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Interaction of hypertritons with nuclei at high energies
~:~
NUCLEAR PHYSICSA
ELSEVIER
Nuclear Physics A 632 (1998) 624-632
Interaction of hypertritons with nuclei at high energies M.V. Evlanov a, A.M. Sokolov a, V.K. Tartakovsky a, S.A. Khorozov b'l, J. Lukstins b a Institute for Nuclear Research, Prosp. Nauki, 47, 252028 Kiev, Ukraine b Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia
Received 12 November 1997; revised 22 December 1997; accepted 20 January 1998
Abstract Calculations of the integral cross sections of the Coulomb and diffraction interactions of hypertritons are performed. The possibilities to determine the hypertriton binding energy with a higher precision, to refine information on the A d interaction and other details of the hypertriton structure and interaction by means of the suggested measurements are also shown. @ 1998 Elsevier Science B.V. PACS: 24.10.Ht; 25.10.+s Keywords: Hypertriton-nuclei interactions; Calculated cross sections; Binding energy
1. Introduction At present, the planning and beginning of experimental work on generating the beams of relativistic hypernuclei containing A hyperons (and above all, beams of hypertritons) afford a good opportunity for making more careful investigations of collisions between incident hypernuclei and ordinary nuclei, the structure of hypernuclei and the interaction between A-hyperon and nucleons. The existing hypernuclear research programme [ 1-3] of the new Dubna Nuclotron accelerator considers the possibility to study the interaction of hypernuclei. Indeed, in these experiments, hypernuclei will be produced by the excitation of beam ions. It is emphasized that the momentum of the hypernucleus is slightly smaller than that of 1Work supported by Russian Fundamental Science Foundation Grant 96-02-19165. 0375-9474/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98) 00 1 16-X
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
625
the incident ion. Such a high energy hypernucleus decays at a large distance from the production point, so one can investigate hypernuclear interactions with a medium. The suggested experimental approach stimulates theoretical calculations predicting the hypernuclear interaction properties and the structure of hypernuclei. In this study, we consider the interaction of intermediate and high energy hypertritons with various absorbing nuclei using the diffraction approximation which is well applicable to the indicated energies. The hypertriton is an extremely loose hypernucleus with an anomalously low binding energy (e = 0.13 i 0.05 MeV [4]) with respect to disintegration into a A-hyperon and a deuteron. Owing to this, the radius of the hypertriton considerably exceeds those of the heaviest target nuclei. For this reason, the ~H hypernucleus can be treated, first of all, as a two-cluster system consisting of a A-hyperon (first) cluster and a deuteron (second) cluster, which is assumed to be a point-like particle as a first approximation. Special attention is given to obtaining such dependencies which are very sensitive to the value of hypertriton binding energy e measured so far with large errors. In this context, some recommendations for carrying out experiments on the determination of the value of e with a sufficient accuracy, are proposed. In particular, the process of two-particle dissociation of the hypertriton in the Coulomb field of the target nucleus, which is proved to be most sensitive to the value of e, is considered in detail. Such effect could be revealed under the measurements. It should be noted that this effect at low energies has been considered by Bohm and Wysotzki [5]
2. Integral cross sections of diffraction interaction of hypertritons with nuclei in the framework of the two-cluster model As known [6-9], the integral cross sections of various processes of the diffraction interaction of any incident two-cluster nucleus (the hypertriton can be considered as such a particle) with a nuclear target depend on the incident nucleus structure through its form factor
@(q) = f
d r e-iqr~p2(r) ,
a,(0)
= 1,
( 1)
where q~(r) is the wave function of a relative motion of clusters in the ground state (we assume that it corresponds to the S-state) and hq is the momentum transfer. For a black spherical nucleus, the total cross section o't, the integral cross sections of the stripping process o-s and absorption o-a are defined by the following simple expressions:
IJ 0
x = qR, X
(2)
M. ~ Evlanov et a l . / N u c l e a r Physics A 632 (1998) 624-632
626
O"s = "/7"R2
dx -f-@
1 -- 2
Jr(x)
,
(3)
0 O ' a = 21-O't - -
2O-s,
(4)
where R is the interaction radius of each of two (point-like) clusters with the target nucleus. The integral cross sections of diffraction elastic scattering o-e and dissociation o-x of two-cluster nucleus (hypertriton) by absorbing nucleus can be written in the form: Of3
0 oo
27r
2"ul/dyy@(y)IIt''aq~ Ji(I/31x-yl) Ji(I/32x l&x ++yl) Yl 0
~t-
'
(5)
0
~,
(6)
where /31 = I - / 3 2 = M I / ( M 1 + M2), Ml = MA and M2 = Md are the masses of the first (A-byperon) and second (deuteron) clusters, and
1131,2x ± y l = [/31,2x 2 2 i 2/31,2xycos~ + y2] 1/2 Figs. 1 and 2 present the calculated results of the integral cross sections o-t, O'e, O"dN , o-~, and O'a in barns as functions of hypertriton binding energy e at A = 238 and the target nucleus mass number A at e = 0.13 MeV, respectively, using the simple wave function of the relative motion of the A-hyperon and deuteron in the hypertriton
~f~ e ...... ~p(r) = V ~
r
1 '
OlA = h 2 V ~ '
M AMd
# = MA + md
(7)
As the cross sections o-t, O'dN and Os decrease, the cross sections O-e and O-a increase with increasing e. All five integral cross sections increase monotonously with increasing A. The hypertriton structure essentially affects the behaviour of all cross sections ( 2 ) - ( 6 ) as functions of the interaction range R, deviating from the simple R 2 law. The cross sections o't, Or N and O's depend on R approximately as R 16, and the cross sections O'e and o'a behave as R 2z. A qualitative dependence of all five cross sections on e (Fig. 1) is the same for medium and light target nuclei. Note that relative values of all these cross sections for the incident hypertriton differ greatly from the corresponding ratios of these cross sections for the incident deuteron, which are primarily due to large dimensions (low binding energies) of the hypertriton.
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
627
9 6 7 6 5 4 3
(~e
2 1 00
I
i
i
i
0.05
0.1
0.15
0.2
0.25
E, MeV Fig. 1. Integral cross sections of various processes o f the diffraction nuclear interactions o f ! i l l hypernuclei with 2381! 92 - nuclei as functions o f the hypertriton binding energy e.
= 8 t.
~7 6
j
F
O.s
I
I
I
I
50
100
150
200
'~d
250
A Fig. 2. Calculated integral cross sections of the diffraction nuclear interaction of hypertritons as functions of the target nucleus mass n u m b e r A at e = 0.13 MeV.
628
ll4. V Evlanov et al./Nuclear Physics A 632 (1998) 624-632
3. Dissociation of the hypertriton into a A-hyperon and a deuteron in the nuclear Coulomb field The integral cross section of Coulomb disintegration in the nuclear target field of the incident two-cluster nucleus, the first cluster of which is electrically neutral and the second one is charged (as in the hypertriton) and has charge z, can be presented in the following way [8]
o-c=8~n2R 2 F ( x ~ )
In
dxlnxdF(x)
1
j,
(8)
.~'llli n ,)
oo
F(x)=
1
fi02
dr JI
(Y~') ff 2in
-
,
n
by'
(9)
where v is a relative velocity of the incident two-cluster nucleus and the nuclear target with charge Z. The value of xmin, related to a minimum momentum transfer hqmin according to Xmi, = Rq,~,, is found from physical considerations and is equal to [8,9]
Xmin =
eR(
v2) ½ h--v 1 - 7 max(l,4n) .
(10)
It often appears (for example, at relativistic energies) that .,Vmin << l. Thus, for hypertritons colliding with uranium nuclei at a kinetic energy of E = 27 GeV and e = 0.13 MeV, we have Xmin ~ 1.63 x 10 -3. In such cases, one can retain the first term,which dominates in magnitude in the expression (8), substituting F(xmin) in it for F ( 0 ) = fll{r2)/3R 2, where (r 2) = fdrr2~2(r). The integral in (8), converging at Xmin --+ 0, can be replaced by the approximate expression 1~12(r2) In f12(r2} 6R 2 R2
obtained making allowance for large dimensions of the hypertriton ((r 2) >> R2). As a result, we get from ( 8 ) - ( 9 ) , in a good approximation, the following expression of the hypertriton Coulomb dissociation cross section in the explicit form 1/2
"
hv (1 -
8¢rn2 2 2
o-c = - ~ - f l l ( r
)In
c2j
4 n\ e f l l ( r 2 ) l / 2
,
4 n > 1.
(11)
For 4n < 1, the factor 4n under the logarithm sign should be replaced by unity. According to formulas ( 8 ) - ( 1 1 ) , taking into account minimum momentum transfer hqmin = (h/R)xmin at high energies leads to deviations from a simple dependence of the hypertriton Coulomb dissociation cross section o-ac ~ Z 2, which has been obtained in a series of papers [ 10-12]. The resulting reduction of the exponent of the target-nucleus charge (in our case, o-ac ~ Z 1"92) is approximately confirmed experimentally in studying the collisions of complex charged particles at intermediate energies [ 13,14].
M. V Evlanov et al./Nuclear Physics A 632 (1998) 624-632
629
= 70 t~
t3 60 U
50 40 30
ra
20 10
00
~u
0.05
0.1
0.15
0'.2
0.25
e, MeV Fig. 3. Theoretical dependencies of the integral cross sections o-c of the ~IH Coulomb dissociation on the binding energy e for various target nuclei at an energy E of 17 GeV.
Fig. 3 shows the dependencies of the hypertriton Coulomb dissociation cross section o "c (in barns) on 3AH binding energy e for various nuclear targets at the incident energy of hypernuclei E = 17 GeV. The cross sections are calculated by formulas ( 8 ) - ( 1 0 ) using the wave function ( 7 ) . The calculation by formula (11) gives nearly the same results as in all cases when the values of Xmin are very small. As can be seen from Fig. 3, the ~ H Coulomb dissociation cross section o -c proves to be highly sensitive to the hypertriton binding energy e, especially in the range of small values of e, where o-c becomes as large as several tens of barns for the heaviest nuclear targets, considerably exceeding the integral cross sections of diffraction nuclear processes. Thus, as mentioned above, it makes possible determining the ~ H binding energy e with a sufficient accuracy (especially if e turns out to be small) by measuring the cross section o-c. As the Coulomb dissociation cross section o-c for heavy target nuclei considerably exceeds the diffraction nuclear dissociation cross section o-x, for light and medium mass nuclei we have observed the reverse pattern: o-~ > crc. It should be noted that in all cases, as our calculations show, one can neglect the interference between the diffraction and Coulomb mechanisms of hypertriton dissociation. The cross section o-c for the Coulomb dissociation of hypertritons on nuclei was also calculated by exact formulas ( 8 ) - ( 1 0 ) as a function of projectile-hypertriton kinetic energy E at various BAH binding energies and for various target nuclei.As an illustration, Table 1 shows the results of these calculations for a fixed binding energy of e = 0.13 MeV. It is seen that the dependence of o-c on E has a similar qualitative character for various target nuclei: with increasing the kinetic energy of incident hypertritons, the cross section o -c in the non-relativistic region (v << c) decreases approximately in
M.V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
630
Table 1 The integral cross section ~rC (in barns) as a function of the projectile-hypertriton kinetic energy E for various target nuclei at a fixed 31H binding energy e = 0.13 MeV E (GeV) 0.1 0.2 0.4 0.7 1 2 4
23811 92 w
181Ta 73
21.859 14.067 8.944 6.408 5.331 4.072 3.545
14.875 9.303 5.851 4.180 3.476 2.654 2,309
~Cu 2.867 1.741 1.092 0.782 0.650 0.495 0.422
27 13A1
E (GeV)
0.689 0.419 0.250 0.171 0.139 0.102 0.086
6 8 !2 17 27 50 75
23811 92 ~
181T.~ 73 "~
3.455 3.459 3.533 3.643 3.833 4.132 4.346
2.248 2.248 2.293 2.361 2.480 2.669 2.806
~Cu 0.406 0.403 0.408 0.418 0.437 0.469 0.497
27A1 13 0.083 0.083 0.083 0.085 0.089 0.096 0.102
inverse proportion to the projectile energy Z2
E ln,
then it passes through a broad minimum, and in the ultrarelativistic region (v --~ c) this cross section slightly increases approximately according to a logarithmic law E
o -c ~ In
( M A q- Mcl)C 2) "
The indicated explicit dependencies of the cross section o-c on E in non-relativistic and relativistic ranges arise from the exact and asymptotic formulas for o -c listed above. Note that while the 3AH Coulomb dissociation cross section o"c depends on E rather greatly, the integral cross sections of nuclear diffraction depend slightly on incident particle energy E.
4. H y p e r t r i t o n structure influence on the integral cross sections of interaction with nuclei
Here, we study the influence of the form of internal wave functions q~(r) of the hypertriton, which is treated as a two-point -like cluster system consisting of a Ahyperon and a deuteron, on the behaviour of the aforementioned integral cross sections. They were calculated so far using only the 3AH one-parametric wave function (7) corresponding to the assumption of zero-range nuclear forces between the A-hyperon and deuteron in the 3AH hypernucleus (r0 = 0). Now, we use as ~p(r) two two-parametric wave functions of the hypertriton, normalized to unity, for the Ad Hulthen potential /OIA~A(OIA q- t~A) e . . . . . .
~H(r) = V
2"rr(/3A
-
-
Or'A)2
e-[3,tr
r
'
/3A 1 ~ r0,
(12)
and for the square well of the Ad-forces of depth V0 and width r0 (k = v/2/z(V0 - e ) )
M.V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
631
Table 2 Ratios of cross sections calculated with Hulthen (12) and Gauss (14) wave functions to cross sections calculated with exponential wave function (7) (the left and the right columns, respectively) Nuclei
eft
(re
¢rN
~raC
O's
rra
~SU
1.03
1.13
1.01
1.13
1.10
l.ll
1.15
1.16
1.13
1.49
0.93
~2Ta
1.03
1.13
1.01
1.13
1.10
1.14
1.15
1.16
1.13
1.50
0.92
0.74 0.71
2(~Cu
1.04
1.16
1.01
1.13
1.10
1.20
1.15
1.[6
1.13
1.51
0.90
0.60
~7A1
1.04
1.17
1.00
1.09
1.10
1.29
[.16
1.17
1.14
1.52
0.87
0.51
sin kr V/2 g~ro(r) =
- - ,r e cel(r°-r) sin k r o - - ,
OlA
,77"(1 + oemr0)
r < r0, r >~ ro,
(13)
r
as well as, for comparison, the one-parametric wave function in the Gaussian form ~ 6 ( r ) = (~rR 2)(-3/4) exp
-
,
(14)
w h e r e RN is the parameter characterizing the incident particle size. If /3A j --~ 0 and r0 --~ 0, then functions (12) and (13) turn to function (7). Parameters ]~a for function (12) and r0 for function (13) (or V0, because r0 and V0 are related by k cot kr0 --- - a ) characterizing the Ad-interaction were in agreement with the results of a series of works [4,15-17], where the AN- and Ad-interactions were studied. Parameter RN w a s found from the equality of the rms radii for both wave functions (12) and (14). Optimum values obtained for the parameters of wave functions ( 1 2 ) - ( 1 4 ) are the following
/3A= 1.45 fm - t ,
1/0=20MeV ( r 0 = l . 9 5 fro),
1
,~- 1-0.0061fm
2R2N
-2.
(15)
However, some uncertainties remain in determining the parameters /3A, V0 (or ro),RN. For this reason, the measurements of the cross sections calculated with wave functions ( 1 2 ) - ( 1 4 ) can reveal more realistic values of these parameters. We have calculated the integral cross sections ( 2 ) - ( 6 ) and ( 8 ) - ( 1 0 ) for four target nuclei: uranium, tantalum, copper, and aluminum at e = 0.13 MeV and E = 27 GeV for the functions ( 1 2 ) - ( 1 4 ) with the parameters (15). Since the cross sections for the functions ( 12)-(13) proved to be rather close, the calculated ratios of the cross sections for the Hulthen function (12) in the left columns and for the Gauss function (14) in the right columns to the corresponding cross sections obtained earlier for the function (7) are listed in resulting Table 2. Taking into account the finite range of the Ad-forces in the hypertriton (/3A I and r0 4: 0) the cross section o-t increases only by 3-4%, the cross section O'e does not practically change, the cross section o-u increases approximately by 10%, and the cross section o-~ by 13-14%, the cross section O-a decreases by 6-13%. A relative change of O'a,
632
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
unlike the other nuclear integral cross sections, depends strongly on the target-nucleus mass number; the Coulomb dissociation cross section o-o c increases by 14-16%. It should be noted that the differential cross sections of the considered processes can depend on the hypertriton structure more noticeably. As to rather considerable differences of the cross sections in the case of using the Gauss function (14) and the function (7), it may be connected with wrong asymptotics of the Gauss function, so the asymptotics of the functions affects the cross sections significantly.
5. Conclusion i) The present theoretical studies show that a more precise experimental determination of the hypertriton binding energy e is possible by measuring various cross sections of the interaction of high-energy hypertritons with nuclei and, in particular, the 3 H Coulomb dissociation cross section o-ac. ii) The calculations show that the measurements of various cross sections of hypertriton-nucleus interaction would make it possible to obtain additional information on A d and AN-interactions and to refine the internal wave function of the ~ H hypernucleus. iii) The calculations indicate deviations of the integral cross sections of the hypertriton diffraction interaction with nuclei from a simple dependence on the interaction radius ( ~ R 2) that could be verified in the experiment. The ~H Coulomb dissociation cross section deviates from a simple dependence ( ~ Z 2) on the charge of the target nucleus, as well. This is approximately confirmed by the experiments for other incident nuclei and requires a further experimental study. iv) The formalism developed and used here has rather a common character and can be applied to an analysis of the interaction of other incident hypernuclei with nuclei in the diffraction approximation.
References Ill 12] 131 [4] [5] [6] 17l 181 [9] 10]
A.M. Baldin and A.1. Malakhov, Nucl. Phys. A 566 (1994) 61 lc. S.A. Avramenkoet al., JINR Rapid Commun. 5[68] (1994) 14. S.A. Avramenkoet al., Nucl. Phys. A 585 (1995) 91c. M, Juric et al., Nucl. Phys. B 52 (1973) 1. G. Bofim and F. Wysotzki, Nucl. Phys. B 15 (1970) 628. A.G. Sitenko, Theory of Nuclear Reactions (World Scientific, Singapore, 1990). M,V. Evlanov, A.M. Sokolov and V.K. Tartakovsky, Yad. Fiz. 59 (1996) 679. MV. Evlanov, A.M. Sokolov and V.K. Tartakovsky, Yad. Fiz. 60 (1997) 444. M.V. Evlanov, A.M. Sokolov, V.K. Tartakovsky, JINR Rapid Commun.4[78] (1996) 33. N.N. Kolesnikovand R.V. Vedrinsky,JETF 47 (1964) 1740; Yad. Fiz. 3 (1966) 946; Nuovo Cim. 43 (1966) 1. I 1] R.V. Vedrinsky,Yad. Fiz. 3 (1966) 672. 121 V.L. Lyuboshitz, Yad. Fiz. 51 (1990) 1013. 131 K. Soutome et al., Prog. Theor. Phys. 87 (1992) 599; 88 (1992) 703. 14] T. Kobayashiet al., Phys. Lett. B 232 (1989) 51. 151 K. Miyagava et al., Phys. Rev. C 51 (1995) 2905. 161 N.M. Petrov, Yad. Fiz. 48 (1988) 50. 17[ I.R. Afnan and B.F. Gibson, Phys. Rev. C 41 (1990) 2787.
NUCLEAR PHYSICSA
ELSEVIER
Nuclear Physics A 632 (1998) 624-632
Interaction of hypertritons with nuclei at high energies M.V. Evlanov a, A.M. Sokolov a, V.K. Tartakovsky a, S.A. Khorozov b'l, J. Lukstins b a Institute for Nuclear Research, Prosp. Nauki, 47, 252028 Kiev, Ukraine b Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980 Russia
Received 12 November 1997; revised 22 December 1997; accepted 20 January 1998
Abstract Calculations of the integral cross sections of the Coulomb and diffraction interactions of hypertritons are performed. The possibilities to determine the hypertriton binding energy with a higher precision, to refine information on the A d interaction and other details of the hypertriton structure and interaction by means of the suggested measurements are also shown. @ 1998 Elsevier Science B.V. PACS: 24.10.Ht; 25.10.+s Keywords: Hypertriton-nuclei interactions; Calculated cross sections; Binding energy
1. Introduction At present, the planning and beginning of experimental work on generating the beams of relativistic hypernuclei containing A hyperons (and above all, beams of hypertritons) afford a good opportunity for making more careful investigations of collisions between incident hypernuclei and ordinary nuclei, the structure of hypernuclei and the interaction between A-hyperon and nucleons. The existing hypernuclear research programme [ 1-3] of the new Dubna Nuclotron accelerator considers the possibility to study the interaction of hypernuclei. Indeed, in these experiments, hypernuclei will be produced by the excitation of beam ions. It is emphasized that the momentum of the hypernucleus is slightly smaller than that of 1Work supported by Russian Fundamental Science Foundation Grant 96-02-19165. 0375-9474/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII S0375-9474(98) 00 1 16-X
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
625
the incident ion. Such a high energy hypernucleus decays at a large distance from the production point, so one can investigate hypernuclear interactions with a medium. The suggested experimental approach stimulates theoretical calculations predicting the hypernuclear interaction properties and the structure of hypernuclei. In this study, we consider the interaction of intermediate and high energy hypertritons with various absorbing nuclei using the diffraction approximation which is well applicable to the indicated energies. The hypertriton is an extremely loose hypernucleus with an anomalously low binding energy (e = 0.13 i 0.05 MeV [4]) with respect to disintegration into a A-hyperon and a deuteron. Owing to this, the radius of the hypertriton considerably exceeds those of the heaviest target nuclei. For this reason, the ~H hypernucleus can be treated, first of all, as a two-cluster system consisting of a A-hyperon (first) cluster and a deuteron (second) cluster, which is assumed to be a point-like particle as a first approximation. Special attention is given to obtaining such dependencies which are very sensitive to the value of hypertriton binding energy e measured so far with large errors. In this context, some recommendations for carrying out experiments on the determination of the value of e with a sufficient accuracy, are proposed. In particular, the process of two-particle dissociation of the hypertriton in the Coulomb field of the target nucleus, which is proved to be most sensitive to the value of e, is considered in detail. Such effect could be revealed under the measurements. It should be noted that this effect at low energies has been considered by Bohm and Wysotzki [5]
2. Integral cross sections of diffraction interaction of hypertritons with nuclei in the framework of the two-cluster model As known [6-9], the integral cross sections of various processes of the diffraction interaction of any incident two-cluster nucleus (the hypertriton can be considered as such a particle) with a nuclear target depend on the incident nucleus structure through its form factor
@(q) = f
d r e-iqr~p2(r) ,
a,(0)
= 1,
( 1)
where q~(r) is the wave function of a relative motion of clusters in the ground state (we assume that it corresponds to the S-state) and hq is the momentum transfer. For a black spherical nucleus, the total cross section o't, the integral cross sections of the stripping process o-s and absorption o-a are defined by the following simple expressions:
IJ 0
x = qR, X
(2)
M. ~ Evlanov et a l . / N u c l e a r Physics A 632 (1998) 624-632
626
O"s = "/7"R2
dx -f-@
1 -- 2
Jr(x)
,
(3)
0 O ' a = 21-O't - -
2O-s,
(4)
where R is the interaction radius of each of two (point-like) clusters with the target nucleus. The integral cross sections of diffraction elastic scattering o-e and dissociation o-x of two-cluster nucleus (hypertriton) by absorbing nucleus can be written in the form: Of3
0 oo
27r
2"ul/dyy@(y)IIt''aq~ Ji(I/31x-yl) Ji(I/32x l&x ++yl) Yl 0
~t-
'
(5)
0
~,
(6)
where /31 = I - / 3 2 = M I / ( M 1 + M2), Ml = MA and M2 = Md are the masses of the first (A-byperon) and second (deuteron) clusters, and
1131,2x ± y l = [/31,2x 2 2 i 2/31,2xycos~ + y2] 1/2 Figs. 1 and 2 present the calculated results of the integral cross sections o-t, O'e, O"dN , o-~, and O'a in barns as functions of hypertriton binding energy e at A = 238 and the target nucleus mass number A at e = 0.13 MeV, respectively, using the simple wave function of the relative motion of the A-hyperon and deuteron in the hypertriton
~f~ e ...... ~p(r) = V ~
r
1 '
OlA = h 2 V ~ '
M AMd
# = MA + md
(7)
As the cross sections o-t, O'dN and Os decrease, the cross sections O-e and O-a increase with increasing e. All five integral cross sections increase monotonously with increasing A. The hypertriton structure essentially affects the behaviour of all cross sections ( 2 ) - ( 6 ) as functions of the interaction range R, deviating from the simple R 2 law. The cross sections o't, Or N and O's depend on R approximately as R 16, and the cross sections O'e and o'a behave as R 2z. A qualitative dependence of all five cross sections on e (Fig. 1) is the same for medium and light target nuclei. Note that relative values of all these cross sections for the incident hypertriton differ greatly from the corresponding ratios of these cross sections for the incident deuteron, which are primarily due to large dimensions (low binding energies) of the hypertriton.
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
627
9 6 7 6 5 4 3
(~e
2 1 00
I
i
i
i
0.05
0.1
0.15
0.2
0.25
E, MeV Fig. 1. Integral cross sections of various processes o f the diffraction nuclear interactions o f ! i l l hypernuclei with 2381! 92 - nuclei as functions o f the hypertriton binding energy e.
= 8 t.
~7 6
j
F
O.s
I
I
I
I
50
100
150
200
'~d
250
A Fig. 2. Calculated integral cross sections of the diffraction nuclear interaction of hypertritons as functions of the target nucleus mass n u m b e r A at e = 0.13 MeV.
628
ll4. V Evlanov et al./Nuclear Physics A 632 (1998) 624-632
3. Dissociation of the hypertriton into a A-hyperon and a deuteron in the nuclear Coulomb field The integral cross section of Coulomb disintegration in the nuclear target field of the incident two-cluster nucleus, the first cluster of which is electrically neutral and the second one is charged (as in the hypertriton) and has charge z, can be presented in the following way [8]
o-c=8~n2R 2 F ( x ~ )
In
dxlnxdF(x)
1
j,
(8)
.~'llli n ,)
oo
F(x)=
1
fi02
dr JI
(Y~') ff 2in
-
,
n
by'
(9)
where v is a relative velocity of the incident two-cluster nucleus and the nuclear target with charge Z. The value of xmin, related to a minimum momentum transfer hqmin according to Xmi, = Rq,~,, is found from physical considerations and is equal to [8,9]
Xmin =
eR(
v2) ½ h--v 1 - 7 max(l,4n) .
(10)
It often appears (for example, at relativistic energies) that .,Vmin << l. Thus, for hypertritons colliding with uranium nuclei at a kinetic energy of E = 27 GeV and e = 0.13 MeV, we have Xmin ~ 1.63 x 10 -3. In such cases, one can retain the first term,which dominates in magnitude in the expression (8), substituting F(xmin) in it for F ( 0 ) = fll{r2)/3R 2, where (r 2) = fdrr2~2(r). The integral in (8), converging at Xmin --+ 0, can be replaced by the approximate expression 1~12(r2) In f12(r2} 6R 2 R2
obtained making allowance for large dimensions of the hypertriton ((r 2) >> R2). As a result, we get from ( 8 ) - ( 9 ) , in a good approximation, the following expression of the hypertriton Coulomb dissociation cross section in the explicit form 1/2
"
hv (1 -
8¢rn2 2 2
o-c = - ~ - f l l ( r
)In
c2j
4 n\ e f l l ( r 2 ) l / 2
,
4 n > 1.
(11)
For 4n < 1, the factor 4n under the logarithm sign should be replaced by unity. According to formulas ( 8 ) - ( 1 1 ) , taking into account minimum momentum transfer hqmin = (h/R)xmin at high energies leads to deviations from a simple dependence of the hypertriton Coulomb dissociation cross section o-ac ~ Z 2, which has been obtained in a series of papers [ 10-12]. The resulting reduction of the exponent of the target-nucleus charge (in our case, o-ac ~ Z 1"92) is approximately confirmed experimentally in studying the collisions of complex charged particles at intermediate energies [ 13,14].
M. V Evlanov et al./Nuclear Physics A 632 (1998) 624-632
629
= 70 t~
t3 60 U
50 40 30
ra
20 10
00
~u
0.05
0.1
0.15
0'.2
0.25
e, MeV Fig. 3. Theoretical dependencies of the integral cross sections o-c of the ~IH Coulomb dissociation on the binding energy e for various target nuclei at an energy E of 17 GeV.
Fig. 3 shows the dependencies of the hypertriton Coulomb dissociation cross section o "c (in barns) on 3AH binding energy e for various nuclear targets at the incident energy of hypernuclei E = 17 GeV. The cross sections are calculated by formulas ( 8 ) - ( 1 0 ) using the wave function ( 7 ) . The calculation by formula (11) gives nearly the same results as in all cases when the values of Xmin are very small. As can be seen from Fig. 3, the ~ H Coulomb dissociation cross section o -c proves to be highly sensitive to the hypertriton binding energy e, especially in the range of small values of e, where o-c becomes as large as several tens of barns for the heaviest nuclear targets, considerably exceeding the integral cross sections of diffraction nuclear processes. Thus, as mentioned above, it makes possible determining the ~ H binding energy e with a sufficient accuracy (especially if e turns out to be small) by measuring the cross section o-c. As the Coulomb dissociation cross section o-c for heavy target nuclei considerably exceeds the diffraction nuclear dissociation cross section o-x, for light and medium mass nuclei we have observed the reverse pattern: o-~ > crc. It should be noted that in all cases, as our calculations show, one can neglect the interference between the diffraction and Coulomb mechanisms of hypertriton dissociation. The cross section o-c for the Coulomb dissociation of hypertritons on nuclei was also calculated by exact formulas ( 8 ) - ( 1 0 ) as a function of projectile-hypertriton kinetic energy E at various BAH binding energies and for various target nuclei.As an illustration, Table 1 shows the results of these calculations for a fixed binding energy of e = 0.13 MeV. It is seen that the dependence of o-c on E has a similar qualitative character for various target nuclei: with increasing the kinetic energy of incident hypertritons, the cross section o -c in the non-relativistic region (v << c) decreases approximately in
M.V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
630
Table 1 The integral cross section ~rC (in barns) as a function of the projectile-hypertriton kinetic energy E for various target nuclei at a fixed 31H binding energy e = 0.13 MeV E (GeV) 0.1 0.2 0.4 0.7 1 2 4
23811 92 w
181Ta 73
21.859 14.067 8.944 6.408 5.331 4.072 3.545
14.875 9.303 5.851 4.180 3.476 2.654 2,309
~Cu 2.867 1.741 1.092 0.782 0.650 0.495 0.422
27 13A1
E (GeV)
0.689 0.419 0.250 0.171 0.139 0.102 0.086
6 8 !2 17 27 50 75
23811 92 ~
181T.~ 73 "~
3.455 3.459 3.533 3.643 3.833 4.132 4.346
2.248 2.248 2.293 2.361 2.480 2.669 2.806
~Cu 0.406 0.403 0.408 0.418 0.437 0.469 0.497
27A1 13 0.083 0.083 0.083 0.085 0.089 0.096 0.102
inverse proportion to the projectile energy Z2
E ln,
then it passes through a broad minimum, and in the ultrarelativistic region (v --~ c) this cross section slightly increases approximately according to a logarithmic law E
o -c ~ In
( M A q- Mcl)C 2) "
The indicated explicit dependencies of the cross section o-c on E in non-relativistic and relativistic ranges arise from the exact and asymptotic formulas for o -c listed above. Note that while the 3AH Coulomb dissociation cross section o"c depends on E rather greatly, the integral cross sections of nuclear diffraction depend slightly on incident particle energy E.
4. H y p e r t r i t o n structure influence on the integral cross sections of interaction with nuclei
Here, we study the influence of the form of internal wave functions q~(r) of the hypertriton, which is treated as a two-point -like cluster system consisting of a Ahyperon and a deuteron, on the behaviour of the aforementioned integral cross sections. They were calculated so far using only the 3AH one-parametric wave function (7) corresponding to the assumption of zero-range nuclear forces between the A-hyperon and deuteron in the 3AH hypernucleus (r0 = 0). Now, we use as ~p(r) two two-parametric wave functions of the hypertriton, normalized to unity, for the Ad Hulthen potential /OIA~A(OIA q- t~A) e . . . . . .
~H(r) = V
2"rr(/3A
-
-
Or'A)2
e-[3,tr
r
'
/3A 1 ~ r0,
(12)
and for the square well of the Ad-forces of depth V0 and width r0 (k = v/2/z(V0 - e ) )
M.V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
631
Table 2 Ratios of cross sections calculated with Hulthen (12) and Gauss (14) wave functions to cross sections calculated with exponential wave function (7) (the left and the right columns, respectively) Nuclei
eft
(re
¢rN
~raC
O's
rra
~SU
1.03
1.13
1.01
1.13
1.10
l.ll
1.15
1.16
1.13
1.49
0.93
~2Ta
1.03
1.13
1.01
1.13
1.10
1.14
1.15
1.16
1.13
1.50
0.92
0.74 0.71
2(~Cu
1.04
1.16
1.01
1.13
1.10
1.20
1.15
1.[6
1.13
1.51
0.90
0.60
~7A1
1.04
1.17
1.00
1.09
1.10
1.29
[.16
1.17
1.14
1.52
0.87
0.51
sin kr V/2 g~ro(r) =
- - ,r e cel(r°-r) sin k r o - - ,
OlA
,77"(1 + oemr0)
r < r0, r >~ ro,
(13)
r
as well as, for comparison, the one-parametric wave function in the Gaussian form ~ 6 ( r ) = (~rR 2)(-3/4) exp
-
,
(14)
w h e r e RN is the parameter characterizing the incident particle size. If /3A j --~ 0 and r0 --~ 0, then functions (12) and (13) turn to function (7). Parameters ]~a for function (12) and r0 for function (13) (or V0, because r0 and V0 are related by k cot kr0 --- - a ) characterizing the Ad-interaction were in agreement with the results of a series of works [4,15-17], where the AN- and Ad-interactions were studied. Parameter RN w a s found from the equality of the rms radii for both wave functions (12) and (14). Optimum values obtained for the parameters of wave functions ( 1 2 ) - ( 1 4 ) are the following
/3A= 1.45 fm - t ,
1/0=20MeV ( r 0 = l . 9 5 fro),
1
,~- 1-0.0061fm
2R2N
-2.
(15)
However, some uncertainties remain in determining the parameters /3A, V0 (or ro),RN. For this reason, the measurements of the cross sections calculated with wave functions ( 1 2 ) - ( 1 4 ) can reveal more realistic values of these parameters. We have calculated the integral cross sections ( 2 ) - ( 6 ) and ( 8 ) - ( 1 0 ) for four target nuclei: uranium, tantalum, copper, and aluminum at e = 0.13 MeV and E = 27 GeV for the functions ( 1 2 ) - ( 1 4 ) with the parameters (15). Since the cross sections for the functions ( 12)-(13) proved to be rather close, the calculated ratios of the cross sections for the Hulthen function (12) in the left columns and for the Gauss function (14) in the right columns to the corresponding cross sections obtained earlier for the function (7) are listed in resulting Table 2. Taking into account the finite range of the Ad-forces in the hypertriton (/3A I and r0 4: 0) the cross section o-t increases only by 3-4%, the cross section O'e does not practically change, the cross section o-u increases approximately by 10%, and the cross section o-~ by 13-14%, the cross section O-a decreases by 6-13%. A relative change of O'a,
632
M. V. Evlanov et al./Nuclear Physics A 632 (1998) 624-632
unlike the other nuclear integral cross sections, depends strongly on the target-nucleus mass number; the Coulomb dissociation cross section o-o c increases by 14-16%. It should be noted that the differential cross sections of the considered processes can depend on the hypertriton structure more noticeably. As to rather considerable differences of the cross sections in the case of using the Gauss function (14) and the function (7), it may be connected with wrong asymptotics of the Gauss function, so the asymptotics of the functions affects the cross sections significantly.
5. Conclusion i) The present theoretical studies show that a more precise experimental determination of the hypertriton binding energy e is possible by measuring various cross sections of the interaction of high-energy hypertritons with nuclei and, in particular, the 3 H Coulomb dissociation cross section o-ac. ii) The calculations show that the measurements of various cross sections of hypertriton-nucleus interaction would make it possible to obtain additional information on A d and AN-interactions and to refine the internal wave function of the ~ H hypernucleus. iii) The calculations indicate deviations of the integral cross sections of the hypertriton diffraction interaction with nuclei from a simple dependence on the interaction radius ( ~ R 2) that could be verified in the experiment. The ~H Coulomb dissociation cross section deviates from a simple dependence ( ~ Z 2) on the charge of the target nucleus, as well. This is approximately confirmed by the experiments for other incident nuclei and requires a further experimental study. iv) The formalism developed and used here has rather a common character and can be applied to an analysis of the interaction of other incident hypernuclei with nuclei in the diffraction approximation.
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