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Total nucleon-helium cross section near 147 MeV
Nuclear Physics 59 (1964) 253--256; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
T O T A L N U C L E O N - H E L I U M C R O S S S E C T I O N N E A R 147 M e V J. N. PALMIERI Department of Physics, Oberlin College, Oberlin, Ohio, USA
and Cyclotron Laboratory, Harvard University, Cambridge, Massachusetts, USA t and R. GOLOSKIE Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts, USA, and
Cyclotron Laboratory, Harvard University, Cambridge, Massachusetts, USA t Received 28 May 1964 Abstract: Protons with energies of 141 and 154 MeV were scattered by a target containing liquid
helium. The cross section was measured for incident protons not to leave the target within a cone (half-angle 6°) in the forward direction. At an average proton energy of 147 MeV the cross section was 113.4~ 0.8 rob. This datum along with proton-helium differential cross section and polarization data at 147 MeV were analysed using the impulse approximation and yielded a value of the total neutron-heliumcross section of 114.5± 1.0 rob. E[
NUCLEAR REACTIONS He4(p,p), Ep = 141, 154 MeV; measured a. He4(n, n), En = 136, 149 MeV; deduced a.
1. In~oducfion
T h e scattering amplitude for n e u t r o n s to be scattered by a spin-zero nucleus has the f o r m 1) fN(q) = yN(q)+hN(q)sin 0 a" n,
(1)
where q is the m o m e n t u m lost by the n e u t r o n , gN(q) and hr,(q) are complex functions o f q, 0 is the cm scattering angle, a is the Pauli spin matrix, a n d n is a u n i t vector n o r m a l to the scattering plane tt; the subscript N is used to indicate that only n u c l e a r effects are included, the magnetic m o m e n t interaction being ignored. F o r the scattering of p r o t o n s electromagnetic interactions m u s t be included. The C o u l o m b scattering a m p l i t u d e can be defined as f c ( q ) = g c ( q ) + h c ( q ) sin 0 ~" n; t This work was supported by the U.S. Office of Naval Research. tt The Basle convention has been used to assign n the direction ktn× kout. 253
(2)
254
J. N . P A L M I E R I
AND
R.
expressions for gc(q) and hc(q) are given in ref. ing by a spin-zero nucleus is then given by 1)
GOLOSKIE
2). The amplitude for proton
f(q) = z(q)fN(q)+ fc(g),
scatter-
(3)
where z(q) takes account of the modification offN(q) because of the Coulomb field; an expression for z(q) is given in ref. 2). From the optical theorem 1) the imaginary part of gN(0) can be easily related to the total neutron cross section a T gNI(O) = kaT/4rc,
(4)
where k is the wave number of the incident neutron in the cm system. Each of the other experimentally observable properties of nucleon-nucleus scattering can also be expressed 1) in terms off(q). It has been shown 2 - , ) that f(q) can be determined (apart from an angle-independent phase) for small scattering angles (0 < 15°) from a knowledge of aT and the elastic differential cross section dap/df2 and polarization Pp(O) in proton-nucleus scattering in this angular range. The analysis is performed by considering the real and imaginary parts of gN(q) and hN(q) to be functions of two adjustable parameters each. Thus for the real part of gN(q) we write gNR(q) = gNR(O)Fc(q)(1--tlgq2),
(5)
where Fc(q) is a form factor (assumed to be the same as the Coulomb form factor) 2), and gNR(0) and t/R are the adjustable parameters. Analogous equations can be written for gNl(q), hNR(q) and hNi(q). The eight parameters are then adjusted to provide the best least-squares fit to the experimental data 2). If a T is known sufficiently well, it will almost by itself determine gnu(0), thereby reducing by one the number of free parameters to be determined by the other data. The object of this experiment was to provide such a value of a T for helium, although a monochromatic neutron beam was not available. The procedure was to measure the "total" proton-helium cross section and then to "correct" it for the difference between neutron- and proton-helium scattering.
2. The Experiment The total proton-helium cross section axp is defined as the cross section for an incident proton not to leave the helium target in the forward direction in a cone with half-angle ft. The angle fl was selected to be in the Coulomb interference region of the p-helium differential cross section and was 5.95 ° (laboratory system). The experimental method was identical to that reported earlier 5) for studying the p-p total cross section. The external unpolarized proton beam of the Harvard University Cyclotron was focussed and collimated to produce a 0.3 cm diam. beam in the experimental area. This beam passed through two 0.7 cm diam. plastic scintillation
NUCLEON-HELIUM CROSS-SECTION
255
detectors (no. 1 and 2), separated by 123 cm, and then entered the target of liquid helium which was 15 cm long. Protons leaving the target within the cone, whose halfangle was fl and whose axis coincided with the beam axis, were detected by a third scintillation counter (3), which was 15 cm in diam. and 73 cm behind the centre of the target. For measurements of the background scattering due to air and the windows of the target chamber, the helium target was replaced by a dummy target. Details of this operation and precise values of the experimental parameters are given in ref. 5). Aside from small corrections, the fraction of particles scattered by the helium target is given by 1122 "~, where 12 is the number of protons counted in coincidence by counters 1 and 2, and 123 represents the number of protons detected in 1 and 2 but not in 3 during the same period. Corrections to this ratio must be made for deadtime losses in the scalers (less than 1 ~o) and for background scattering (about 25 of the helium effect). A multiple scattering correction s) was made to the contribution to aTp from elastic scattering alone, as determined by a numerical integration of the p-helium differential cross section dap/d f2 at 147 MeV, ref. 6); this correction required the subtraction of 0.3 +__0.1 mb from the measured aTp. All the corrections are discussed in detail in ref. 5). TABLE 1 Proton-helium total cross section
Energy of protons (MeV)
arp
(mb)
enter
leave
average
154
144
149
112.4±0.8
141
131
136
117.04-0.8
Using formulae (5) and (6) of reL 5), aTp and its uncertainty were calculated. The results are given in table 1. The energy of the protons entering the target was determined by a range measurement s, 7); the energy lost in the target was calculated from the target thickness. (In making the various calculations, the following parameters were used: Avogadro's number = 6.025 x 1023 mole-1; density s) of liquid helium = 0.1255 g/cm3; atomic weight of helium = 4.00.). 3. D e t e r m i n a t i o n
of a T
Under certain conditions the total neutron cross section can be obtained from the total proton cross section. We assume charge independence (including the fact that the inelastic scattering will be the same for both nucleons). Furthermore inelastic scattering at angles smaller than fl is assumed to contribute insignificantly to a T. Therefore we can write O'T = O'Tp-~- an, p "{- O'n, sraall,
(6)
256
J. N . P A L M I E R / A N D R .
GOLOSKIE
where an, v is the difference between the neutron- and proton-helium contributions to the total cross section in the angular range of fl to 180 ° (lab), and an, st*an is the contribution to a T due to elastic scattering of neutrons at angles smaller than 8. Therefore or,, p = 2~
(dan/dO - dcrp/dO) sin 0 dO,
(7)
(do'n/d~) sin 0 dO,
(8)
o
crn, stun = 27r
fl
where 0o is the centre-of-mass angle corresponding to/~, and dan/dO is the neutronhelium differential cross section at 147 MeV. While experimental values of dcrp/dO are available, dan~dOhas not been measured at this energy. Therefore the integrals in eqs. (7) and (8) were evaluated using theoretical values of these quantities obtained in the following way. Experimental s) values of dap/dO and Pp at 147 MeV were used with an estimated value of a T to determine preliminary values of the eight parameters defined by expressions like eq. (5). (Details of this fitting procedure are given in ref. 2)). For this purpose a T was taken as 118+5 rob, about 5 mb greater than 113.2+0.8 mb, which is the value of avp obtained by linearly interpolating between the results in table 1. The preliminary parameters were used to calculate do-p/dO and d a j d O , so that the integrals in eqs. (7) and (8) could be evaluated numerically. (The upper limit on the integral for o-n, p was actually taken as 22 °, beyond which angle the contribution would have been negligible.) The parameters were then redetermined using the new value of a T obtained from eq. (6). The new parameters yielded values of an, p and o". . . . . 11 which were essentially unchanged from those calculated from the first set of parameters, and were 0.8 +0.4 and 0.5 + 0.4 rob, respectively. Therefore the value of a T at 147 MeV is (113.2+0.8+0.5) or 114.5+1.0 mb. If it is assumed that an, p and a . . . . . i1 are the same at the other energies in table 1 as at 147 MeV, then a r may be determined at 149 MeV and 136 MeV as 113.7 and 118.3 rob, respectively. We thank Professor A. M. Cormack of Tufts University, Professor A. H. Cromer of Northeastern University and Mr. A. M. Koehler and the staff of the Harvard Cyclotron Laboratory for their assistance during the course of this experiment.
References 1) 2) 3) 4) 5) 6) 7) 8)
L. Wolfenstein, Annual review of nuclear science, Vol. 6 (Annual Reviews, Palo Alto) 1956,p. 43 A. H. Cromer and J. N. Palmieri, Ann. of. Phys., to be published H.A. Bethe, Ann. of Phys. 3 (1958) 190 A. H. Cromer, Phys. Rev. 113 (1959) 1607 R. Goloskie and J. N. Palmieri, Nuclear Physics 55 (1964) 463 A. M. Cormack, J. N. Palmieri, N. F. Ramsey and R. Wilson, Phys. Rev. 115 (1959) 599 W. A. Aron, B. G. Hoffman, and F. C. Williams, U.S. Govt. Printing Office 0-335880 (1955) International critical tables, Vol. 1 (McGraw-Hill Book Co., New York, 1926)
T O T A L N U C L E O N - H E L I U M C R O S S S E C T I O N N E A R 147 M e V J. N. PALMIERI Department of Physics, Oberlin College, Oberlin, Ohio, USA
and Cyclotron Laboratory, Harvard University, Cambridge, Massachusetts, USA t and R. GOLOSKIE Department of Physics, Worcester Polytechnic Institute, Worcester, Massachusetts, USA, and
Cyclotron Laboratory, Harvard University, Cambridge, Massachusetts, USA t Received 28 May 1964 Abstract: Protons with energies of 141 and 154 MeV were scattered by a target containing liquid
helium. The cross section was measured for incident protons not to leave the target within a cone (half-angle 6°) in the forward direction. At an average proton energy of 147 MeV the cross section was 113.4~ 0.8 rob. This datum along with proton-helium differential cross section and polarization data at 147 MeV were analysed using the impulse approximation and yielded a value of the total neutron-heliumcross section of 114.5± 1.0 rob. E[
NUCLEAR REACTIONS He4(p,p), Ep = 141, 154 MeV; measured a. He4(n, n), En = 136, 149 MeV; deduced a.
1. In~oducfion
T h e scattering amplitude for n e u t r o n s to be scattered by a spin-zero nucleus has the f o r m 1) fN(q) = yN(q)+hN(q)sin 0 a" n,
(1)
where q is the m o m e n t u m lost by the n e u t r o n , gN(q) and hr,(q) are complex functions o f q, 0 is the cm scattering angle, a is the Pauli spin matrix, a n d n is a u n i t vector n o r m a l to the scattering plane tt; the subscript N is used to indicate that only n u c l e a r effects are included, the magnetic m o m e n t interaction being ignored. F o r the scattering of p r o t o n s electromagnetic interactions m u s t be included. The C o u l o m b scattering a m p l i t u d e can be defined as f c ( q ) = g c ( q ) + h c ( q ) sin 0 ~" n; t This work was supported by the U.S. Office of Naval Research. tt The Basle convention has been used to assign n the direction ktn× kout. 253
(2)
254
J. N . P A L M I E R I
AND
R.
expressions for gc(q) and hc(q) are given in ref. ing by a spin-zero nucleus is then given by 1)
GOLOSKIE
2). The amplitude for proton
f(q) = z(q)fN(q)+ fc(g),
scatter-
(3)
where z(q) takes account of the modification offN(q) because of the Coulomb field; an expression for z(q) is given in ref. 2). From the optical theorem 1) the imaginary part of gN(0) can be easily related to the total neutron cross section a T gNI(O) = kaT/4rc,
(4)
where k is the wave number of the incident neutron in the cm system. Each of the other experimentally observable properties of nucleon-nucleus scattering can also be expressed 1) in terms off(q). It has been shown 2 - , ) that f(q) can be determined (apart from an angle-independent phase) for small scattering angles (0 < 15°) from a knowledge of aT and the elastic differential cross section dap/df2 and polarization Pp(O) in proton-nucleus scattering in this angular range. The analysis is performed by considering the real and imaginary parts of gN(q) and hN(q) to be functions of two adjustable parameters each. Thus for the real part of gN(q) we write gNR(q) = gNR(O)Fc(q)(1--tlgq2),
(5)
where Fc(q) is a form factor (assumed to be the same as the Coulomb form factor) 2), and gNR(0) and t/R are the adjustable parameters. Analogous equations can be written for gNl(q), hNR(q) and hNi(q). The eight parameters are then adjusted to provide the best least-squares fit to the experimental data 2). If a T is known sufficiently well, it will almost by itself determine gnu(0), thereby reducing by one the number of free parameters to be determined by the other data. The object of this experiment was to provide such a value of a T for helium, although a monochromatic neutron beam was not available. The procedure was to measure the "total" proton-helium cross section and then to "correct" it for the difference between neutron- and proton-helium scattering.
2. The Experiment The total proton-helium cross section axp is defined as the cross section for an incident proton not to leave the helium target in the forward direction in a cone with half-angle ft. The angle fl was selected to be in the Coulomb interference region of the p-helium differential cross section and was 5.95 ° (laboratory system). The experimental method was identical to that reported earlier 5) for studying the p-p total cross section. The external unpolarized proton beam of the Harvard University Cyclotron was focussed and collimated to produce a 0.3 cm diam. beam in the experimental area. This beam passed through two 0.7 cm diam. plastic scintillation
NUCLEON-HELIUM CROSS-SECTION
255
detectors (no. 1 and 2), separated by 123 cm, and then entered the target of liquid helium which was 15 cm long. Protons leaving the target within the cone, whose halfangle was fl and whose axis coincided with the beam axis, were detected by a third scintillation counter (3), which was 15 cm in diam. and 73 cm behind the centre of the target. For measurements of the background scattering due to air and the windows of the target chamber, the helium target was replaced by a dummy target. Details of this operation and precise values of the experimental parameters are given in ref. 5). Aside from small corrections, the fraction of particles scattered by the helium target is given by 1122 "~, where 12 is the number of protons counted in coincidence by counters 1 and 2, and 123 represents the number of protons detected in 1 and 2 but not in 3 during the same period. Corrections to this ratio must be made for deadtime losses in the scalers (less than 1 ~o) and for background scattering (about 25 of the helium effect). A multiple scattering correction s) was made to the contribution to aTp from elastic scattering alone, as determined by a numerical integration of the p-helium differential cross section dap/d f2 at 147 MeV, ref. 6); this correction required the subtraction of 0.3 +__0.1 mb from the measured aTp. All the corrections are discussed in detail in ref. 5). TABLE 1 Proton-helium total cross section
Energy of protons (MeV)
arp
(mb)
enter
leave
average
154
144
149
112.4±0.8
141
131
136
117.04-0.8
Using formulae (5) and (6) of reL 5), aTp and its uncertainty were calculated. The results are given in table 1. The energy of the protons entering the target was determined by a range measurement s, 7); the energy lost in the target was calculated from the target thickness. (In making the various calculations, the following parameters were used: Avogadro's number = 6.025 x 1023 mole-1; density s) of liquid helium = 0.1255 g/cm3; atomic weight of helium = 4.00.). 3. D e t e r m i n a t i o n
of a T
Under certain conditions the total neutron cross section can be obtained from the total proton cross section. We assume charge independence (including the fact that the inelastic scattering will be the same for both nucleons). Furthermore inelastic scattering at angles smaller than fl is assumed to contribute insignificantly to a T. Therefore we can write O'T = O'Tp-~- an, p "{- O'n, sraall,
(6)
256
J. N . P A L M I E R / A N D R .
GOLOSKIE
where an, v is the difference between the neutron- and proton-helium contributions to the total cross section in the angular range of fl to 180 ° (lab), and an, st*an is the contribution to a T due to elastic scattering of neutrons at angles smaller than 8. Therefore or,, p = 2~
(dan/dO - dcrp/dO) sin 0 dO,
(7)
(do'n/d~) sin 0 dO,
(8)
o
crn, stun = 27r
fl
where 0o is the centre-of-mass angle corresponding to/~, and dan/dO is the neutronhelium differential cross section at 147 MeV. While experimental values of dcrp/dO are available, dan~dOhas not been measured at this energy. Therefore the integrals in eqs. (7) and (8) were evaluated using theoretical values of these quantities obtained in the following way. Experimental s) values of dap/dO and Pp at 147 MeV were used with an estimated value of a T to determine preliminary values of the eight parameters defined by expressions like eq. (5). (Details of this fitting procedure are given in ref. 2)). For this purpose a T was taken as 118+5 rob, about 5 mb greater than 113.2+0.8 mb, which is the value of avp obtained by linearly interpolating between the results in table 1. The preliminary parameters were used to calculate do-p/dO and d a j d O , so that the integrals in eqs. (7) and (8) could be evaluated numerically. (The upper limit on the integral for o-n, p was actually taken as 22 °, beyond which angle the contribution would have been negligible.) The parameters were then redetermined using the new value of a T obtained from eq. (6). The new parameters yielded values of an, p and o". . . . . 11 which were essentially unchanged from those calculated from the first set of parameters, and were 0.8 +0.4 and 0.5 + 0.4 rob, respectively. Therefore the value of a T at 147 MeV is (113.2+0.8+0.5) or 114.5+1.0 mb. If it is assumed that an, p and a . . . . . i1 are the same at the other energies in table 1 as at 147 MeV, then a r may be determined at 149 MeV and 136 MeV as 113.7 and 118.3 rob, respectively. We thank Professor A. M. Cormack of Tufts University, Professor A. H. Cromer of Northeastern University and Mr. A. M. Koehler and the staff of the Harvard Cyclotron Laboratory for their assistance during the course of this experiment.
References 1) 2) 3) 4) 5) 6) 7) 8)
L. Wolfenstein, Annual review of nuclear science, Vol. 6 (Annual Reviews, Palo Alto) 1956,p. 43 A. H. Cromer and J. N. Palmieri, Ann. of. Phys., to be published H.A. Bethe, Ann. of Phys. 3 (1958) 190 A. H. Cromer, Phys. Rev. 113 (1959) 1607 R. Goloskie and J. N. Palmieri, Nuclear Physics 55 (1964) 463 A. M. Cormack, J. N. Palmieri, N. F. Ramsey and R. Wilson, Phys. Rev. 115 (1959) 599 W. A. Aron, B. G. Hoffman, and F. C. Williams, U.S. Govt. Printing Office 0-335880 (1955) International critical tables, Vol. 1 (McGraw-Hill Book Co., New York, 1926)