear theory (IAEA, 1967) ch. 5 7) C. Mahaux and H. A. Weidenmiiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) p. 12 8) B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271 9) P. D. Kunz, University of Colorado, unpublished 10) B. Duelli, F. Hinterberger, G. Mairle, U. Schmidt-Rohr, P. Turek and G. Wagner, Phys. Lett. 23 (1966) 485 11) G. C. Ball and J. Cerny, Phys. Rev. 177 (1968) 1466 12) M. Sakai and K. Kubo, Nucl. Phys. A185 (1972) 219 13) D. Hartwig, dissertation, Univ. of Karlsruhe, 1971 14) H. Doubre et aL, Phys. Lett. 29B (1969) 355 15) C. A. Englebrecht and H. A. Weidenmiiller, Nucl. Phys. AI84 (1972) 385