Volume 39B, number 5 PHYSICS L E T T E R S 29 May 1972

DO M U L T I N E U T R O N S R E A L L Y E X I S T ?

A. I. BAZ and V. N. BRAGIN Kurchatov Institute of Atomic Energy, Moscow, USSR

Received 10March 1972

It is argued, that nuclear multineutrons, if they exist at all, are rather heavy nuclei, which contain at [east 100 nuetrons.

Do bound s ta tes in neut ron sys tems real ly ex- is t? We give a posi t ive answer , because due to gravi ta t ion, any number A of neut rons does have bound states . But the mul t ineu t rons a re not those the nuc lea r phys ic i s t thinks of. The point is that the the binding energy of the "gravi ta t ional mul t i - neu t rons" is too smal l and the radius is too large. Namely, the s t ra ight forward calcula t ion with the he lp of the K-ha rmon ic s method (see below) gives us:

E A ~ 0.19 ~ G ~ A T / 3 ~ 10-68AT/3eV;

h 2 R A ~ 2.2 __m3GA -1 /3 ~ 1025"4-1/3c m.

What can be said about "nuclear mul t ineutrons" , binding energ ies and dens i t ies of which are on the nuc lea r sca les (E ~ MeV, R ~ fm). As we will see in a moment , it i s imposs ib le to prove their exis tence or nonexis tence on theoret ica l grounds at the p re sen t stage. But we can say confidently that if "nuclear mul t ineu t rons" do exist , the min imum number of neut rons A in such a neutron sys tem is ra the r large:

We have invest igated bound s tates in the sys - t em of A neut rons in o rder to unders tand what conclus ions about "nuclear mul t tneu t rons" can be drawn from modern neu t ron -neu t ron poten- t ia ls data. We a s sumed that:

1) This sys tem is descr ibed by the nonre l a - t iv is t ic SchrSdtnger equation.

2) An in te rac t ion of the ith and j th neut rons is a sum of s inglet and t r ip le t cen t ra l potent ia ls :

V(ij) : V31(rijlPs= 0 + Va3(~j)Ps= 1

^

where Ps is a pro jec t ion opera tor to the s ta tes with total spin s=l or s=O.

If we a s sume a guass ian shape of the poten- t ia ls :

3 V31(r) = ~ V k exp{-½(a-~) 2}

k=l

6

V33(r) = k:4 ~ Vk exp(-½ (~k)2}

we have p a r a m e t e r s V k and a k ready for us in refs. [1-5]. We t r ied five different sets of ne u t r on - ne u t r on potent ia ls in all . The p a r a m e t e r s of the potent ia ls a re given in table 1.

We used Kmin-approxtmat ton of K - ha r mon i c s method [6](the method of hyperspher ica l func- tions) to solve the many neutron Schri~dinger equation:

A A ^ {- h2 ~ A i + ~V(ij:~ - E } @(1.. .A)=0.

2rn i=l i>j

As is known, this method has a var ia t iona l proper ty . Thus we obtained the lower bound for the binding energy for every set of potential . For the sake of s impl ic i ty we confined the number of neut rons A to the d i sc re te set:

A = l ( n + l ) ( n + 2 ) in+3); n =0 , 1, 2 , . . . ,

that i s A = 2, 8, 20, 40, 70, 112, 168.

Our f indings a re as follows: 1) The potent ia ls II, III (table 1) do not form

bound s ta tes , i r r e spec t ive of the value ofA. 2) The potent ia ls I, IV, V, form bound s tates

t fA >1112; bound s ta tes do not exist i r a < 112. The binding energ ies and the radi i of the

l tghtest mul t ineu t rons (withA =112 and A=168)

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Volume 39B, n u m b e r 5 P H Y S I C S L E T T E R S

Table 1 The p a r a m e t e r s of potent ia ls .

29 May 1972

Number of k 1 2 3 4 5 6 Ref. Iootentials

I V K (MeV) 144.86 -83.34 - 16.67 -28.97 - [11

a- K (fro) 0.59 1.13 - 1.13 0.59 -

II Vii {MeV) 144.86 -83.34 - 644.0 - - [2]

a/e(fm) 0.59 1.13 - 0.46 - -

/If ~ i (MeV) 120.0 -61.3 - 65.0 - - [3]

aK(fm) 0.68 1.46 - 1.64 - -

IV VK(MeV~ 880.0 -70.0 -21.0 - - - [4]

aK(fm) 0.3045 0.885 1.02 -

V ~f (MeV) 560.0 -390.7 -1.501 9.335 -1.37 0.1663 [5]

a K (fro) 0.57 0.73 2.27 0.84 1~49 2.27

Table 2 The binding e n e r g i e s and radii of the [ ightest mul t i -

neu t rons .

Number of Po ten t ia l s n e u t r o n s A I IV V

112 EA (MeV) 313.9 ~:99.4 518.4

RA(fm ) 12.11 11.78 l l .71

168 E A (MeV) 3827 5760 5697

R A (fro) 12.80 12.62 12.52

a r e g i v e n in t a b l e 2.

T h e r e i s no r e a s o n to p r e f e r one of t he s e t s of

t h e p o t e n t i a l s s h o w n a b o v e to o t h e r s . T h u s w e h a v e a r r i v e d a t the c o n c l u s i o n tha t m o d e r n k n o w - l e d g e of the n e u t r o n - n e u t r o n i n t e r a c t i o n d o e s no t a l l o w u s to s a y if " n u c l e a r m u l t i n e u t r o n s " do

e x i s t o r no t . H o w e v e r , t h e c a l c u l a t i o n s s h o w

t h a t if " n u c l e a r m u l t i n e u t r o n s " e x i s t t hey c o n s i s t

of a l a r g e n u m b e r of n e u t r o n s A ~ 1 1 2 .

R e f e r e n c e s

[1] A.B. VoIkov, Nuc I .Phys . 7,t (1965) 33. [2] A . I . Baz and M.V. Zhukov, Proc . 2nd P rob lem Symp. on

N u c l . p h y s . , Novos ib i rsk , 1970. [3] A. I . Baz, et al., J E T P Le t t e r s , 12 (1970) 151. [4] I1. E i k e m e i e r and H. Hackenbroich, Leit. Phys . 195

(1966) 412. [5] D. Gogny, P. P i r e s and R. de T o u r r i e l , lohys. L e t t e r s

32B (1970) 591. [6] A. M. Badalyan, A. Calegero and Yu. A. Simonov,

Nuov. Cim. 68A (1970) 572.

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