By using the method of hyperspherical functions within the appropriate for this method K_{\min} approximation, the simple three-cluster model for description of the ground state and the continuous spectrum states of \6He is developed. It is shown that many properties of \6He (its large rms radius and large values of the matrix elements of electromagnetic transitions from the ground state into the continuous spectrum) follow from the fact that the potential energy of \6He system decreases very slowly (as \rho^{-3}) and the binding energy is small

Filippov, G F

Korennov, S V

Rybkin, I Yu

CERN Document Server

Phenomenological Three-Cluster Model of
6
He
G. F. Filippov, I. Yu. Rybkin, S. V. Korennov
Bogolyubov Institute for Theoretical Physics 252143, Kiev-143, Ukraine
(To appear in Yad. Fiz.)
Abstract
By using the method of hyperspherical functions within the appropriate for this
method K
min
approximation, the simple three-cluster model for description of the
ground state and the continuous spectrum states of
6
He is developed. It is shown that
many properties of
6
He (its large rms radius and large values of the matrix elements of
electromagnetic transitions from the ground state into the continuous spectrum) follow
from the fact that the potential energy of
6
He system decreases very slowly (as 
 3
)
and the binding energy is small.
1 Introduction
The
6
He nucleus is an example of a three-cluster system whose lowest threshold (+n+n) is a
three-particle one. Such systems have a number of remarkable properties determined mainly
by two factors|the Pauli principle and the character of the potential energy dependence
due to the three-body structure.
The Pauli principle imposes essential restrictions on the wave function of a system allow-
ing it to contain only the components antisymmetric with respect to nucleon permutation.
Thus, if we are using the expansion over the harmonic-oscillator basis, we must solve the
problem of excluding the states forbidden by the Pauli principle [1]. This, in particular,
leads to the fact that the simplest wave function of the 0
+
-state of
6
He obtained with the
translation-invariant shell model (it coincides with the lowest oscillator-basis function) is
a superposition of states with three-particle hypermomentum K = 0 ( 5%) and K = 2
( 95%). This function generates an innite set of states diering by a number of oscillator
quanta and having almost the same ratio of K = 0 and K = 2 components. The approxi-
mation using only this set of functions can be appropriately called the K
min
-approximation.
Other basis states contain components with K > 2 whose weight is small, as shown in [2].
Notice that for 2
+
-states, wave functions of the K
min
-approximation contain only K = 2
components, while for the 1
 
-states they are the superpositions of K = 1 (
>

25%) and
K = 3 (
<

75%) components.
The second characteristic feature of systems under consideration is the slow decrease of
their potential energy for large values of the hyperradius . As known (see for example
The research described in this publication was made possible in part by Grant No. U48000 from the
International Science Foundation. The work was also supported in part by the Ukrainian State Committee
for Science and Technology
1
[3]), in three-body systems, the potential energy has the asymptotical behaviour const=
3
as
!1, even if the two-body forces acting between any two of constituent particles are short-
range. This fact leads to two important results. First, the boundary of a system is extremely
loose and the rms radius is signicantly larger compared with that of those neighbouring
nuclei whose three-particle threshold is much higher than the binary one. Second, the phase
shift of the three-body scattering rises sharply at low energies (it is proportional to
p
E
as E ! 0) which directly aects the behaviour of the electromagnetic transitions matrix
elements between the ground state of
6
He and its continuous spectrum.
The aim of the present paper is to give a detailed quantitative analysis of general regu-
larities in
6
He structure, based on a simple model taking into account the main features of
a loosely-bound system with the lowest three-particle threshold.
2 The Model
The allowed basis functions for states with zero angular momentum '
n
(L = 0) of the
K
min
-approximation have the following form
'
n
(L = 0) = A
0
(n)'
n0
()u
0
(!) +A
2
(n)'
n2
()u
2
(!); (1)
where the superposition coecients are
A
0
(n) =
s
4(n + 1)
29n + 104
; A
2
(n) =
s
25(n + 4)
29n + 104
; (2)
u
K
(!) are the hyperspherical functions,
'
nK
() = N
nk

K
e
 
2
=2
L
K+2
n
(
2
) (3)
are the normalized hyperradial harmonic-oscillator basis functions, L

n
(
2
) are the Laguerre
polynomials, N
nk
are the normalizing coecients, n = 0; 1; : : : Therefore, the wave function
of the L = 0 state in the K
min
-approximation is reduced to the expansion
	(L = 0) =
1
X
n=0
C
n
'
n
(L = 0): (4)
For all n, the contribution of K = 2 states is signicantly greater than that of the K = 0
states
0:86 < A
2
2
(n) < 0:96; A
2
0
(n) = 1  A
2
2
(n);
hence, in a simple model, the contribution of K = 0 states can be neglected. Then, for the
expansion (4) the following relations hold
	(L = 0) =
1
X
n=0
C
n
'
n2
()u
2
() = ()u
2
(
) =
()

5=2
u
2
(
): (5)
The function () should satisfy the one-dimensional Schrodinger equation
h
2
2m
"
 
d
2

d
2
+

K +
3
2

K +
5
2



2
#
+ V () = E; (6)
2
where m is the nucleon mass, K is the hypermomentum (K = 2 for L = 0), V () is the
eective three-body potential.
The equation (6) is a starting point for the subsequent discussion. It was solved by a
numerical integration over  from zero to a certain suciently large cuto radius 
max
. The
eective potential is modelled by a function
V () =
V
0
1 +


a

3
; (7)
having correct asymptotics and without a singularity at  = 0. To reproduce the states with
zero angular momentum, we have chosen the parameters V
0
and a such that the equation (6)
would give experimental values for the
6
He binding energy and the rms radius, E =  E =
 0:97 MeV,
q
hr
2
i = 2:57 Fm. The appropriate values of the parameters are
V
0
=  87 MeV; a = 3:073 Fm: (8)
With these values we have calculated the wave function of the
6
He ground state and 0
+
-states
of its continuous spectrum.
3 The Ground State of
6
He
The wave function of the ground state of
6
He together with the eective potential V () are
presented in Fig. 1. The horizontal line below the -axis corresponds to the ground state
energy, its intersection with the potential energy curve marks the classical turning point.
The vertical line separates the values of  less than
q
h
2
i = 5:59 Fm. The wave function
falls as 
K+5=2
for small , due to the strong kinematical barrier, while for large , the long-
range character of the potential V () leads to the slow decrease of the wave function and,
respectively, to a signicant diuseness of the boundary of a nuclear system.
As known, loosely-bound binary systems with a short-range potential can be rather well
approximated outside the potential range by the exponential function
() '
p
2 exp( r);  =
s
2m
h
2
E ; (9)
where E is the bound state energy, r is the radial variable of the binary channel. The formal
criterion of validity of such an approximation (the zero-range approximation) is the smallness
of a ratio of the potential range and the cluster radii to the system radius 1=
p
2 expressed
in terms of binding energy. It seems useful to compare the exact wave function obtained
after the solution of Eq. (6) with the approximation given by Eq. (9) (for   0:22 Fm
 1
,
dashed line in Fig. 1). It can be seen that the nucleon system is much more loose than it is
predicted by the wave function (9). The reason for that is not only that the function ()
at small  behaves as 
K+5=2
as  ! 0, but rst of all that the eective potential energy
decreases slower than the exponential, while the formula (9) is obtained by supposing that
the potential is short-range. As a result, the estimates of the square of hyperradius mean
value
q
h
2
i based on (9) are about three times less than the experimental value.
Of course, for suciently large  the function () has the asymptotics
() ' C
p
K
4
() ' C exp( ); (10)
3
where K
4
() is the Macdonald function, and the value of the coecient C (it is related to
the so-called nuclear vertex constants, see [4]) is considerably less than that of
p
2
C  0:12 Fm
 1=2
<
p
2  0:66 Fm
 1=2
: (11)
This asymptotics is denoted by short-dashed line in Fig. 1.
4 0
+
-States of the Continuous Spectrum
Of the special interest are the solutions of Eq. (6) for the continuous spectrum for rela-
tively low over-threshold energies. In particular, attention must be paid to the question
of the behaviour of the three-to-three scattering phase  as a function of energy in a po-
tential eld with the asymptotical behaviour const=
3
and the powerful kinematic barrier
(h
2
=2m)(63=4
2
) corresponding to K = 2. According to the estimates [5, 6], at low energies,
the phase shift, as is also demonstrated by our calculations, is proportional to k (or
p
E)
 ' Ak + : : : ; A = 3:95 Fm: (12)
The low-energy values of tan  calculated with dierent values of the cuto radius 
max
are
presented as functions of k in Fig. 2. As seen, 
max
should be rather large, at least 1000 Fm
for energies below 10 keV. The obtained value of A gives a rather reasonable prediction for
the ground-state energy of
6
He
E
approx:
=
h
2
2m
1
A
2
 1:3 MeV; (cf: E
exp:
= 0:97 MeV): (13)
The three-body phase shift  calculated with dierent 
max
as functions of energy E
in an interval up to 5 MeV are presented in Fig. 3. As seen, for reliable calculations for
medium-energy region ( 1 MeV) 
max
may be taken about 50 Fm and only for the lowest
energies it must be increased. The phase shift rises steeply from zero to values exceeding
90

for E  2:5 MeV and then slowly goes down. When the phase shift is near 90

the
rst maximum of the wave function moves closer to zero so that the matrix element of the
isoscalar transition from the ground state into the continuous spectrum increases (see Fig. 4).
This matrix element reaches its maximal value for E  1:3 MeV which could directly aect
the electrodisintegration cross-section of
6
He.
5 2
+
-States of the Continuous Spectrum
The K
min
-approximation satisfying the Pauli principle for the 2
+
-states of
6
He contain only
the hyperspherical function with K = 2. Therefore, to obtain the wave functions () of
2
+
-states, we again turn to the equation (6) with K = 2 but with the other parameters of
the potential V ().
The
6
He nucleus has no 2
+
bound state but it has a 2
+
-resonance at the energy of
0.8220.025 MeV with the width 0.1130.020 MeV [7]. The energy and the width of the
resonance can be reproduced with the following parameters of potential:
V
0
=  92 MeV; a = 2:834 Fm: (14)
4
The wave function () of the resonance state and the potential energy with the param-
eters (14) are presented in Fig. 5. For small  the resonance wave function behaves similarly
to the wave function of the ground state (dashed line in Fig. 5) but for larger , in the
asymptotical region it, as it should be, oscillates.
The scattering phase shift in the 2
+
-state obtained by solving the equation (6) with the
new potential parameters is presented in Fig. 6. At the energy of about 1 MeV the phase
shift has a typical behaviour for a resonance region.
We have also calculated the matrix element of the operator of the isoscalar E2 transition
from the ground state of
6
He to the 2
+
-states of its continuous spectrum. The dependence
of this matrix element upon the energy is presented in Fig. 7. The narrow peak observed
for the energy about 0:8 MeV again demonstrates the small width of 2
+
resonance and the
large value of the matrix element.
6 Conclusion
The simple three-cluster model based on the phenomenological long-range potential with
the Pauli principle taken into account has allowed us to reveal a number of regularities both
for the weakly-coupled ground state of
6
He and for the states of its continuous spectrum
for the relatively low (up to a few MeV) energies. The large value of the rms radius in the
ground state can be explained by the considerable diuseness of the wave function caused
by the slowly-decreasing eective potential. The asymptotic region, where the wave function
decreases exponentially begins only at the hyperradius values greater than 15{20 Fm.
The phase shift of elastic scattering (3 ! 3) is proportional to k or
p
E at low energies
which leads to the sharp maximum of the matrix element of the isoscalar transition from the
ground state to the 0
+
-states of the continuous spectrum. Finally, our calculations predict
a considerable enhancing of the probability of the radiative capture of two neutrons by the
alpha-particle for the energy corresponding to the 2
+
-resonance of
6
He.
We thank Dr. A. A. Korsheninnikov for stimulating discussions.
References
[1] G. F. Filippov, A. V. Nesterov, I. Yu. Rybkin, S. V. Korennov, Fiz. Elem. Chastits. At.
Yadra 25, 1347 (1994).
[2] M. V. Zhukov, B. V. Danilin, D. V. Fedorov et al., Phys. Rep. 231, 151 (1993).
[3] A. M. Badalyan, Yu. A. Simonov, Yad. Fiz. 3, 1032 (1066).
[4] L. D. Blokhintsev, I. Borbely, E. I. Dolinsky, Yad. Fiz. 8, 1189 (1977).
[5] B. R. Levy, J. B. Keller, J. Math. Phys. 4, 54 (1963).
[6] L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nauka, Moscow, 1989)
[7] F. Ajzenberg-Selov, Nucl. Phys. A490, 1 (1988).
5
Figure 1: The wave function of
6
He ground state, its asymptotic behaviour (Eqs. (9) and
(10)) and the eective potential. The ground-state energy is marked by a horizontal line,
the vertical line corresponds to  =
q
h
2
i = 5:59 Fm).
Figure 2: Low-energy behaviour of tan  as a function of k.
6
Figure 3: The scattering phase shift in the 0
+
-state.
Figure 4: The matrix element of the isoscalar transition from the ground state to the 0
+
states of the continuous spectrum. (Results for 
max
= 50 Fm and 100 Fm are almost
identical.)
7
Figure 5: The wave function of the 2
+
resonance and the ground state (scaled) and the
eective potential in the 2
+
state
Figure 6: The scattering phase shift in the 2
+
-state.
8
Figure 7: The matrix element of the isoscalar E2 transition from the ground state to the
2
+
states of the continuous spectrum. (Results for 
max
= 50 Fm and 100 Fm are almost
identical.)
9


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