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Possible existence of 10He as narrow three-body resonance
NUCLEAR PHYSICS A
Nuclear Physics A559 (1993) 208-220 North-Holland
Possible existence of loHe as narrow three-body resonance A.A. Korsheninnikov,
B.V. Danilin, M.V. Zhukov
The Kurchatov Institute, 123182 Moscow, Russian Federation
Received 13 January 1993
Abstract Theoretical hyperspherical studies of the exotic nucleus “He in the framework of the n + n + 8He model are performed. The results of calculations show that the ground state of “He can exist as a narrow three-body resonance with energy - 1 MeV above the three-particle threshold.
1. Introduction In studies of light neutron-drip-line the investigations of the “He nucleus represent one of the most interesting problems. This nucleus continues the series of helium isotopes, which have such fascinating features as “neutron halo” [l-8], the theoretically predicted exotic spatial structure [4,1,2,5], and peculiar behavior of binding energies called a “helium anomaly” [6]. On being observed, the “He nucleus would have the maximal neutron excess (per one proton) among known nuclei. Since “He can be considered as the “Li nucleus without one proton in the 9Li core, an analogy is seen between “He and “Li, while the latter gives a sample of very specific properties of neutron-rich systems [7,8]. In addition, in calculations [9] it was found that “He can have very peculiar multipole excited states attributed to the neutron excess. A number of experimental attempts to find “He [lo-161 have demonstrated that this nucleus should be unbound. For example, in ref. [16] the upper limit on the bound “He production rate in the 180 beam fragmentation was obtained, which is about 1000 times lower than the value expected from the smooth trend of yields of neutron-rich H, He and Li isotopes measured for the same fragmentation Correspondence Federation.
to: Dr. A.A. Korsheninnikov,
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The Kurchatov Institute,
Moscow 123182, Russian
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AA. Korsheninnikov et al. / Three-body resonance
209
process. Estimates of the Garvey-Kelson type of extrapolation predicted also unbound “He [17-201. Notice that unstable “He has not yet been observed. From the experimental point of view the question about the existence of “He as an unbound resonance is closely connected with the question on a possible width of this state. Unstable loHe should decay into three particles, ‘He + n + n, because the subsystems ‘He + n and n f n have no bound states. From the view point of the simplest two-body model, which relates to the decay into ‘He and point-like dineutron, “He could not exist as a well-pronounced resonance, because in this process the centrifugal barrier is absent (I = 0) and the width should be large (several MeV). At the same time, the mentioned two-particle approach can be to some extent misleading, since three-particle systems are known to have peculiarities going beyond the simple two-particle problem (see, e.g., ref. [21]). To clarify the situation it is necessary to study the “He nucleus taking into account the three-body continuum effects. In this paper we present the results of the calculations for the energy of possible resonance “He (ground state) and for the width for the “He decay into the three-body continuum, “He +8He + n + n. The results indicate the definite possibility for “He to be a narrow resonance. The width calculations are carried out using the three-body generalization [22] of the method, which was successfully used to solve the coupled-channel resonant problem for two-particle decays [23] based on Feshbach’s formal theory of resonances [24]. To describe motion in the ‘He + n + n system we are solving the three-body Schrodinger equation expanding the ‘He + n + n wave function in the hyperspherical harmonics basis. The paper is organized as follows. In the next section we discuss the pairwise interactions ‘He + n and n + n and some physical assumptions on the “He structure. In sect. 3 we describe the theoretical method for investigation of the decay into the three-body continuum. In sect. 4 the results of calculations are considered. The main conclusions are given in the summary.
2. The n +*He and n + n interactions From the view point of the conventional shell model two valence neutrons in “He should be located at the OP~,~ shell because of OS,,, and Op,,, shells are occupied in ‘He. Thus in the ‘He + n potential the Osl,* and Op,,, states must be forbidden. For such a situation in the three-body problem two ways are known to treat the Pauli principle between valence nucleons and nucleons from a corenucleus. One way is to project out the occupied states [4,5,25] from the three-body wave function and another to include a repulsive core in corresponding partial components of the core-nucleon potential [1,2,51. In ref. [51 it was shown that both approaches give similar results for loosely bound three-particle system. In this
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Table 1 Parameters of the n +‘He potential components, given in MeV Variant
u
T/cfd = U2fidexp(-r2/&
QL~l
KLal
UIf-Pauli
20
I1
50
-5
-33 -33
12 13
50 50 50
-10 -15 -20
-33 -33 -33
LS
50
0
r. = 3.4 fm, U2Lvd are
15 10 5 0
paper we use the Pauli-repulsive core for the 8H + n interaction at the Osl,* and Ops,* states. For the latter we make use of the sign-inverse spin-orbit-type potential, which pushes out the Opa,* state far into continuum, while the position of the Op,,, state is preserved. The potential for Op,,, was chosen to reproduce the energy of the 9He ground state. Note that the above-mentioned picture corresponds to the limiting case of very strong jj-coupling. For nuclei in the middle of the Op shell the case of strong LS-coupling is also possible, which means strong mixing of the OP~,~ states caused by interaction between nucleons in the Op shell. In such a situation the effective n + ‘He spin-orbit forces are very small. The corresponding picture was investigated in details in refs. [26,271 for the “Li nucleus. We can take into account these two extreme (jj and LS) as well as intermediate cases by varying the repulsive core in the spin-orbit potential for Op-state (as was mentioned above, the OP~,~ state position is fixed). Parameters of the used total are presented in table 1. The central potential% 6 + 8He = CA = s,p&&al Pauli, part of the potential includes s, p and d partial components. For the potential components we take the gaussian shape with the width parameter, r. = 3.4 fm, being the ‘He matter radius, 2.5 fm 13,281,increased by 0.9 fm. The latter value, which can be attributed to a neutron, is chosen to be the same as a difference between the a-particle matter radius and the width of the n + (Ygaussian potential reproducing the experimental n + (Yphase shifts [21. The strength of the s-state component (table 1) is typical for the s-state Pauli-repulsive core [2,5]. When we change this component from 50 to 30 MeV, the results of calculations are stable, e.g., the “He resonance energy changes only by 0.06 MeV. The varied components of the p-state potential (table 1) correspond to the conserved strength for the p 1,2 potential, U(p,,,) = -20 MeV. Being calculated for this potential, the phase shift of the n +*He scattering is presented in fig. 1. The phase shift achieves the value 1/27-r at the n-8He relative energy En_nHe= 1.15 MeV. This resonant energy is in agreement with the experimental values for 9He, which are known from the ‘Be(r-, r+)‘He reaction, EgHe = 1.13 f 0.1 MeV 1201, from the 9Be(13C, 130j9He reaction, EgHe = 1.16 + 0.1 MeV 1291, and from the
vff-
A.A. Korsheninnikou et al. / Three-body resonance
0
Fig.
1
2 Xc,,
3 MeV
4
211
5
1. The phase-shift energy dependence for the p-state n+*He scattering. In the insert corresponding n + *He potentials are shown (nuclear one and its sum with centrifugal barrier).
the
9Be(‘4C, 140)9He reaction, ,?&ne= 1.21(8) MeV [30]. If we vary the ~r,~ potential strength in the limits allowed by the mentioned experimental data on EyHe, the change of the final is small in comparison with the changes caused by the Pauli-repulsive core variance (table 1). The above-mentioned experiments are very complicated because of low cross section for the 9He population ( N 40 nb/sr) and only the low statistics manifestations on the possible existence of the 9He excited states were obtained, while the angular distributions could not be measured and J” and 9He* could not be found experimentally. To estimate the influence of 9He* we consider the most statistically significant peculiarity in the experimental distributions [20,29] at E& - 4 MeV while assuming for the state J” = 2 + (shell-model calculation [31]; the comparison with predictions of other shell-model calculations is discussed in ref. [20]). The d-state potential, which reproduces the energy position of such a peculiarity in the n + ‘He scattering cross section (the phase shift does not achieve l/277, so it is not a resonance in a sense>, is also presented in table 1 though the influence of this interaction on final conclusions of this paper is negligible. For the nn interaction we use the realistic Gogny-Pires-de Tourreil potential [32], which includes repulsion at small distance, spin-orbit and tensor forces.
3. Theoretical approach for the decay into the three-body continuum 3.1. Form of the 8He + n + II wave function The asymptotics of the wave function, which describes the motion in a three-body system, is more complicated than in a two-particle problem. In the three-particle case the asymptotics has three components [33]. One of them corresponds to the bound state in a two-particle subsystem; in the case under consideration this
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A.A. Korsheninnikov et al. / Three-body resonance
asymptotics does not exist since the subsystems 8He + n and n + n have no bound states. The second type of asymptotics, which corresponds to two-particle rescattering on the energy-shell, i.e., on large distance from the third particle, is not predominant in the 8He + n + n system due to the ‘He resonance and virtual state of two neutrons are very short-lived states (at the same time, the final state interaction at the resonance ‘He and the nn virtual state is mainly taken into account in the wave function expansion used below). The predominant type is the third type of the three-body asymptotics, the genuinely three-particle form ‘I2 In the latter formula x2 = 2mE/h2 (m is nucleon mass, E is the N expGp)/p . energy of the three-body system) and p2 =.x;_~~ +x;_~ is a permutation invariant variable connected with the reduced Jacobi coordinates (A.. . are the mass numbers of the particles)
Ai-jk =
Ai(Aj
AjAk
+Ak)
Aj_k = A, +A,.
Ai+Aj+Ak’
It should be stressed here that the genuinely three-particle asymptotics differs drastically from the simple model discussed in the Introduction for the decay into 8He and point-like di-neutron; the latter case would correspond to the above-mentioned first type of Merkur’ev’s asymptotics (with bound binary subsystem) having different functional dependence [331. In such a case, when long-lived binary subsystems are absent and the genuinely three-particle asymptotics plays a primal role, it is convenient to present the 8He + n + n wave function $,,,,,(x~_~~, x~_~), which is an “active” part of the total wave function
(the exponent describes the center-of-mass motion; II,,, is the ‘He intrinsic wave function), in the following form of expansion in basis of hyperspherical harmonics
where
{fly)) = {e(‘),
p Sin eci), Xi_jk
nj-k,
ni_jk)
= p COS eci), nj_kxj_k/xj,,
are
hyperspherical ni_jk
=xi_jk/Xi-jk);
variables XKLS
are
(xj-k wQCn-
=
A.A. Korsheninnikov et al. / Three-body resonance
213
dent hyperradial functions; 0, and 0, are spin and isospin functions (5’ is the summary spin of two neutrons); the quantum number of hypermomentum K= Ij_j,+lj_,+2n (n=O, 1,2, . ..) is connected with orbital momenta between particles “j” and “k” and between their center of mass and particle “7; the total angular momentum, J, is the “He spin, J = MJ = 0; due to the positive parity of “He, in eq. (3) at J = 0 we have li_jk = Zj_+ Hyperspherical harmonics have the form
=
~~-jkfJ--*( tIci)) C ( li_jkml,_,m'ILM)Y~-jk(
ni_jk)Ykj-k(
nj_k),
mm’
4:-jkI/-k(,g(i)) =
NKLM
N
KLM
cos't -jke(')
sinl,-ke(i)
p
:~:;_i$1’_;;;;‘2’(
cos 289
)
(2K+4)(~(K-Ijk-lj_jk))!(f(K+l,-k+’i-jk)+1)! =
l/2
r(3(K-lj_k+li_jk+3))r(~(K+Ij-k-‘i-jk+3))
’ I
(4) where (Iml’m’ ) LM) are Clebsch-Gordan coefficients; I$, are spherical harmonics; Pf’7b) are Jacobi polynomials; r are gamma functions. With respect to two-neutron permutation it is convenient to consider the wave function (3) in the coordinates {i - jk, j - k) = {‘He-nn, nn}. Then antisymmetry makes the value 1~n~_~,,= I,, to be even at S = 0 and odd at S = 1. The wave function (3), being considered at the boundary condition for the hyperradial functions ,y&& + 03)N exp( -xp), represents a case of “quasibound” wave function because it corresponds to the approximation, when “He at the energy above three-particle threshold is treated as a bound state. An improved appraoch was formulated in ref. [22], where the genuinely three-body asymptotics was rewritten including in the formula the centrifugal barrier effects and then, using close analogy with formulas for two-particle scattering, the boundary condition was derived, ,ygLS(p + m>m~~/~N~+~(p), w h’lc h IS . analogous to the singlechannel problem at the resonant value of phase shift (above, N, is the Neumann function). At this boundary condition the wave function (3) represents a “resonant” wave function because it corresponds to the finite-range resonant state (at p RI. In ref. [221 it has been demonstrated that the resonant wave function can be successfully used to calculate the width for three-particle decays. In eq. (3) the explicit form of the hyperradial functions xiLS(p) is still unknown. To find them we should solve a dynamic equation.
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3.2. Three-body Schriidinger equation
The three-particle
wave function (3) is a solution of the Schrodinger
equation
(5) with V being the pair interactions between two neutrons and between one neutron and 8He. The kinetic energy can be written in the hyperspherical variables with the operator of hypermomentum, d, which relates to the hyperspherical harmonics (4)
(6) If we insert eq. (3) into eq. (5) using eq. (6), multiply the result by the function, which is in brackets [ * - * 1, in eq. (3), and then integrate over d@, we get the following set of second-order differential equations for the hyperradial functions XL&):
PI -E1xk( I KLS,KLS( LI =-K’L’S’I’I’ cah vg~;r,!,#( p);;YLrst( p) P
-__
a2
--
2m
ap2
(K + %K + $1 + J.I,,l,I, P2
P>
(7)
(in the sum the term with K = K’, L = S = L’ = S’, I= 1’ is omitted). In eq. (7) the matrix elements Vk’i&,L,S, ( p > = J[Yi!L, 8 O,,]~J[Y~~ @ Os]J=O da:) can be easily calculated for that component of the potential, which depends on the coordinate xi-k being chosen for Y~#&?. Two other components dependend on x~_~ and xi-j can be calculated after rotation of the hyperspherical harmonics Yki to the corresponding coordinates with the help of Reynal-Revai coefficients, Y&&2l’)) = C,,( I’l’ ( u>,,Y&(n~)> [34,35]. As it is seen from (7), the three-body effective centrifugal barrier, V,, N {K + $)(K + $1/p”, does not vanish even if K = 0, i.e., if orbital momenta lj_k and li_jk are equal to zero. This is one of the most important differences between the three-body picture and the simply two-body model for the decay into *He and point-like di-neutron, discussed in the Introduction. The hyperradial functions xsLS satisfying eq. (7) are regular at the origin, + X~LS(P + 0) u P K+5/2. As it was mentioned above, for the asymptotics xiLs(p 03) N exp( - xp) we get finally the quasibound wave function (3), while the asymptotics x&Jp + 0) N ~‘/~lV~+~(p) g ives the resonant wave function.
A.A. Korsheninnikou et al. / Three-body resonance
215
3.3. Width of the three-body decaying state
The three-body Schrddinger equation in the form of the coupled channel set (7) is equivalent to the case of many-channel scattering in a deformed field. The latter problem was studied in ref. [23] for two-particle decays of a resonant state and the corresponding expression for a width was derived. This expression can be rewritten for the three-body case in the following form [32]:
(8) where l-$$ and r are partial and total widths of the three-particle decaying state. In eq. (8) the following notations have been used: (i) x$!$,,&) are hyperradial components of the resonant wave function. They are normalized in the range from the origin to the matching hyperradius, R,
c /“IX$&) I2dP= 1.
KLSl,I,
0
(9)
(ii) &$&) = iK\IXPJK+2(xp> are Bessel functions regular at the origin from the six-dimensional plane wave expansion (29TP3
eXp(iqj_krj_k
+ iqi_jkri-jk)
(iii) In eq. (10) and qi_jk are reduced Jacobi momenta conjugated to the coordinates (1) (p . . . are momenta of particles) qj_k
(11)
(iv)In eq. (10) {n;(i)) = (P(r), ?z:__~,nF_jk) are hyperspherical momentum variables, qj_k = x sin W(l), qi_jk = x COS ml), ny_k = qj_k/qj_k, n;_jk = qi_jk/qi_jk, x2 = qi2_jk+ qJ’_k = 2mE/h*. An element of volume in the six-dimensional momentum space (three-particle phase space) is dr = df&_jk dqj_k = X5dX d0;(r)
(12)
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Table 2 Partial norms of the resonant wave function components for the n +sHe potentials from table 1. The last column gives a sample for the quasibound wave function Variant of the n + ‘He potential, VCr&tra,, ~,~_rauli (MeV) Resonant state Wave function components K
L=S
I,, = 18He-nn
0 2 2 4 4 4 6 6 6 6 8 8 8 8
0 0 1 0 0 1 0 0 1 1 0 0 1 1
0 0 1 2 0 1 2 0 3 1 2 0 3 1
Quasibound
0, 20
5, 10
1.40 41.37 52.71 0.50 0.91 0.28 0.69 0.02 1.84 0.06 0.02 0.15 0.03 0.031
1.27 44.16 50.02 0.48 1.02 0.27 0.70 0.03 1.78 0.06 0.02 0.16 0.03 0.006
15,5 +“%I’-““=‘(%)
15, 5
KL-S
1.35 64.62 29.20 0.48 1.86 0.17 0.84 0.07 1.11 0.04 0.02 0.24 0.02 0.003
2.39 64.82 27.75 0.40 2.12 0.17 0.84 0.08 1.09 0.04 0.02 0.25 0.02 0.004
In other words, the formula (8) for the width corresponds like in the two-particle case to the subdivision of the wave function space into two parts, the smooth continuum, I&-, and the finite-range function of the resonance, JIn, embedded into the continuum. As was demonstrated in ref. [22], to calculate the width, r N I (+c I I/ l1cIn) I ‘, th ese two functions can be approximated with the help of the six-dimensional plane wave (for chargeless particle) and the resonant wave function, discussed in previous paragraphs. In ref. [22] the comparative analysis of this method, three-body scattering and three-body decay approaches was done and close agreement between the results of all three methods for the width calculations was found. 4. Results of calculations To solve numerically eqs. (7) the hypermomentum values, K, from 0 up to K = KnZ!Xshould be considered, where K,, is large enough to stabilize the wave function and the width. For the finally chosen K,,, = 8 such a convergence is reached as is seen from tables 2 and 3. In table 2 we present the partial norms, N~~~~~n”=‘, obtained for the wave function components in eq. (3), N~~~~~““=’ = 10”I ,y$!?‘=‘(p) I 2dp CR is the matching hyperradius, 20 fm). In table 2 as an example the results for the three potential sets from table 1 are shown for the resonant wave function calculations as well as the result for the quasibound wave function (last column in table 2).
A.A. Korsheninnikou et al. / Three-body resonance
217
Table 3 The partial and total widths of ‘“He for the n +‘He potentials from table 1 The n + ‘He potential, V&,,, & v,5p,,,i(MeV) 0, 20
Wave function components K
L=S
bm = be-m
0
0
0
2 2 4 4 4 6 6 6 6 8 8 8 8
0 1 0 0 1 0 0 1 1 0 0 1 1
0 1 2 0 1 2 0 3 1 2 0 3 1
Total width (keV)
5, 15 ,~;‘1p-““==l
119 83 115 2.7 x 10-l 1.8 x 10-4 3.3 x 10-z 1.3 x 10-3 6.5 x 1O-6 4.2 x 1O-3 1.5 x 10-4 1.8 x 10-7 6.7 x 1O-7 2.3 x 1O-7 1.7 x 10-s 318
106 84 103 2.2 x 10-l 1.3 x 10-j 2.8 x 10-2 1.2 x 10-3 2.5 x 1O-6 3.5 x 10-z 1.2 x 10-4 1.5 x 10-7 6.1 x lo-’ 1.8 x 10-7 1.3 x 10-s 293
15,5 (keV)
47 72 35 4.2 x 9.5 x 6.2 x 3.9 x 4.7 x 5.4 x 1.9 x 3.4 x 1.9 x 1.9 x 1.4 x
lo-* 10-3 1O-3 10-4 1o-7 1o-4 1o-5 10-s 10-7 1o-8 10-9
155
As is seen in table 2, in all cases the components at K = 2 are predominant and exhaust = 94% of the total wave function norm. The higher-order terms at K > 2 are strongly suppressed due to the increased of the three-body centrifugal barrier, Kb = (K + $XK + $1,while the component at K = 0 in spite of the lowest value of the centrifugal barrier is restrained by the Pauli-repulsive core being the consequence of the OS,,~ state occupation. The predominant role of the single hypermomentum value was previously found for the states of the A = 6 nuclei also [1,2]. This suggests that the hypermomentum K is a “good” quantum number for many applications reflecting specific symmetries in the three-particle system. Table 2 shows that the variance of the Pauli-repulsive spin-orbit-type potential changes the relative intensities of two components at K = 2. In the case of the potential set with the maximum value I/lsP_Pauli = 20 MeV the component with L = S = 1 (53%) is more intensive than the component with L = S = 0 (41%). This is in agreement with the simplest estimation in the frame of jj-coupling picture for two neutrons at the Op,,, states, which gives the weights : for S = 1 and f for S = 0 (the squared values of the corresponding 9j-symbols). The ratio of the norms for two components at K = 2 becomes opposite when we reduce l$,P_rauli(table 2) reflecting the configuration mixing due to the nn-interaction. It is interesting to compare the results of calculations for the resonant wave function with the quasibound-state approximation, which is usually used for estimations. The last two columns in table 2 demonstrate that the structures of these wave functions (for the same potentials) are very close.
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A.A. Korsheninnikov et al. / Three-body resonance
In table 3 we show the partial and total widths of “He, which were calculated with the help of eq. (8) based on the resonant wave functions (in table 3 as an example the results for the three potential sets from table 1 are presented). It is seen, that the width is completely determined by the lowest partial components at K = 0, 2. The role of terms at K > 2 is negligible due to the higher centrifugal barrier. Though the wave function components for K = 0 are suppressed in the internal region (table l), the partial width with K = 0 contributes significantly to the total width (table 3) because this decay channel is the most preferable from the view point of the centrifugal barrier. The partial widths for the decay channels at K = 2, in spite of higher centrifugal barrier, and enhanced by the predominance of the corresponding components in the wave function. As is illustrated in table 3, last row, the “He nucleus can be a narrow state. The results of calculations for all potential sets from table 1 give an estimation for the total width rloHe - 150-300 KeV. This small value can be explained physically being proportional to the three-body phase volume - E2, which is significantly lower (for small energy) than the two-body one - E112. The results of calculations at 0 6 K G 8 for different potential sets from table 1 allow one to suggest the “He state position at the energy EloHe - 0.7-0.9 MeV above the n + n +‘He threshold. Just when convergence for the wave function components and the partial widths is very fast in the hyperspherical harmonics method, while convergence for the state energy is slower because the energy is a small difference between two large values, potential and kinetic energies. Using the asymptotic exponential law for K,,, + 03 [36,37], which is known for gaussian potentials, we estimate an uncertainty of the above-mentioned value, EloHe - 0.70.9 MeV, as SE - 0.2-0.3 MeV. Thus from the above calculations it is seen that the loHe nucleus can exist as a well-pronounced state with the width rloHe - 150-300 keV at energy less than 1
K=Z.L=S=l
K=O.L=S=O , . .
0
3 P,
Fig. 2. Hyperradial corresponding
dependence three-body
9
6
12
fm
of the wave function components at K = 2 and 0. In the insert potentials are shown including the three-body centrifugal barrier.
the
A.A. Korsheninnikov et al. / Three-body resonance
219
MeV above the three-particle threshold. From the physical point of view the origin of resonant states is a wave pocket in one of the partial potentials as in the two-particle problem. For example, earlier investigations of the A = 6 nuclei [38] showed a pocket corresponding to well-known resonances with J” = O+ and J” = 2+ (T = 1). For the case of loHe the wave pocket exists in the K = 2 partial components as illustrated in fig. 2. 5. Summary
The experimental and theoretical investigations of the “He nucleus are very interesting from various points of view. In particular, this nucleus is situated just near the neutron drip-line. The conventional shell model allows one to consider “He as a double-closed-shell neutron-rich nucleus. So the “He nucleus can provide an answer about the validity of shell-model prescriptions near the neutron drip-line. Since “He can be considered as the “Li nucleus without one proton in the 9Li core, it is easy to estimate, in the framework of the three-body approach, the energy of the ground state of “He . As a first approximation this energy can be evaluated as 2E,(n + ‘He) - Epairing. From the “Li binding energy the pairing energy of valence neutrons is EpairingN 1.3 MeV, so we get for the “He resonance the energy less than 1 MeV. This estimation is in agreement with the strict three-body calculations performed in this article, which give for the “He resonance energy 0.7-0.9 MeV. To calculate the energy and width of possible resonance “He (ground state) we have investigated the exotic “He nucleus in the n + n + *He model, which takes into account the three-body continuum for the decay “He + n + n +‘He. The results of calculations suggest that the “He resonance at energy less than 1 MeV above the three-particle threshold has a width rloHe N 150-300 keV. The latter value is an order of magnitude less than the value given by the two-body estimation for the decay into ‘He and point- like di-neutron. So three-body continuum dynamics supports the existence of the “He resonance. Experimental searching for the “He resonance is badly needed. As is known from many-body calculations, the stability of such systems as loHe depends on fine details of the nucleon-nucleon interaction and particularly on the interaction in the p-state (see, e.g., ref. [39]). So any definite experimental information on the “He energy should be useful. One of the authors, A.A.K., is thankful for the hospitality to RIKEN, where part of this work was done. References [l] B.V. Danilin, M.V. Zhukov, A.A. Korsheninnikov, L.V. Chulkov and V.D. Efros, Sov. J. Nucl. Phys. 48 (1988) 766; 49 (1989) 217; 49 (1989) 223; M.V. Zhukov, L.V. Chulkov, B.V. Danilin and A.A. Korsheninnikov, Nucl. Phys. A533 (1991) 428
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et al. / Three-body resonance
[2] B.V. Danilin, M.V. Zhukov, A.A. Korsheninnikov and L.V. Chulkov, Sov. J. Nucl. Phys. 53 (1991) 45 [3] I. Tanihata, D. Hirata, T. Kobayashi, S. Shimoura, K. Sugimoto and H. Toki, Phys. Lett. B289 (1992) 261 [4] V.I. Kukulin, V.M. Krasnopol’sky, V.T. Voronchev and P.B. Sazonov, Nucl. Phys. A453 (1986) 365 [5] J.M. Bang, I.J. Thompson, M.V. Zhukov, B.V. Danilin, D.V. Fedorov and J.S. Vaagen, Preprint of Niels Bohr Institute, Denmark, NBI-91-31, 1991; submitted to Phys. Reports [6] A.A. Ogloblin and Yu.E. Penionzhkevich, in Treatise on heavy-ion science, ed. D.A. Bromley (Plenum, New York, 1989) v.8, p. 2612 [7] I. Tanihata, in Treatise on heavy-ion science, ed. D.A. Bromley (Plenum, New York, 1989) v.8, p. 443 [8] T. Kobayashi, Nucl. Phys. A538 (1992) 343 [9] H. Sagava, Nucl. Phys. A538 (1992) 619~ [lo] A.M. Poskanzer, S.W. Cosper, E.K. Hyde and J. Cerny, Phys. Rev. Lett. 17 (1966) 1271 [ll] S.L. Whetstone, Jr., and T.D. Thomas, Phys. Rev. 154 (1967) 1174 [12] S.W. Cosper, J. Cerny and R.C. Gatti, Phys. Rev. 154 (1967) 1193 [13] A.G. Artuhk, V.V. Avdeichikov, G.F. Gridnev, V.L. Mikheev, V.V. Volkov and J. Wilczynski, Nucl. Phys. Al68 (1971) 321 [14] G.G. Beznogikh, N.K. Zhidkov, L.F. Kirillova, V.A. Nikitin, P.V. Nomokonov, V.V. Avdeichikov, Yu.A. Mirin, V.S. Oplavin, V.D. Maisyukov, Yu.V. Maslennikov, A.P. Shevchenko, A. Buyak and M. Shavlovski, JETP Lett. 30 (1979) 323 [15] Yu.Ts. Oganesyan, Yu.E. Penionzhkevich, E. Gierlik, R. Kalpakchieva, T. Pawlat, C. Borcea, A.V. Belozerov, Yu.P. Kharitonov, S.P. Tretvakova, V.G. Subbotin, SM. Lukvanov, N.V. Pronin and A.A. Bykov, JETP Lett. 36 (1982) 129 [16] J. Stevenson, B.A. Brown, Y. Chen, J. Clayton, E. Kashy, D. Mikolas, J. Nolen, M. Samuel, B. Sherrill, J.S. Winfield, Z.Q. Xie, R.E. Julies and W.A. Richter, Phys. Rev. C37 (1988) 2220 [17] C. Thibault and R. Klapish, Phys. Rev. C6 (1972) 1509 [18] N.A. Jelley, J. Cerny, D.P. Stahel and K.H. Wilcox, Phys. Rev. Cl1 (1975) 2049 [19] J. Janecke, At. Data Nucl. Data Tables 17 (1976) 455 [20] Kamal K. Seth. M. Artuso, D. Barlow, S. Iversen, M. Kaletka, H. Nann, B. Parker and R. Soundranayagam, Phys. Rev. Lett. 58 (1987) 1930 [21] V.M.Efimov, Comm. Nucl. Part. Phys. 19 (1990) 271 [22] B.V. Danilin, Izv. Akad. Nauk SSSR (ser. fiz.) 54 (1990) 2212 [23] J. Bang, F.A. Gareev, I.V. Puzynin and R.M. Jamalejev, Nucl. Phys. A261 (1976) 59 [24] H. Feshback, Ann. of Phys. 5 (1958) 357; 19 (1962) 283; 43 (1967) 410 [25] K. Kato and K. Ikeda, preprint of Niigata University, Japan, 1992 [26] A.C. Hayes, Phys. Lett. B254 (1991) 15 [27] J.M. Bang and I.J. Thompson, Phys. Lett. B279 (1992) 210 [28] I. Tanihata, T. Kobayashi, 0. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda and H. Sato, Phys. Lett. B206 (1988) 592 [29] H.G. Bohlen, Proc. Int. Symp. on structure and reactions of unstable nuclei, Niigata, Japan, 17-19 June 1991, ed. K. Ikeda and Y. Suzuki, p. 83 [30] H.G. Bohlen, preliminary results, private communication [31] N.A.F.M. Poppelier, L.D. Wood and P.W.M. Glaudemans, Phys. Lett. B157 (1985) 120 [32] D. Gogny, P. Pires and R. de Tourreil, Phys. Lett. B32 (1970) 591 [33] S.P. Merkur’ev, Sov. J. Nucl. Phys. 19 (1974) 222 [34] J. Raynal and J. Revai, Nuovo Cim. A68 (1970) 612 [35] Ya.A. Smorodinskii and V.D. Efros, Sov. J. Nucl. Phys. 17 (1973) 107 [36] A.M. Badalyan and Yu.A. Simonov, Sov. J. Nucl. Phys. 3 (1966) 755 [37] T.R. Schneider, Phys. Lett. B40 (1972) 439 [38] B.V. Danilin and M.V. Zhukov, Sov. J. Nucl. Phys. 56 (1993) 123 [39] AI. Baz, V.F. Demin and M.V. Zhukov, Sov. J. Nucl. Phys. 9 (1969) 1184
Nuclear Physics A559 (1993) 208-220 North-Holland
Possible existence of loHe as narrow three-body resonance A.A. Korsheninnikov,
B.V. Danilin, M.V. Zhukov
The Kurchatov Institute, 123182 Moscow, Russian Federation
Received 13 January 1993
Abstract Theoretical hyperspherical studies of the exotic nucleus “He in the framework of the n + n + 8He model are performed. The results of calculations show that the ground state of “He can exist as a narrow three-body resonance with energy - 1 MeV above the three-particle threshold.
1. Introduction In studies of light neutron-drip-line the investigations of the “He nucleus represent one of the most interesting problems. This nucleus continues the series of helium isotopes, which have such fascinating features as “neutron halo” [l-8], the theoretically predicted exotic spatial structure [4,1,2,5], and peculiar behavior of binding energies called a “helium anomaly” [6]. On being observed, the “He nucleus would have the maximal neutron excess (per one proton) among known nuclei. Since “He can be considered as the “Li nucleus without one proton in the 9Li core, an analogy is seen between “He and “Li, while the latter gives a sample of very specific properties of neutron-rich systems [7,8]. In addition, in calculations [9] it was found that “He can have very peculiar multipole excited states attributed to the neutron excess. A number of experimental attempts to find “He [lo-161 have demonstrated that this nucleus should be unbound. For example, in ref. [16] the upper limit on the bound “He production rate in the 180 beam fragmentation was obtained, which is about 1000 times lower than the value expected from the smooth trend of yields of neutron-rich H, He and Li isotopes measured for the same fragmentation Correspondence Federation.
to: Dr. A.A. Korsheninnikov,
0375-9474/93/$06.00
The Kurchatov Institute,
Moscow 123182, Russian
0 1993 - Elsevier Science Publishers B.V. AII rights reserved
AA. Korsheninnikov et al. / Three-body resonance
209
process. Estimates of the Garvey-Kelson type of extrapolation predicted also unbound “He [17-201. Notice that unstable “He has not yet been observed. From the experimental point of view the question about the existence of “He as an unbound resonance is closely connected with the question on a possible width of this state. Unstable loHe should decay into three particles, ‘He + n + n, because the subsystems ‘He + n and n f n have no bound states. From the view point of the simplest two-body model, which relates to the decay into ‘He and point-like dineutron, “He could not exist as a well-pronounced resonance, because in this process the centrifugal barrier is absent (I = 0) and the width should be large (several MeV). At the same time, the mentioned two-particle approach can be to some extent misleading, since three-particle systems are known to have peculiarities going beyond the simple two-particle problem (see, e.g., ref. [21]). To clarify the situation it is necessary to study the “He nucleus taking into account the three-body continuum effects. In this paper we present the results of the calculations for the energy of possible resonance “He (ground state) and for the width for the “He decay into the three-body continuum, “He +8He + n + n. The results indicate the definite possibility for “He to be a narrow resonance. The width calculations are carried out using the three-body generalization [22] of the method, which was successfully used to solve the coupled-channel resonant problem for two-particle decays [23] based on Feshbach’s formal theory of resonances [24]. To describe motion in the ‘He + n + n system we are solving the three-body Schrodinger equation expanding the ‘He + n + n wave function in the hyperspherical harmonics basis. The paper is organized as follows. In the next section we discuss the pairwise interactions ‘He + n and n + n and some physical assumptions on the “He structure. In sect. 3 we describe the theoretical method for investigation of the decay into the three-body continuum. In sect. 4 the results of calculations are considered. The main conclusions are given in the summary.
2. The n +*He and n + n interactions From the view point of the conventional shell model two valence neutrons in “He should be located at the OP~,~ shell because of OS,,, and Op,,, shells are occupied in ‘He. Thus in the ‘He + n potential the Osl,* and Op,,, states must be forbidden. For such a situation in the three-body problem two ways are known to treat the Pauli principle between valence nucleons and nucleons from a corenucleus. One way is to project out the occupied states [4,5,25] from the three-body wave function and another to include a repulsive core in corresponding partial components of the core-nucleon potential [1,2,51. In ref. [51 it was shown that both approaches give similar results for loosely bound three-particle system. In this
A.A. Korsheninnikov et al. / Three-body resonance
210
Table 1 Parameters of the n +‘He potential components, given in MeV Variant
u
T/cfd = U2fidexp(-r2/&
QL~l
KLal
UIf-Pauli
20
I1
50
-5
-33 -33
12 13
50 50 50
-10 -15 -20
-33 -33 -33
LS
50
0
r. = 3.4 fm, U2Lvd are
15 10 5 0
paper we use the Pauli-repulsive core for the 8H + n interaction at the Osl,* and Ops,* states. For the latter we make use of the sign-inverse spin-orbit-type potential, which pushes out the Opa,* state far into continuum, while the position of the Op,,, state is preserved. The potential for Op,,, was chosen to reproduce the energy of the 9He ground state. Note that the above-mentioned picture corresponds to the limiting case of very strong jj-coupling. For nuclei in the middle of the Op shell the case of strong LS-coupling is also possible, which means strong mixing of the OP~,~ states caused by interaction between nucleons in the Op shell. In such a situation the effective n + ‘He spin-orbit forces are very small. The corresponding picture was investigated in details in refs. [26,271 for the “Li nucleus. We can take into account these two extreme (jj and LS) as well as intermediate cases by varying the repulsive core in the spin-orbit potential for Op-state (as was mentioned above, the OP~,~ state position is fixed). Parameters of the used total are presented in table 1. The central potential% 6 + 8He = CA = s,p&&al Pauli, part of the potential includes s, p and d partial components. For the potential components we take the gaussian shape with the width parameter, r. = 3.4 fm, being the ‘He matter radius, 2.5 fm 13,281,increased by 0.9 fm. The latter value, which can be attributed to a neutron, is chosen to be the same as a difference between the a-particle matter radius and the width of the n + (Ygaussian potential reproducing the experimental n + (Yphase shifts [21. The strength of the s-state component (table 1) is typical for the s-state Pauli-repulsive core [2,5]. When we change this component from 50 to 30 MeV, the results of calculations are stable, e.g., the “He resonance energy changes only by 0.06 MeV. The varied components of the p-state potential (table 1) correspond to the conserved strength for the p 1,2 potential, U(p,,,) = -20 MeV. Being calculated for this potential, the phase shift of the n +*He scattering is presented in fig. 1. The phase shift achieves the value 1/27-r at the n-8He relative energy En_nHe= 1.15 MeV. This resonant energy is in agreement with the experimental values for 9He, which are known from the ‘Be(r-, r+)‘He reaction, EgHe = 1.13 f 0.1 MeV 1201, from the 9Be(13C, 130j9He reaction, EgHe = 1.16 + 0.1 MeV 1291, and from the
vff-
A.A. Korsheninnikou et al. / Three-body resonance
0
Fig.
1
2 Xc,,
3 MeV
4
211
5
1. The phase-shift energy dependence for the p-state n+*He scattering. In the insert corresponding n + *He potentials are shown (nuclear one and its sum with centrifugal barrier).
the
9Be(‘4C, 140)9He reaction, ,?&ne= 1.21(8) MeV [30]. If we vary the ~r,~ potential strength in the limits allowed by the mentioned experimental data on EyHe, the change of the final is small in comparison with the changes caused by the Pauli-repulsive core variance (table 1). The above-mentioned experiments are very complicated because of low cross section for the 9He population ( N 40 nb/sr) and only the low statistics manifestations on the possible existence of the 9He excited states were obtained, while the angular distributions could not be measured and J” and 9He* could not be found experimentally. To estimate the influence of 9He* we consider the most statistically significant peculiarity in the experimental distributions [20,29] at E& - 4 MeV while assuming for the state J” = 2 + (shell-model calculation [31]; the comparison with predictions of other shell-model calculations is discussed in ref. [20]). The d-state potential, which reproduces the energy position of such a peculiarity in the n + ‘He scattering cross section (the phase shift does not achieve l/277, so it is not a resonance in a sense>, is also presented in table 1 though the influence of this interaction on final conclusions of this paper is negligible. For the nn interaction we use the realistic Gogny-Pires-de Tourreil potential [32], which includes repulsion at small distance, spin-orbit and tensor forces.
3. Theoretical approach for the decay into the three-body continuum 3.1. Form of the 8He + n + II wave function The asymptotics of the wave function, which describes the motion in a three-body system, is more complicated than in a two-particle problem. In the three-particle case the asymptotics has three components [33]. One of them corresponds to the bound state in a two-particle subsystem; in the case under consideration this
212
A.A. Korsheninnikov et al. / Three-body resonance
asymptotics does not exist since the subsystems 8He + n and n + n have no bound states. The second type of asymptotics, which corresponds to two-particle rescattering on the energy-shell, i.e., on large distance from the third particle, is not predominant in the 8He + n + n system due to the ‘He resonance and virtual state of two neutrons are very short-lived states (at the same time, the final state interaction at the resonance ‘He and the nn virtual state is mainly taken into account in the wave function expansion used below). The predominant type is the third type of the three-body asymptotics, the genuinely three-particle form ‘I2 In the latter formula x2 = 2mE/h2 (m is nucleon mass, E is the N expGp)/p . energy of the three-body system) and p2 =.x;_~~ +x;_~ is a permutation invariant variable connected with the reduced Jacobi coordinates (A.. . are the mass numbers of the particles)
Ai-jk =
Ai(Aj
AjAk
+Ak)
Aj_k = A, +A,.
Ai+Aj+Ak’
It should be stressed here that the genuinely three-particle asymptotics differs drastically from the simple model discussed in the Introduction for the decay into 8He and point-like di-neutron; the latter case would correspond to the above-mentioned first type of Merkur’ev’s asymptotics (with bound binary subsystem) having different functional dependence [331. In such a case, when long-lived binary subsystems are absent and the genuinely three-particle asymptotics plays a primal role, it is convenient to present the 8He + n + n wave function $,,,,,(x~_~~, x~_~), which is an “active” part of the total wave function
(the exponent describes the center-of-mass motion; II,,, is the ‘He intrinsic wave function), in the following form of expansion in basis of hyperspherical harmonics
where
{fly)) = {e(‘),
p Sin eci), Xi_jk
nj-k,
ni_jk)
= p COS eci), nj_kxj_k/xj,,
are
hyperspherical ni_jk
=xi_jk/Xi-jk);
variables XKLS
are
(xj-k wQCn-
=
A.A. Korsheninnikov et al. / Three-body resonance
213
dent hyperradial functions; 0, and 0, are spin and isospin functions (5’ is the summary spin of two neutrons); the quantum number of hypermomentum K= Ij_j,+lj_,+2n (n=O, 1,2, . ..) is connected with orbital momenta between particles “j” and “k” and between their center of mass and particle “7; the total angular momentum, J, is the “He spin, J = MJ = 0; due to the positive parity of “He, in eq. (3) at J = 0 we have li_jk = Zj_+ Hyperspherical harmonics have the form
=
~~-jkfJ--*( tIci)) C ( li_jkml,_,m'ILM)Y~-jk(
ni_jk)Ykj-k(
nj_k),
mm’
4:-jkI/-k(,g(i)) =
NKLM
N
KLM
cos't -jke(')
sinl,-ke(i)
p
:~:;_i$1’_;;;;‘2’(
cos 289
)
(2K+4)(~(K-Ijk-lj_jk))!(f(K+l,-k+’i-jk)+1)! =
l/2
r(3(K-lj_k+li_jk+3))r(~(K+Ij-k-‘i-jk+3))
’ I
(4) where (Iml’m’ ) LM) are Clebsch-Gordan coefficients; I$, are spherical harmonics; Pf’7b) are Jacobi polynomials; r are gamma functions. With respect to two-neutron permutation it is convenient to consider the wave function (3) in the coordinates {i - jk, j - k) = {‘He-nn, nn}. Then antisymmetry makes the value 1~n~_~,,= I,, to be even at S = 0 and odd at S = 1. The wave function (3), being considered at the boundary condition for the hyperradial functions ,y&& + 03)N exp( -xp), represents a case of “quasibound” wave function because it corresponds to the approximation, when “He at the energy above three-particle threshold is treated as a bound state. An improved appraoch was formulated in ref. [22], where the genuinely three-body asymptotics was rewritten including in the formula the centrifugal barrier effects and then, using close analogy with formulas for two-particle scattering, the boundary condition was derived, ,ygLS(p + m>m~~/~N~+~(p), w h’lc h IS . analogous to the singlechannel problem at the resonant value of phase shift (above, N, is the Neumann function). At this boundary condition the wave function (3) represents a “resonant” wave function because it corresponds to the finite-range resonant state (at p
214
A.A. Korsheninnikov et al. / Three-body resonance
3.2. Three-body Schriidinger equation
The three-particle
wave function (3) is a solution of the Schrodinger
equation
(5) with V being the pair interactions between two neutrons and between one neutron and 8He. The kinetic energy can be written in the hyperspherical variables with the operator of hypermomentum, d, which relates to the hyperspherical harmonics (4)
(6) If we insert eq. (3) into eq. (5) using eq. (6), multiply the result by the function, which is in brackets [ * - * 1, in eq. (3), and then integrate over d@, we get the following set of second-order differential equations for the hyperradial functions XL&):
PI -E1xk( I KLS,KLS( LI =-K’L’S’I’I’ cah vg~;r,!,#( p);;YLrst( p) P
-__
a2
--
2m
ap2
(K + %K + $1 + J.I,,l,I, P2
P>
(7)
(in the sum the term with K = K’, L = S = L’ = S’, I= 1’ is omitted). In eq. (7) the matrix elements Vk’i&,L,S, ( p > = J[Yi!L, 8 O,,]~J[Y~~ @ Os]J=O da:) can be easily calculated for that component of the potential, which depends on the coordinate xi-k being chosen for Y~#&?. Two other components dependend on x~_~ and xi-j can be calculated after rotation of the hyperspherical harmonics Yki to the corresponding coordinates with the help of Reynal-Revai coefficients, Y&&2l’)) = C,,( I’l’ ( u>,,Y&(n~)> [34,35]. As it is seen from (7), the three-body effective centrifugal barrier, V,, N {K + $)(K + $1/p”, does not vanish even if K = 0, i.e., if orbital momenta lj_k and li_jk are equal to zero. This is one of the most important differences between the three-body picture and the simply two-body model for the decay into *He and point-like di-neutron, discussed in the Introduction. The hyperradial functions xsLS satisfying eq. (7) are regular at the origin, + X~LS(P + 0) u P K+5/2. As it was mentioned above, for the asymptotics xiLs(p 03) N exp( - xp) we get finally the quasibound wave function (3), while the asymptotics x&Jp + 0) N ~‘/~lV~+~(p) g ives the resonant wave function.
A.A. Korsheninnikou et al. / Three-body resonance
215
3.3. Width of the three-body decaying state
The three-body Schrddinger equation in the form of the coupled channel set (7) is equivalent to the case of many-channel scattering in a deformed field. The latter problem was studied in ref. [23] for two-particle decays of a resonant state and the corresponding expression for a width was derived. This expression can be rewritten for the three-body case in the following form [32]:
(8) where l-$$ and r are partial and total widths of the three-particle decaying state. In eq. (8) the following notations have been used: (i) x$!$,,&) are hyperradial components of the resonant wave function. They are normalized in the range from the origin to the matching hyperradius, R,
c /“IX$&) I2dP= 1.
KLSl,I,
0
(9)
(ii) &$&) = iK\IXPJK+2(xp> are Bessel functions regular at the origin from the six-dimensional plane wave expansion (29TP3
eXp(iqj_krj_k
+ iqi_jkri-jk)
(iii) In eq. (10) and qi_jk are reduced Jacobi momenta conjugated to the coordinates (1) (p . . . are momenta of particles) qj_k
(11)
(iv)In eq. (10) {n;(i)) = (P(r), ?z:__~,nF_jk) are hyperspherical momentum variables, qj_k = x sin W(l), qi_jk = x COS ml), ny_k = qj_k/qj_k, n;_jk = qi_jk/qi_jk, x2 = qi2_jk+ qJ’_k = 2mE/h*. An element of volume in the six-dimensional momentum space (three-particle phase space) is dr = df&_jk dqj_k = X5dX d0;(r)
(12)
A.A. Korsheninnikov et al. / Three-body resonance
216
Table 2 Partial norms of the resonant wave function components for the n +sHe potentials from table 1. The last column gives a sample for the quasibound wave function Variant of the n + ‘He potential, VCr&tra,, ~,~_rauli (MeV) Resonant state Wave function components K
L=S
I,, = 18He-nn
0 2 2 4 4 4 6 6 6 6 8 8 8 8
0 0 1 0 0 1 0 0 1 1 0 0 1 1
0 0 1 2 0 1 2 0 3 1 2 0 3 1
Quasibound
0, 20
5, 10
1.40 41.37 52.71 0.50 0.91 0.28 0.69 0.02 1.84 0.06 0.02 0.15 0.03 0.031
1.27 44.16 50.02 0.48 1.02 0.27 0.70 0.03 1.78 0.06 0.02 0.16 0.03 0.006
15,5 +“%I’-““=‘(%)
15, 5
KL-S
1.35 64.62 29.20 0.48 1.86 0.17 0.84 0.07 1.11 0.04 0.02 0.24 0.02 0.003
2.39 64.82 27.75 0.40 2.12 0.17 0.84 0.08 1.09 0.04 0.02 0.25 0.02 0.004
In other words, the formula (8) for the width corresponds like in the two-particle case to the subdivision of the wave function space into two parts, the smooth continuum, I&-, and the finite-range function of the resonance, JIn, embedded into the continuum. As was demonstrated in ref. [22], to calculate the width, r N I (+c I I/ l1cIn) I ‘, th ese two functions can be approximated with the help of the six-dimensional plane wave (for chargeless particle) and the resonant wave function, discussed in previous paragraphs. In ref. [22] the comparative analysis of this method, three-body scattering and three-body decay approaches was done and close agreement between the results of all three methods for the width calculations was found. 4. Results of calculations To solve numerically eqs. (7) the hypermomentum values, K, from 0 up to K = KnZ!Xshould be considered, where K,, is large enough to stabilize the wave function and the width. For the finally chosen K,,, = 8 such a convergence is reached as is seen from tables 2 and 3. In table 2 we present the partial norms, N~~~~~n”=‘, obtained for the wave function components in eq. (3), N~~~~~““=’ = 10”I ,y$!?‘=‘(p) I 2dp CR is the matching hyperradius, 20 fm). In table 2 as an example the results for the three potential sets from table 1 are shown for the resonant wave function calculations as well as the result for the quasibound wave function (last column in table 2).
A.A. Korsheninnikou et al. / Three-body resonance
217
Table 3 The partial and total widths of ‘“He for the n +‘He potentials from table 1 The n + ‘He potential, V&,,, & v,5p,,,i(MeV) 0, 20
Wave function components K
L=S
bm = be-m
0
0
0
2 2 4 4 4 6 6 6 6 8 8 8 8
0 1 0 0 1 0 0 1 1 0 0 1 1
0 1 2 0 1 2 0 3 1 2 0 3 1
Total width (keV)
5, 15 ,~;‘1p-““==l
119 83 115 2.7 x 10-l 1.8 x 10-4 3.3 x 10-z 1.3 x 10-3 6.5 x 1O-6 4.2 x 1O-3 1.5 x 10-4 1.8 x 10-7 6.7 x 1O-7 2.3 x 1O-7 1.7 x 10-s 318
106 84 103 2.2 x 10-l 1.3 x 10-j 2.8 x 10-2 1.2 x 10-3 2.5 x 1O-6 3.5 x 10-z 1.2 x 10-4 1.5 x 10-7 6.1 x lo-’ 1.8 x 10-7 1.3 x 10-s 293
15,5 (keV)
47 72 35 4.2 x 9.5 x 6.2 x 3.9 x 4.7 x 5.4 x 1.9 x 3.4 x 1.9 x 1.9 x 1.4 x
lo-* 10-3 1O-3 10-4 1o-7 1o-4 1o-5 10-s 10-7 1o-8 10-9
155
As is seen in table 2, in all cases the components at K = 2 are predominant and exhaust = 94% of the total wave function norm. The higher-order terms at K > 2 are strongly suppressed due to the increased of the three-body centrifugal barrier, Kb = (K + $XK + $1,while the component at K = 0 in spite of the lowest value of the centrifugal barrier is restrained by the Pauli-repulsive core being the consequence of the OS,,~ state occupation. The predominant role of the single hypermomentum value was previously found for the states of the A = 6 nuclei also [1,2]. This suggests that the hypermomentum K is a “good” quantum number for many applications reflecting specific symmetries in the three-particle system. Table 2 shows that the variance of the Pauli-repulsive spin-orbit-type potential changes the relative intensities of two components at K = 2. In the case of the potential set with the maximum value I/lsP_Pauli = 20 MeV the component with L = S = 1 (53%) is more intensive than the component with L = S = 0 (41%). This is in agreement with the simplest estimation in the frame of jj-coupling picture for two neutrons at the Op,,, states, which gives the weights : for S = 1 and f for S = 0 (the squared values of the corresponding 9j-symbols). The ratio of the norms for two components at K = 2 becomes opposite when we reduce l$,P_rauli(table 2) reflecting the configuration mixing due to the nn-interaction. It is interesting to compare the results of calculations for the resonant wave function with the quasibound-state approximation, which is usually used for estimations. The last two columns in table 2 demonstrate that the structures of these wave functions (for the same potentials) are very close.
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A.A. Korsheninnikov et al. / Three-body resonance
In table 3 we show the partial and total widths of “He, which were calculated with the help of eq. (8) based on the resonant wave functions (in table 3 as an example the results for the three potential sets from table 1 are presented). It is seen, that the width is completely determined by the lowest partial components at K = 0, 2. The role of terms at K > 2 is negligible due to the higher centrifugal barrier. Though the wave function components for K = 0 are suppressed in the internal region (table l), the partial width with K = 0 contributes significantly to the total width (table 3) because this decay channel is the most preferable from the view point of the centrifugal barrier. The partial widths for the decay channels at K = 2, in spite of higher centrifugal barrier, and enhanced by the predominance of the corresponding components in the wave function. As is illustrated in table 3, last row, the “He nucleus can be a narrow state. The results of calculations for all potential sets from table 1 give an estimation for the total width rloHe - 150-300 KeV. This small value can be explained physically being proportional to the three-body phase volume - E2, which is significantly lower (for small energy) than the two-body one - E112. The results of calculations at 0 6 K G 8 for different potential sets from table 1 allow one to suggest the “He state position at the energy EloHe - 0.7-0.9 MeV above the n + n +‘He threshold. Just when convergence for the wave function components and the partial widths is very fast in the hyperspherical harmonics method, while convergence for the state energy is slower because the energy is a small difference between two large values, potential and kinetic energies. Using the asymptotic exponential law for K,,, + 03 [36,37], which is known for gaussian potentials, we estimate an uncertainty of the above-mentioned value, EloHe - 0.70.9 MeV, as SE - 0.2-0.3 MeV. Thus from the above calculations it is seen that the loHe nucleus can exist as a well-pronounced state with the width rloHe - 150-300 keV at energy less than 1
K=Z.L=S=l
K=O.L=S=O , . .
0
3 P,
Fig. 2. Hyperradial corresponding
dependence three-body
9
6
12
fm
of the wave function components at K = 2 and 0. In the insert potentials are shown including the three-body centrifugal barrier.
the
A.A. Korsheninnikov et al. / Three-body resonance
219
MeV above the three-particle threshold. From the physical point of view the origin of resonant states is a wave pocket in one of the partial potentials as in the two-particle problem. For example, earlier investigations of the A = 6 nuclei [38] showed a pocket corresponding to well-known resonances with J” = O+ and J” = 2+ (T = 1). For the case of loHe the wave pocket exists in the K = 2 partial components as illustrated in fig. 2. 5. Summary
The experimental and theoretical investigations of the “He nucleus are very interesting from various points of view. In particular, this nucleus is situated just near the neutron drip-line. The conventional shell model allows one to consider “He as a double-closed-shell neutron-rich nucleus. So the “He nucleus can provide an answer about the validity of shell-model prescriptions near the neutron drip-line. Since “He can be considered as the “Li nucleus without one proton in the 9Li core, it is easy to estimate, in the framework of the three-body approach, the energy of the ground state of “He . As a first approximation this energy can be evaluated as 2E,(n + ‘He) - Epairing. From the “Li binding energy the pairing energy of valence neutrons is EpairingN 1.3 MeV, so we get for the “He resonance the energy less than 1 MeV. This estimation is in agreement with the strict three-body calculations performed in this article, which give for the “He resonance energy 0.7-0.9 MeV. To calculate the energy and width of possible resonance “He (ground state) we have investigated the exotic “He nucleus in the n + n + *He model, which takes into account the three-body continuum for the decay “He + n + n +‘He. The results of calculations suggest that the “He resonance at energy less than 1 MeV above the three-particle threshold has a width rloHe N 150-300 keV. The latter value is an order of magnitude less than the value given by the two-body estimation for the decay into ‘He and point- like di-neutron. So three-body continuum dynamics supports the existence of the “He resonance. Experimental searching for the “He resonance is badly needed. As is known from many-body calculations, the stability of such systems as loHe depends on fine details of the nucleon-nucleon interaction and particularly on the interaction in the p-state (see, e.g., ref. [39]). So any definite experimental information on the “He energy should be useful. One of the authors, A.A.K., is thankful for the hospitality to RIKEN, where part of this work was done. References [l] B.V. Danilin, M.V. Zhukov, A.A. Korsheninnikov, L.V. Chulkov and V.D. Efros, Sov. J. Nucl. Phys. 48 (1988) 766; 49 (1989) 217; 49 (1989) 223; M.V. Zhukov, L.V. Chulkov, B.V. Danilin and A.A. Korsheninnikov, Nucl. Phys. A533 (1991) 428
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