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New statistical function for the angular distribution of evaporation residues produced by heavy ions
Nuclear Instruments and Methods in Physics Research A 352 (1995)
663-664
Letter to the Editor
ELSEVIER
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SectionA
New statistical function for the angular distribution of evaporation residues produced by heavy ions J. Rigol Fleroc Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow District, Russian Federation
Received
4 May 1994;
revised form received 1 September 1994
Abstract A new statistical function has been found to simulate the angular distributions of evaporation residues produced by heavy-ions . Experimental results are compared with the calculated ones.
The problem of the angular distribution of evaporation residues (EVR) with heavy-ions induced reaction has been studied for many years [1]. However, the interest in this problem has been increasing due to the development of new electromagnetic separators for the study of the evaporation residues from heavy-ion induced reactions [2-7]. The acceptance of electromagnetic separators depends on the widths of the angular distribution of the EVRs recoiled out from the target. For thick targets, the angular distribution depends fundamentally on the multiple scattering of the EVRs by the nuclei of the target . But, for thin targets (when the range of EVRs in this target is much larger than the target thickness), the neutron evaporation can also play an important role . Experimentally, the angular distribution Rexp(0,) is determined by counting the number of nuclei impinging on some disks whose radii are defined by the expression L* sin(B,), where L is the distance from the target to the disk and 0, is a certain mean angle (expressed in degrees here) between the central trajectory of the beam and the direction of the central circle inside the disk "i". The number of counts on each disk depends on the angle (through sin (B,)) and on certain other functions of 0. The experimental mean squared angle (02 )exp can be determined using the function Rexp(B, ), by the expression : )exp
= Y_
0,2R ex ,(
Bi ) .
It is well known that the parameter <02> is very important, because under the assumption of isotropic neutron evaporation it is possible to obtain an expression which establishes a relation between this value and some parameters of the studied reaction . In the first approximation, this relation takes the form : (0 2 ) = 2XEn/( 3AbEb), 0168-9002/95/$09.50 © 1995 Elsevier SSDI0168-9002(94)01127-3
(2)
where X is the number of emitted neutrons, E is the mean energy of these neutrons, Ab and Eb are the mass number and the laboratory bombarding energy of the projectiles, respectively . Usually [8] the function R(B) is modelled using the Gaussian function combined with the function sin(9) . Nevertheless, with this combination, one can not obtain a mean square value of 0 corresponding to the experimental one because, for the normal distribution, the mean square value coincides with the variance. This detail explains the discrepancy in the expression of the ( 02 ) that is observed in some works [9,10] . In this work a new statistical function has been found which describes the experimental results in natural form and gives a better concordance between the experimental and calculated values . This function is determined only by one parameter, namely, the ( 0 2 ) value: /. R(O) =360/(rr(B )) sin(6) e-BZ
Z
The function R(6) satisfies the following conditions :
Io90.R(B) d0= 1,
(4)
(B Z) =
(5)
I0 2 R(6) d0 .
The integral is taken to 90°, however, the function R(B) goes rapidly down to zero for values of 0 taken three or four times of the value ( B) . In Figs . 1 and 2, the angular distributions [9,10] of the nuclei after evaporation of X neutrons are compared with those calculated ones using experimental parameter )exp
In Fig. 1, the angular distributions of the residual nuclei 149g .l.b and 149,151Dy produced by the reactions [9]: 1) 146Nd + "B(104 MeV) -> 149gTb + 8n,
Science B.V . All rights reserved
J. Rigol /NucL Instr. and Meth. in Phys. Res. A 352 (1995) 663-664
664
ANGULAR DISTRIBUTIONS OF EVRs
the angular distributions when in the experiment all particles arriving under the angle 0 are detected . It is very
020 R(0)
useful also for the evaluation of the acceptance of collecting the EVR of determined heavy-ion induced reaction, using electromagnetic separators.
016 012 008
Acknowledgements
004
I would like to thank Prof. Yu . Ts . Oganessian who made it possible for me to do this work and Drs. Yu. A. Lazarev and Yu . V. Lobanov for their interest, valuable
000
0
3
6
9
12
15 @(Deg )
help and encouragement.
Fig. 1. Angular distributions of EVRs. 2) 140
References
149Dy MeV) -> + 7n, +12 151Dy 3)144Nd C(94 MeV) + 5n, Ce
+160(111
are shown. The experimental results (crosses, circles and triangles, respectively) are compared with the calculated ones (solid lines) obtained by using the function R(0.
In Fig. 2, the experimental [11] angular distribution of
the nuclei with mass number A = 125 obtained from the natAg reaction + 22 Ne are compared with that calculated
using the Gaussian distribution (point-dashed) [11] and
with that calculated using the function R(0) (solid line).
We stress the fact that there is no fitting of any kind,
but only the calculation of angular distributions using the function R(0) with experimental parameters ( 0,2 )exp The function R(0) is very simple and it describes the
process in the natural form . Nevertheless the origin of the coefficient (360/-tr) is not quite clear yet. But, in any case, it was very nice to find it .
The function R(0) is particularly useful for describing ANGULAR DISTRIBUTION FOR EVRs WITH A=125
0 25 R(0)
-Calc with function R(O)
020
- - Calc with Gauss function[l ++++Expenmental resultsli
015
I I
l 1
010 005 000
0
3
6
9
12
15 0(Deg )
Fig. 2. Angular distribution for EVRs with A = 125.
[1] P.F . Donovan, B.G . Harvey, W.H . Wade, Phys . Rev. 119 (1960) 218. [2] V.A. Karnaukhov, L. Rubinskaya, G. Ter Akopyan, V. Titov, V.A. Chugreev . JINR, P13-4454, Dubna (1969); and I. Bacho, D.D . Bogdanov, Sh . Dorotsi, V.A . Kamaukhov, L.A. Petrov, G.M . TerAkopyan, JINR, P13-4453, Dubna (1969) . [3] Yu .Ts. Oganessian, Yu .B . Lobanov, A.G . Popeko, J. Rigol, F. Sh. Abdulin, V.V . Bekhterev, G.G . Gulbekian, A.A. Ledovskoi, V.N. Melnikov, S.P. Tretyakova, Yu . P. Haritonov. Yu.S . Tsyganov, V.A. Chugreev, Proc. Int. School-Seminar on Heavy Ion Physics, JINR, Report, D7-90142, Dubna (1990) p. 44 . [41 Yu . A. Lazarev, Yu . V. Lobanov, A.N. Mezentsev, Yu . Ts . Oganessian, V.G . Subbotin, V.K . Utyonkov, F.Sh. Abdulin, V.V . Bekhterev, S. Iliev, I.V. Kolesov, A.N. Polyakov, I.M . Sedykh, LV . Shirakovsky, A.M. Sukhov, Yu. S. Tsyganov, V.E . Zhuchko. Proc . Int. School-Semina r on Heavy Ion Physics, vol. 2, Dubna (1993) p. 497. [5] G. Munzenberg, W. Faust, S. Hofmann, P. Armbruster, K. Guttner, H. Ewald, Nucl. Instr. and Meth . 161 (1979) 65 . [6] A. Ghiorso, S. Yashita, M.E . Leino, L. Frank, J. Kalwins, P. Armbruster, J.P . Doufour, P.K. Lemmertz, Nucl . Instr. and Meth . A 269 (1988) 192. A.V . Yeremin, A.N . Andreyev, D.D . Bogdanov, V.I . Chepigin, V.A . Gorshkov, A.I . Ivanenko, A.P . Kabachenko, L.A . Rubinskaya, E.M . Smimova, S .V . Stepantsov, E.N . Voronkov, G.M . Ter Akopyan, Nucl . Instr. and Meth . A 274 (l989) 528. [81 R.N . Sagaidak, JINR, D7-89-551, Dubna (1989). [9] G.N. Simonoff, J.M . Alexander, Phys. Rev. 133 (1964) B-104. [10] M. Dahlinger, W. Bonin, E. Kankeleit, H. Backe, Nucl . Instr. and Meth . 219 (1984) 513. [11] R.N . Sagaidak, Yu. V. Lobanov, Yu. Ts. Oganessian, A.G . Popeko, E.A . Cherepanov, Scientific Report 1989-1990, E791-75, FLNR, JINR, Dubna (1991) 109.
663-664
Letter to the Editor
ELSEVIER
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH SectionA
New statistical function for the angular distribution of evaporation residues produced by heavy ions J. Rigol Fleroc Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow District, Russian Federation
Received
4 May 1994;
revised form received 1 September 1994
Abstract A new statistical function has been found to simulate the angular distributions of evaporation residues produced by heavy-ions . Experimental results are compared with the calculated ones.
The problem of the angular distribution of evaporation residues (EVR) with heavy-ions induced reaction has been studied for many years [1]. However, the interest in this problem has been increasing due to the development of new electromagnetic separators for the study of the evaporation residues from heavy-ion induced reactions [2-7]. The acceptance of electromagnetic separators depends on the widths of the angular distribution of the EVRs recoiled out from the target. For thick targets, the angular distribution depends fundamentally on the multiple scattering of the EVRs by the nuclei of the target . But, for thin targets (when the range of EVRs in this target is much larger than the target thickness), the neutron evaporation can also play an important role . Experimentally, the angular distribution Rexp(0,) is determined by counting the number of nuclei impinging on some disks whose radii are defined by the expression L* sin(B,), where L is the distance from the target to the disk and 0, is a certain mean angle (expressed in degrees here) between the central trajectory of the beam and the direction of the central circle inside the disk "i". The number of counts on each disk depends on the angle (through sin (B,)) and on certain other functions of 0. The experimental mean squared angle (02 )exp can be determined using the function Rexp(B, ), by the expression : )exp
= Y_
0,2R ex ,(
Bi ) .
It is well known that the parameter <02> is very important, because under the assumption of isotropic neutron evaporation it is possible to obtain an expression which establishes a relation between this value and some parameters of the studied reaction . In the first approximation, this relation takes the form : (0 2 ) = 2XEn/( 3AbEb), 0168-9002/95/$09.50 © 1995 Elsevier SSDI0168-9002(94)01127-3
(2)
where X is the number of emitted neutrons, E is the mean energy of these neutrons, Ab and Eb are the mass number and the laboratory bombarding energy of the projectiles, respectively . Usually [8] the function R(B) is modelled using the Gaussian function combined with the function sin(9) . Nevertheless, with this combination, one can not obtain a mean square value of 0 corresponding to the experimental one because, for the normal distribution, the mean square value coincides with the variance. This detail explains the discrepancy in the expression of the ( 02 ) that is observed in some works [9,10] . In this work a new statistical function has been found which describes the experimental results in natural form and gives a better concordance between the experimental and calculated values . This function is determined only by one parameter, namely, the ( 0 2 ) value: /. R(O) =360/(rr(B )) sin(6) e-BZ
Z
The function R(6) satisfies the following conditions :
Io90.R(B) d0= 1,
(4)
(B Z) =
(5)
I0 2 R(6) d0 .
The integral is taken to 90°, however, the function R(B) goes rapidly down to zero for values of 0 taken three or four times of the value ( B) . In Figs . 1 and 2, the angular distributions [9,10] of the nuclei after evaporation of X neutrons are compared with those calculated ones using experimental parameter )exp
In Fig. 1, the angular distributions of the residual nuclei 149g .l.b and 149,151Dy produced by the reactions [9]: 1) 146Nd + "B(104 MeV) -> 149gTb + 8n,
Science B.V . All rights reserved
J. Rigol /NucL Instr. and Meth. in Phys. Res. A 352 (1995) 663-664
664
ANGULAR DISTRIBUTIONS OF EVRs
the angular distributions when in the experiment all particles arriving under the angle 0 are detected . It is very
020 R(0)
useful also for the evaluation of the acceptance of collecting the EVR of determined heavy-ion induced reaction, using electromagnetic separators.
016 012 008
Acknowledgements
004
I would like to thank Prof. Yu . Ts . Oganessian who made it possible for me to do this work and Drs. Yu. A. Lazarev and Yu . V. Lobanov for their interest, valuable
000
0
3
6
9
12
15 @(Deg )
help and encouragement.
Fig. 1. Angular distributions of EVRs. 2) 140
References
149Dy MeV) -> + 7n, +12 151Dy 3)144Nd C(94 MeV) + 5n, Ce
+160(111
are shown. The experimental results (crosses, circles and triangles, respectively) are compared with the calculated ones (solid lines) obtained by using the function R(0.
In Fig. 2, the experimental [11] angular distribution of
the nuclei with mass number A = 125 obtained from the natAg reaction + 22 Ne are compared with that calculated
using the Gaussian distribution (point-dashed) [11] and
with that calculated using the function R(0) (solid line).
We stress the fact that there is no fitting of any kind,
but only the calculation of angular distributions using the function R(0) with experimental parameters ( 0,2 )exp The function R(0) is very simple and it describes the
process in the natural form . Nevertheless the origin of the coefficient (360/-tr) is not quite clear yet. But, in any case, it was very nice to find it .
The function R(0) is particularly useful for describing ANGULAR DISTRIBUTION FOR EVRs WITH A=125
0 25 R(0)
-Calc with function R(O)
020
- - Calc with Gauss function[l ++++Expenmental resultsli
015
I I
l 1
010 005 000
0
3
6
9
12
15 0(Deg )
Fig. 2. Angular distribution for EVRs with A = 125.
[1] P.F . Donovan, B.G . Harvey, W.H . Wade, Phys . Rev. 119 (1960) 218. [2] V.A. Karnaukhov, L. Rubinskaya, G. Ter Akopyan, V. Titov, V.A. Chugreev . JINR, P13-4454, Dubna (1969); and I. Bacho, D.D . Bogdanov, Sh . Dorotsi, V.A . Kamaukhov, L.A. Petrov, G.M . TerAkopyan, JINR, P13-4453, Dubna (1969) . [3] Yu .Ts. Oganessian, Yu .B . Lobanov, A.G . Popeko, J. Rigol, F. Sh. Abdulin, V.V . Bekhterev, G.G . Gulbekian, A.A. Ledovskoi, V.N. Melnikov, S.P. Tretyakova, Yu . P. Haritonov. Yu.S . Tsyganov, V.A. Chugreev, Proc. Int. School-Seminar on Heavy Ion Physics, JINR, Report, D7-90142, Dubna (1990) p. 44 . [41 Yu . A. Lazarev, Yu . V. Lobanov, A.N. Mezentsev, Yu . Ts . Oganessian, V.G . Subbotin, V.K . Utyonkov, F.Sh. Abdulin, V.V . Bekhterev, S. Iliev, I.V. Kolesov, A.N. Polyakov, I.M . Sedykh, LV . Shirakovsky, A.M. Sukhov, Yu. S. Tsyganov, V.E . Zhuchko. Proc . Int. School-Semina r on Heavy Ion Physics, vol. 2, Dubna (1993) p. 497. [5] G. Munzenberg, W. Faust, S. Hofmann, P. Armbruster, K. Guttner, H. Ewald, Nucl. Instr. and Meth . 161 (1979) 65 . [6] A. Ghiorso, S. Yashita, M.E . Leino, L. Frank, J. Kalwins, P. Armbruster, J.P . Doufour, P.K. Lemmertz, Nucl . Instr. and Meth . A 269 (1988) 192. A.V . Yeremin, A.N . Andreyev, D.D . Bogdanov, V.I . Chepigin, V.A . Gorshkov, A.I . Ivanenko, A.P . Kabachenko, L.A . Rubinskaya, E.M . Smimova, S .V . Stepantsov, E.N . Voronkov, G.M . Ter Akopyan, Nucl . Instr. and Meth . A 274 (l989) 528. [81 R.N . Sagaidak, JINR, D7-89-551, Dubna (1989). [9] G.N. Simonoff, J.M . Alexander, Phys. Rev. 133 (1964) B-104. [10] M. Dahlinger, W. Bonin, E. Kankeleit, H. Backe, Nucl . Instr. and Meth . 219 (1984) 513. [11] R.N . Sagaidak, Yu. V. Lobanov, Yu. Ts. Oganessian, A.G . Popeko, E.A . Cherepanov, Scientific Report 1989-1990, E791-75, FLNR, JINR, Dubna (1991) 109.