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The elastic scattering of 12.0 MeV He3 particles by nuclei
PHYSICS
Volume 11, number 4
15 August 1964
LETTERS
Performing the calculation one gets : for protons in this 3- state 3.34 n.m., and for neutrons -0.27 n.m. Thus the T = 0 state of 018 has the magnetic moment 1.53 n.m. This result was to be expected3) since the spin parts of the total wave function should not be very much changed in the excitation and thus the magnetic moment should depend only on the magnetic moment of the orbital angular momenta. The effective gyromagnetic ratio should
then be 0.5, and the magnetic moment of the excited state about 1.50 n.m.
References 1) J.P.Elliott and B.H.Flowers, Proc.Phys.Soc.A242 (1957) 57. 2) G. E. Brown and M. Boleterli, Phys . Rev. Letters 3 (1959)472. 3) B. R. Mottelson, private communication.
***t*
THE
ELASTIC
SCATTERING
OF
12.0
MeV
He3
PARTICLES
BY NUCLEI
*
J. L. YNTEMA and B. ZEIDMAN Argonne
National Laboratory,
Argonne,
Illinois
and R. H. BASSEL Oak Ridge NatiMzal Laboratoy!,
Oak Ridge,
Tennessee
Received 28 June 1964 The optical-model parameters used in the entrance and exit channels of distorted-wave calculations affect both the calculated shape of the angular distribution and the theoretical cross section. The information on the variation of the optical-model parameters with particle energy and with mass is not well known in the case of He3 particles. It was felt necessary in connection with several experiments carried out at Argonne to obtain reasonable parameters in the vicinity of the energies of the outgoing particles in (d, He3) reactions and of the incident particles in (He3, cr) and (He3, d) reactions. The elastic scattering of He3 by Mg24, $ahO, Sc45 Ti46, Ti4*, Ti50, V, Fe54, Ni5*, Zn6 , and Zr8d has been measured at an incident energy of 12.00 MeV. The He3 particles were accelerated with the ANL tandem Van de Ti46, Ti4*, Ti50, Fe54, Ni5*, Graaff. The 324, Zn84, and Zr8 targets were prepared from isotopically enriched material. All the targets were in the form of self-supporting foils **. The targets were at an angle of 45O with respect to the beam direction. The points measured from 200 to * Research sponsored by the U. S. Atomic Energy Commission, in Part, under contract with the Union Carbide Corporation. ** We are indebtedto F. Karasek for the preparation of the targets. 302
90’ were obtained in a transmission arrangement and the ones from 90° to 120° in a reflection a.rrangement. The cross sections at 90° obtained with the two different geometrical arrangements are in good agreement. The target thickness was obtained as the weight per unit area. Usually the targets are sufficiently uniform to yield a value with an error of less than 2%. In some cases, with targets with a thickness less than 0.8 mg/cm2, corrections had to be made. The weight of the Ca in the Ca target was obtained after the experiment by chemical methods to eliminate the error due to the presence of oxygen. In the case of zirconium, a correction’of about 8% was made to compensate for the presence of absorbed gases in the target. Both the distance between the centre of scattering and the detector and the area of the defining aperture of the detector unit are known to within less than 0.2%. The angle is known to within less than O.l” and the angle subtended by the aperture of the detector unit was less than 0.3O. The current was integrated with a system similar to the one designed by Hamler et al. I) and is accurate within less than 1%. The He3 particles were detected with a surface-barrier silicon detector; in some cases in which the error caused by CYparticles from the (He3,@) reaction might be bothersome, they were also de-
Volume 11, number 4
PHYSICS
15 August 1964
LETTERS
8c.m.
Fig. 1.
tected with a (dZZ/dx)-E counter telescope. The resolution width of the system was about 80 keV for the transmission arrangement and about 150 keV for the reflection case. The absolute cross sections were calculated with the aid of a computer program. The rms errors in the absolute cross sections vary from 3% to 6%. These data have been analyzed with an optical model. Because the differential cross sections show little structure as a function of angle, it is possible to obtain good fits with potentials having a variety of shapes. We shall report on calculations with a tential which is consistent with one found for He$ scattering at higher bombarding energies 3). In this model the real and imaginary potentials are allowed to have different shapes, in particular, V(Y) = - vo{l + exd(r - y. Af)/4)-’ and W(r) = -Wo(l + exp[(r - Y~A&&~
.
Calculations were performed with the Cak Ridge automatic search code “Hunter” 3) in which selected parameters are varied until the quantity
is minimized. We have taken the point of view that the parameters should vary smoothly as a function of target and have biased the parameters accordingly. We find that it is possible to fit angular distributions for all targets, excepting Mg34 and Ca40, with the same potential, whose parameters
are listed in the table, along with the predicted reaction cross sections. The angular distributions computed with this potential are compared with the data in fig. 1. Table 1 Optid-model parameters for the scattering of 12 MeV He3. In all cases r. = 1.07 fm, a = 0.754 fm, yw = 1.81 fm, and aW = 0.592 fm. Target
w24 Ca40 SC45 Ti46 Ti48
96.4 96.4 96.4 106.3 96.4 96.4 96.4
yz Fe54 Ni58 Zn64 Zrso
96.4 96.4 96.4 96.4 96.4
14.75 14.61 14.5 7.43 10 10 10
0 4 8 0 0 0 0
985 985 985 709 812 767 789
10 10 10 10 10
0 0 0 0 0
810 780 630 558 506 200
If the geometrical arameters (yo, a, Yw, a$ for the Mg34 and Ca4 B potentials are constrained to agree with those used for the other targets, it is possible to obtain a good fit to the scattering from Ca40 by allowing a deeper real well and a shallower imaginary well. That a somewhat deeper imaginary well is required for ~g34 than for the other nuclei may, perhaps, reflect dependence on nuclear structure. These parameters are listed in the table, and the corresponding angular distributions are included in fig. 1. In view of the usual difficulties associated with analyses of light nuclei, these discrepancies are not taken too seriously. 303
PHYSICS
Volume 11, number 4
15 August 1964
LETTERS
This analysis has shown that it is possible to explain the main features of 12 MeV He3 scattering with a potential that varies smoothly with atomic mass. This potential has many features in common with the o tical potential found for deuteron scattering 4y, In particular, the optical model for both projectiles requires “weak” but long-range imaginary potentials. This indicates that surface absorption is important, a conclusion similar to that suggested by Hodgson and collaborators 5) which, however, is based on very deep imaginary wells of different shape. A more detailed discussion of the optical model for He8 ions is in preparation.
Fig. 2. To determine whether or not spin-orbit coupling would improve the detailed fit for Mg84, calculations were performed with a Thomastype spin-orbit well based on the parameters of the real well. With spin-orbit depths, Vs of 4 and 8 MeV (pion mass units), no significant changes in fit were found and only minor changes in the imaginary well depth were caused. On the basis of these results, scattering cannot establish the necessity for a spin-orbit force; a measurement of the polarization would seem necessary. Fig. 2 shows the predicted polarization calculated with a spin-orbit strength of 8 MeV.
1) W. Brookshier, W. Ramler and R . Benaroya, unpublished 2) R.H.Bassel, G.R.SatchlerandR.M.Drisko, tobe published. 3) R. M. Drisko, unpublished. 4) E. C. Halbert, Nucl. Phya. , to be published. C. and F.G.Perey, Phys.Rev.132 (1963) 755; L. L. Lee, J. P.Schiffer, G. R. Satchler, R. M.Drisko and R.H.Bassel, tobe publishedin Phys.Rev. 5) P. E. Hodgson, The optical model of elastic scattering (Cxford University Press, London, 1963) and refs. therein.
*****
THE
VARIATION
OF ENERGY
GAP
IN ROTATIONAL
STATES
OF NUCLEI
K. Y. CHAN and J. G. VALATIN Department University
of Mathematical
Physics,
of Birmingham,
England
Received 6 July 1964 Nuclear rotation affects the strength of the pairing correlation 1). The effect has been investigated in a simple model, with a deformed shell model potential rotating with angular velocity a, and a pairing force independent of rotation. The dependence of the gap parameter A on 52results from a gap equation which is obtained in assuming A as a function of a and performing an additional perturbation calculation 8) in the rotational coupling. The equation can be written in the compact form
$=~&+ny$, K
where E, = (eK2 + A2)$, and
304
cK
cKt+A2 (2) EKEK’
is the effective moment of inertia, (without correction terms which would give an explicit dependence on 52). A similar relationship between effective moment of inertia and gap equation can be derived to all orders in the angular velocity. In a deformed oscillator shell model potential, the double sums practically reduce to single sums and can be evaluated with little numerical work. Fig. 1 shows the resulting A = A(Q) curves for &f,0** The variation of neutron and proton gaps, An and Ap, is calculated separately, assuming pairing correlations only between similar particles, A harmonic oscillator potential with frequencies
Volume 11, number 4
15 August 1964
LETTERS
Performing the calculation one gets : for protons in this 3- state 3.34 n.m., and for neutrons -0.27 n.m. Thus the T = 0 state of 018 has the magnetic moment 1.53 n.m. This result was to be expected3) since the spin parts of the total wave function should not be very much changed in the excitation and thus the magnetic moment should depend only on the magnetic moment of the orbital angular momenta. The effective gyromagnetic ratio should
then be 0.5, and the magnetic moment of the excited state about 1.50 n.m.
References 1) J.P.Elliott and B.H.Flowers, Proc.Phys.Soc.A242 (1957) 57. 2) G. E. Brown and M. Boleterli, Phys . Rev. Letters 3 (1959)472. 3) B. R. Mottelson, private communication.
***t*
THE
ELASTIC
SCATTERING
OF
12.0
MeV
He3
PARTICLES
BY NUCLEI
*
J. L. YNTEMA and B. ZEIDMAN Argonne
National Laboratory,
Argonne,
Illinois
and R. H. BASSEL Oak Ridge NatiMzal Laboratoy!,
Oak Ridge,
Tennessee
Received 28 June 1964 The optical-model parameters used in the entrance and exit channels of distorted-wave calculations affect both the calculated shape of the angular distribution and the theoretical cross section. The information on the variation of the optical-model parameters with particle energy and with mass is not well known in the case of He3 particles. It was felt necessary in connection with several experiments carried out at Argonne to obtain reasonable parameters in the vicinity of the energies of the outgoing particles in (d, He3) reactions and of the incident particles in (He3, cr) and (He3, d) reactions. The elastic scattering of He3 by Mg24, $ahO, Sc45 Ti46, Ti4*, Ti50, V, Fe54, Ni5*, Zn6 , and Zr8d has been measured at an incident energy of 12.00 MeV. The He3 particles were accelerated with the ANL tandem Van de Ti46, Ti4*, Ti50, Fe54, Ni5*, Graaff. The 324, Zn84, and Zr8 targets were prepared from isotopically enriched material. All the targets were in the form of self-supporting foils **. The targets were at an angle of 45O with respect to the beam direction. The points measured from 200 to * Research sponsored by the U. S. Atomic Energy Commission, in Part, under contract with the Union Carbide Corporation. ** We are indebtedto F. Karasek for the preparation of the targets. 302
90’ were obtained in a transmission arrangement and the ones from 90° to 120° in a reflection a.rrangement. The cross sections at 90° obtained with the two different geometrical arrangements are in good agreement. The target thickness was obtained as the weight per unit area. Usually the targets are sufficiently uniform to yield a value with an error of less than 2%. In some cases, with targets with a thickness less than 0.8 mg/cm2, corrections had to be made. The weight of the Ca in the Ca target was obtained after the experiment by chemical methods to eliminate the error due to the presence of oxygen. In the case of zirconium, a correction’of about 8% was made to compensate for the presence of absorbed gases in the target. Both the distance between the centre of scattering and the detector and the area of the defining aperture of the detector unit are known to within less than 0.2%. The angle is known to within less than O.l” and the angle subtended by the aperture of the detector unit was less than 0.3O. The current was integrated with a system similar to the one designed by Hamler et al. I) and is accurate within less than 1%. The He3 particles were detected with a surface-barrier silicon detector; in some cases in which the error caused by CYparticles from the (He3,@) reaction might be bothersome, they were also de-
Volume 11, number 4
PHYSICS
15 August 1964
LETTERS
8c.m.
Fig. 1.
tected with a (dZZ/dx)-E counter telescope. The resolution width of the system was about 80 keV for the transmission arrangement and about 150 keV for the reflection case. The absolute cross sections were calculated with the aid of a computer program. The rms errors in the absolute cross sections vary from 3% to 6%. These data have been analyzed with an optical model. Because the differential cross sections show little structure as a function of angle, it is possible to obtain good fits with potentials having a variety of shapes. We shall report on calculations with a tential which is consistent with one found for He$ scattering at higher bombarding energies 3). In this model the real and imaginary potentials are allowed to have different shapes, in particular, V(Y) = - vo{l + exd(r - y. Af)/4)-’ and W(r) = -Wo(l + exp[(r - Y~A&&~
.
Calculations were performed with the Cak Ridge automatic search code “Hunter” 3) in which selected parameters are varied until the quantity
is minimized. We have taken the point of view that the parameters should vary smoothly as a function of target and have biased the parameters accordingly. We find that it is possible to fit angular distributions for all targets, excepting Mg34 and Ca40, with the same potential, whose parameters
are listed in the table, along with the predicted reaction cross sections. The angular distributions computed with this potential are compared with the data in fig. 1. Table 1 Optid-model parameters for the scattering of 12 MeV He3. In all cases r. = 1.07 fm, a = 0.754 fm, yw = 1.81 fm, and aW = 0.592 fm. Target
w24 Ca40 SC45 Ti46 Ti48
96.4 96.4 96.4 106.3 96.4 96.4 96.4
yz Fe54 Ni58 Zn64 Zrso
96.4 96.4 96.4 96.4 96.4
14.75 14.61 14.5 7.43 10 10 10
0 4 8 0 0 0 0
985 985 985 709 812 767 789
10 10 10 10 10
0 0 0 0 0
810 780 630 558 506 200
If the geometrical arameters (yo, a, Yw, a$ for the Mg34 and Ca4 B potentials are constrained to agree with those used for the other targets, it is possible to obtain a good fit to the scattering from Ca40 by allowing a deeper real well and a shallower imaginary well. That a somewhat deeper imaginary well is required for ~g34 than for the other nuclei may, perhaps, reflect dependence on nuclear structure. These parameters are listed in the table, and the corresponding angular distributions are included in fig. 1. In view of the usual difficulties associated with analyses of light nuclei, these discrepancies are not taken too seriously. 303
PHYSICS
Volume 11, number 4
15 August 1964
LETTERS
This analysis has shown that it is possible to explain the main features of 12 MeV He3 scattering with a potential that varies smoothly with atomic mass. This potential has many features in common with the o tical potential found for deuteron scattering 4y, In particular, the optical model for both projectiles requires “weak” but long-range imaginary potentials. This indicates that surface absorption is important, a conclusion similar to that suggested by Hodgson and collaborators 5) which, however, is based on very deep imaginary wells of different shape. A more detailed discussion of the optical model for He8 ions is in preparation.
Fig. 2. To determine whether or not spin-orbit coupling would improve the detailed fit for Mg84, calculations were performed with a Thomastype spin-orbit well based on the parameters of the real well. With spin-orbit depths, Vs of 4 and 8 MeV (pion mass units), no significant changes in fit were found and only minor changes in the imaginary well depth were caused. On the basis of these results, scattering cannot establish the necessity for a spin-orbit force; a measurement of the polarization would seem necessary. Fig. 2 shows the predicted polarization calculated with a spin-orbit strength of 8 MeV.
1) W. Brookshier, W. Ramler and R . Benaroya, unpublished 2) R.H.Bassel, G.R.SatchlerandR.M.Drisko, tobe published. 3) R. M. Drisko, unpublished. 4) E. C. Halbert, Nucl. Phya. , to be published. C. and F.G.Perey, Phys.Rev.132 (1963) 755; L. L. Lee, J. P.Schiffer, G. R. Satchler, R. M.Drisko and R.H.Bassel, tobe publishedin Phys.Rev. 5) P. E. Hodgson, The optical model of elastic scattering (Cxford University Press, London, 1963) and refs. therein.
*****
THE
VARIATION
OF ENERGY
GAP
IN ROTATIONAL
STATES
OF NUCLEI
K. Y. CHAN and J. G. VALATIN Department University
of Mathematical
Physics,
of Birmingham,
England
Received 6 July 1964 Nuclear rotation affects the strength of the pairing correlation 1). The effect has been investigated in a simple model, with a deformed shell model potential rotating with angular velocity a, and a pairing force independent of rotation. The dependence of the gap parameter A on 52results from a gap equation which is obtained in assuming A as a function of a and performing an additional perturbation calculation 8) in the rotational coupling. The equation can be written in the compact form
$=~&+ny$, K
where E, = (eK2 + A2)$, and
304
cK
cKt+A2 (2) EKEK’
is the effective moment of inertia, (without correction terms which would give an explicit dependence on 52). A similar relationship between effective moment of inertia and gap equation can be derived to all orders in the angular velocity. In a deformed oscillator shell model potential, the double sums practically reduce to single sums and can be evaluated with little numerical work. Fig. 1 shows the resulting A = A(Q) curves for &f,0** The variation of neutron and proton gaps, An and Ap, is calculated separately, assuming pairing correlations only between similar particles, A harmonic oscillator potential with frequencies