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Test of correlation expansions for inelastic scattering of protons from 16O and 40Ca in Glauber theory
Nuclear Physics A37S (1982) 45369 Q North-Holland Publishing Company
TEST OF CORRELATION EXPANSIONS FOR INELASTIC SCATTERING OF PROTONS FROM 160 AND '°Ca IN GLAUBER THEORY DAVID R . HARRINGTON and V . TUTUNJIAN
Serin Physics Laboratory, Rutgers University, PO Box 849, Piscataway, New Jersey 08854, USA Received 3 August 1981 Abstract : Two different versions of a correlation expansion for the A-body nuclear transition density required to evaluate the Glauber amplitude for inelastic proton-nucleus scattering are tested . Antisymmetrized oscillator wave functions, containing only Pauli correlations, are used to calculate the "exact" amplitude as well as various teams in the correlation expansions for the excitation of the 3 - (6.13 MeV) state of r60 and the 5 - (4.49 MeV) state of 4°Ca . The leading term in both expansions, which ignores all correlations and corresponds to the Glauber theory version of the DWIA, leads to errors which are larger than present experimental errors, especially et large momentum transfers . In one version of the correlation expansion, due to Alkhazov tt aL, the first-order correction contains both elastic and inelastic two-body correlations and leads to satisfactory results . In the other version, used by Abgrall et aL, the first-order correction contains only inelastic two-body conrelations . In this cane the first- and second-order corrections are needed to obtain accuracy comparable to that of the latest experiments.
1. Introduction
Medium energy proton-nucleus scattering has already greatly extended our knowledge of nuclear structure, and more high quality experimental data are expected in the future . For elastic scattering the optical limit of ICMT t) or Glauber theory 2) relates the differential cross section directly to the single-nucleon densities. There are fairly small but significant corrections to the optical limit, mainly from two-nucleon correlations 3), but it seems that these can be estimated with suflïcient accuracy to fully exploit the accuracy of existing experimental data `) once the spin dependence of the nucleon-nucleon scattering amplitude is fixed s). For inelastic scattering the analogue of the optical limit is the distorted wave impulse approximation iDWIA) t) which ignores correlations in the nuclear excitedstate wave function as well as those in the ground state. Here again corrections can be obtained from a correlation expansion 6~9) of the full A-body transition density, but in this case the expansion is not unique . If, for example, one groups terms so that the expansion is in the order of correlations, then the leading correction to the DWIA will be the sum of two terms, one resulting from pair correlations in the inelastic channel, the other from those in the elastic channel 6). An alternate approach is to group terms in a way which, apart from self-correlation terms, yields 453
454
D.R . Harrington, V. Tutunjian / Correlation espansions
an expansion in the number of inelastic steps'). Now the leading correction to the DWIA will include two-step processes plus a self-correlation term . The accuracy of the DWIA and the convergence of the correlation expansion in the two approaches has not yet been adequately studied, although some work has
been done in thïs direction. For example, Manayenkov 9) has compared the Glauber theory version of the DWIA, which he calls the single inelastic collision approximation, with the complete Glauber theory for inelastic proton scattering from'ZC and 160 .
More recently Abgrall et al.') have shown that the DWIA is quite inaccurate in describing some of the rotational excitations of the deformed 12C nucleus, but including two-step processes improves matters considerably . Even more recently Alkhazov 1°) has studied the effect of pair correlations on the excitation of 1Z C and ss Ni . The work of Manayenkov 9) shows that the DWIA is fairly accurate for nuclei with only short-range correlations and one might guess that the correlation expansion would converge rapidly for these nuclei . One of the present authors has shown s), however, that this is not the case : the correction to the DWIA from two-step processes did not systematically improve the accuracy of the inelastic profile function . Since the errors introduced by the DWIA, although relatively small for nuclei without long-range correlations, are still larger than the errors in the experimental data, it seemed worthwhile investigating these points further. In this paper we again use the antisyrhmetrized oscillator model of ref. s) to study inelastic scattering in Glauber theory . We extend and improve the work there in several ways, however. In the first place in this paper we calculate the differential cross sections for exciting "physical" excited states so that we can make a more detailed comparison of theoretical and experimental errors . We also do calculations for 160 as well as for °°Ca to see whether our results have any significant Adependence, and we do not make the zero-range approximation used in ref. 8). More importantly, we compare the expansion in the order of correlations to that in the number of inelastic steps to determine which converges more rapidly to the exact result . The body of the paper begins in sect . 2 by reviewing and comparing the two correlation expansions for inelastic scattering in Glauber theory . The antisymmetrized oscillator model for the wave functions is, introduced in sect . 3; these wave functions include only short-range Pauli correlations but have~the advantage that the exact Glauber theory scattering amplitude as well as all the terms, in the correlation expansions can be evaluated. The general expressions are specialized to the two nuclei '60 and a° Ca in sect. 4, and in particular to the excitation of the 3 - and 5- states, respectively . These states were chosen because they have unique wave functions in our model and the corresponding physical states are known to be strongly excited. The results of the "exact" and the approximate calculations for both versions of the correlation expansion are presented and discussed in sect . 5 . Although we also compare our results with experiment we are in no way trying
D.R. Harringwn, V. Tutunjlan / Conzlation expansions
455
to fit experimental data at this stage: we have left out Coulomb and spin-dependence effects and our wave functions cannot represent really accurately the single-particle ll .i2) transition densities. Studies of inelastic, electron scattering have shown, for example, that ground-state correations associated with particle-hole pairs have a large effect on the magnitudes of the one-body transition densities. The purpose of the comparison with experimental data is to verify that our model is at least qualitatively accurate and to compare the sizes of the discrepancies between exact and approximate calculations with experimental errors at the same scattering angles . Our paper concludes in sect. 6 with a review and discussion of our results. 2. Glsnber theory and the rnrrels8on ezpsnsion In Glauber theory 2) the nucleon-nucleus scattering amplitude is given in the impact parameter representation by
c-~ .br"o(b) d e b ~ Fno(q) = 2~r ~
(1)
where n and 0 label the final and initial nuclear states, respectively, and q is the momentum transfer . The nuclear profile function
w rno(~%)-f%n0-(/ll ~ (1-Yi)~0)~ i~l where A is the mass number of the nucleus and
Yi = Y(b - bi) is the profile function for elastic scattering from the ith nucleon, located at bi, z;, defined so that, in analogy with eq . (1), the amplitude for scattering from a single nucleon is
Our scattering amplitude is normalized so that the differential cross section = ~FRo(q)~Z (5) dd and the optical theorem is
Q
Ptue = 4~r Im Foo(0) ~
~
(6)
Glauber theory is known to be quite accurate for medium-energy scattering and 1314 the corrections to it are well understood ' ). Unfortunately, except for very light nuclei, it is difficult to evaluate the matrix element in eq . (2) exactly because it involves an A-body operator . If one ignores all correations in the nuclear wave
456
D.R .
Harrington, V. Tutunjian / Correlation expansions
functions one obtains the Glauber theory version of the optical limit. For elastic scattering corrections to this limit have been discussed in refs . 3'6). For inelastic scattering corrections to this limit can be obtained from the correlation expansion of the transition density pnô ) ( 1, 2, . . . , A) 6) : Pnô (1, 2, . . . , A) = Yr t, (1, 2, . . . , A)Yro(1, 2, . . . , A)
(7)
where C~ZÔ (i~ I) = PRö(t~ 1) - P~lô(l)Pôö (1) - Pnö(1)Pôö (~)
(9)
is the transition density pair correlation function, Cnö (~~ 1+ k) = P~ö (1~ 1+ k) - Cnö (~~ 1)Pôö (k) - CnZÔ (1+ k)Pôö (~) - Cnô (i~ k)Pôô (1) - P~ö(~)Pôö(I+k) - Pnö(1)Pôö (l. k) - Pnö(k)Pôö(i.l)
(1~)
is the transition density three-particle correlation function, and the symbol (A-m )/ P00 1i~2~ " . .,A/1,t2, . . .,t,n)
represents the ground-state (A - m)-body density, a function of the A - m variables which remain when the variables i,, iZ, . . . , i, are eliminated .
The expansion (8) is asymmetric in the initial and final states since it involves Poo but not p , but this is necessary on practical grounds, since much more is known about ground-state densities than about the density of excited states . In fact the main reason for inelastic scattering experiments is to study the properties of nuclear excited states . If the ground-state densities were all known we could stop here with only one version of the correlation expansion. We can hope to know, however, only the single-particle density and a few low-order ground-state correlation functions so we are forced to use a correlation expansion for the ground state densities in eq . (8) : N ploNO)(1~2 . . . .,N)= ~ Pôö(1)+~ Côô) (hl) ~ Pôö(k) l-1
+ ~ where
i
1
Côö (j.1~ k)
k~l,
~ Pôö (1)+ . . . ~
lft,j,k
Côö (~~ 1) = Pôö (~~ 1) - Pôö (i)Pôö (1)
(11) (I2)
D.R.
457
Harrington, V. Tutunjian / Correlation expansions
is the ground-state pair correlation function and Côö (~~j, k)=Pôö(~~1. k)-Pôö(~,j)Pôö(k) - Pôö (~, k)pôö (j) -Pôö (1~ k)Pôô (1)+2pôä (1)Pôä (j)Pôö (k)
(13),
is the ground-state three,-particle correlation function . Substituting (11) in (8) and dropping terms which are products of correlation functions, one obtains the following expansion for p ;,ô Pôô (1,2, . . .,A)=~pnô(1)ln Pôö(j) i
t#1
+ ~ Côô (j, k) j
j
jI 1 #i.i.k
pôô (!)
m #i,j,k " I
+ ~ Cnö (h ]) ~ Pôö (k)+ ~ CKö (h j+ k) ~ Pôö (1)+ . . . " t
(14)
At this stage the ambiguity as to the grouping of terms arises . Alkhazov et al. e) use the order of correlation to group terms and take (A) I II III " " " , Pn0 (ls
2, " " " , A) = pn0 +PRo
+PnO +
where Pn0(lr2r " " " ,A)=LPnÔ(t) ~ PÔÖ(I) . i
j#t
Pnô(1 .2~ " . " ,A)=~ Cn2ô(hj) ~ Pôö(k)+~PRö(1) ~ Côö(j.k) ~ Pnö(!) . !
PKö (1,
2, " . . , A) _
~
i
#i,j,k
l.k#i
Cnö (h j, k) ~ Pôö (!) 1#i.j,k
~+~t Pnlô(i) j
m#i"j,kl
Pôö (m) "
Abgrall et al.'), on the other hand, use the number of inelastic steps as a guide to grouping terms and take (A) I II III " " " ~ " " Pn0
(1e2~
.,A)=pn0+Pn0+Pn0+
with Pn0(l f `s . . . ,
A) _ ~ Cj,ZÔ (tf ~) i
~
k ;ü,j
Poo (k)
s
D.R. Harrington, V. Tutunjian / Correlation expansions
458
PnI0~i+2, " . .,t9) =
~ Cn30ct .I+k)
i
+~Pnö~j) i
~
i
!
(3)
Cn0 (t+ 1+
lC) =
(3)
Pn0 (tr 1+
(2)
P00(l)
Côö (1, k)
_ ~ Cn3ô {i. 1. k)
~
1#l,j,k
~
Isl,i,k
~
l~l,j,k
Pôö cl)
Pôö cl) +
(1)
lC) - Pn0 (tr 1)Po0 ck)
- Pnö(k+ j)Pôö (1)-P~ZÔ(1. k)Pôô (i)+Pnö(j)Pôö (1)Pôö (k) +P~ô(k)Pôö(j)Pôö(1)+Pnö(1)Pôö(k)PôôO) "
Note that the terms containing the ground-state two-body correlations have cancelled so that p ;,ö can be expressed in terms of pnô, pRô, pnô, and pôö : higher-order ground-state densities do not appear. Using the symmetry of the density functions, the two corresponding expansions for the profile function l'no are the version of Alkhazov et al., I II III rno = rno +r n o +rno + . . .
r
and the version of Abgrall et al., I II III rn0`rn0 +i n0 +TnO + . . "
The leading term in both expansions rno = A(nIYI~)L1 - (o~Y~o)l "-I
is the Glauber theory version of the DWIA, with ground-state correlation effects ignored in the "distorting" factor. The leading correlation corrections are rnô
=-iA(A-i)L(nIYIYZIo)-2(nlYlo>(olYlo>l[i-(o~Y~o>lA-2,
cis)'
rnlo,o = zf1cA - i)cA - 2)(nlYlo>[(oIYIYZIo)-(olYlo>Zl[i-(olYlo)~~ -'
ci~)
where contains the leading ground-state correlation corrections to the distorting factor . The next higher-order corrections are l'Rö =6A(A - i)cA-2){(nIYIYzYsIO)
-3(n~YIY2I0)(o~Y~o)+3(n~Y~O)(~~Y~O) Z}~1 - (olYlo)lA -3 , III
III
II
III
rno = rno - rn0,0 +rn0,0+
cis)
D.R. Harrington, V. Tutunjian / Correlation expansions
459
where rnIO.o =-bt1(A-1)(A-2)(A-3)(n~Y~o) x L(DIYiYzY3I 0) -3 (~IYlY2I~)(~IYI~)+2(~~Y~O) 3 ~ x L1-(~~Y~~)lA-a
(19)
contains the correction to the distorting factor due to three-particle correlations in the ground state. If I';,`ô .o is negligible, as suggested by the results of ref. 3), then the two expansions are identical if three terms are kept . They differ simply in that Alkhazov et al. add rn`o .o to l'n`o and subtract it from Tn`ô. To a given order, then, the Alkhazov et al. (ABV) expansion equals the Abgrall et al . (ALMC) expansion plus an additional term which corrects the distorting factor for ground-state correlations . In the ALMC approach this term is not included because it would include contributions with a higher number of inelastic steps than the other one. For example, we can write
rnlo.o
=zA(A-2)(nIYI~)
x CA
~ (~IYIm)(m~Y~O) +( o~Y~~)Z-(~~YZ~O)][I-(olYlo)) "-',
mf0
(ao)
where the ~m) are anti-symmetrized nuclear excited states, showing that T `o .o contains three-step contributions proportional to (n~y~0)(O~y~m)(m~y~0). A similar transformation gives rnIO=-iA 2{ E (nIYImXm~Y~~) m~n,0 +(nIYI~)L(nIYIn)-(A-2)A_i(~~Y~~)~ - (n~YZ~~) } {1-(D~Y~~)}A-2,
(21)
which contains at most two-step contributions. Clearly the two expansions will give identical results if enough terms are included . To a given order the ABV expansion treats more accurately the ground-state correlations which modify the distortions due to elastic scattering. In the ALMC expansion only the one-body ground-state density is included explicitly in the term (O~y~O) . The ground state correlations are hidden in matrix elements such as (nIYlYzY3I~) which depend only on the transition density. If the ground-state correlation corrections to the distorting factor are relatively large the ABV expansion will obviously .be more useful, while if the degree of smallness of a term is determined mainly by the maximum number of inelastic steps it contains, the ALMC expansion would be more appropriate. It might of course happen that one expansion converges more rapidly for one class of nuclei and the other for another class. In the remainder of this paper we shall test the convergence of the two expansions using nuclear models with only short-range Pauli correlations.
460
D.R . Harrington, V. Tutunjian / Correlation expansions
3. OscWator model We shall assume that the states ~0) and ~n) are simple superpositions of antisymmetrized products of single-particle states (i.e. Slater determinants) labelled by sets of A single-particle quantum numbers {a} = a,, aZ, . . . , aA and {ß} _ I+I, ßz, . . . , ßA " Then we can use the well-known result that, for {a} ~ {ß},
where G({ß}, {a}) is an A x A matrix with elements given by one-body matrix elements :
To evaluate the terms in the two versions of the correlation expansion for T we can use the usual expressions for the matrix elements of one-, two-, and three-body operators between Slater determinants . For definiteness we shall consider here only the case where ~{ß}) is a lp-lh state ~p, h) relative to a fixed ground state ~0) _ ~{a }), i.e. where a I = h, is the time reverse of h and ßI = p, but ß; = a,, for j = 2, 3, . . . , A. Then A
A(o~Y~o) _ ~ (arI Ylar) f-I
rph = (PIYIh)L1 -(~IYI~)lA-I ,
(24)
and for the ALMC expansion r~, °~ E (~IYIa~XarIYIh ) -(~IYIhXaIYIo>~L1-(o~Y~o)lA -Z r I
and _
r~ =
A
E (pIYIa~XatIYIaIXaIIYIh)
i.l-1
A
-~(olYlo> ;_EI (plYlar>(atlYlh>+z(~IYIh> [(A+2)(O~Y~~)Z- .~ (arIYIaI)(aIIYIar), }[1 - (~~ Y~~)}A-3 . til-I
For the ABV expansion
II r+II II rpk = rPh +TPh.O .
(25)
(26)
D.R .
Harrington, V. Tutunjian / Correlation expansions
461 .
where
rrti.o = (PIYIh)[z(A - 2)(D~y~O)Z -1 (A-2) E (~ :lyla>>(atlyl~~>~[1-(olylo)~A-3 . 2A ,, j _1
(27)
I'c`> in terms of the same single-particle matrix elements These expressions give the required to calculate l'r,, exactly from eqs. (22) and (23) . To simplify the calculation of these matrix eleménts we shall take the singleparticle spatial wave functions to be harmonic oscillator eigenfunctions, ignore the spin- and isospin-dependence of y, and take y(b)=
_vA Z 2~r
e
Azb=
(28)
where ~ _ ~(1- ~r1) ,
(29)
with v the total cross section and ~ the ratio of the real to imaginary part of the forward scattering amplitude for nucleon-nucleon scattering . The single-particle matrix elements can then all be evaluated analytically . The calculations are further simplified if we use a Cartesian basis for the oscillator states, using the quantum numbers (nx, n n=), for then, since y is independent of z,, the matrix G is diagonal in n= as well as in spin and isospin. In ref. a) an additional simplifying assumption was made : that the range of the nucleon-nucleon interaction is negligible compared to the distance over which the wave functions change appreciably. In this short- (or zero-) range approximation, equivalent to the limit A -> °o,
so that the matrix G is factorizable as well as block diagonal and the expressions l'c~> become quite simple . In this paper we shall not make this for l' and the approximation except to test its accuracy by comparing it with the exact finite-range results obtained by keeping A finite . 4. Angelar momentum eigenstetes In sect. 3 we introduced a basis of Cartesian oscillator states since these were l'c'~. Physical states of course have definite most convenient for calculating T and the angular momentum so we must construct them from linear combinations of the Cartesian oscillator states . We shall consider here only the two nuclei'60 and °° Ca
462
D.R . Harrington, V. Tutunjian / Correlation expansions
for which the model ground states consist of closed oscillator shells and which have total spin S = 0 and total isospin I = 0. Because we ignore the spin and isospin dependence of y only states with S = 0 and I = 0 can be excited and it is most convenient to use the LS coupling scheme with the total angular momentum J equal to the total orbital angular momentum L. The particle-hole states of definite angular momentum are then (31)
= I (npfp)(ntifti)JM) , I P~M> -
where the n's and l's are the principal and orbital angular momentum quantum numbers of the single-particle oscillator states . These states can be expanded in the Cartesian basis so that the physical states can be expanded directly in terms of the Cartesian states : I P~M) _ ~ I npnk )a (np, n,, ; P~M) , n
nti
(32)
where the coefficients "a" are combinations of Clebsch-Gordan coefficients and the coefficients relating the Cartesian and angular momentum eigenstates of the oscillator . Then, in an obvious notation, rphlM = ~ a * (np, Rh ; Pj1JM)l'nnnh
~
wo nh
(32a)
so that the profile functions for the excitation of physical states are just linear combinations of those for the excitation of states in the Cartesian basis. If we choose the axis of angular momentum quantization in the z-direction, perpendicular to the impact parameter plane, then rpklM(b) = e
-iM
~~p~M(b)
~
where ¢ is the azimuthal angle giving the direction of b. The reduced profile function T depends only on the magnitude of b and rphJ-M(b)
-
(-1)Jrph7M(b) "
(34)
Furthermore, a simple parity argument shows that I'p,,.n,.r vanishes if lp + 1,, +M is odd so that for lfi~ excitations only odd values of M contribute . Because of the simple ~-dependence of l'pr,, the corresponding scattering amplitude can be written FphrM(9)= ( where Fpti.tM(q) =
-j)
-rMmF. pti.rM(q),
(35)
JM(gb)l'p,,,,M(b)b db .
(36)
M-1 e
~ J0
D.R.
Hanington, V. Tutunjian / ComlaHon expansions
463
In an experiment in which all M-values of the final nuclear state are accepted one has dg
rz~r =
~rK (q) ~ 2
=4~rK (q) 2
d R
M
Fn
~~
hrM~ 2 ,
(37)
where K(q)=exp ( ZRZ/4A), with R the oscillator parameter, is the usual factor which for antisymmetrized oscillator wave functions corrects exactly for the centerof-mass constraint. The factor of 4 in (37) comes from the spin and isospin sums if we interpret Fv,~n,.r as the amplitude for any particular spin and isospin of the particle and any particular spin and isospin for the hole for which the amplitude does not vanish, e.g. for ms(p) = ms(h') = mI(p) = mI(h ) = z . g
S. Results We have evaluated the expressions above for the differential cross sections for the excitation of the lowest 3- state of '60 and the lowest 5 - state of a°Ca by protons near 1 GeV. These are "stretched" states in that they have the highest possible J = L for 1~~, S = 0 states and their wave functions are thus uniquely determined ïn the simple shell model which we use. We have taken ~ = 43 mb, ~ _ -0.24, and ~l =1 .514 fm-1 for the parameters describing the nucleon-nucleon interaction near 1 GeV and used Ro =1 .82 fm and R~ =1 .98 fm for the scales of the oscillator wave functions. The differential cross sections calculated "exactly" and in the DWIA are shown in figs . 1 and 2 along with the differential cross section including the leading correlation correction of the ALMC expansion (the other corrected curves are not shown because they are very close to the exact curves). In each case we have also isplotted some recent experimental data ") in order to show its accuracy and range in momentum transfer. (We wish to emphasize again that for reasons given in the introduction we cannot expect our model calculations to agree in detail with experiment.) The relative errors of the different approximate calculations are shown '). in figs . 3 and 4 along with an indication of the size of the experimental errors' 3-l The corresponding results for the integrated cross sections for fixed values of M, w~IMl=2
dwkrlM~ dz q, d 9
are shown in table 1 along with the total integrated cross sections, ww ° I~I
wti .ilMl .
These cross sections are of course dominated by small momentum transfers and
464
D.R . Harrington, V. Tutunjian / Conelation expansions
N
E i E
v
b v o~ 0
Fig. 1 . The differential cross section for the excitation of the 3 - (6 .13 MeV) state of 16 0 by protons near 1 GeV. The solid line shows the results of full Glauber calculations, the dashed line the DWIA, the dash~otted line the DWIA with the first-order corcelation correction of the ALMC expansion. The experimental data for 0.8 GeV protons are taken from ref. 1s).
thus do not reflect the large relative errors seen in the differential cross sections at large momentum transfers. Furthermore, positive and negative errors in the differential cross sections can cancel in the integrated cross sections further reducing the errors of the integrated cross sections . In our model the DWIA is quite accurate in the region of the first diffraction peak but is in error by as much as 40% near the diffraction dips for '60 and by as much as 80% for a°Ca. Adding the first correlation correction of the ABV expansion reduces the errors to less than 12% for °°Ca and less than 7% for '60. The convergence of the ALMC expansion turns out to be slower, especially at large momentum transfers. In this case adding the first correction to the DWIA still gives errors ranging from 30 to 40% in the q2 = 10-15 fm -z region, where the experimental errors are 10-15% l'). Including the second correction reduces the errors to less than 12% for qz < 18 fm -Z, with negative errors which tend to increase with increasing momentum transfer, whereas the errors in the corrected DWIA in
D.R . Harringtoa, V. Tutunjian / Correlation expansions
465
0 .0
N
t(fm 2) Fig. 2. The differential cross section for the excitation of the 5 - (4 .49 MeV) state of `° Ca by protons near 1 GeV. The solid line shows the results of full Glauber calculations, the dashed line the DWIA, the dash-dotted line the DWIA with the first-order correlation correction of the ALMC expansion. The experimental data for 0.8 GeV protons are taken from ref. l').
the ABV expansion have an oscillatory character. It appears then that the ABV approach is the more efficient one ftir nuclei which can be described by our model. Moreover, since the leading correction of the ABV expansion includes a subset of three-step processes, we conclude that if one wants to reduce the theoretical errors to the level of experimental errors one will have to go beyond two-step processes. In ref. 8) the short-range approximation (SRA) was used to simplify the I'c'y. Comparing expressions for T and the the resulting differential cross sections calculated above to those we have found that the SRA for inelastic scattering leads to results which are too large by factors of 4 to 6 in the 10-15 fm -z momentum transfer squared range and is thus useless as far as calculating the absolute value of the differential cross sections is concerned. The relative errors of the DWIA and its corrected versions are, however, given fairly accurately in magnitude, but not in detailed shape, by the SRA .
466
D.R. Harrington, V. Tutunjian / Correlation expansions
40A
2Q0
0 .0
Ij -20.0
-40A 0.0
5 .0
10 .0
t(frn 2 )
15.0
20.0
Fig. 3. Percent relative error of the approximate calculations of the differential cross section for the excitation of the 3 - (6 .13 MeV) state of ' 6 0. The dashed line is for the DWIA, the dash-dotted line for the DWIA with the first correlation correction of the ALMC expansion, the dash~ouble dotted line is for the DWIA with the first- and second-order corrections of the ALMC expansion and the dotted line is for the DWIA with the first-order correlation correction of the ABV expansion. The is). shaded region gives a rough indication of the percent experimental errors from ref.
6. Condasion We have used antisymmetrized oscillator-model wave functions for the ground states and the "stretched" lttw states of t60 and `°Ca for which the full Glauber amplitude as well as the leading terms in two versions of a correlation expansion of that amplitude can be evaluated rather simply . The first tenor in each expansion is the Glauber theory version of the DWIA and has the convenient property that only one-body ground-state and transition densities are needed for its evaluation. If the ground-state density is known from elastic scattering experiments then the transition density could be obtained directly from the inelastic scattering amplitude if the DWIA were accurate . Unfortunately, our calculations indicate that at large momenttun transfers the errors introduced by the DWIA are large compared to the errors of available experimental data . Therefore it will be necessary to include correlation corrections to the DWIA.
D.R .
Narrington, V. Tutunjian l Correlation expansions
467
80 .0
t (fR1
2)
Fig. 4. Percent relative error of the approximate calculations of the differential cross section for the excitation of the 5 - (4 .49 MeV) state of `°Ca . The dashed line is for the DWIA, the dash-dotted line for the DWIA with the first correlation correction of the ALMC expansion, the dash~ouble dotted line for the DWIA with the first- and second-order corrections of the ALMC expansion and the dotted line is for the DWIA with the first-order correlation correction of the ABV expansion. The shaded region gives a rough indication of the percent experimental errors from ref. ") .
Two different inelastic correlation expansions have been suggested. We have found that with the ABV expansion 6) the first correlation correction, which includes both transition and ground state two-body correlations, is sufficient to reduce the theoretical errors to the magnitude of experimental errors, while with the ALMC expansion') both the first- and second-order corrections are required to reach the same level of accuracy. The results for 160 and <°Ca are qualitatively the same except that the errors are generally somewhat larger for °°Ca. We have shown, therefore, that for nuclei with only short-range correlations the ABV expansion is more efficient than the ALMC expansion. In the ABV expansion only l'no and l'~°=r~ô+l'~to,°, as defined in eqs. (15) and (17), need to be calculated to obtain theoretical errors comparable to experimental errors . An approximate method for calculating the ground state two-body oorrelations required for l'~ô.° was tested in ref. a) acid seems to work quite well. In the absence of a
468
D.R. Harrington, V. Tutunjian / Coneiation expansions TABLE 1
The integrated differential cross sections for exciting the 3- (6 .13 MeV) state of 160 and the 5(4 .49 MeV) state of 4° Ca by protons near 1 GeV in the full Glauber theory, in the DWIA, in the DWIA with first-order corrections in the ABV expansion, and in the DWIA with first- and with firstand sewnd-order wrrections in the ALMC expansion Integrated cross sections (mb) ao~ 5(4.49 MeV)
Exact DWIA DWIA+I (ABV) DWIA+I (ALMC) DWIA+II (ALMC)
160
3 - (6 .13 MeV)
~M~ =1
~M~ = 3
~M~ = 5
total
~M~ =1
~M~ = 3
total
0.0302 0.0327 0.0311 0.0361 0.0308
O.076ß 0.0794 0.0783 0.0896 0.0782
0.809 0.695 0.801 0.834 0.814
0.916 0.807 0.910 0.960 0.923
0.319 0.327 0.323 0.347 0.320
2.01 1 .78 2.00 2.05 2.01
2.33 2.11 2.32 2.40 2.33
The integrated aoss sections are given for the different possible values of the magnitude of the magnetic quantum number ~M~ together with the sum of these cross sections .
detailed nuclear model it seems more difficult to estimate the two-body transition correlation term l'~to. In preliminary calculations we have found the short-range approximations used to simplify î;,ô.o not as accurate when applied to l'nto because of the more rapid variation of the excited-state wave functions . This problem must be studied further before a satisfactory practical prescription for l';,ô can be given. One obvious question which we have not answered here is the rate of convergence of the two correlation expansions for the excitation of collective states in nuclei with long-range correlations. The results of ref.') show that the first correction term in the ALMC expansion improves the accuracy, but the first term in the ABV expansion, which might have done an even better job, was not calculated : cléarly additional calculations are required before this point is settled. The authors would like to thank Prof. R.J. Plano for permission to use the Rutgers Bubble Chamber Group's PDP-10 computer, Prof. C. Glashausser for pointing out the existence of the data of ref. "), and Prof. G. Igo for permission to use these data . References 1) 2) 3) 4) 5)
A.K. Kerman, H. McManus and R.M . Thaler, Ann. of Phya . 8 (1959) 551 R.J . Glauber, High energy physics and nuclear structure, ed. S. Devons (Plenum, NY, 1975) p. 207 D.R . Harrington and G.K . Varma, Nucl . Phys. A306 (1978) 477 L. Ray et al., Phys . Rev. C18 (1978) 2641 S.J. Wallace, High energy proton scattering, to be published in Advances in nuclear physics, cd . E. Vogt and J. Negele . 6) G.D. Alkhazov, S.L. Belostotsky and A.A. Vorobyev, Phys. Reporta 42C (1978) 89 7) Y. Abgrall, J. Labarsouque, B. Morand and E. Laurier, Nucl . Phys. A316 (1979) 389
D .R . Hanington, V. Tutunjian / Conelation expansions
469
8) D.R. Harrington, Nucl. Phys . A343 (1980) 417 9) S.I . Manayenkov, Sov. J. Nucl. Phys. 21(1975) 51 10) G.D . Alkhazov, Effects of nucleon correlations and multi-step transitions in the cross séctions for inelastic fast proton scattering on nuclei, Leningrad preprint 599 11) V. Gillet and M.A. Melkanoff, Phys. Rev. 133B (1964) 1190 12) J.C . Bergstrom et al., Phys . Rev. Lett . 24 (1970) 152 13) D.R . Harrington and G.K . Varma, Phys . Lett . 74B (1978) 316 14) S.J. Wallace, Phys. Rev. C12 (1975) 179 15) G.S . Adams et al., Phys. Rev. Lett . 43 (1979) 421 16) G.S. Adams et al., Phys . Rev. C21 (1980) 2485 17) E. Bleszynski et al., UCLA preprint
TEST OF CORRELATION EXPANSIONS FOR INELASTIC SCATTERING OF PROTONS FROM 160 AND '°Ca IN GLAUBER THEORY DAVID R . HARRINGTON and V . TUTUNJIAN
Serin Physics Laboratory, Rutgers University, PO Box 849, Piscataway, New Jersey 08854, USA Received 3 August 1981 Abstract : Two different versions of a correlation expansion for the A-body nuclear transition density required to evaluate the Glauber amplitude for inelastic proton-nucleus scattering are tested . Antisymmetrized oscillator wave functions, containing only Pauli correlations, are used to calculate the "exact" amplitude as well as various teams in the correlation expansions for the excitation of the 3 - (6.13 MeV) state of r60 and the 5 - (4.49 MeV) state of 4°Ca . The leading term in both expansions, which ignores all correlations and corresponds to the Glauber theory version of the DWIA, leads to errors which are larger than present experimental errors, especially et large momentum transfers . In one version of the correlation expansion, due to Alkhazov tt aL, the first-order correction contains both elastic and inelastic two-body correlations and leads to satisfactory results . In the other version, used by Abgrall et aL, the first-order correction contains only inelastic two-body conrelations . In this cane the first- and second-order corrections are needed to obtain accuracy comparable to that of the latest experiments.
1. Introduction
Medium energy proton-nucleus scattering has already greatly extended our knowledge of nuclear structure, and more high quality experimental data are expected in the future . For elastic scattering the optical limit of ICMT t) or Glauber theory 2) relates the differential cross section directly to the single-nucleon densities. There are fairly small but significant corrections to the optical limit, mainly from two-nucleon correlations 3), but it seems that these can be estimated with suflïcient accuracy to fully exploit the accuracy of existing experimental data `) once the spin dependence of the nucleon-nucleon scattering amplitude is fixed s). For inelastic scattering the analogue of the optical limit is the distorted wave impulse approximation iDWIA) t) which ignores correlations in the nuclear excitedstate wave function as well as those in the ground state. Here again corrections can be obtained from a correlation expansion 6~9) of the full A-body transition density, but in this case the expansion is not unique . If, for example, one groups terms so that the expansion is in the order of correlations, then the leading correction to the DWIA will be the sum of two terms, one resulting from pair correlations in the inelastic channel, the other from those in the elastic channel 6). An alternate approach is to group terms in a way which, apart from self-correlation terms, yields 453
454
D.R . Harrington, V. Tutunjian / Correlation espansions
an expansion in the number of inelastic steps'). Now the leading correction to the DWIA will include two-step processes plus a self-correlation term . The accuracy of the DWIA and the convergence of the correlation expansion in the two approaches has not yet been adequately studied, although some work has
been done in thïs direction. For example, Manayenkov 9) has compared the Glauber theory version of the DWIA, which he calls the single inelastic collision approximation, with the complete Glauber theory for inelastic proton scattering from'ZC and 160 .
More recently Abgrall et al.') have shown that the DWIA is quite inaccurate in describing some of the rotational excitations of the deformed 12C nucleus, but including two-step processes improves matters considerably . Even more recently Alkhazov 1°) has studied the effect of pair correlations on the excitation of 1Z C and ss Ni . The work of Manayenkov 9) shows that the DWIA is fairly accurate for nuclei with only short-range correlations and one might guess that the correlation expansion would converge rapidly for these nuclei . One of the present authors has shown s), however, that this is not the case : the correction to the DWIA from two-step processes did not systematically improve the accuracy of the inelastic profile function . Since the errors introduced by the DWIA, although relatively small for nuclei without long-range correlations, are still larger than the errors in the experimental data, it seemed worthwhile investigating these points further. In this paper we again use the antisyrhmetrized oscillator model of ref. s) to study inelastic scattering in Glauber theory . We extend and improve the work there in several ways, however. In the first place in this paper we calculate the differential cross sections for exciting "physical" excited states so that we can make a more detailed comparison of theoretical and experimental errors . We also do calculations for 160 as well as for °°Ca to see whether our results have any significant Adependence, and we do not make the zero-range approximation used in ref. 8). More importantly, we compare the expansion in the order of correlations to that in the number of inelastic steps to determine which converges more rapidly to the exact result . The body of the paper begins in sect . 2 by reviewing and comparing the two correlation expansions for inelastic scattering in Glauber theory . The antisymmetrized oscillator model for the wave functions is, introduced in sect . 3; these wave functions include only short-range Pauli correlations but have~the advantage that the exact Glauber theory scattering amplitude as well as all the terms, in the correlation expansions can be evaluated. The general expressions are specialized to the two nuclei '60 and a° Ca in sect. 4, and in particular to the excitation of the 3 - and 5- states, respectively . These states were chosen because they have unique wave functions in our model and the corresponding physical states are known to be strongly excited. The results of the "exact" and the approximate calculations for both versions of the correlation expansion are presented and discussed in sect . 5 . Although we also compare our results with experiment we are in no way trying
D.R. Harringwn, V. Tutunjlan / Conzlation expansions
455
to fit experimental data at this stage: we have left out Coulomb and spin-dependence effects and our wave functions cannot represent really accurately the single-particle ll .i2) transition densities. Studies of inelastic, electron scattering have shown, for example, that ground-state correations associated with particle-hole pairs have a large effect on the magnitudes of the one-body transition densities. The purpose of the comparison with experimental data is to verify that our model is at least qualitatively accurate and to compare the sizes of the discrepancies between exact and approximate calculations with experimental errors at the same scattering angles . Our paper concludes in sect. 6 with a review and discussion of our results. 2. Glsnber theory and the rnrrels8on ezpsnsion In Glauber theory 2) the nucleon-nucleus scattering amplitude is given in the impact parameter representation by
c-~ .br"o(b) d e b ~ Fno(q) = 2~r ~
(1)
where n and 0 label the final and initial nuclear states, respectively, and q is the momentum transfer . The nuclear profile function
w rno(~%)-f%n0-(/ll ~ (1-Yi)~0)~ i~l where A is the mass number of the nucleus and
Yi = Y(b - bi) is the profile function for elastic scattering from the ith nucleon, located at bi, z;, defined so that, in analogy with eq . (1), the amplitude for scattering from a single nucleon is
Our scattering amplitude is normalized so that the differential cross section = ~FRo(q)~Z (5) dd and the optical theorem is
Q
Ptue = 4~r Im Foo(0) ~
~
(6)
Glauber theory is known to be quite accurate for medium-energy scattering and 1314 the corrections to it are well understood ' ). Unfortunately, except for very light nuclei, it is difficult to evaluate the matrix element in eq . (2) exactly because it involves an A-body operator . If one ignores all correations in the nuclear wave
456
D.R .
Harrington, V. Tutunjian / Correlation expansions
functions one obtains the Glauber theory version of the optical limit. For elastic scattering corrections to this limit have been discussed in refs . 3'6). For inelastic scattering corrections to this limit can be obtained from the correlation expansion of the transition density pnô ) ( 1, 2, . . . , A) 6) : Pnô (1, 2, . . . , A) = Yr t, (1, 2, . . . , A)Yro(1, 2, . . . , A)
(7)
where C~ZÔ (i~ I) = PRö(t~ 1) - P~lô(l)Pôö (1) - Pnö(1)Pôö (~)
(9)
is the transition density pair correlation function, Cnö (~~ 1+ k) = P~ö (1~ 1+ k) - Cnö (~~ 1)Pôö (k) - CnZÔ (1+ k)Pôö (~) - Cnô (i~ k)Pôô (1) - P~ö(~)Pôö(I+k) - Pnö(1)Pôö (l. k) - Pnö(k)Pôö(i.l)
(1~)
is the transition density three-particle correlation function, and the symbol (A-m )/ P00 1i~2~ " . .,A/1,t2, . . .,t,n)
represents the ground-state (A - m)-body density, a function of the A - m variables which remain when the variables i,, iZ, . . . , i, are eliminated .
The expansion (8) is asymmetric in the initial and final states since it involves Poo but not p , but this is necessary on practical grounds, since much more is known about ground-state densities than about the density of excited states . In fact the main reason for inelastic scattering experiments is to study the properties of nuclear excited states . If the ground-state densities were all known we could stop here with only one version of the correlation expansion. We can hope to know, however, only the single-particle density and a few low-order ground-state correlation functions so we are forced to use a correlation expansion for the ground state densities in eq . (8) : N ploNO)(1~2 . . . .,N)= ~ Pôö(1)+~ Côô) (hl) ~ Pôö(k) l-1
+ ~ where
i
1
Côö (j.1~ k)
k~l,
~ Pôö (1)+ . . . ~
lft,j,k
Côö (~~ 1) = Pôö (~~ 1) - Pôö (i)Pôö (1)
(11) (I2)
D.R.
457
Harrington, V. Tutunjian / Correlation expansions
is the ground-state pair correlation function and Côö (~~j, k)=Pôö(~~1. k)-Pôö(~,j)Pôö(k) - Pôö (~, k)pôö (j) -Pôö (1~ k)Pôô (1)+2pôä (1)Pôä (j)Pôö (k)
(13),
is the ground-state three,-particle correlation function . Substituting (11) in (8) and dropping terms which are products of correlation functions, one obtains the following expansion for p ;,ô Pôô (1,2, . . .,A)=~pnô(1)ln Pôö(j) i
t#1
+ ~ Côô (j, k) j
j
jI 1 #i.i.k
pôô (!)
m #i,j,k " I
+ ~ Cnö (h ]) ~ Pôö (k)+ ~ CKö (h j+ k) ~ Pôö (1)+ . . . " t
(14)
At this stage the ambiguity as to the grouping of terms arises . Alkhazov et al. e) use the order of correlation to group terms and take (A) I II III " " " , Pn0 (ls
2, " " " , A) = pn0 +PRo
+PnO +
where Pn0(lr2r " " " ,A)=LPnÔ(t) ~ PÔÖ(I) . i
j#t
Pnô(1 .2~ " . " ,A)=~ Cn2ô(hj) ~ Pôö(k)+~PRö(1) ~ Côö(j.k) ~ Pnö(!) . !
PKö (1,
2, " . . , A) _
~
i
#i,j,k
l.k#i
Cnö (h j, k) ~ Pôö (!) 1#i.j,k
~+~t Pnlô(i) j
m#i"j,kl
Pôö (m) "
Abgrall et al.'), on the other hand, use the number of inelastic steps as a guide to grouping terms and take (A) I II III " " " ~ " " Pn0
(1e2~
.,A)=pn0+Pn0+Pn0+
with Pn0(l f `s . . . ,
A) _ ~ Cj,ZÔ (tf ~) i
~
k ;ü,j
Poo (k)
s
D.R. Harrington, V. Tutunjian / Correlation expansions
458
PnI0~i+2, " . .,t9) =
~ Cn30ct .I+k)
i
+~Pnö~j) i
~
i
!
(3)
Cn0 (t+ 1+
lC) =
(3)
Pn0 (tr 1+
(2)
P00(l)
Côö (1, k)
_ ~ Cn3ô {i. 1. k)
~
1#l,j,k
~
Isl,i,k
~
l~l,j,k
Pôö cl)
Pôö cl) +
(1)
lC) - Pn0 (tr 1)Po0 ck)
- Pnö(k+ j)Pôö (1)-P~ZÔ(1. k)Pôô (i)+Pnö(j)Pôö (1)Pôö (k) +P~ô(k)Pôö(j)Pôö(1)+Pnö(1)Pôö(k)PôôO) "
Note that the terms containing the ground-state two-body correlations have cancelled so that p ;,ö can be expressed in terms of pnô, pRô, pnô, and pôö : higher-order ground-state densities do not appear. Using the symmetry of the density functions, the two corresponding expansions for the profile function l'no are the version of Alkhazov et al., I II III rno = rno +r n o +rno + . . .
r
and the version of Abgrall et al., I II III rn0`rn0 +i n0 +TnO + . . "
The leading term in both expansions rno = A(nIYI~)L1 - (o~Y~o)l "-I
is the Glauber theory version of the DWIA, with ground-state correlation effects ignored in the "distorting" factor. The leading correlation corrections are rnô
=-iA(A-i)L(nIYIYZIo)-2(nlYlo>(olYlo>l[i-(o~Y~o>lA-2,
cis)'
rnlo,o = zf1cA - i)cA - 2)(nlYlo>[(oIYIYZIo)-(olYlo>Zl[i-(olYlo)~~ -'
ci~)
where contains the leading ground-state correlation corrections to the distorting factor . The next higher-order corrections are l'Rö =6A(A - i)cA-2){(nIYIYzYsIO)
-3(n~YIY2I0)(o~Y~o)+3(n~Y~O)(~~Y~O) Z}~1 - (olYlo)lA -3 , III
III
II
III
rno = rno - rn0,0 +rn0,0+
cis)
D.R. Harrington, V. Tutunjian / Correlation expansions
459
where rnIO.o =-bt1(A-1)(A-2)(A-3)(n~Y~o) x L(DIYiYzY3I 0) -3 (~IYlY2I~)(~IYI~)+2(~~Y~O) 3 ~ x L1-(~~Y~~)lA-a
(19)
contains the correction to the distorting factor due to three-particle correlations in the ground state. If I';,`ô .o is negligible, as suggested by the results of ref. 3), then the two expansions are identical if three terms are kept . They differ simply in that Alkhazov et al. add rn`o .o to l'n`o and subtract it from Tn`ô. To a given order, then, the Alkhazov et al. (ABV) expansion equals the Abgrall et al . (ALMC) expansion plus an additional term which corrects the distorting factor for ground-state correlations . In the ALMC approach this term is not included because it would include contributions with a higher number of inelastic steps than the other one. For example, we can write
rnlo.o
=zA(A-2)(nIYI~)
x CA
~ (~IYIm)(m~Y~O) +( o~Y~~)Z-(~~YZ~O)][I-(olYlo)) "-',
mf0
(ao)
where the ~m) are anti-symmetrized nuclear excited states, showing that T `o .o contains three-step contributions proportional to (n~y~0)(O~y~m)(m~y~0). A similar transformation gives rnIO=-iA 2{ E (nIYImXm~Y~~) m~n,0 +(nIYI~)L(nIYIn)-(A-2)A_i(~~Y~~)~ - (n~YZ~~) } {1-(D~Y~~)}A-2,
(21)
which contains at most two-step contributions. Clearly the two expansions will give identical results if enough terms are included . To a given order the ABV expansion treats more accurately the ground-state correlations which modify the distortions due to elastic scattering. In the ALMC expansion only the one-body ground-state density is included explicitly in the term (O~y~O) . The ground state correlations are hidden in matrix elements such as (nIYlYzY3I~) which depend only on the transition density. If the ground-state correlation corrections to the distorting factor are relatively large the ABV expansion will obviously .be more useful, while if the degree of smallness of a term is determined mainly by the maximum number of inelastic steps it contains, the ALMC expansion would be more appropriate. It might of course happen that one expansion converges more rapidly for one class of nuclei and the other for another class. In the remainder of this paper we shall test the convergence of the two expansions using nuclear models with only short-range Pauli correlations.
460
D.R . Harrington, V. Tutunjian / Correlation expansions
3. OscWator model We shall assume that the states ~0) and ~n) are simple superpositions of antisymmetrized products of single-particle states (i.e. Slater determinants) labelled by sets of A single-particle quantum numbers {a} = a,, aZ, . . . , aA and {ß} _ I+I, ßz, . . . , ßA " Then we can use the well-known result that, for {a} ~ {ß},
where G({ß}, {a}) is an A x A matrix with elements given by one-body matrix elements :
To evaluate the terms in the two versions of the correlation expansion for T we can use the usual expressions for the matrix elements of one-, two-, and three-body operators between Slater determinants . For definiteness we shall consider here only the case where ~{ß}) is a lp-lh state ~p, h) relative to a fixed ground state ~0) _ ~{a }), i.e. where a I = h, is the time reverse of h and ßI = p, but ß; = a,, for j = 2, 3, . . . , A. Then A
A(o~Y~o) _ ~ (arI Ylar) f-I
rph = (PIYIh)L1 -(~IYI~)lA-I ,
(24)
and for the ALMC expansion r~, °~ E (~IYIa~XarIYIh ) -(~IYIhXaIYIo>~L1-(o~Y~o)lA -Z r I
and _
r~ =
A
E (pIYIa~XatIYIaIXaIIYIh)
i.l-1
A
-~(olYlo> ;_EI (plYlar>(atlYlh>+z(~IYIh> [(A+2)(O~Y~~)Z- .~ (arIYIaI)(aIIYIar), }[1 - (~~ Y~~)}A-3 . til-I
For the ABV expansion
II r+II II rpk = rPh +TPh.O .
(25)
(26)
D.R .
Harrington, V. Tutunjian / Correlation expansions
461 .
where
rrti.o = (PIYIh)[z(A - 2)(D~y~O)Z -1 (A-2) E (~ :lyla>>(atlyl~~>~[1-(olylo)~A-3 . 2A ,, j _1
(27)
I'c`> in terms of the same single-particle matrix elements These expressions give the required to calculate l'r,, exactly from eqs. (22) and (23) . To simplify the calculation of these matrix eleménts we shall take the singleparticle spatial wave functions to be harmonic oscillator eigenfunctions, ignore the spin- and isospin-dependence of y, and take y(b)=
_vA Z 2~r
e
Azb=
(28)
where ~ _ ~(1- ~r1) ,
(29)
with v the total cross section and ~ the ratio of the real to imaginary part of the forward scattering amplitude for nucleon-nucleon scattering . The single-particle matrix elements can then all be evaluated analytically . The calculations are further simplified if we use a Cartesian basis for the oscillator states, using the quantum numbers (nx, n n=), for then, since y is independent of z,, the matrix G is diagonal in n= as well as in spin and isospin. In ref. a) an additional simplifying assumption was made : that the range of the nucleon-nucleon interaction is negligible compared to the distance over which the wave functions change appreciably. In this short- (or zero-) range approximation, equivalent to the limit A -> °o,
so that the matrix G is factorizable as well as block diagonal and the expressions l'c~> become quite simple . In this paper we shall not make this for l' and the approximation except to test its accuracy by comparing it with the exact finite-range results obtained by keeping A finite . 4. Angelar momentum eigenstetes In sect. 3 we introduced a basis of Cartesian oscillator states since these were l'c'~. Physical states of course have definite most convenient for calculating T and the angular momentum so we must construct them from linear combinations of the Cartesian oscillator states . We shall consider here only the two nuclei'60 and °° Ca
462
D.R . Harrington, V. Tutunjian / Correlation expansions
for which the model ground states consist of closed oscillator shells and which have total spin S = 0 and total isospin I = 0. Because we ignore the spin and isospin dependence of y only states with S = 0 and I = 0 can be excited and it is most convenient to use the LS coupling scheme with the total angular momentum J equal to the total orbital angular momentum L. The particle-hole states of definite angular momentum are then (31)
= I (npfp)(ntifti)JM) , I P~M> -
where the n's and l's are the principal and orbital angular momentum quantum numbers of the single-particle oscillator states . These states can be expanded in the Cartesian basis so that the physical states can be expanded directly in terms of the Cartesian states : I P~M) _ ~ I npnk )a (np, n,, ; P~M) , n
nti
(32)
where the coefficients "a" are combinations of Clebsch-Gordan coefficients and the coefficients relating the Cartesian and angular momentum eigenstates of the oscillator . Then, in an obvious notation, rphlM = ~ a * (np, Rh ; Pj1JM)l'nnnh
~
wo nh
(32a)
so that the profile functions for the excitation of physical states are just linear combinations of those for the excitation of states in the Cartesian basis. If we choose the axis of angular momentum quantization in the z-direction, perpendicular to the impact parameter plane, then rpklM(b) = e
-iM
~~p~M(b)
~
where ¢ is the azimuthal angle giving the direction of b. The reduced profile function T depends only on the magnitude of b and rphJ-M(b)
-
(-1)Jrph7M(b) "
(34)
Furthermore, a simple parity argument shows that I'p,,.n,.r vanishes if lp + 1,, +M is odd so that for lfi~ excitations only odd values of M contribute . Because of the simple ~-dependence of l'pr,, the corresponding scattering amplitude can be written FphrM(9)= ( where Fpti.tM(q) =
-j)
-rMmF. pti.rM(q),
(35)
JM(gb)l'p,,,,M(b)b db .
(36)
M-1 e
~ J0
D.R.
Hanington, V. Tutunjian / ComlaHon expansions
463
In an experiment in which all M-values of the final nuclear state are accepted one has dg
rz~r =
~rK (q) ~ 2
=4~rK (q) 2
d R
M
Fn
~~
hrM~ 2 ,
(37)
where K(q)=exp ( ZRZ/4A), with R the oscillator parameter, is the usual factor which for antisymmetrized oscillator wave functions corrects exactly for the centerof-mass constraint. The factor of 4 in (37) comes from the spin and isospin sums if we interpret Fv,~n,.r as the amplitude for any particular spin and isospin of the particle and any particular spin and isospin for the hole for which the amplitude does not vanish, e.g. for ms(p) = ms(h') = mI(p) = mI(h ) = z . g
S. Results We have evaluated the expressions above for the differential cross sections for the excitation of the lowest 3- state of '60 and the lowest 5 - state of a°Ca by protons near 1 GeV. These are "stretched" states in that they have the highest possible J = L for 1~~, S = 0 states and their wave functions are thus uniquely determined ïn the simple shell model which we use. We have taken ~ = 43 mb, ~ _ -0.24, and ~l =1 .514 fm-1 for the parameters describing the nucleon-nucleon interaction near 1 GeV and used Ro =1 .82 fm and R~ =1 .98 fm for the scales of the oscillator wave functions. The differential cross sections calculated "exactly" and in the DWIA are shown in figs . 1 and 2 along with the differential cross section including the leading correlation correction of the ALMC expansion (the other corrected curves are not shown because they are very close to the exact curves). In each case we have also isplotted some recent experimental data ") in order to show its accuracy and range in momentum transfer. (We wish to emphasize again that for reasons given in the introduction we cannot expect our model calculations to agree in detail with experiment.) The relative errors of the different approximate calculations are shown '). in figs . 3 and 4 along with an indication of the size of the experimental errors' 3-l The corresponding results for the integrated cross sections for fixed values of M, w~IMl=2
dwkrlM~ dz q, d 9
are shown in table 1 along with the total integrated cross sections, ww ° I~I
wti .ilMl .
These cross sections are of course dominated by small momentum transfers and
464
D.R . Harrington, V. Tutunjian / Conelation expansions
N
E i E
v
b v o~ 0
Fig. 1 . The differential cross section for the excitation of the 3 - (6 .13 MeV) state of 16 0 by protons near 1 GeV. The solid line shows the results of full Glauber calculations, the dashed line the DWIA, the dash~otted line the DWIA with the first-order corcelation correction of the ALMC expansion. The experimental data for 0.8 GeV protons are taken from ref. 1s).
thus do not reflect the large relative errors seen in the differential cross sections at large momentum transfers. Furthermore, positive and negative errors in the differential cross sections can cancel in the integrated cross sections further reducing the errors of the integrated cross sections . In our model the DWIA is quite accurate in the region of the first diffraction peak but is in error by as much as 40% near the diffraction dips for '60 and by as much as 80% for a°Ca. Adding the first correlation correction of the ABV expansion reduces the errors to less than 12% for °°Ca and less than 7% for '60. The convergence of the ALMC expansion turns out to be slower, especially at large momentum transfers. In this case adding the first correction to the DWIA still gives errors ranging from 30 to 40% in the q2 = 10-15 fm -z region, where the experimental errors are 10-15% l'). Including the second correction reduces the errors to less than 12% for qz < 18 fm -Z, with negative errors which tend to increase with increasing momentum transfer, whereas the errors in the corrected DWIA in
D.R . Harringtoa, V. Tutunjian / Correlation expansions
465
0 .0
N
t(fm 2) Fig. 2. The differential cross section for the excitation of the 5 - (4 .49 MeV) state of `° Ca by protons near 1 GeV. The solid line shows the results of full Glauber calculations, the dashed line the DWIA, the dash-dotted line the DWIA with the first-order correlation correction of the ALMC expansion. The experimental data for 0.8 GeV protons are taken from ref. l').
the ABV expansion have an oscillatory character. It appears then that the ABV approach is the more efficient one ftir nuclei which can be described by our model. Moreover, since the leading correction of the ABV expansion includes a subset of three-step processes, we conclude that if one wants to reduce the theoretical errors to the level of experimental errors one will have to go beyond two-step processes. In ref. 8) the short-range approximation (SRA) was used to simplify the I'c'y. Comparing expressions for T and the the resulting differential cross sections calculated above to those we have found that the SRA for inelastic scattering leads to results which are too large by factors of 4 to 6 in the 10-15 fm -z momentum transfer squared range and is thus useless as far as calculating the absolute value of the differential cross sections is concerned. The relative errors of the DWIA and its corrected versions are, however, given fairly accurately in magnitude, but not in detailed shape, by the SRA .
466
D.R. Harrington, V. Tutunjian / Correlation expansions
40A
2Q0
0 .0
Ij -20.0
-40A 0.0
5 .0
10 .0
t(frn 2 )
15.0
20.0
Fig. 3. Percent relative error of the approximate calculations of the differential cross section for the excitation of the 3 - (6 .13 MeV) state of ' 6 0. The dashed line is for the DWIA, the dash-dotted line for the DWIA with the first correlation correction of the ALMC expansion, the dash~ouble dotted line is for the DWIA with the first- and second-order corrections of the ALMC expansion and the dotted line is for the DWIA with the first-order correlation correction of the ABV expansion. The is). shaded region gives a rough indication of the percent experimental errors from ref.
6. Condasion We have used antisymmetrized oscillator-model wave functions for the ground states and the "stretched" lttw states of t60 and `°Ca for which the full Glauber amplitude as well as the leading terms in two versions of a correlation expansion of that amplitude can be evaluated rather simply . The first tenor in each expansion is the Glauber theory version of the DWIA and has the convenient property that only one-body ground-state and transition densities are needed for its evaluation. If the ground-state density is known from elastic scattering experiments then the transition density could be obtained directly from the inelastic scattering amplitude if the DWIA were accurate . Unfortunately, our calculations indicate that at large momenttun transfers the errors introduced by the DWIA are large compared to the errors of available experimental data . Therefore it will be necessary to include correlation corrections to the DWIA.
D.R .
Narrington, V. Tutunjian l Correlation expansions
467
80 .0
t (fR1
2)
Fig. 4. Percent relative error of the approximate calculations of the differential cross section for the excitation of the 5 - (4 .49 MeV) state of `°Ca . The dashed line is for the DWIA, the dash-dotted line for the DWIA with the first correlation correction of the ALMC expansion, the dash~ouble dotted line for the DWIA with the first- and second-order corrections of the ALMC expansion and the dotted line is for the DWIA with the first-order correlation correction of the ABV expansion. The shaded region gives a rough indication of the percent experimental errors from ref. ") .
Two different inelastic correlation expansions have been suggested. We have found that with the ABV expansion 6) the first correlation correction, which includes both transition and ground state two-body correlations, is sufficient to reduce the theoretical errors to the magnitude of experimental errors, while with the ALMC expansion') both the first- and second-order corrections are required to reach the same level of accuracy. The results for 160 and <°Ca are qualitatively the same except that the errors are generally somewhat larger for °°Ca. We have shown, therefore, that for nuclei with only short-range correlations the ABV expansion is more efficient than the ALMC expansion. In the ABV expansion only l'no and l'~°=r~ô+l'~to,°, as defined in eqs. (15) and (17), need to be calculated to obtain theoretical errors comparable to experimental errors . An approximate method for calculating the ground state two-body oorrelations required for l'~ô.° was tested in ref. a) acid seems to work quite well. In the absence of a
468
D.R. Harrington, V. Tutunjian / Coneiation expansions TABLE 1
The integrated differential cross sections for exciting the 3- (6 .13 MeV) state of 160 and the 5(4 .49 MeV) state of 4° Ca by protons near 1 GeV in the full Glauber theory, in the DWIA, in the DWIA with first-order corrections in the ABV expansion, and in the DWIA with first- and with firstand sewnd-order wrrections in the ALMC expansion Integrated cross sections (mb) ao~ 5(4.49 MeV)
Exact DWIA DWIA+I (ABV) DWIA+I (ALMC) DWIA+II (ALMC)
160
3 - (6 .13 MeV)
~M~ =1
~M~ = 3
~M~ = 5
total
~M~ =1
~M~ = 3
total
0.0302 0.0327 0.0311 0.0361 0.0308
O.076ß 0.0794 0.0783 0.0896 0.0782
0.809 0.695 0.801 0.834 0.814
0.916 0.807 0.910 0.960 0.923
0.319 0.327 0.323 0.347 0.320
2.01 1 .78 2.00 2.05 2.01
2.33 2.11 2.32 2.40 2.33
The integrated aoss sections are given for the different possible values of the magnitude of the magnetic quantum number ~M~ together with the sum of these cross sections .
detailed nuclear model it seems more difficult to estimate the two-body transition correlation term l'~to. In preliminary calculations we have found the short-range approximations used to simplify î;,ô.o not as accurate when applied to l'nto because of the more rapid variation of the excited-state wave functions . This problem must be studied further before a satisfactory practical prescription for l';,ô can be given. One obvious question which we have not answered here is the rate of convergence of the two correlation expansions for the excitation of collective states in nuclei with long-range correlations. The results of ref.') show that the first correction term in the ALMC expansion improves the accuracy, but the first term in the ABV expansion, which might have done an even better job, was not calculated : cléarly additional calculations are required before this point is settled. The authors would like to thank Prof. R.J. Plano for permission to use the Rutgers Bubble Chamber Group's PDP-10 computer, Prof. C. Glashausser for pointing out the existence of the data of ref. "), and Prof. G. Igo for permission to use these data . References 1) 2) 3) 4) 5)
A.K. Kerman, H. McManus and R.M . Thaler, Ann. of Phya . 8 (1959) 551 R.J . Glauber, High energy physics and nuclear structure, ed. S. Devons (Plenum, NY, 1975) p. 207 D.R . Harrington and G.K . Varma, Nucl . Phys. A306 (1978) 477 L. Ray et al., Phys . Rev. C18 (1978) 2641 S.J. Wallace, High energy proton scattering, to be published in Advances in nuclear physics, cd . E. Vogt and J. Negele . 6) G.D. Alkhazov, S.L. Belostotsky and A.A. Vorobyev, Phys. Reporta 42C (1978) 89 7) Y. Abgrall, J. Labarsouque, B. Morand and E. Laurier, Nucl . Phys. A316 (1979) 389
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8) D.R. Harrington, Nucl. Phys . A343 (1980) 417 9) S.I . Manayenkov, Sov. J. Nucl. Phys. 21(1975) 51 10) G.D . Alkhazov, Effects of nucleon correlations and multi-step transitions in the cross séctions for inelastic fast proton scattering on nuclei, Leningrad preprint 599 11) V. Gillet and M.A. Melkanoff, Phys. Rev. 133B (1964) 1190 12) J.C . Bergstrom et al., Phys . Rev. Lett . 24 (1970) 152 13) D.R . Harrington and G.K . Varma, Phys . Lett . 74B (1978) 316 14) S.J. Wallace, Phys. Rev. C12 (1975) 179 15) G.S . Adams et al., Phys. Rev. Lett . 43 (1979) 421 16) G.S. Adams et al., Phys . Rev. C21 (1980) 2485 17) E. Bleszynski et al., UCLA preprint