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Quasielastic cluster knock-out reactions and the microscopic cluster model
Nuclear Physics @ North-Holland
A446 (1985) 703-726 Publishing Company
QUASIELASTIC CLUSTER KNOCK-OUT REACTIONS AND THE MICROSCOPIC CLUSTER MODEL R. BECK+ Kemforschungszentrum
and
F. DICKMANN+
Karlsruhe, Institut fiir Kernphysik III, P.O. B. 3640, Karlsruhe, Federal Republic of Germany R.G.
LOVAS
Kernforschungszentrum
Karlsruhe, Institut fiir Kemphysik III, P.O.B. 3640, Karlsruhe, Federal Republic of Germany* and Institute of Nuclear Research, Debrecen, P.O.B. 51, H-4001, Hungary Received 20 September 1984 (Revised 3 June 1985)
Abstract: The spectroscopic information contained in quasielastic cluster knock-out reactions is examined with a microscopic approach to the impulse approximation. It is shown that, because of the Pauli principle, the extracted spectroscopic factor is distinct from the probability (or amount) ofclustering. A formalism is elaborated for the calculation of these quantities in a generator-coordinate model for a superposition of different clusterizations. This formalism is used to study the (Y+d clustering properties of “Li described as a superposition of the (Y+ d and ‘He + p systems. The mod.el predicts the spectroscopic factor (amount of clustering) to be 1.04 (0.97) and 1.01 (0.94) for the ground state and first excited state, respectively. The calculated spectroscopic amplitude, as a function of the ad relative momentum, is in good agreement with those extracted from high-energy 6Li(p, pd)a and 6Li(cr, 2Lu)d experiments.
1. Introduction Quasielastic
cluster
knock-out
reactions
offer a powerful
clusterization properties of light nuclei. In this paper from the viewpoint of the microscopic cluster model. already been calculated in the resonating group (RG) there are both conceptual and technical problems in
tool for investigating
we shall study these properties Although these properties have method by several authors lW4), the application of microscopic
theories. First of all, the nature of the spectroscopic information needs clarifying. The conventional view is that it is the momentum density or, equivalently, the density of the intercluster relative motion that is extracted. In fact, the model transition amplitude contains, instead of a probability density of this kind, an overlap of the wave function of the target with those of the separate clusters. This quantity is sometimes called the fragmentation amplitude, reduced-width amplitude or spectroscopic amplitude. Although this quantity is not a probability amplitude 5), in practical ’ Deceased. * IAEA research
fellowship
in 1984. 703
R. Beck et al. / Quasielastic cluster knock-out
704
applications square, dubiosity,
it is invariably
the
so-called
attempts
antisymmetrical
regarded
spectroscopic
were made recently
framework
as a relative factor,
wave function,
as a probability.
to reformulate
“). In the new theory
and
its norm
To eliminate
the reaction
these conventional
theory
this
in a fully
quantities
are
replaced with new ones, to which probability meaning can be ascribed. Since in the well-established derivation of the cluster knock-out model ‘) the treatment of antisymmetrization is unclear, we shall reconsider this problem in this paper+. The microscopic models applied so far to the interpretation of knock-out experiments all contain a single clusterization. Such a restriction obviously tends to overestimate the role of the chosen clusterization and might lead to unrealistic, large values of the spectroscopic factors. Therefore, it is desirable to employ wave functions which contain other clusterizations as well. In this paper, we investigate the spectroscopic amplitude and related quantities of 6Li in the cy+d break-up channel. For this nucleus it has been demonstrated already that it is insufficient to consider only the cr + d clusterization. In their study of elastic (Y+ d scattering, Hackenbroich et al. “) also included the 3He+ t, 5He(5He*) + p and ‘Li(‘Li*) + n clusterizations, where the (unstable) nuclei 5He and ‘Li are both in their 2 “ground” and 4 excited states. For the ground state of 6Li, Krivec and Mihailovic “) showed that the inclusion of the ‘He+p and 5Li+n clusterizations lowers the energy by 1.1 MeV as compared to a pure (Y+d clusterization, if one of the effective interactions of Volkov lo) is employed. In the present investigation we adopt a simplified version of this model (taking the LY+d and ‘He+p clusterizations only) and study the influence of the second clusterization on the (Y+ d spectroscopic amplitude. In sect. 2 we rederive the transition amplitude in the distorted-wave impulse approximation (DWIA), carefully keeping track of the antisymmetrization. In sect. 3 we develop a formalism for extracting spectroscopic coordinate (GC) wave function which may contain
information from a generator several clusterizations (with
intrinsic orbital angular momenta, as in the case of ‘He). present and discuss our results for 6Li. 2. Fragmentation The conventional definition of a nucleus A into fragments
Finally,
in sect. 4 we
amplitude and transition amplitude of the fragmentation B and b is
amplitude
for the partitioning
+ The rigorous microscopic aspect of antisymmetrization as well as its loose treatment usual in phenomenological reaction models have been recently reviewed by Jackson 45). Her monograph contains exhaustive references to original papers. We thank one of the referees for drawing our attention to this work.
R. Beck et al. / Quasielastic
clusrer knock-out
705
(2.1)
where 4 and 8 are internal
wave functions
and coordinates,
respectively,
rBb is the
vector connecting the two centres of mass (cm.), and the intercluster antisymmetrizer dBb is defined as de, = C, ( -)“lP1, with the summation running over all intercluster permutations Pi, the parities of which being denoted by pi. The subscript x of #Q stands for the set of all quantum numbers. In the alternative formalism of Fliessbach expression, which, in a limiting case, reduces
“) the amplitude g enters to the “new amplitude”
into
G(r) =(rlii-“zlg), where a is an integral
operator
whose kernel
an
(2.2) A(r, r’) is
A( r, r’) =
It can be shown easily (see e.g. subsect. 3.4) that, when #A is a pure B + b two-cluster wave function, then (g/&‘/g) = (GIG) = 1, so that it is G(r) that may be considered a relative wave function, i.e. a probability amplitude. Whether or not we deal with a pure two-cluster configuration, we can introduce a spectroscopic factor pertinent to each of the amplitudes,
s = klg)
9
S=(GlG). From
the above
discussion
it is clear that
(2.4a) (2.4b)
S may be considered
a probability
of
finding A in the Bb cluster configuration, whereas s may not. As was mentioned in sect. 1, the appearance of one or the other of the fragmentation amplitudes in the reaction model depends vitally on the treatment of antisymmetrization. It is just in this respect that the presentation of the DWIA model by Chant and Roos ‘) lacks clarity. They start with an A(a, a’b)B transition amplitude of the form (+r’l V,,j+k”) [eq. (2) of ref. ‘)I, where both functions are antisymmetrized, and that is not the correctly antisymmetrized amplitude. Furthermore, in the manipulation ensuing [in eq. (4) of ref. 7)], they apparently commute dBb with the t-matrix operator, which is not justifiable. Because of these dubious steps, one is compelled to reconsider their formalism before saying anything about the fragmentation amplitude involved. That is what we are going to do now. We shall follow the 1Qgic of the derivation of Chant and Roos ‘) but try to avoid the same pitfalls.
706
R. Beck et al. / Quasielastic
In a fully antisymmetrical framework A(a, a’b)B may be written as
cluster knock-out
the transition
amplitude
for the process
x (+!-‘(B,b, a’)1Vba(i)i~Aa$‘+‘(A, a)),
(2.5)
where V,,,(i) is the interaction of fragments b and a in the ith permutation of the coordinates, $‘+‘(+$-‘) is a wave function belonging to the full hamiltonian H (to H - vb,( i)) with outgoing (ingoing) boundary condition that includes a plane wave in the entrance (exit) channel, and the summation runs over all interfragment permutations. The wave functions are understood to be antisymmetrical within the fragments throughout this section. They are normalized asymptotically as in usual (non-antisymmetrical) scattering theory. Eq. (2.5) is just formula (16.48) in Newton’s book ‘*) specified for the reaction A(a, a’b)B. It is derived there with full rigour, but its correctness can be made plausible in very simple terms as well. Since both dAa$(+) and xi ( -)p$I-’ V,,( i) are fully antisymmetrical, the (A+ a)!/B! b!a! terms of the sum in eq. (2.5) are equal. Therefore, eq. (2.5) can be written as
with (+‘-‘I vb, denoting any one of (+iP’I vb,( i). Let us consider eq. (2.6a). The bra-ket is the transition amplitude from the correct antisymmetrical initial state * . . ]1’2.dAa~(+) to a “final state” 4i-j belonging to any one of labellings i. Apart [ from kinematical factors, the cross section of the antisymmetrized theory must be equal to the sum, over all labellings i, of the modulus squares of such amplitudes. Since these amplitudes do not depend on i, the summation reduces to a multiplication by the number of labellings, (A + a)!/( B! b ! a !), in the cross section, or by [(A+ “* in the amplitude as is indeed so in eq. (2.6a). a)!/(B!b!u!)] Anticipating the model in which B is a mere spectator, one can neglect the terms that arise from particle exchange between B and a [ref. ‘)I. Hence eq. (2.6b) reduces to 9 a+a'bz By formal
definition
9
a+a’b
b, a’)1&&%&“+‘(A,a)) .
one can now introduce
Y-’a+a'b where
(&‘(B,
=
I*) a t-matrix
operator
(2.7) as follows:
I/* =
(4’-‘(B 3b , a’)lfl.&d(+)(A,
+‘+‘(A, a) is the wave function,
belonging
a)>,
to the hamiltonian
(2.8) H - V,,, with
701
R. Beck et al. / Quasielastic cluster knock-out
outgoing
boundary
condition
that includes
We can then assume ‘) that both 4’-‘(B, wave functions
4 and relative
a plane
wave in the entrance
b, a’) and +‘+‘(A, a) are products
wave functions
(or distorted
channel. of internal
waves, DW) q: (2.9)
Here the set of quantum numbers a’ is the same as that denoted by a except that the spin projection Ma is replaced by ML. The approximation in eq. (2.9) amounts to neglecting other channels. It is now convenient to introduce, for +A, the parentage expansion (2.10) To keep the model manageable, it is reasonable to assume ‘) that after substitution of eq. (2.10) in eq. (2.9) the 6 # b terms of the expansion give negligible contributions unless 6 differs from b only in M6 # Mb. (Since, however, the M6 sum contains just one non-zero term, we shall drop that sum for simplicity.) Moreover, we also assume ‘) that t does not act on &, i.e. B is a passive spectator. Then the integration over the coordinates B can be carried out, and we get 9 Now both &; relative momenta:
a+a’b’=
(2.11)
(~b’~a’7)~~~ltl~ba{~b~agA(Bb)77k+d}>.
can be expanded
and g,(,,,~~~’
d3k;,
d3kbb d3kA, d3kgb(&)al&,,
1
(4b’h’
x (kBb(gA(Bb))(kAa(7]afd’>
x Iddba{+b4a
exp
If we introduce the coordinate commutes with the antisymmetrizer
in terms
ii(kAa
exp
’ rAa+kBb
[i(kia
of eigenstates
of the
kb,,)
’ rAa+
%b
. %b)l
(2.12)
’ hb)l)).
r B,a+b= rB - (ar, + brb)( a + b)-‘, which obviously tiba, we may rewrite eq. (2.12) in the following
way: F-a+a’b’ = [ . * ’ Xid
I(~b’d%‘exp exP
Xdba{+b+a
{i[(%-kB/(A+a))
{i[(kB-kB/(A+a)) exp
’ rB,a+b+kk,
’ rbal}
’ rB,a+bl}
tikba
(2.13)
’ rba)}),
where k = k, + kb + k, is the total momentum. The next step is the impulse approximation, which assumes that the r-operator in eq. (2.13) is independent of the coordinate Then the integration over this variable can be carried out, yielding the rB,a+b, S-function (2r)3@k,+b-
k:+b) = (2n)3%ks-
kk) ,
(2.14)
708
R. Beck et al. / Quasielastic
which expresses
the conservation
cluster knock-out
of the c.m. momentum
of the a + b system,
/c~+~=
k, + k,,, or, alternatively, shows that the momentum of the spectator B is unchanged. After carrying out the integration over Q,~+~, we assume ‘) that the remaining t-matrix element varies sufficiently slowly with the momenta so that its arguments may be replaced by their asymptotic values, k b: and kg:. The remaining integration over the momenta may then be easily carried out by inserting the identities (2r)36(kL-
kB) =
d3rBb exp [i(kb,+
d3rbs s(r,,.J
1=
k&,-
yk,,-
kBb) . rBb] ,
* r,J,
exp [i(-yk,,+&)
(2.15)
I and using the relations (kL,+kb,)
. rBb+ k&, * r,,,= kba. rBa+ k& * rBb,
( $+,a + km,) ’ where
y = B/A. 9
a+a’b’
rBb +
kAa
’ rba =
kAa
’ rAa +
kBb
’ rBb
>
(2.16)
Thus we obtain =
($b’$a’
exp
(&it
I-
x J
’ rba)l
tldba{+b$a
exp
(ikci
’ rba)>>
r
d3 rba
s(rba>
d3rBb
v&?trBay
rBb)gA(Bb)(rBb)17~~(rAa),
t2.17)
J
which is identical to eq. (9) of ref. ‘) except that here the proper antisymmetrization of the t-matrix is displayed explicitly. Eq. (2.17) shows that the transition amplitude of a properly antisymmetrized DWIA for the knock-out process contains the old spectroscopic amplitude gA(Bb). It remains to see, however, whether the expression obtained could be transformed into the form of Fliessbach’s model. In his alternative model for direct cluster reactions Fliessbach “) assumes that the removal of particle b from B can be treated separately from the rest of the process. It is pointed out elsewhere 13) that such a separation cannot be carried out for a general knock-out reaction. In the DWIA, however, the transition amplitude may be reduced to this form, and therefore, it is instructive to seek conditions for the “new amplitude” GA(Bb) of eq. (2.2) to appear. The required “) form of the transition amplitude can be obtained by adopting the approximation ‘) n&L = n&,)( r&n&,)( rBb): S- a-a’b’=
(2.18a)
~&+k~‘2~kA(Bb))
= (,&‘lul
GA(Bb))
from eq. (2.17)
,
(2.18b)
where the notations v(r) = (I tl)[R~~‘2x~~‘( r)]*A,f”x&‘( yr) and ,y = A”2q have been introduced. The motivation “) for expressing n in terms of x is that it is x that is normalized as a scattering wave function in a non-antisymmetrical theory.
R. Beck ef al. / Quus~eZasficcluster knock-auf
The approximate
relation
(2.1Sb) becomes
const. This complicated function conditions are the following: exp [ i(-kL,-+
an exact equality
709
if and only if v(r) =
may be constant in the plane-wave limit, and the (i) ~~~/2x~;)=x&;a), a,~“x~~‘xr:,‘, and (ii)
ykAa) 5 r] = 1, i.e. -ykAa= kba. Condition
(i) is met if kb, and ykAa are
high enough, but condition (ii) is not satisfied in usual experiments; indeed, ykA, kh, is not even constant within a single experiment. Rather, I-ykAa- kL,l may be large enough for d~~‘*z~g,~~~~^- vg,(.,, to be assumed. Whether or not this is the case, however, the approximation (2.18b) is not valid and the “new amplitude” G AfBhj does not appear. From these considerations we have learned that the cluster knock-out analyses provide information on the conventional spectroscopic amplitude and factor, g,&_(Bb) and SA(Bbh rather than GAcBbl and SAtBbf. Although the formal properties of ga(sb) interpretation, it is obvious that these and SA(Bb) do not facilitate a probability quantities are also related to the cluster structure of A. In the following the nuclear-structure aspect of these quantities will be elaborated on, and the relationship of sA(&) to SA(Bb) will be studied. 3. The generator coordinate representation In this section involved in in eq. (2-l), a test state, model, and, state on the
we develop
a formalism
for the computation
of the quantities
the spectroscopy of quasielastic cluster knock-out reactions. As is seen the fragmentation amplitude is the overlap of the nuclear state 4A with which is a pure two-cluster state. We describe nucleus A in the GC for the sake of consistency, we opt for a formalism that treats the test same footing. However, in the test state the relative motion is described
by a &function, which is very inconvenient to treat in the GC formalism. To circumvent this difficulty, one can start, instead, with the overlap of the 6Li wave function with the product of the LYand d wave functions generated by oscillator wells fixed relative to each other, but the relationship of this amplitude to g(r) is rather involved 14). It may thus be more convenient to construct first the Fourier transform g(k) of g(r), which is the overlap of &,+ with the antisymmetrical wave function of two non-interacting ciusters. We can then obtain g(r) by an inverse Fourier transformation. Since a plane-wave analysis of the knock-out reaction provides directly g”(k) rather than g(r), this version of “GC spectroscopy” is particularly suited to such analyses. The nuclear wave functions considered consist of superpositions of various two-cluster wave functions. The GC wave function for the clusterization Bi + bi has the form (3.la) with the generator
function
710
R. Beck et al. / Quasielastic cluster knock-out
where s = S,,, - S,, and b$&, + BiSs, = 0. The functions @,, ( QB,) are antisymmetrized and normalized products of the lowest-lying single-particle states corresponding to oscillator potentials the same parameter
centred about S,,(S,t). All oscillator wells are characterized by /3 = mm/h, with m and w being the nucleon mass and the
oscillator frequency, respectively. In subsect. 3.1 we construct the wave function of the non-interacting clusters. In subsect. 3.2 we formulate the (quasi-) bound-state problem of +,+ Subsects. 3.3 and 3.4 describe the calculation of the fragmentation amplitudes and of the spectroscopic factors as well as amounts of clusterings, respectively.
3.1. THE CC WAVE
FUNCTION
OF NON-INTERACTING
CLUSTERS
The GC wave functions of non-interacting clusters have already been studied since they are involved, as the asymptotic states, in the GC description of scattering. Such asymptotic states may be constructed by solving the asymptotic Hill-Wheeler (HW) equation 15) or by expressing the corresponding RG wave function in GC terms 16-18). The equivalence of the two treatments is non-trivial. Here we treat the problem by solving the asymptotic HW equation and discuss the equivalence of the two treatments. The asymptotic
HW equation ,d’s’J;(S
J
reads ~‘)&(s)]H,,-E,~&~(s~))=0,
(3.2)
where Ha, contains no interaction between bi and Bi. If we assume that the clusters b, and B1 contain nucleons in relative OS orbits only, then the matrix elements for the fragmentation channel i = 1 are given by (&1(s)j&I(~‘))=exp[-$~(s-s’)2],
(3.3a)
(~~(s)lH,,-E,I~,(s’))/(~~(s)l~~(s’)) (3.3b) where pi is the reduced mass and pi = /?biBi/A is the reduced size parameter. In eqs. (3.2) and (3.3b) the energy Ek is the sum of the intrinsic energies of the non-interacting clusters and the kinetic energy (hk)‘/2pi of their relative motion. Eq. (3.2) is in fact the Schrodinger equation of a free particle of mass ki and energy (hk)‘/2pi in the GC representation. In a spherical basis, G,(s) = s-‘CLM @i( Ls) YfM( ;), f( k, s) = s-’ CLMfL( k, s) YLM (s1), and it can be shown 15) that the regular (irregular) solutions of this equation are spherical Bessel (Neumann) functions 19), s-‘fyg)(
k, s) = c( k)j,( ks) ,
(3.4a)
s-‘f;rr)(
k, s) = c( k)y,( ks) ,
(3.4b)
711
R. Beck et al. / Quasielastic cluster knock-out
where we have dropped in constructing be determined GC amplitudes
the subscript
i = 1. Unlike
in the description
of scattering
15),
g(k) it is important to know the normalization factor c(k). It may by calculating the wronskian 16) of the GC functions (3.1) with the (3.4a, b) and comparing it with the standard result 20) w = h2/2p,k.
In the GC representation,
the wronskian
:[fl”“‘(‘k,
k, s’)] .
(3.6)
is symmetrical under the interchange of s and s’, brackets is antisymmetrical. Thus any integration in the two-dimensional integral gives zero. We may by its asymptotic limit (s, s’+ 00) and obtain from
J J ds’
0
k, s)fp’(
cc
m
X
is
s)fl’“‘( k, s’) -f;“‘(
The matrix element in eq. (3.6) whereas the term in the square over a finite symmetrical region therefore replace the integrand eqs. (3.3), (3.4) and (3.6)
(3.5)
ds exp [-~P:(s-s’)~]
sin [k(s’-s)]
0
(3.7) where the integration
over s was performed
is important in the GCM is a consequence energy operator over the space of stationary (3.5) and (3.7), we find
first. The fact that the order of integration of the nonhermiticity scattering states. Upon
of the kinetic comparing eqs.
(3.8) From the regular solution (3.4a) we may now construct plane-wave solutions eq. (3.2) to obtain GC functions ]k) which describe two clusters moving relative each other with momentum k:
IW= J d3s~3n)-~/~~k, s)lo,(s)),
of to
(3.9a)
where (3.9b) which agrees with the result “) obtained by the conventional method. The method of calculating the asymptotic GC amplitude presented here may easily be generalized
R. Beck et al. / Quasielastic cluster knock-out
712
Ha, in eq. (3.2) still contains
to the case where the protons.
In that case, however,
the Coulomb
the result is different+
interaction
between
from that obtainable
from
the RG formalism. Owing to the antisymmetrization,
the normalized
free cluster
eq. (3.9), do not constitute an orthonormal set. For example, the Q + d fragmentation, the overlap of two generator functions (@,(s)]@,(s’))=
i
wave functions
]k),
when i = 1 denotes Q,(s) is given by *‘)
v’,‘,‘exp [-~p;(s2+sr2)+pw(l~)s.
s’],
(3.10a)
i=l
where V’,:’= 1 )
ui;’ = -2
u’l:’ = 1 3 (3.10b)
(1) _Z w11-3, Using this result together
w(2) -I 11 -6,
w’l;’ ZZ- f .
with eq. (3.9), we find
(k]k’)=S(k-k’)
where
(r$’ = $li f pw\\). The terms
i = 2,3 in eqs. (3.10a)
and
(3.11) causing
the
departure from orthonormalization arise from particle exchange. These terms, in contrast, do not contribute to the “asymptotic normalization”, expressed by the wronskian (3.6). 3.2. THE
GENERATOR
For a (quasi-) orbital momenta form 2’,22)
COORDINATE
WAVE
FUNCTION
FOR
BOUND
CLUSTERS
bound system, composed of several clusterizations i of intrinsic Iim, the unprojected GC wave function (3.1) can be written in the
!P =C 1 sf d~i~(l,mis,)~i(limis,), i I.%1 I ’ The Coulomb
interaction
way: The intracluster Z, (Z,)
is the number
(CI)
CI results
in the cluster ,,,c,,,
(Q’(s)l+
matrix
+M
element
(3.3b)
in the following
(2p/n)” to the energy E,, where KY b (B). The intercluster CI leads to the matrix element
lx, -x,l~‘l4(s’))
=2Z,Z,(s+s’l-’ where erf is the error function
the “asymptotic”
in the contribution
of protons (4(s)l
changes
(3.12)
erf (~Jpls+s’l)(i,(s)li,(s’)),
19). If the intercluster l&,(s’)) =ZZ,Z,(s+s’l-’
It is easy to see that only in this approximation of two charged point-like particles.
Cl is approximated
by Z,Z,/r,
then one gets
erf (tJ~ls+s’l)(l,(s)i~,(s’)).
is the HW, eq. (3.2), identical
to the Schriidinger
equation
R. Beck et al. / Quasielastic
where the sum runs over a discrete Qi into partial
cluster knock-out
set [a] of values
713
for the GC’s si. If we expand
waves @,(l,m,s;) =
s;’ c @i(L,M,zimisi) Yf&jii)
)
(3.13a)
L,V we may construct coordinates:
22) a spherical
tensor
@,([LiZi]cL?s/HSi) = 1
of rank
of the particle
.Y in the space
CDi(LiMilimiSi)(LiMi,
ZifTlilL?A) .
(3.13b)
Ml?
For the GC amplitude
in eq. (3.12), we use a similar J;(bWi)
=
sL’
C
f;tLiMirimJi)
expansion: YL,M,(ii)
(3.14a)
I
Of,
and introduce (3.14b) Substituting the inverted form of the expansions (3.13b) and (3.14b) into eqs. (3.13a) and (3.14a) and inserting these into eq. (3.12), we obtain a decomposition of !P in terms of functions of definite %U and parity n: (3.15) where the sum over Li and Z, is restricted problem of projecting the wave function
by the parity relation ( -)‘Lr+‘~) = n. The onto the values of the orbital
@,(f,m,s,)
angular momentum Z’A is thus reduced to an analysis of its tensor properties in the space of the GC’s s,. In practical calculations it is advantageous to perform this analysis on the matrix elements rather than on the wave functions. As an example, we consider
the overlap
matrix
element
of the wave functions
@, and
Q2, where
Q2 is the generator function of the clusterization 5He + p. In this clusterization, there is a neutron occupying an I= 1 orbit. Using the algebraic programming system REDUCE-2 [ref. “)I, we obtain *‘) for the overlap (@,(s)lQ2(lm’s’))=
2 u\i:?VylmJJ~s) i=l Xexp[-&3's2+/3's'2)+@v'i's~ I 2
12
s’]
(3.16a)
,
where &’
= - ($)“2,
(1) -I WI2
-3,
v(l;) = (ST)‘/*,
w(2) 12 -
-;,
(3.16b)
and 9,, denotes a solid spherical harmonic function 24)_Using the expansion (3.13a) for the bra and the ket in eq. (3.16a), it is straightforward to construct the spherical
R. Beck et al. / Quasielastic cluster knock-out
714
tensor
(3.13b),
and the result is
(3.17) Further examples of overlaps and matrix elements of the hamiltonian (containing a central, spin-orbit and tensor interaction) may be found in ref. *‘). By following the usual course, it is then simple to couple the nulceon spins to eigenvalues Iv and to couple the corresponding eigenstate with ?P(.JZA) to eigenstates ?P([Z1]JM) of the total angular momentum. The GC amplitudes J;([ L,Z,]P&) are determined variationally, by solving the coupled HW equations SE=G[(~lHIW)/(WI~)]=O,
(3.18)
where the hamiltonian has the form H = -( h2/2m)Ci VT +Cicj VP In the calculations reported in this paper, the nucleon-nucleon interaction Vj is approximated by the effective forces of Volkov lo) and Brink 25), and the Coulomb interaction is neglected. For the nucleon mass in 6Li, we use m =938.926 MeV/c*. Finally, we sum up what techniques we used in the structure calculation. First, the matrix elements of the GC functions a,(s) were calculated analytically with resort to algebraic programming. The result is expressed in terms of exponentials (some of them multiplied by polynomials in s and s’) depending on quadratic forms in s and s’. The coefficients specifying these functions were provided by a REDUCE program. The GC amplitude specifying the (quasi-) bound state of the system was then obtained 3.3. THE
a+d
by solving
the HW equation.
FRAGMENTATION
AMPLITUDE
IN 6Li
Having constructed the GC wave function of 6Li, we now proceed to discuss the spectroscopic information contained in it. The momentum-space (Y+ d fragmentation amplitude g”(k) is nothing but the overlap of the 6Li wave function ‘P([Z, 1= l]JM) with the wave function of cx and d [spins 1, = 0, (Id, vd) = (1, v)] in a relative plane-wave state of momentum k. We can expand g(k) as g”(k)=(kvIW([cYl]JM)) = ; (kl ~(~JU(=%
‘5
1 +‘W
~d;p(k)Y~~(R)(-i)-iPJ4rr(~~,
lv(JM),
(3.19)
where the total orbital angular momentum %t% comes entirely from the relative motion of the (Y+d cluster. In order to extract g&(k) from eq. (3.19), we introduce
R. Beck et al. / Quasielastic cluster knock-out
the GC representation,
715
eqs. (3.9) and (3.15), and obtain
(3.20) where we have introduced the notation i= (2LS 1)“2. The function &(k) is not identical but is related to the probability amplitude of the intercluster relative momentum just as gY(r) is related to that of the relative position. In the following we shall call I&(k)l” the fragmentation strength. It is interesting to note that this function contains the gaussian factor exp {-Ak’/[b(Ab)p]}, which is rather similar to the momentum distribution -exp [-i(hk/a)*] predicted by statistical models 26), where (T = CTJ(A - b)b/(A - l)]“‘= 114 MeV/c. With the value /? =0.48 fm-2, adopted in this work (see next section), we obtain (T = 112 MeV/c. The Z-wave fragmentation amplitude g,,(r) is related to the function &(k) by the integral transformation l/2 gY(r)=
If we substitute
&4(k)
1
0 7r
J
N
dk k2jy( kr)&(
k) .
(3.21a)
0
from eq. (3.20), we find
+2JFf;
i2( “02 :,
3
,x1 ~2_1-2([~2113~2)
1 (3.21b) where iy is a modified
spherical
Bessel function
19).
R. Beck et al. J Quasielastic cluster knock-out
716 3.4. SPECTROSCOPIC
FACTOR
AND
AMOUNT
OF CLUSTERING
In our model the schematic formula for the spectroscopic specified for the LY+ d clusterization as
factor,
eq. (2.4a),
is
a? s9=1[
Y
d3k ](k~\ V([Y’l]JM))1*
dk k21&(k)1’.
= 47r I0
If we use eq. (3.20) and carry out the integration,
(3.22a)
we find
s9 = &‘) + &*) + $2) )
(3.22b)
where
xcij @=4:Z$
(i) 011
(j).
v11 lz
(2P2 TWll
1 L*
ii2(-,Z @
~~~vCJ~w(‘)j
XC ij
(i)
(i)
w11
)
s,s; ,
1
11
Lz
0 CL2
(2p2 p:Wll
(i)
0 l
(i)
w12
(3.22~)
0 “>
SlS2
lS& Sffi(%,) ,’
I
7
(3.22d)
)
For later convenience, we have introduced the quantities @, tip;‘, which here are identical to v’,‘: and w(,i,‘),respectively. Because of the nonorthogonality of the vectors Ik), as expressed by eq. (3.11), the spectroscopic factor So must not be interpreted as the probability of finding, in 6Li, an (Y+ d cluster, which may also be called the amount of clustering. This probability may be defined as the norm of the “new” amplitude G(r). Here we resort to another definition, which can be visualized in geometrical terms, and later on show the equivalence of the two definitions.
717
R. Beck et al. / Quasielastic cluster knock-out
vector )P) which may be expressed
To find that part of the (normalized) of the (nonorthogonal)
basis
vectors
i.e. we determine .
IU’)-~d3kT(k)lk),
Ik), we minimize the function
T(k)
d3kr(k)lk)ll
=O.
W-J
the length
in terms
of the vector
by the variational
equation (3.23a)
The minimum value of the norm square (( ]I measures the size of the component, of )V), that is orthogonal to the clusterization spanned by {(k)}. It is therefore judicious to define the amount of clustering represented by {Jk)} as (3.24a) where T(k) is a solution of eq. (3.23a). Obviously, 0~ SS 1. On carrying out the variation in eq. (3.23a) and using the function g(k), eq. (3.19), we find that T(k) obeys the integral equation g(k)
J J
=
while S is given by S=
d3k’ T(k’)(klk’)
,
(3.23b)
d3kg*(k)l-(k).
(3.24b)
The quantities g(k) and T(k) are the covariant and contravariant components of the vector I P) in the basis Jk). Eq. (3.24b) expresses that the amount of clustering is the length square of that part of 1?P) that lies in the space spanned by the vectors
IQ. To show that eqs. (3.23b) and (3.24b) are consistent with our earlier definition, eq. (2.4b), ofthe amount of clustering, we substitute the formal solution of eq. (3.23b), T(k)
into eq. (3.24b),
=
with the result that S =
J
d3k’(k(k’)-Ii(
(3.23~)
J
d3k d3k’ g(k)*(k(k')-'g(k')
,
(3.24~)
which is nothing but eq. (2.4b), written in the momentum representation. We shall now recast eq. (3.23b) into a form suitable for GC wave functions. the 1.h.s. we use eq. (3.15) to express the r.h.s. we use the expansion
tY in terms of the generator
r(k2iA)=~ci(k=YM)=~
1
s;
f LM,b,[s,l
X~i(kfim,a)
i;(klimis;)=(kJ~,(Iim;s,))+
J
Qti On
J
d?;f;([ L,&?.&)
YL,M,(S:)(LiM,, Z,mil~~)
In this way eq. (3.23b) becomes
functions
On
.
d3k’~i(k’limPi)(k)k’)‘)
(3.25)
(3.26)
718
R. Beck et al. / Quasielastic cluster knock-out
where (k( k’)’ = S (k - k’) - (k( k’). It is important to notice that in eq. (3.26) the different clusterizations i are not coupled. If there are two s-wave clusters in clusterization i = 1, for i = 1 eq. (3.26) can be solved
analytically,
with the result
P,(k,s)=(r/3{))3’4exp(-ik*s-k2/2P;). Substituting
this formula
(3.27a)
into eq. (3.25), we can prove that
d3kT,(kZ&)(k) = ‘P, . (3.27b) J‘ Because of eq. (3.24a) this implies that, for a single-clusterization model, S = 1 holds as it should. [It is, however, not justifiable to expect 27,28) that the spectroscopic factor sZ, eq. (3.22b), also obeys such a simple rule!] Furthermore, for a twoclusterization model, eqs. (3.24a) and (3.27b) imply
d3kT2(k)(!P,Jk).
(3.28)
Solving eq. (3.26) for i = 2 by iteration, and substituting the result as well as jk) of eq. (3.9) and the i = 2 term of eq. (3.15) into eq. (3.28), we obtain
X
-(~2([L211~~2)]~2([L;ll~~;))+ IF C $P(n, 4,. . . , in> , n=O I
(3.29)
i,...i.
where the n-summation comes from the terms of p2 obtained via the iterative solution, and S($) is obtained from sy) of eq. (3.22e) by redefining fi$ and 3%’ as ),
(3.30a)
(3.30b) In the actual calculation sum over n in eq. (3.29) taken into account. (The 3 to 4, was less than 5 x
to be discussed in the next section it turned out that the is rapidly converging, so that only a few terms have to be relative change in S,, obtained by increasing max (n) from 10e4.)
4. Results and discussion In the previous section we expressed the quantities involved in cluster reaction spectroscopy in a calculable form. Now we show their application to the (Y+ d clustering in 6Li.
R. Beck et al. / Quasielastic cluster knock-out 4.1. SOLUTION
OF THE
HILL-WHEELER
EQUATION
FOR
719
6Li
We solve the coupled HW equations (3.18) for the ground (9 = 0) and first-excited (2 = 2) state in 6Li, using the truncated trial function [cf. eq. (3.15)]
+ 2 f*([L,
= 1, 1, = l]&)CD2([L2
= 1, 12= l]Z&J
.
(4.1)
[%I
(The nucleon spins are coupled to I = 1. The interactions between the J-values allowed.) While for the ground state possibility, for the excited state the choice (4.1) amounts l2 = 1 component. The unstable excited state was treated as the variational stability condition 29) being approximately
used do not distinguish L, = 1, Z2= 1 is the only to neglecting the Lz=3, a quasibound state, with satisfied. The results for
the effective nucleon-nucleon interactions number 1 and 2 of Volkov lo) (“Volkov 1” and “Volkov 2”) and that of Brink 25) (“Brink”) are listed in table 1. For both the ground and the excited state, the minimum values of the energies are obtained with p = 0.48 fin-*. Before comparing our results with experimental data, we should mention that the splitting of the triplet (9 = 2, I = 1) state in 6Li into three levels J” = 3+, 2+ and l+, with the excitation energies “) 2.185, 4.31 and 5.65 MeV, can, of course, not be reproduced with a spin-independent force. Therefore, we compare the calculated excitation energy with the average value of the experimental excitation energies, E,, = 3.6 MeV. Although the calculated ground-state binding energy is too small, the a +d separation energy and the excitation energy of the first excited state are in qualitative agreement with experiment. From the point of view of the (Y+ d fragmentation this is very important. The reasonability of the separation energy values indicates that the minor binding-energy
defect of 6Li is likely to arise from the failure
of the forces
to bind the individual clusters properly. In fact, the cu-particle binding energies are also reasonable, and so the blame is to be put solely on the deuteron. From table 1 it is seen that the coupling of the ‘He+p clusterization lowers the ground-state energy
by -0.6
MeV. It is interesting
to compare
our results
for the ground-state
energy (-26.54 MeV) and excitation energy (3.25 MeV), obtained with the Volkov 1 interaction, to a recent calculation 3’) by Filippov et al., who found -26.11 MeV and 4.08 MeV, respectively, using the same force and an expansion of the wave function in terms of hyperspherical functions. For the case of the Volkov 2 force, we may compare our result for the ground state energy (-28.08 MeV) with the one (-28.3 MeV) obtained by Krivec and MihailoviC 9), who considered the three coupled clusterizations (Y+ d, ‘He+ p and ‘Li+n. We conclude that, as far as the ground-state energy is concerned, the inclusion of the third clusterization is only of minor importance.
R. Beck et al. / Quasielasric cluster knock-out
720
TABLE
1
Energies and rms radii r of the ground state and first excited state of 6Li in the (Y+ d and in the {a +d, ‘He-tp} models with the Volkov 1, Volkov 2 and Brink interactions Volkov Ground
Es., “1 [Mevl
a+d a+d,‘He+p
1
state *) -25.91 -26.54
Volkov 2
-21.46 -28.08
Brink
-25.71 -26.26
a+d a+d,‘He+p
1.15 1.78
2.29
0.55 1.10
a+d a+d,‘He+p
2.64 2.68
2.61 2.65
2.77 2.78
1.61
First excited stare ‘? a+d a+d,5He+p
3.54 3.25
3.60 3.21
3.55 3.35
atd a+d,‘He+p
2.15 2.72
2.69 2.68
3.01 2.90
“) Binding energy. b, Separation energy. ‘) Excitation energy. d, Experimental values from ref. 30): Es,, = -31.9929 MeV; E, = 1.4735 MeV; 2.51 f 0.1 fm; E,, = 3.6 MeV, weighted average for the Z’= 2 triplet.
r=
In table 1 we also give the results for the rms charge radius. The reasonability of the cluster-model description is also shown by the fact that the ground-state radius is well reproduced.
In all cases the rms radius
of the 9? = 2 state is larger than that
of the .Z = 0 ground state. This may be interpreted as a centrifugal stretching effect and is in accordance with the findings of Kanada et al. 32) From table 1 one can infer that this stretching effect is caused by the (Y+d clusterization only, and the ‘He + p clusterization tends to weaken the magnitude of stretching. Indeed, for the 9 = 0 state, the coupling of the 5He+ p to the (Y+ d clusterization results in an increase of the rms radius, while for the 9 = 2 state, the coupling causes a decrease.
4.2. SPECTROSCOPIC
INFORMATION
ON 6Li
With the wave function of eq. (4.1), the function I&(k)l’ for the cy+d fragmentation can be readily computed by using eq. (3.20). In fig. 1 we compare l&,(k)l’ for the ground state as predicted by the Volkov 2 force with experimental data extracted from a plane-wave analysis 33). As is seen from this figure, the effect of switching on the 5He+p clusterization (full curve) is to lower l&(k)l’ for ks 0.3 fin-’ as
R. Beck et al. / Quasielastic cluster knock-out
5
721
4 (a, 2U) 700 MeV
k(fm-‘1 Fig. 1. 6Li + a + d momentum-space fragmentation strengths. The data are extracted “) from 6Li(p, pd)cr experiments at 590 MeV and from 6Li(cy, 2a)d experiments at 700 MeV. The theoretical curves are obtained from the {cr+d, 5He+p} cluster model (solid line) and the a +d cluster model (dashed line).
compared to the case of the pure (Y+ d clusterization (broken curve), thus improving the agreement with the experimental data. For k* 0.4 fin-’ both models give the same
result.
In this region,
compared to the data. To illuminate the quality
however, of agreement
we should point out what are in the theoretical model. For approximation is questionable of ~~o(k)~2 from experiment. Nevertheless,
the predicted
distribution
is too small
as
one can expect from such a comparison,
the uncertainties in the experimental data as well as momenta k larger than -0.4 fm-‘, the plane-wave ‘“) and the DWIA should be used for the extraction Thus one should not expect any better agreement.
the PWIA does have the advantage
that it is free from the inevitable
arbitrariness associated with the optical potentials that enter into the DWIA. On the other hand, the specific distortion 32) of the deuteron at small distances from the a-particle might not be fully taken into account by admixing the ‘He+p clusterization, giving rise to uncertainties in l&(k)l’ at large momenta k. It is instructive to study the fragmentation amplitude g,(r), eq. (3.21b), which is displayed in fig. 2. This curve shows a node at r = 1.7 fm, which is required 34) by the Pauli principle. The result that there is practically no difference between the spectroscopic amplitude of the coupled clusterizations (solid curve) and that of the
122
I?. Beck et al. / Quasielastic
-0.1 0 Fig. 2. Fragmentation
amplitudes
I 2
I 4
I G r(fm)
cluster knock-out
I 8
(6Li + a + d) for the 6Li ground as in fig. 1.
I 10
12
state obtained
from the cluster models
pure (Y+ d clusterization (dashed curve) for distances r c 4 fm reflects the effect of the Pauli principle, which reduces greatly the differences 35) between apparently different cluster wave functions. Finally, we note that the three-body model of 6Li (a-particle plus two nucleons) introduced by Lehmann and Mamta Rajan 36), predicts an effective cr +d wave function whose shape is rather similar (except for large distances) to the fragmentation amplitude g,(r) obtained in this paper. If we now apply our model to the first excited state (2 = 2) in 6Li and calculate the function Jg2( k)12 for the Volkov 2 force, we find the result shown in fig. 3. Again, the dashed (solid) curve represents the result for the pure (Y+ d (coupled) clusterization( Each of the curves shows a peak at E= 0.5 fm-‘. In a classical picture, we may relate the kinetic energy ( hE)2/2p, = 3.9 MeV to the excitation energy, E,, = 3.6 MeV. This interpretation is consistent with that fact that, for the ground state, the peak in the momentum distribution, which is probably close to I&(k)12, occurs at k = 0. From a comparison of figs. 1 and 3, it is seen that the difference in I&( k)(’ between the model of the coupled clusterizations and of the pure (Y+ d clusterization is more pronounced for the 2 = 2 state than for the Z’= 0 state. This result may be understood in terms of the centrifugal stretching of the Z? = 2 state, which obviously diminishes the effect of the Pauli principle. The spectroscopic amplitude g2(r) for the 58 = 2 state is depicted in fig. 4. Here one finds both in the coupled-clusterization (solid curve) and single-clusterization
R. Beck ef al. / Quasielastic
,I
0.Y
cluster knock-out
723
.\ :
:
,
Y -
2s
-0.2
0.1
.
0.c 0
1.0
0.5
1.5
k (fin-’ I Fig. 3. Comparison
of the ‘Li*+ a +d momentum-space fragmentation strengths state in 6Li obtained from the cluster models as in fig. 1.
0 Fig. 4. Fragmentation
amplitudes
2
il
6 r(fml
8
10
for the first excited
12
j6Li* + cy+ d) for the first excited state in ‘Li obtained models as in fig. 1.
from the cluster
R. Beck et al. f Quasielastic cluster knock-out
724
(dashed curve) case that there is no node and the curves are shifted to larger distances as compared to the ground state, again indicating a centrifugal stretching. Finally, calculated
we discuss
the spectroscopic
with our wave functions
factor sY and the amount of clustering S, for both the ground and excited state. The results
obtained with the forces Volkov 1, Volkov 2 and Brink are given in table 2. All spectroscopic factors sY are larger than unity, showing that they must not be interpreted as probabilities. The coupling of the 5He+p clusterization results in a small reduction of sY. The experimentally deduced spectroscopic factors for the 6Li ground state range from 0.04 (3~0.12) [ref. “)I to 1.75 (kO.5) [ref. “)I with many intermediate values 4~27~28~33~38~42), depending on the reaction and on the method of extraction. The data 33) given in table 2 are extracted from (p, pd) reactions at 590 MeV and from (a, 2cz) reactions at 700 MeV, using the PWIA. We prefer these data to those 33) taken at the lower energies of 156 and 64 MeV, respectively, because at higher energies the impulse approximation seems better justified. The calculated spectroscopic factors are slightly smaller than both the experimental values 33) and the result obtained by Kurdyumov et al. 43) from an (Y+ d harmonic-oscillator cluster model.
TABLE 2 Alpha spectroscopic factor s and the amount of u-clustering S of the ground state and first excited state of 6Li in the a + d, {n + d, 5He + p} models with the Volkov 1, Volkov 2 and Brink interactions Volkov
cu+d
a+d,5He+p
o+d
a+d,sHe+p
1
s
Ground state “) 1.071
S
1
Volkov 2
Brink
1.073
1.059
1
1
s
1.038
1.041
1.033
S
0.967
0.969
0.972
1.069
1.046
1
1
s
First excited stare 1.065
s
1
s
1.010
1.013
1.006
s
0.940
0.942
0.952
“) s= 1.08: (p, pd) experiment at 590 MeV [ref.33)]; experiment at 700 MeV [ref. “)I; s = 1.05: harmonic-oscillator lation 43).
s= 1.05*0.12: cluster-model
(a,2a) calcu-
The amount of LY+d clustering S,,, which is unity for the pure a + d model, is reduced for the case of the coupled clusterizations. For the 3 = 2 state, both the spectroscopic factors s2 and amounts of (Y+ d clustering S2 are slightly smaller than
R. Beck et al. / Quasielastic cluster knock-out
725
the corresponding quantities of the Y = 0 state. This is consistent with our earlier finding (see figs. 1 and 3) that the second clusterization (‘He+p) is more important for the 2 = 2 than for the .Z = 0 state. We can sum up the results of this paper as follows. We have shown how to apply the viewpoint of the microscopic cluster model to the description of quasielastic cluster knock-out reactions. It was pointed out that, because of the Pauli principle, the measured spectroscopic factor differs from the amount of clustering, and only the latter quantity may be interpreted as a probability. To illustrate these considerations, we chose a version, of the GCM, that describes nuclei in terms of various clusterizations with the internal motion of the clusters only disturbed by the other nucleons through the Pauli principle. It was found that this GCM is well suited not only to describing the ingredients of these reactions but to illustrating the general concepts as well. The (Y+ d fragmentation properties of 6Li are reproduced by this model reasonably well. The amount of clustering does differ from the spectroscopic factor though by a few percent only. We would like to mention that the application, to this problem, of our more recent model 44), which assumes one pair of breafhing clusters rather than several frozen clusterizations, is under way. One of the authors (R.G.L.) wishes to thank Professor G. Schatz and the Kernforschungszentrum Karlsruhe for their kind hospitality and the IAEA for a fellowship.
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22) R. Beck and F. Dickmann, Kernforschungszentrum Karlsruhe, report KfK-3402, 1982 23) A.C. Hearn, REDUCE 2 user’s manual (University of Utah, 1973) 24) A.R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957) 25) D.M. Brink, Proc. lnt. School of Physics, Enrico Fermi, course XXXVI (1966) p. 247 26) AS. Goldhaber, Phys. Lett. 53B (1974) 306 27) J.W. Watson, H.G. Pugh, P.G. Roos, D.A. Goldberg, R.A.J. Riddle and D.I. Bonbright, Nucl. Phys. A172 (1971) 513 28) P.G. Roos, N.S. Chant, A.A. Cowley, D.A. Goldberg, H.D. Holmgren and R. Woody III, Phys. Rev. Cl5 (1977) 69 29) R.G. Lovas and M.A. Nagarajan, J. of Phys. Al5 (1982) 2383 30) F. Ajzenberg-Selove, Nucl. Phys. A413 (1984) 1 31) G.F. Filippov, V.S. Vasilevskii and S.P. Kruchinin, Izv. Akad. Nauk SSSR (ser. fiz.) 44 (1980) 2313 32) H. Kanada, T. Kaneko and Y.C. Tang, Nucl. Phys. A389 (1982) 285 33) W. Dollhopf, C.F. Perdrisat, P. Kitching and W.C. Olsen, Phys. Lett. 58B (1975) 425 34) Y.C. Tang, K. Wildermuth and L.D. Perlstein, Phys. Rev. 123 (1961) 548 35) K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977) 36) D.R. Lehmann and M. Rajan, Phys. Rev. C25 (1982) 2743 37) K. B%hr, T. Becker, O.M. Bilaniuk and R. Jahr, Phys. Rev. 178 (1969) 1706 38) M. Jain, P.G. Roos, H.G. Pugh and H.D. Holmgren, Nucl. Phys. A153 (1970) 49 39) C. Ruhla, M. Riou, M. Gusakow, J.C. Jacmart, M. Liu and L. Valentin, Phys. Lett. 6 (1963) 282 40) P. Kitching, W.C. Olsen, H.S. Sherif, W. Dollhopf, C. Lunke, CF. Perdrisat, J.R. Priest and W.K. Roberts, Phys. Rev. Cl1 (1975) 420 41) D. Albrecht, M. Csatl&, J. Erii, Z. Fodor, I. Hernyes, Hongsung Mu, B.A. Khomenko, N.N. Khovanskij, P. Koncz, Z.V. Krumstein, Yu. P. Merekov, V.I. Petrukhin, Z. Seres and L. Vegh, Nucl. Phys. A338 (1980) 477 42) J. Cook, Nucl. Phys. A417 (1984) 477 43) I.V. Kurdyumov, V.G. Neudatchin and Yu.F. Smirnov, Phys. Lett. 31B (1970) 426 44) R. Beck, F. Dickmann and A.T. Kruppa, Phys. Rev. C30 (1984) 1044 45) D.F. Jackson, in Clustering phenomena in nuclei, ed. K. Wildermuth and P. Kramer, vol. 3 (Vieweg, Braunschweig, 1983) p. 157
A446 (1985) 703-726 Publishing Company
QUASIELASTIC CLUSTER KNOCK-OUT REACTIONS AND THE MICROSCOPIC CLUSTER MODEL R. BECK+ Kemforschungszentrum
and
F. DICKMANN+
Karlsruhe, Institut fiir Kernphysik III, P.O. B. 3640, Karlsruhe, Federal Republic of Germany R.G.
LOVAS
Kernforschungszentrum
Karlsruhe, Institut fiir Kemphysik III, P.O.B. 3640, Karlsruhe, Federal Republic of Germany* and Institute of Nuclear Research, Debrecen, P.O.B. 51, H-4001, Hungary Received 20 September 1984 (Revised 3 June 1985)
Abstract: The spectroscopic information contained in quasielastic cluster knock-out reactions is examined with a microscopic approach to the impulse approximation. It is shown that, because of the Pauli principle, the extracted spectroscopic factor is distinct from the probability (or amount) ofclustering. A formalism is elaborated for the calculation of these quantities in a generator-coordinate model for a superposition of different clusterizations. This formalism is used to study the (Y+d clustering properties of “Li described as a superposition of the (Y+ d and ‘He + p systems. The mod.el predicts the spectroscopic factor (amount of clustering) to be 1.04 (0.97) and 1.01 (0.94) for the ground state and first excited state, respectively. The calculated spectroscopic amplitude, as a function of the ad relative momentum, is in good agreement with those extracted from high-energy 6Li(p, pd)a and 6Li(cr, 2Lu)d experiments.
1. Introduction Quasielastic
cluster
knock-out
reactions
offer a powerful
clusterization properties of light nuclei. In this paper from the viewpoint of the microscopic cluster model. already been calculated in the resonating group (RG) there are both conceptual and technical problems in
tool for investigating
we shall study these properties Although these properties have method by several authors lW4), the application of microscopic
theories. First of all, the nature of the spectroscopic information needs clarifying. The conventional view is that it is the momentum density or, equivalently, the density of the intercluster relative motion that is extracted. In fact, the model transition amplitude contains, instead of a probability density of this kind, an overlap of the wave function of the target with those of the separate clusters. This quantity is sometimes called the fragmentation amplitude, reduced-width amplitude or spectroscopic amplitude. Although this quantity is not a probability amplitude 5), in practical ’ Deceased. * IAEA research
fellowship
in 1984. 703
R. Beck et al. / Quasielastic cluster knock-out
704
applications square, dubiosity,
it is invariably
the
so-called
attempts
antisymmetrical
regarded
spectroscopic
were made recently
framework
as a relative factor,
wave function,
as a probability.
to reformulate
“). In the new theory
and
its norm
To eliminate
the reaction
these conventional
theory
this
in a fully
quantities
are
replaced with new ones, to which probability meaning can be ascribed. Since in the well-established derivation of the cluster knock-out model ‘) the treatment of antisymmetrization is unclear, we shall reconsider this problem in this paper+. The microscopic models applied so far to the interpretation of knock-out experiments all contain a single clusterization. Such a restriction obviously tends to overestimate the role of the chosen clusterization and might lead to unrealistic, large values of the spectroscopic factors. Therefore, it is desirable to employ wave functions which contain other clusterizations as well. In this paper, we investigate the spectroscopic amplitude and related quantities of 6Li in the cy+d break-up channel. For this nucleus it has been demonstrated already that it is insufficient to consider only the cr + d clusterization. In their study of elastic (Y+ d scattering, Hackenbroich et al. “) also included the 3He+ t, 5He(5He*) + p and ‘Li(‘Li*) + n clusterizations, where the (unstable) nuclei 5He and ‘Li are both in their 2 “ground” and 4 excited states. For the ground state of 6Li, Krivec and Mihailovic “) showed that the inclusion of the ‘He+p and 5Li+n clusterizations lowers the energy by 1.1 MeV as compared to a pure (Y+d clusterization, if one of the effective interactions of Volkov lo) is employed. In the present investigation we adopt a simplified version of this model (taking the LY+d and ‘He+p clusterizations only) and study the influence of the second clusterization on the (Y+ d spectroscopic amplitude. In sect. 2 we rederive the transition amplitude in the distorted-wave impulse approximation (DWIA), carefully keeping track of the antisymmetrization. In sect. 3 we develop a formalism for extracting spectroscopic coordinate (GC) wave function which may contain
information from a generator several clusterizations (with
intrinsic orbital angular momenta, as in the case of ‘He). present and discuss our results for 6Li. 2. Fragmentation The conventional definition of a nucleus A into fragments
Finally,
in sect. 4 we
amplitude and transition amplitude of the fragmentation B and b is
amplitude
for the partitioning
+ The rigorous microscopic aspect of antisymmetrization as well as its loose treatment usual in phenomenological reaction models have been recently reviewed by Jackson 45). Her monograph contains exhaustive references to original papers. We thank one of the referees for drawing our attention to this work.
R. Beck et al. / Quasielastic
clusrer knock-out
705
(2.1)
where 4 and 8 are internal
wave functions
and coordinates,
respectively,
rBb is the
vector connecting the two centres of mass (cm.), and the intercluster antisymmetrizer dBb is defined as de, = C, ( -)“lP1, with the summation running over all intercluster permutations Pi, the parities of which being denoted by pi. The subscript x of #Q stands for the set of all quantum numbers. In the alternative formalism of Fliessbach expression, which, in a limiting case, reduces
“) the amplitude g enters to the “new amplitude”
into
G(r) =(rlii-“zlg), where a is an integral
operator
whose kernel
an
(2.2) A(r, r’) is
A( r, r’) =
It can be shown easily (see e.g. subsect. 3.4) that, when #A is a pure B + b two-cluster wave function, then (g/&‘/g) = (GIG) = 1, so that it is G(r) that may be considered a relative wave function, i.e. a probability amplitude. Whether or not we deal with a pure two-cluster configuration, we can introduce a spectroscopic factor pertinent to each of the amplitudes,
s = klg)
9
S=(GlG). From
the above
discussion
it is clear that
(2.4a) (2.4b)
S may be considered
a probability
of
finding A in the Bb cluster configuration, whereas s may not. As was mentioned in sect. 1, the appearance of one or the other of the fragmentation amplitudes in the reaction model depends vitally on the treatment of antisymmetrization. It is just in this respect that the presentation of the DWIA model by Chant and Roos ‘) lacks clarity. They start with an A(a, a’b)B transition amplitude of the form (+r’l V,,j+k”) [eq. (2) of ref. ‘)I, where both functions are antisymmetrized, and that is not the correctly antisymmetrized amplitude. Furthermore, in the manipulation ensuing [in eq. (4) of ref. 7)], they apparently commute dBb with the t-matrix operator, which is not justifiable. Because of these dubious steps, one is compelled to reconsider their formalism before saying anything about the fragmentation amplitude involved. That is what we are going to do now. We shall follow the 1Qgic of the derivation of Chant and Roos ‘) but try to avoid the same pitfalls.
706
R. Beck et al. / Quasielastic
In a fully antisymmetrical framework A(a, a’b)B may be written as
cluster knock-out
the transition
amplitude
for the process
x (+!-‘(B,b, a’)1Vba(i)i~Aa$‘+‘(A, a)),
(2.5)
where V,,,(i) is the interaction of fragments b and a in the ith permutation of the coordinates, $‘+‘(+$-‘) is a wave function belonging to the full hamiltonian H (to H - vb,( i)) with outgoing (ingoing) boundary condition that includes a plane wave in the entrance (exit) channel, and the summation runs over all interfragment permutations. The wave functions are understood to be antisymmetrical within the fragments throughout this section. They are normalized asymptotically as in usual (non-antisymmetrical) scattering theory. Eq. (2.5) is just formula (16.48) in Newton’s book ‘*) specified for the reaction A(a, a’b)B. It is derived there with full rigour, but its correctness can be made plausible in very simple terms as well. Since both dAa$(+) and xi ( -)p$I-’ V,,( i) are fully antisymmetrical, the (A+ a)!/B! b!a! terms of the sum in eq. (2.5) are equal. Therefore, eq. (2.5) can be written as
with (+‘-‘I vb, denoting any one of (+iP’I vb,( i). Let us consider eq. (2.6a). The bra-ket is the transition amplitude from the correct antisymmetrical initial state * . . ]1’2.dAa~(+) to a “final state” 4i-j belonging to any one of labellings i. Apart [ from kinematical factors, the cross section of the antisymmetrized theory must be equal to the sum, over all labellings i, of the modulus squares of such amplitudes. Since these amplitudes do not depend on i, the summation reduces to a multiplication by the number of labellings, (A + a)!/( B! b ! a !), in the cross section, or by [(A+ “* in the amplitude as is indeed so in eq. (2.6a). a)!/(B!b!u!)] Anticipating the model in which B is a mere spectator, one can neglect the terms that arise from particle exchange between B and a [ref. ‘)I. Hence eq. (2.6b) reduces to 9 a+a'bz By formal
definition
9
a+a’b
b, a’)1&&%&“+‘(A,a)) .
one can now introduce
Y-’a+a'b where
(&‘(B,
=
I*) a t-matrix
operator
(2.7) as follows:
I/* =
(4’-‘(B 3b , a’)lfl.&d(+)(A,
+‘+‘(A, a) is the wave function,
belonging
a)>,
to the hamiltonian
(2.8) H - V,,, with
701
R. Beck et al. / Quasielastic cluster knock-out
outgoing
boundary
condition
that includes
We can then assume ‘) that both 4’-‘(B, wave functions
4 and relative
a plane
wave in the entrance
b, a’) and +‘+‘(A, a) are products
wave functions
(or distorted
channel. of internal
waves, DW) q: (2.9)
Here the set of quantum numbers a’ is the same as that denoted by a except that the spin projection Ma is replaced by ML. The approximation in eq. (2.9) amounts to neglecting other channels. It is now convenient to introduce, for +A, the parentage expansion (2.10) To keep the model manageable, it is reasonable to assume ‘) that after substitution of eq. (2.10) in eq. (2.9) the 6 # b terms of the expansion give negligible contributions unless 6 differs from b only in M6 # Mb. (Since, however, the M6 sum contains just one non-zero term, we shall drop that sum for simplicity.) Moreover, we also assume ‘) that t does not act on &, i.e. B is a passive spectator. Then the integration over the coordinates B can be carried out, and we get 9 Now both &; relative momenta:
a+a’b’=
(2.11)
(~b’~a’7)~~~ltl~ba{~b~agA(Bb)77k+d}>.
can be expanded
and g,(,,,~~~’
d3k;,
d3kbb d3kA, d3kgb(&)al&,,
1
(4b’h’
x (kBb(gA(Bb))(kAa(7]afd’>
x Iddba{+b4a
exp
If we introduce the coordinate commutes with the antisymmetrizer
in terms
ii(kAa
exp
’ rAa+kBb
[i(kia
of eigenstates
of the
kb,,)
’ rAa+
%b
. %b)l
(2.12)
’ hb)l)).
r B,a+b= rB - (ar, + brb)( a + b)-‘, which obviously tiba, we may rewrite eq. (2.12) in the following
way: F-a+a’b’ = [ . * ’ Xid
I(~b’d%‘exp exP
Xdba{+b+a
{i[(%-kB/(A+a))
{i[(kB-kB/(A+a)) exp
’ rB,a+b+kk,
’ rbal}
’ rB,a+bl}
tikba
(2.13)
’ rba)}),
where k = k, + kb + k, is the total momentum. The next step is the impulse approximation, which assumes that the r-operator in eq. (2.13) is independent of the coordinate Then the integration over this variable can be carried out, yielding the rB,a+b, S-function (2r)3@k,+b-
k:+b) = (2n)3%ks-
kk) ,
(2.14)
708
R. Beck et al. / Quasielastic
which expresses
the conservation
cluster knock-out
of the c.m. momentum
of the a + b system,
/c~+~=
k, + k,,, or, alternatively, shows that the momentum of the spectator B is unchanged. After carrying out the integration over Q,~+~, we assume ‘) that the remaining t-matrix element varies sufficiently slowly with the momenta so that its arguments may be replaced by their asymptotic values, k b: and kg:. The remaining integration over the momenta may then be easily carried out by inserting the identities (2r)36(kL-
kB) =
d3rBb exp [i(kb,+
d3rbs s(r,,.J
1=
k&,-
yk,,-
kBb) . rBb] ,
* r,J,
exp [i(-yk,,+&)
(2.15)
I and using the relations (kL,+kb,)
. rBb+ k&, * r,,,= kba. rBa+ k& * rBb,
( $+,a + km,) ’ where
y = B/A. 9
a+a’b’
rBb +
kAa
’ rba =
kAa
’ rAa +
kBb
’ rBb
>
(2.16)
Thus we obtain =
($b’$a’
exp
(&it
I-
x J
’ rba)l
tldba{+b$a
exp
(ikci
’ rba)>>
r
d3 rba
s(rba>
d3rBb
v&?trBay
rBb)gA(Bb)(rBb)17~~(rAa),
t2.17)
J
which is identical to eq. (9) of ref. ‘) except that here the proper antisymmetrization of the t-matrix is displayed explicitly. Eq. (2.17) shows that the transition amplitude of a properly antisymmetrized DWIA for the knock-out process contains the old spectroscopic amplitude gA(Bb). It remains to see, however, whether the expression obtained could be transformed into the form of Fliessbach’s model. In his alternative model for direct cluster reactions Fliessbach “) assumes that the removal of particle b from B can be treated separately from the rest of the process. It is pointed out elsewhere 13) that such a separation cannot be carried out for a general knock-out reaction. In the DWIA, however, the transition amplitude may be reduced to this form, and therefore, it is instructive to seek conditions for the “new amplitude” GA(Bb) of eq. (2.2) to appear. The required “) form of the transition amplitude can be obtained by adopting the approximation ‘) n&L = n&,)( r&n&,)( rBb): S- a-a’b’=
(2.18a)
~&+k~‘2~kA(Bb))
= (,&‘lul
GA(Bb))
from eq. (2.17)
,
(2.18b)
where the notations v(r) = (I tl)[R~~‘2x~~‘( r)]*A,f”x&‘( yr) and ,y = A”2q have been introduced. The motivation “) for expressing n in terms of x is that it is x that is normalized as a scattering wave function in a non-antisymmetrical theory.
R. Beck ef al. / Quus~eZasficcluster knock-auf
The approximate
relation
(2.1Sb) becomes
const. This complicated function conditions are the following: exp [ i(-kL,-+
an exact equality
709
if and only if v(r) =
may be constant in the plane-wave limit, and the (i) ~~~/2x~;)=x&;a), a,~“x~~‘xr:,‘, and (ii)
ykAa) 5 r] = 1, i.e. -ykAa= kba. Condition
(i) is met if kb, and ykAa are
high enough, but condition (ii) is not satisfied in usual experiments; indeed, ykA, kh, is not even constant within a single experiment. Rather, I-ykAa- kL,l may be large enough for d~~‘*z~g,~~~~^- vg,(.,, to be assumed. Whether or not this is the case, however, the approximation (2.18b) is not valid and the “new amplitude” G AfBhj does not appear. From these considerations we have learned that the cluster knock-out analyses provide information on the conventional spectroscopic amplitude and factor, g,&_(Bb) and SA(Bbh rather than GAcBbl and SAtBbf. Although the formal properties of ga(sb) interpretation, it is obvious that these and SA(Bb) do not facilitate a probability quantities are also related to the cluster structure of A. In the following the nuclear-structure aspect of these quantities will be elaborated on, and the relationship of sA(&) to SA(Bb) will be studied. 3. The generator coordinate representation In this section involved in in eq. (2-l), a test state, model, and, state on the
we develop
a formalism
for the computation
of the quantities
the spectroscopy of quasielastic cluster knock-out reactions. As is seen the fragmentation amplitude is the overlap of the nuclear state 4A with which is a pure two-cluster state. We describe nucleus A in the GC for the sake of consistency, we opt for a formalism that treats the test same footing. However, in the test state the relative motion is described
by a &function, which is very inconvenient to treat in the GC formalism. To circumvent this difficulty, one can start, instead, with the overlap of the 6Li wave function with the product of the LYand d wave functions generated by oscillator wells fixed relative to each other, but the relationship of this amplitude to g(r) is rather involved 14). It may thus be more convenient to construct first the Fourier transform g(k) of g(r), which is the overlap of &,+ with the antisymmetrical wave function of two non-interacting ciusters. We can then obtain g(r) by an inverse Fourier transformation. Since a plane-wave analysis of the knock-out reaction provides directly g”(k) rather than g(r), this version of “GC spectroscopy” is particularly suited to such analyses. The nuclear wave functions considered consist of superpositions of various two-cluster wave functions. The GC wave function for the clusterization Bi + bi has the form (3.la) with the generator
function
710
R. Beck et al. / Quasielastic cluster knock-out
where s = S,,, - S,, and b$&, + BiSs, = 0. The functions @,, ( QB,) are antisymmetrized and normalized products of the lowest-lying single-particle states corresponding to oscillator potentials the same parameter
centred about S,,(S,t). All oscillator wells are characterized by /3 = mm/h, with m and w being the nucleon mass and the
oscillator frequency, respectively. In subsect. 3.1 we construct the wave function of the non-interacting clusters. In subsect. 3.2 we formulate the (quasi-) bound-state problem of +,+ Subsects. 3.3 and 3.4 describe the calculation of the fragmentation amplitudes and of the spectroscopic factors as well as amounts of clusterings, respectively.
3.1. THE CC WAVE
FUNCTION
OF NON-INTERACTING
CLUSTERS
The GC wave functions of non-interacting clusters have already been studied since they are involved, as the asymptotic states, in the GC description of scattering. Such asymptotic states may be constructed by solving the asymptotic Hill-Wheeler (HW) equation 15) or by expressing the corresponding RG wave function in GC terms 16-18). The equivalence of the two treatments is non-trivial. Here we treat the problem by solving the asymptotic HW equation and discuss the equivalence of the two treatments. The asymptotic
HW equation ,d’s’J;(S
J
reads ~‘)&(s)]H,,-E,~&~(s~))=0,
(3.2)
where Ha, contains no interaction between bi and Bi. If we assume that the clusters b, and B1 contain nucleons in relative OS orbits only, then the matrix elements for the fragmentation channel i = 1 are given by (&1(s)j&I(~‘))=exp[-$~(s-s’)2],
(3.3a)
(~~(s)lH,,-E,I~,(s’))/(~~(s)l~~(s’)) (3.3b) where pi is the reduced mass and pi = /?biBi/A is the reduced size parameter. In eqs. (3.2) and (3.3b) the energy Ek is the sum of the intrinsic energies of the non-interacting clusters and the kinetic energy (hk)‘/2pi of their relative motion. Eq. (3.2) is in fact the Schrodinger equation of a free particle of mass ki and energy (hk)‘/2pi in the GC representation. In a spherical basis, G,(s) = s-‘CLM @i( Ls) YfM( ;), f( k, s) = s-’ CLMfL( k, s) YLM (s1), and it can be shown 15) that the regular (irregular) solutions of this equation are spherical Bessel (Neumann) functions 19), s-‘fyg)(
k, s) = c( k)j,( ks) ,
(3.4a)
s-‘f;rr)(
k, s) = c( k)y,( ks) ,
(3.4b)
711
R. Beck et al. / Quasielastic cluster knock-out
where we have dropped in constructing be determined GC amplitudes
the subscript
i = 1. Unlike
in the description
of scattering
15),
g(k) it is important to know the normalization factor c(k). It may by calculating the wronskian 16) of the GC functions (3.1) with the (3.4a, b) and comparing it with the standard result 20) w = h2/2p,k.
In the GC representation,
the wronskian
:[fl”“‘(‘k,
k, s’)] .
(3.6)
is symmetrical under the interchange of s and s’, brackets is antisymmetrical. Thus any integration in the two-dimensional integral gives zero. We may by its asymptotic limit (s, s’+ 00) and obtain from
J J ds’
0
k, s)fp’(
cc
m
X
is
s)fl’“‘( k, s’) -f;“‘(
The matrix element in eq. (3.6) whereas the term in the square over a finite symmetrical region therefore replace the integrand eqs. (3.3), (3.4) and (3.6)
(3.5)
ds exp [-~P:(s-s’)~]
sin [k(s’-s)]
0
(3.7) where the integration
over s was performed
is important in the GCM is a consequence energy operator over the space of stationary (3.5) and (3.7), we find
first. The fact that the order of integration of the nonhermiticity scattering states. Upon
of the kinetic comparing eqs.
(3.8) From the regular solution (3.4a) we may now construct plane-wave solutions eq. (3.2) to obtain GC functions ]k) which describe two clusters moving relative each other with momentum k:
IW= J d3s~3n)-~/~~k, s)lo,(s)),
of to
(3.9a)
where (3.9b) which agrees with the result “) obtained by the conventional method. The method of calculating the asymptotic GC amplitude presented here may easily be generalized
R. Beck et al. / Quasielastic cluster knock-out
712
Ha, in eq. (3.2) still contains
to the case where the protons.
In that case, however,
the Coulomb
the result is different+
interaction
between
from that obtainable
from
the RG formalism. Owing to the antisymmetrization,
the normalized
free cluster
eq. (3.9), do not constitute an orthonormal set. For example, the Q + d fragmentation, the overlap of two generator functions (@,(s)]@,(s’))=
i
wave functions
]k),
when i = 1 denotes Q,(s) is given by *‘)
v’,‘,‘exp [-~p;(s2+sr2)+pw(l~)s.
s’],
(3.10a)
i=l
where V’,:’= 1 )
ui;’ = -2
u’l:’ = 1 3 (3.10b)
(1) _Z w11-3, Using this result together
w(2) -I 11 -6,
w’l;’ ZZ- f .
with eq. (3.9), we find
(k]k’)=S(k-k’)
where
(r$’ = $li f pw\\). The terms
i = 2,3 in eqs. (3.10a)
and
(3.11) causing
the
departure from orthonormalization arise from particle exchange. These terms, in contrast, do not contribute to the “asymptotic normalization”, expressed by the wronskian (3.6). 3.2. THE
GENERATOR
For a (quasi-) orbital momenta form 2’,22)
COORDINATE
WAVE
FUNCTION
FOR
BOUND
CLUSTERS
bound system, composed of several clusterizations i of intrinsic Iim, the unprojected GC wave function (3.1) can be written in the
!P =C 1 sf d~i~(l,mis,)~i(limis,), i I.%1 I ’ The Coulomb
interaction
way: The intracluster Z, (Z,)
is the number
(CI)
CI results
in the cluster ,,,c,,,
(Q’(s)l+
matrix
+M
element
(3.3b)
in the following
(2p/n)” to the energy E,, where KY b (B). The intercluster CI leads to the matrix element
lx, -x,l~‘l4(s’))
=2Z,Z,(s+s’l-’ where erf is the error function
the “asymptotic”
in the contribution
of protons (4(s)l
changes
(3.12)
erf (~Jpls+s’l)(i,(s)li,(s’)),
19). If the intercluster l&,(s’)) =ZZ,Z,(s+s’l-’
It is easy to see that only in this approximation of two charged point-like particles.
Cl is approximated
by Z,Z,/r,
then one gets
erf (tJ~ls+s’l)(l,(s)i~,(s’)).
is the HW, eq. (3.2), identical
to the Schriidinger
equation
R. Beck et al. / Quasielastic
where the sum runs over a discrete Qi into partial
cluster knock-out
set [a] of values
713
for the GC’s si. If we expand
waves @,(l,m,s;) =
s;’ c @i(L,M,zimisi) Yf&jii)
)
(3.13a)
L,V we may construct coordinates:
22) a spherical
tensor
@,([LiZi]cL?s/HSi) = 1
of rank
of the particle
.Y in the space
CDi(LiMilimiSi)(LiMi,
ZifTlilL?A) .
(3.13b)
Ml?
For the GC amplitude
in eq. (3.12), we use a similar J;(bWi)
=
sL’
C
f;tLiMirimJi)
expansion: YL,M,(ii)
(3.14a)
I
Of,
and introduce (3.14b) Substituting the inverted form of the expansions (3.13b) and (3.14b) into eqs. (3.13a) and (3.14a) and inserting these into eq. (3.12), we obtain a decomposition of !P in terms of functions of definite %U and parity n: (3.15) where the sum over Li and Z, is restricted problem of projecting the wave function
by the parity relation ( -)‘Lr+‘~) = n. The onto the values of the orbital
@,(f,m,s,)
angular momentum Z’A is thus reduced to an analysis of its tensor properties in the space of the GC’s s,. In practical calculations it is advantageous to perform this analysis on the matrix elements rather than on the wave functions. As an example, we consider
the overlap
matrix
element
of the wave functions
@, and
Q2, where
Q2 is the generator function of the clusterization 5He + p. In this clusterization, there is a neutron occupying an I= 1 orbit. Using the algebraic programming system REDUCE-2 [ref. “)I, we obtain *‘) for the overlap (@,(s)lQ2(lm’s’))=
2 u\i:?VylmJJ~s) i=l Xexp[-&3's2+/3's'2)+@v'i's~ I 2
12
s’]
(3.16a)
,
where &’
= - ($)“2,
(1) -I WI2
-3,
v(l;) = (ST)‘/*,
w(2) 12 -
-;,
(3.16b)
and 9,, denotes a solid spherical harmonic function 24)_Using the expansion (3.13a) for the bra and the ket in eq. (3.16a), it is straightforward to construct the spherical
R. Beck et al. / Quasielastic cluster knock-out
714
tensor
(3.13b),
and the result is
(3.17) Further examples of overlaps and matrix elements of the hamiltonian (containing a central, spin-orbit and tensor interaction) may be found in ref. *‘). By following the usual course, it is then simple to couple the nulceon spins to eigenvalues Iv and to couple the corresponding eigenstate with ?P(.JZA) to eigenstates ?P([Z1]JM) of the total angular momentum. The GC amplitudes J;([ L,Z,]P&) are determined variationally, by solving the coupled HW equations SE=G[(~lHIW)/(WI~)]=O,
(3.18)
where the hamiltonian has the form H = -( h2/2m)Ci VT +Cicj VP In the calculations reported in this paper, the nucleon-nucleon interaction Vj is approximated by the effective forces of Volkov lo) and Brink 25), and the Coulomb interaction is neglected. For the nucleon mass in 6Li, we use m =938.926 MeV/c*. Finally, we sum up what techniques we used in the structure calculation. First, the matrix elements of the GC functions a,(s) were calculated analytically with resort to algebraic programming. The result is expressed in terms of exponentials (some of them multiplied by polynomials in s and s’) depending on quadratic forms in s and s’. The coefficients specifying these functions were provided by a REDUCE program. The GC amplitude specifying the (quasi-) bound state of the system was then obtained 3.3. THE
a+d
by solving
the HW equation.
FRAGMENTATION
AMPLITUDE
IN 6Li
Having constructed the GC wave function of 6Li, we now proceed to discuss the spectroscopic information contained in it. The momentum-space (Y+ d fragmentation amplitude g”(k) is nothing but the overlap of the 6Li wave function ‘P([Z, 1= l]JM) with the wave function of cx and d [spins 1, = 0, (Id, vd) = (1, v)] in a relative plane-wave state of momentum k. We can expand g(k) as g”(k)=(kvIW([cYl]JM)) = ; (kl ~(~JU(=%
‘5
1 +‘W
~d;p(k)Y~~(R)(-i)-iPJ4rr(~~,
lv(JM),
(3.19)
where the total orbital angular momentum %t% comes entirely from the relative motion of the (Y+d cluster. In order to extract g&(k) from eq. (3.19), we introduce
R. Beck et al. / Quasielastic cluster knock-out
the GC representation,
715
eqs. (3.9) and (3.15), and obtain
(3.20) where we have introduced the notation i= (2LS 1)“2. The function &(k) is not identical but is related to the probability amplitude of the intercluster relative momentum just as gY(r) is related to that of the relative position. In the following we shall call I&(k)l” the fragmentation strength. It is interesting to note that this function contains the gaussian factor exp {-Ak’/[b(Ab)p]}, which is rather similar to the momentum distribution -exp [-i(hk/a)*] predicted by statistical models 26), where (T = CTJ(A - b)b/(A - l)]“‘= 114 MeV/c. With the value /? =0.48 fm-2, adopted in this work (see next section), we obtain (T = 112 MeV/c. The Z-wave fragmentation amplitude g,,(r) is related to the function &(k) by the integral transformation l/2 gY(r)=
If we substitute
&4(k)
1
0 7r
J
N
dk k2jy( kr)&(
k) .
(3.21a)
0
from eq. (3.20), we find
+2JFf;
i2( “02 :,
3
,x1 ~2_1-2([~2113~2)
1 (3.21b) where iy is a modified
spherical
Bessel function
19).
R. Beck et al. J Quasielastic cluster knock-out
716 3.4. SPECTROSCOPIC
FACTOR
AND
AMOUNT
OF CLUSTERING
In our model the schematic formula for the spectroscopic specified for the LY+ d clusterization as
factor,
eq. (2.4a),
is
a? s9=1[
Y
d3k ](k~\ V([Y’l]JM))1*
dk k21&(k)1’.
= 47r I0
If we use eq. (3.20) and carry out the integration,
(3.22a)
we find
s9 = &‘) + &*) + $2) )
(3.22b)
where
xcij @=4:Z$
(i) 011
(j).
v11 lz
(2P2 TWll
1 L*
ii2(-,Z @
~~~vCJ~w(‘)j
XC ij
(i)
(i)
w11
)
s,s; ,
1
11
Lz
0 CL2
(2p2 p:Wll
(i)
0 l
(i)
w12
(3.22~)
0 “>
SlS2
lS& Sffi(%,) ,’
I
7
(3.22d)
)
For later convenience, we have introduced the quantities @, tip;‘, which here are identical to v’,‘: and w(,i,‘),respectively. Because of the nonorthogonality of the vectors Ik), as expressed by eq. (3.11), the spectroscopic factor So must not be interpreted as the probability of finding, in 6Li, an (Y+ d cluster, which may also be called the amount of clustering. This probability may be defined as the norm of the “new” amplitude G(r). Here we resort to another definition, which can be visualized in geometrical terms, and later on show the equivalence of the two definitions.
717
R. Beck et al. / Quasielastic cluster knock-out
vector )P) which may be expressed
To find that part of the (normalized) of the (nonorthogonal)
basis
vectors
i.e. we determine .
IU’)-~d3kT(k)lk),
Ik), we minimize the function
T(k)
d3kr(k)lk)ll
=O.
W-J
the length
in terms
of the vector
by the variational
equation (3.23a)
The minimum value of the norm square (( ]I measures the size of the component, of )V), that is orthogonal to the clusterization spanned by {(k)}. It is therefore judicious to define the amount of clustering represented by {Jk)} as (3.24a) where T(k) is a solution of eq. (3.23a). Obviously, 0~ SS 1. On carrying out the variation in eq. (3.23a) and using the function g(k), eq. (3.19), we find that T(k) obeys the integral equation g(k)
J J
=
while S is given by S=
d3k’ T(k’)(klk’)
,
(3.23b)
d3kg*(k)l-(k).
(3.24b)
The quantities g(k) and T(k) are the covariant and contravariant components of the vector I P) in the basis Jk). Eq. (3.24b) expresses that the amount of clustering is the length square of that part of 1?P) that lies in the space spanned by the vectors
IQ. To show that eqs. (3.23b) and (3.24b) are consistent with our earlier definition, eq. (2.4b), ofthe amount of clustering, we substitute the formal solution of eq. (3.23b), T(k)
into eq. (3.24b),
=
with the result that S =
J
d3k’(k(k’)-Ii(
(3.23~)
J
d3k d3k’ g(k)*(k(k')-'g(k')
,
(3.24~)
which is nothing but eq. (2.4b), written in the momentum representation. We shall now recast eq. (3.23b) into a form suitable for GC wave functions. the 1.h.s. we use eq. (3.15) to express the r.h.s. we use the expansion
tY in terms of the generator
r(k2iA)=~ci(k=YM)=~
1
s;
f LM,b,[s,l
X~i(kfim,a)
i;(klimis;)=(kJ~,(Iim;s,))+
J
Qti On
J
d?;f;([ L,&?.&)
YL,M,(S:)(LiM,, Z,mil~~)
In this way eq. (3.23b) becomes
functions
On
.
d3k’~i(k’limPi)(k)k’)‘)
(3.25)
(3.26)
718
R. Beck et al. / Quasielastic cluster knock-out
where (k( k’)’ = S (k - k’) - (k( k’). It is important to notice that in eq. (3.26) the different clusterizations i are not coupled. If there are two s-wave clusters in clusterization i = 1, for i = 1 eq. (3.26) can be solved
analytically,
with the result
P,(k,s)=(r/3{))3’4exp(-ik*s-k2/2P;). Substituting
this formula
(3.27a)
into eq. (3.25), we can prove that
d3kT,(kZ&)(k) = ‘P, . (3.27b) J‘ Because of eq. (3.24a) this implies that, for a single-clusterization model, S = 1 holds as it should. [It is, however, not justifiable to expect 27,28) that the spectroscopic factor sZ, eq. (3.22b), also obeys such a simple rule!] Furthermore, for a twoclusterization model, eqs. (3.24a) and (3.27b) imply
d3kT2(k)(!P,Jk).
(3.28)
Solving eq. (3.26) for i = 2 by iteration, and substituting the result as well as jk) of eq. (3.9) and the i = 2 term of eq. (3.15) into eq. (3.28), we obtain
X
-(~2([L211~~2)]~2([L;ll~~;))+ IF C $P(n, 4,. . . , in> , n=O I
(3.29)
i,...i.
where the n-summation comes from the terms of p2 obtained via the iterative solution, and S($) is obtained from sy) of eq. (3.22e) by redefining fi$ and 3%’ as ),
(3.30a)
(3.30b) In the actual calculation sum over n in eq. (3.29) taken into account. (The 3 to 4, was less than 5 x
to be discussed in the next section it turned out that the is rapidly converging, so that only a few terms have to be relative change in S,, obtained by increasing max (n) from 10e4.)
4. Results and discussion In the previous section we expressed the quantities involved in cluster reaction spectroscopy in a calculable form. Now we show their application to the (Y+ d clustering in 6Li.
R. Beck et al. / Quasielastic cluster knock-out 4.1. SOLUTION
OF THE
HILL-WHEELER
EQUATION
FOR
719
6Li
We solve the coupled HW equations (3.18) for the ground (9 = 0) and first-excited (2 = 2) state in 6Li, using the truncated trial function [cf. eq. (3.15)]
+ 2 f*([L,
= 1, 1, = l]&)CD2([L2
= 1, 12= l]Z&J
.
(4.1)
[%I
(The nucleon spins are coupled to I = 1. The interactions between the J-values allowed.) While for the ground state possibility, for the excited state the choice (4.1) amounts l2 = 1 component. The unstable excited state was treated as the variational stability condition 29) being approximately
used do not distinguish L, = 1, Z2= 1 is the only to neglecting the Lz=3, a quasibound state, with satisfied. The results for
the effective nucleon-nucleon interactions number 1 and 2 of Volkov lo) (“Volkov 1” and “Volkov 2”) and that of Brink 25) (“Brink”) are listed in table 1. For both the ground and the excited state, the minimum values of the energies are obtained with p = 0.48 fin-*. Before comparing our results with experimental data, we should mention that the splitting of the triplet (9 = 2, I = 1) state in 6Li into three levels J” = 3+, 2+ and l+, with the excitation energies “) 2.185, 4.31 and 5.65 MeV, can, of course, not be reproduced with a spin-independent force. Therefore, we compare the calculated excitation energy with the average value of the experimental excitation energies, E,, = 3.6 MeV. Although the calculated ground-state binding energy is too small, the a +d separation energy and the excitation energy of the first excited state are in qualitative agreement with experiment. From the point of view of the (Y+ d fragmentation this is very important. The reasonability of the separation energy values indicates that the minor binding-energy
defect of 6Li is likely to arise from the failure
of the forces
to bind the individual clusters properly. In fact, the cu-particle binding energies are also reasonable, and so the blame is to be put solely on the deuteron. From table 1 it is seen that the coupling of the ‘He+p clusterization lowers the ground-state energy
by -0.6
MeV. It is interesting
to compare
our results
for the ground-state
energy (-26.54 MeV) and excitation energy (3.25 MeV), obtained with the Volkov 1 interaction, to a recent calculation 3’) by Filippov et al., who found -26.11 MeV and 4.08 MeV, respectively, using the same force and an expansion of the wave function in terms of hyperspherical functions. For the case of the Volkov 2 force, we may compare our result for the ground state energy (-28.08 MeV) with the one (-28.3 MeV) obtained by Krivec and MihailoviC 9), who considered the three coupled clusterizations (Y+ d, ‘He+ p and ‘Li+n. We conclude that, as far as the ground-state energy is concerned, the inclusion of the third clusterization is only of minor importance.
R. Beck et al. / Quasielasric cluster knock-out
720
TABLE
1
Energies and rms radii r of the ground state and first excited state of 6Li in the (Y+ d and in the {a +d, ‘He-tp} models with the Volkov 1, Volkov 2 and Brink interactions Volkov Ground
Es., “1 [Mevl
a+d a+d,‘He+p
1
state *) -25.91 -26.54
Volkov 2
-21.46 -28.08
Brink
-25.71 -26.26
a+d a+d,‘He+p
1.15 1.78
2.29
0.55 1.10
a+d a+d,‘He+p
2.64 2.68
2.61 2.65
2.77 2.78
1.61
First excited stare ‘? a+d a+d,5He+p
3.54 3.25
3.60 3.21
3.55 3.35
atd a+d,‘He+p
2.15 2.72
2.69 2.68
3.01 2.90
“) Binding energy. b, Separation energy. ‘) Excitation energy. d, Experimental values from ref. 30): Es,, = -31.9929 MeV; E, = 1.4735 MeV; 2.51 f 0.1 fm; E,, = 3.6 MeV, weighted average for the Z’= 2 triplet.
r=
In table 1 we also give the results for the rms charge radius. The reasonability of the cluster-model description is also shown by the fact that the ground-state radius is well reproduced.
In all cases the rms radius
of the 9? = 2 state is larger than that
of the .Z = 0 ground state. This may be interpreted as a centrifugal stretching effect and is in accordance with the findings of Kanada et al. 32) From table 1 one can infer that this stretching effect is caused by the (Y+d clusterization only, and the ‘He + p clusterization tends to weaken the magnitude of stretching. Indeed, for the 9 = 0 state, the coupling of the 5He+ p to the (Y+ d clusterization results in an increase of the rms radius, while for the 9 = 2 state, the coupling causes a decrease.
4.2. SPECTROSCOPIC
INFORMATION
ON 6Li
With the wave function of eq. (4.1), the function I&(k)l’ for the cy+d fragmentation can be readily computed by using eq. (3.20). In fig. 1 we compare l&,(k)l’ for the ground state as predicted by the Volkov 2 force with experimental data extracted from a plane-wave analysis 33). As is seen from this figure, the effect of switching on the 5He+p clusterization (full curve) is to lower l&(k)l’ for ks 0.3 fin-’ as
R. Beck et al. / Quasielastic cluster knock-out
5
721
4 (a, 2U) 700 MeV
k(fm-‘1 Fig. 1. 6Li + a + d momentum-space fragmentation strengths. The data are extracted “) from 6Li(p, pd)cr experiments at 590 MeV and from 6Li(cy, 2a)d experiments at 700 MeV. The theoretical curves are obtained from the {cr+d, 5He+p} cluster model (solid line) and the a +d cluster model (dashed line).
compared to the case of the pure (Y+ d clusterization (broken curve), thus improving the agreement with the experimental data. For k* 0.4 fin-’ both models give the same
result.
In this region,
compared to the data. To illuminate the quality
however, of agreement
we should point out what are in the theoretical model. For approximation is questionable of ~~o(k)~2 from experiment. Nevertheless,
the predicted
distribution
is too small
as
one can expect from such a comparison,
the uncertainties in the experimental data as well as momenta k larger than -0.4 fm-‘, the plane-wave ‘“) and the DWIA should be used for the extraction Thus one should not expect any better agreement.
the PWIA does have the advantage
that it is free from the inevitable
arbitrariness associated with the optical potentials that enter into the DWIA. On the other hand, the specific distortion 32) of the deuteron at small distances from the a-particle might not be fully taken into account by admixing the ‘He+p clusterization, giving rise to uncertainties in l&(k)l’ at large momenta k. It is instructive to study the fragmentation amplitude g,(r), eq. (3.21b), which is displayed in fig. 2. This curve shows a node at r = 1.7 fm, which is required 34) by the Pauli principle. The result that there is practically no difference between the spectroscopic amplitude of the coupled clusterizations (solid curve) and that of the
122
I?. Beck et al. / Quasielastic
-0.1 0 Fig. 2. Fragmentation
amplitudes
I 2
I 4
I G r(fm)
cluster knock-out
I 8
(6Li + a + d) for the 6Li ground as in fig. 1.
I 10
12
state obtained
from the cluster models
pure (Y+ d clusterization (dashed curve) for distances r c 4 fm reflects the effect of the Pauli principle, which reduces greatly the differences 35) between apparently different cluster wave functions. Finally, we note that the three-body model of 6Li (a-particle plus two nucleons) introduced by Lehmann and Mamta Rajan 36), predicts an effective cr +d wave function whose shape is rather similar (except for large distances) to the fragmentation amplitude g,(r) obtained in this paper. If we now apply our model to the first excited state (2 = 2) in 6Li and calculate the function Jg2( k)12 for the Volkov 2 force, we find the result shown in fig. 3. Again, the dashed (solid) curve represents the result for the pure (Y+ d (coupled) clusterization( Each of the curves shows a peak at E= 0.5 fm-‘. In a classical picture, we may relate the kinetic energy ( hE)2/2p, = 3.9 MeV to the excitation energy, E,, = 3.6 MeV. This interpretation is consistent with that fact that, for the ground state, the peak in the momentum distribution, which is probably close to I&(k)12, occurs at k = 0. From a comparison of figs. 1 and 3, it is seen that the difference in I&( k)(’ between the model of the coupled clusterizations and of the pure (Y+ d clusterization is more pronounced for the 2 = 2 state than for the Z’= 0 state. This result may be understood in terms of the centrifugal stretching of the Z? = 2 state, which obviously diminishes the effect of the Pauli principle. The spectroscopic amplitude g2(r) for the 58 = 2 state is depicted in fig. 4. Here one finds both in the coupled-clusterization (solid curve) and single-clusterization
R. Beck ef al. / Quasielastic
,I
0.Y
cluster knock-out
723
.\ :
:
,
Y -
2s
-0.2
0.1
.
0.c 0
1.0
0.5
1.5
k (fin-’ I Fig. 3. Comparison
of the ‘Li*+ a +d momentum-space fragmentation strengths state in 6Li obtained from the cluster models as in fig. 1.
0 Fig. 4. Fragmentation
amplitudes
2
il
6 r(fml
8
10
for the first excited
12
j6Li* + cy+ d) for the first excited state in ‘Li obtained models as in fig. 1.
from the cluster
R. Beck et al. f Quasielastic cluster knock-out
724
(dashed curve) case that there is no node and the curves are shifted to larger distances as compared to the ground state, again indicating a centrifugal stretching. Finally, calculated
we discuss
the spectroscopic
with our wave functions
factor sY and the amount of clustering S, for both the ground and excited state. The results
obtained with the forces Volkov 1, Volkov 2 and Brink are given in table 2. All spectroscopic factors sY are larger than unity, showing that they must not be interpreted as probabilities. The coupling of the 5He+p clusterization results in a small reduction of sY. The experimentally deduced spectroscopic factors for the 6Li ground state range from 0.04 (3~0.12) [ref. “)I to 1.75 (kO.5) [ref. “)I with many intermediate values 4~27~28~33~38~42), depending on the reaction and on the method of extraction. The data 33) given in table 2 are extracted from (p, pd) reactions at 590 MeV and from (a, 2cz) reactions at 700 MeV, using the PWIA. We prefer these data to those 33) taken at the lower energies of 156 and 64 MeV, respectively, because at higher energies the impulse approximation seems better justified. The calculated spectroscopic factors are slightly smaller than both the experimental values 33) and the result obtained by Kurdyumov et al. 43) from an (Y+ d harmonic-oscillator cluster model.
TABLE 2 Alpha spectroscopic factor s and the amount of u-clustering S of the ground state and first excited state of 6Li in the a + d, {n + d, 5He + p} models with the Volkov 1, Volkov 2 and Brink interactions Volkov
cu+d
a+d,5He+p
o+d
a+d,sHe+p
1
s
Ground state “) 1.071
S
1
Volkov 2
Brink
1.073
1.059
1
1
s
1.038
1.041
1.033
S
0.967
0.969
0.972
1.069
1.046
1
1
s
First excited stare 1.065
s
1
s
1.010
1.013
1.006
s
0.940
0.942
0.952
“) s= 1.08: (p, pd) experiment at 590 MeV [ref.33)]; experiment at 700 MeV [ref. “)I; s = 1.05: harmonic-oscillator lation 43).
s= 1.05*0.12: cluster-model
(a,2a) calcu-
The amount of LY+d clustering S,,, which is unity for the pure a + d model, is reduced for the case of the coupled clusterizations. For the 3 = 2 state, both the spectroscopic factors s2 and amounts of (Y+ d clustering S2 are slightly smaller than
R. Beck et al. / Quasielastic cluster knock-out
725
the corresponding quantities of the Y = 0 state. This is consistent with our earlier finding (see figs. 1 and 3) that the second clusterization (‘He+p) is more important for the 2 = 2 than for the .Z = 0 state. We can sum up the results of this paper as follows. We have shown how to apply the viewpoint of the microscopic cluster model to the description of quasielastic cluster knock-out reactions. It was pointed out that, because of the Pauli principle, the measured spectroscopic factor differs from the amount of clustering, and only the latter quantity may be interpreted as a probability. To illustrate these considerations, we chose a version, of the GCM, that describes nuclei in terms of various clusterizations with the internal motion of the clusters only disturbed by the other nucleons through the Pauli principle. It was found that this GCM is well suited not only to describing the ingredients of these reactions but to illustrating the general concepts as well. The (Y+ d fragmentation properties of 6Li are reproduced by this model reasonably well. The amount of clustering does differ from the spectroscopic factor though by a few percent only. We would like to mention that the application, to this problem, of our more recent model 44), which assumes one pair of breafhing clusters rather than several frozen clusterizations, is under way. One of the authors (R.G.L.) wishes to thank Professor G. Schatz and the Kernforschungszentrum Karlsruhe for their kind hospitality and the IAEA for a fellowship.
References 1) K. Nagatani, Phys. Lett. 29B (1969) 468 2) A.K. Jain, N. Sarma and B. Banerjee, Nuovo Cim. 62B (1969) 219; Nucl. Phys. Al42 (1970) 330 3) Yu. A. Kudeyarov, I.V. Kurdyumov, V.G. Neudatchin and Yu.F. Smirnov, Nucl. Phys. Al63 (1971) 316 4) A.K. Jain and N. Sarma, Nucl. Phys. Al95 (1972) 566; A233 (1974) 145 5) H. Feshbach, Ann. of Phys. 19 (1962) 287 6) T. Fliessbach, Z. Phys. A288 (1978) 211; 219 7) N.S. Chant and P.G. Roos, Phys. Rev. Cl5 (1977) 57 8) H.H. Hackenbroich, P. Heiss and Le-Chi-Niem, Nucl. Phys. A221 (1974) 461 9) R. Krivec and M.V. Mihailovic, J. of Phys. C8 (1982) 821 10) A.B. Volkov, Nucl. Phys. 74 (1965) 33 11) T. Fliessbach, 2. Phys. A272 (1975) 39 12) R.G. Newton, Scattering theory of waves and particles (McGraw-Hill, New York, 1966) 13) R.G. Lovas, Z. Phys. A, in press 14) R.G. Lovas, K.F. Pal and M.A. Nagarajan, Nucl. Phys. A402 (1983) 141 15) N.B. de Takacsy, Phys. Rev. C5 (1972) 1883; M.A. Nagarajan, private communication 16) M.V. Mihailovic, L.J.B. Goldfarb and M.A. Nagarajan, Nucl. Phys. A273 (1976) 207 17) R. Beck, M.V. Mihailovic and M. PoljSak, Nucl. Phys. A351 (1981) 295 18) M.V. Mihailovic and M. PoljSak, Nucl. Phys. A388 (1982) 317 19) M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover Publications, New York, 1965) 20) A. Messiah, Quantum mechanics, vol. II (North-Holland, Amsterdam, 1962) p. 488 21) R. Beck, F. Dickmann and J. Oehlschliger, Kernforschungszentrum Karlsruhe, report KfK-3539,1983
726
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cluster knock-out
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