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Pion production by 600 MeV protons on light nuclei
Volume 43B, number 1
PHYSICS LETTERS
8 January 1973
P I O N P R O D U C T I O N BY 6 0 0 M e V P R O T O N S O N L I G H T N U C L E I J.M. EISENBERG, R. GUY:[:, J.V. NOBLE and H.J. WEBER Department of Physics, University of Virginia, Charlottesville, Virginia:~ 22901, USA Received 5 October 1972 .
.
~
÷
.
The forward cross section at Ep = 600 MeV is calculated for the reacUon p + (N, Z) ~ (N+I, Z) + ~r using a single-nucleon mechanism, distorted pion and proton wave functions and bound-state nucleon wave functions with realistic asymptotic behavior. Good agreement with experiment is found for 4He, 12C and 14N.
The reaction p +(N, Z) ~ (N+I, Z) +rr + has long been considered highly improbable because of the very large momentum transfers involved. Recently, the spectra of high-momentum pions produced at 0 ° from light nuclei bombarded by 600-MeV protons were measured [1,2]. These spectra exhibit peaks which are indicative of proton capture to discrete final states in the residual nucleus. In case of carbon and nitrogen, preliminary attemps [1] at a theoretical explanation of the data were completely unsuccessful. These calculations assumed that the pion was emitted directly by the incoming proton, which was then captured into a bound state in the residual nulceus. For the captured nucleon, various momentum distributions, such as those appropriate to an harmonic oscillator or a Woods-Saxon potential, were employed in the calculation of the cross sections. A later analysis by the same group [3] was based on the assumption that the positive pions have their origin in the two-nucleon reaction p+p ~ d+Tr+.
Again, harmonic oscillator wave functions were used and distortions of the proton and pion waves were ignored. The results for the reaction p+d -+ t+lr + were rather encouraging; howeverm the predicted cross section for p + 12C -~ 13Cg.s" +/r + at 0 ° was too small by a factor of about 30. These authors also made calculations in a one-nucleon model [4], again employing oscillator wave functions, with results decidedly inferior to those of the two-nucleon model. Reitan [5] studied the 4He and 12C data [1,2] within the framework of a two-nucleon model in which the pion is emitted by one of the nucleons in the filled shells and then rescattered by the incident proton through the (3,3) resonance channel. For 12C, this approach gave a result essentially identical to that of the two-nucleon calculation of ref. [3] ; however, it is still smaller than experiment by a factor of about 30. In the case of the (p, 7r+) reaction on 4He, Reitan's two-nucleon model was more successful, giving forward cross sections in good agreement with experiment. As pointed out in ref. [5], one can in principle obtain some information on the relative importance of the one-nucleon (direct) and two-nucleon mechanisms by searching for negative pions produced by the bombarding protons. These are forbidden in
Permanent address: Department of Physics, University of Wisconsin-Oshkosh, Oshkosh, Wisconsin. ¢-~ Work supported in part by the National Science Foundation. Table 1
Pion and proton optical potential parameters Nucleus
OlrN (fro2) [13]
PlrN [13]
trpN(fm2) [12]
PpN [12]
Ro(fm)
4He 12C 14N 160
2.91 5.0 7.0 2.91
-0.5 -0.5 -0.5 -0.35
3.96 3.96 3.96 3.96
0.12 0.12 0.12 0.12
1.5 2.7 2.7 4.0
20
Volume 43B, number 1
PHYSICS LETTERS
the direct reaction, but should be manifest if the twonucleon process is dominant. Experimental results [6] indicating the absence of substantial numbers of negative pions have motivated us to reconsider the direct one-nucleon model, including distorted wave functions for the proton and pion and wave functions for the final bound-state neutron that are more realsistic than those of an harmonic oscillator. We therefore calculate the process p+(N, Z) ~ (N+I,Z) + ;r+ ,
do_ 1 (q)EEq (2702
X [1+(1
(Ep,p)
pc°sOEEq ~ ' ( f LAl+l I~~1/ J112 "spin q
(
(2)
(Eq,q)
Here and are the total energl"eS and momenta of the incident proton and emitted pion; is the total energy of the residual nucleus; 0 is the angle between p and q; and the spin sum includes an average over the initial spin projection of the proton and a sum over the final spin projection of the residual nucleus. The nuclear matrix element of the transition operator H is written as [4, 7]
EA+1
f (nljmjl(1 + ~ ) --m
o.
m ~" ~ o. V Ip#) , Vn ~" --. q~-m
(3)
where m and M are, respectively, the pion and nucleon masses; ¢r and ~are the nucleon spin and isospin operators; ~is the pion field ~3perator; # is the spin projection of the incoming proton; are the quantum numb'er of the residual bound neutron; and we take = 0.083. The wave functions for the proton and pion are distorted according to the prescription of Glauber [8]. Thus for a proton (+) or pion ( - ) of momentum k we write the distorted wave function as
{nl/m/)
f2/4lr
d/(±)(b,z)~eik'rexp[~fU(±)'(b+~z')dz' 1
(4)
-T-oo
r =b +~z,.b
where is the impact parameter vector and ia a unit vector in the direction of the proton momentum, taken as the Z-axis. In addition, v_. is the relative form:~
Rll(r) = Nll(1 - exp(-r/Ro))l+l exp(-rr)/r ,
(1)
in which the residual nucleus (N+I ,Z) is in a bound (ground or perhaps excited) state and the proton and pion are distorted by nuclear, but not Coulomb forces. The differential cross section in the laboratory system for this reaction is given by
d~2
8 January 1973
(lp, ld) R20(r ) = N20(1 -
exp(-r/Ro) (1 - e x p ( ( s - r ) [ R o ) )
X exp(-Kr)/r,
(2s)
(7)
where r is determined by the separation energies (see table 2), and s is the location of the node. We have also used Woods-Saxon wave functions which give qualitatively similar results. In table 3 we present results of our distorted wave calculations (DWBA) for forward pion production, together with the relevant experimental results of ref. [ 1]. For comparison we give the results for plane pion and proton waves (PWBA). (For 4He, if we assume the neutron to be captured into a slightly bound 1 Pa/2 state Table 2 Parameters for generalized Hulthen wave functions
Nucleus
(n/)/
Ro(fm)
g(fm-1 )
Node
SHe 13C
lP3/2 lpl/2
1.5 3.0
0.25 0.47
13C 13C lSN
2st/2 lds/2 lpl/2
3.0 3.0 3.3
0.288 0.221 0.7
1.52R o
ISN ISN 170
2Sl/2 lds/2 lds/2
3.3 3.3 3.5
0.5 0.5 0.423
1.5R o
:~We use these analytic forms for several reasons: First, the PWBA cross-section can be evaluated in closed form using them; second, because they simplify the contour rotations we employ to evaluate numerically the double integral appearing in the distorted wave expression; and third, because the difference between Exkart wave functions and Woods-Saxon wave functions with the same binding energy is too small to affect the momentum distribution in the region of interest.
21
PHYSICS LETTERS
Volume 43B, number 1
Table 3 Calculated and experimental forward cross sections in the laboratory system. Reaction
reported calculations [1,3, 5], which suggests that the single-nucleon mechanism is quite adequate to account for the (p, rr+) reaction at 600 MeV.
(do/d~2)oo in ~b/sr PWBA
D W B A Experimenta
12C(p, Tr+)13C(lPl/2)
5.0
0.11
12C(p, lr+)13C*(2s1/2)
39.1
0.57
12C(p, 7r+)13C*(ld5/2 )
0.7
0.05
14N(p, n+)lSN(lPl/2)
26.4
0.13
0.13 ± 0.05
14N(p, ~)1SN*(2sl/2)
86.6
0.19 /
14N(p, n+)lSN*(lds/2)
13.6
0.14
0.23 ± O.07
160(p, ~r+)170(lds/2 )
5.3
0.77
0.25 + 0.10
l
J
0.26 ± 0.10
---
a For the excited states, the experimental results are the sums of the excitation functions for states in the region of the first excited state. We quote here the results of a •reanalysis of ref. [1], quoted in ref. [5], which yields values smaller by about a factor of 3 than those originally presented.
(Esep ~ 1.5 MeV), although in fact n o bound state o f 5He exists, the Born cross section is 23.04/ab/sr. : Distortions reduce this to about 0.12/ab/sr, while the experimental value [2] is 26 +- 3 tab/s.r. Presumably this lack o f agreement reflects either the use of bound wave functions or the non resolution o f 1 P3/2 from 1 Pl/2 states or both.) In general the DWBA results o f table 3 yield better agreement with experiment than that o f the previously
22
8 January 1973
We gratefully acknowledge grants for computer work from the Center for Advanced Studies and from the Computer Science Center at the University o f Virginia.
References [1] [2] [3] [4]
J.J. Domingo et al., Phys. Lett. 32B (1970) 309. K. Gabathuler et al., Nucl. Phys. B40 (1972) 32. C.H.Q. Ingram et al., Nucl. Phys. B31 (1971) 331. J. LeTourneux and J.M. Eisenberg, Nucl. Phys. 87 (1966) 331. [5] A. Reitan, Nucl. Phys. B29 (1971) 525; preprint. [6] J. Rohlin et al., Phys. Lett. 40B (1972) 539. [71 W.B. Jones and J.M. Eisenberg, Nucl. Phys. A154 (1970) 49; M.V. BarnhiU, Nucl. Phys. A131 (1969) 106. [ 8] R.J. Glanber, Lectures in theoretical physics, ed. W.E. Brittin and L.G. Dunham (Interscience, New York, 1959). [9] D.S. Koltun, Adv. NucL Phys. 3 (1969) 71. [10] R.H. Bassel and C. Wilkin, Phys. Rev. 174 (1968) 1179 [ 11 ] J.M. Eisenberg and W. Greiner, Excitation mechanisms of the nucleus (North-Holland, Amsterdam, 1970). [12] D.V. Bugg et aL, Phys. Rev. 146 (1966) 980. [13] G. HShler, G. Ebel and J. Giesecke, Z. Physik 180 (1964) 430. [14] ].V. Noble, Phys. Rev. C1 (1970) 1900, and references contained therein.
PHYSICS LETTERS
8 January 1973
P I O N P R O D U C T I O N BY 6 0 0 M e V P R O T O N S O N L I G H T N U C L E I J.M. EISENBERG, R. GUY:[:, J.V. NOBLE and H.J. WEBER Department of Physics, University of Virginia, Charlottesville, Virginia:~ 22901, USA Received 5 October 1972 .
.
~
÷
.
The forward cross section at Ep = 600 MeV is calculated for the reacUon p + (N, Z) ~ (N+I, Z) + ~r using a single-nucleon mechanism, distorted pion and proton wave functions and bound-state nucleon wave functions with realistic asymptotic behavior. Good agreement with experiment is found for 4He, 12C and 14N.
The reaction p +(N, Z) ~ (N+I, Z) +rr + has long been considered highly improbable because of the very large momentum transfers involved. Recently, the spectra of high-momentum pions produced at 0 ° from light nuclei bombarded by 600-MeV protons were measured [1,2]. These spectra exhibit peaks which are indicative of proton capture to discrete final states in the residual nucleus. In case of carbon and nitrogen, preliminary attemps [1] at a theoretical explanation of the data were completely unsuccessful. These calculations assumed that the pion was emitted directly by the incoming proton, which was then captured into a bound state in the residual nulceus. For the captured nucleon, various momentum distributions, such as those appropriate to an harmonic oscillator or a Woods-Saxon potential, were employed in the calculation of the cross sections. A later analysis by the same group [3] was based on the assumption that the positive pions have their origin in the two-nucleon reaction p+p ~ d+Tr+.
Again, harmonic oscillator wave functions were used and distortions of the proton and pion waves were ignored. The results for the reaction p+d -+ t+lr + were rather encouraging; howeverm the predicted cross section for p + 12C -~ 13Cg.s" +/r + at 0 ° was too small by a factor of about 30. These authors also made calculations in a one-nucleon model [4], again employing oscillator wave functions, with results decidedly inferior to those of the two-nucleon model. Reitan [5] studied the 4He and 12C data [1,2] within the framework of a two-nucleon model in which the pion is emitted by one of the nucleons in the filled shells and then rescattered by the incident proton through the (3,3) resonance channel. For 12C, this approach gave a result essentially identical to that of the two-nucleon calculation of ref. [3] ; however, it is still smaller than experiment by a factor of about 30. In the case of the (p, 7r+) reaction on 4He, Reitan's two-nucleon model was more successful, giving forward cross sections in good agreement with experiment. As pointed out in ref. [5], one can in principle obtain some information on the relative importance of the one-nucleon (direct) and two-nucleon mechanisms by searching for negative pions produced by the bombarding protons. These are forbidden in
Permanent address: Department of Physics, University of Wisconsin-Oshkosh, Oshkosh, Wisconsin. ¢-~ Work supported in part by the National Science Foundation. Table 1
Pion and proton optical potential parameters Nucleus
OlrN (fro2) [13]
PlrN [13]
trpN(fm2) [12]
PpN [12]
Ro(fm)
4He 12C 14N 160
2.91 5.0 7.0 2.91
-0.5 -0.5 -0.5 -0.35
3.96 3.96 3.96 3.96
0.12 0.12 0.12 0.12
1.5 2.7 2.7 4.0
20
Volume 43B, number 1
PHYSICS LETTERS
the direct reaction, but should be manifest if the twonucleon process is dominant. Experimental results [6] indicating the absence of substantial numbers of negative pions have motivated us to reconsider the direct one-nucleon model, including distorted wave functions for the proton and pion and wave functions for the final bound-state neutron that are more realsistic than those of an harmonic oscillator. We therefore calculate the process p+(N, Z) ~ (N+I,Z) + ;r+ ,
do_ 1 (q)EEq (2702
X [1+(1
(Ep,p)
pc°sOEEq ~ ' ( f LAl+l I~~1/ J112 "spin q
(
(2)
(Eq,q)
Here and are the total energl"eS and momenta of the incident proton and emitted pion; is the total energy of the residual nucleus; 0 is the angle between p and q; and the spin sum includes an average over the initial spin projection of the proton and a sum over the final spin projection of the residual nucleus. The nuclear matrix element of the transition operator H is written as [4, 7]
EA+1
f (nljmjl(1 + ~ ) --m
o.
m ~" ~ o. V Ip#) , Vn ~" --. q~-m
(3)
where m and M are, respectively, the pion and nucleon masses; ¢r and ~are the nucleon spin and isospin operators; ~is the pion field ~3perator; # is the spin projection of the incoming proton; are the quantum numb'er of the residual bound neutron; and we take = 0.083. The wave functions for the proton and pion are distorted according to the prescription of Glauber [8]. Thus for a proton (+) or pion ( - ) of momentum k we write the distorted wave function as
{nl/m/)
f2/4lr
d/(±)(b,z)~eik'rexp[~fU(±)'(b+~z')dz' 1
(4)
-T-oo
r =b +~z,.b
where is the impact parameter vector and ia a unit vector in the direction of the proton momentum, taken as the Z-axis. In addition, v_. is the relative form:~
Rll(r) = Nll(1 - exp(-r/Ro))l+l exp(-rr)/r ,
(1)
in which the residual nucleus (N+I ,Z) is in a bound (ground or perhaps excited) state and the proton and pion are distorted by nuclear, but not Coulomb forces. The differential cross section in the laboratory system for this reaction is given by
d~2
8 January 1973
(lp, ld) R20(r ) = N20(1 -
exp(-r/Ro) (1 - e x p ( ( s - r ) [ R o ) )
X exp(-Kr)/r,
(2s)
(7)
where r is determined by the separation energies (see table 2), and s is the location of the node. We have also used Woods-Saxon wave functions which give qualitatively similar results. In table 3 we present results of our distorted wave calculations (DWBA) for forward pion production, together with the relevant experimental results of ref. [ 1]. For comparison we give the results for plane pion and proton waves (PWBA). (For 4He, if we assume the neutron to be captured into a slightly bound 1 Pa/2 state Table 2 Parameters for generalized Hulthen wave functions
Nucleus
(n/)/
Ro(fm)
g(fm-1 )
Node
SHe 13C
lP3/2 lpl/2
1.5 3.0
0.25 0.47
13C 13C lSN
2st/2 lds/2 lpl/2
3.0 3.0 3.3
0.288 0.221 0.7
1.52R o
ISN ISN 170
2Sl/2 lds/2 lds/2
3.3 3.3 3.5
0.5 0.5 0.423
1.5R o
:~We use these analytic forms for several reasons: First, the PWBA cross-section can be evaluated in closed form using them; second, because they simplify the contour rotations we employ to evaluate numerically the double integral appearing in the distorted wave expression; and third, because the difference between Exkart wave functions and Woods-Saxon wave functions with the same binding energy is too small to affect the momentum distribution in the region of interest.
21
PHYSICS LETTERS
Volume 43B, number 1
Table 3 Calculated and experimental forward cross sections in the laboratory system. Reaction
reported calculations [1,3, 5], which suggests that the single-nucleon mechanism is quite adequate to account for the (p, rr+) reaction at 600 MeV.
(do/d~2)oo in ~b/sr PWBA
D W B A Experimenta
12C(p, Tr+)13C(lPl/2)
5.0
0.11
12C(p, lr+)13C*(2s1/2)
39.1
0.57
12C(p, 7r+)13C*(ld5/2 )
0.7
0.05
14N(p, n+)lSN(lPl/2)
26.4
0.13
0.13 ± 0.05
14N(p, ~)1SN*(2sl/2)
86.6
0.19 /
14N(p, n+)lSN*(lds/2)
13.6
0.14
0.23 ± O.07
160(p, ~r+)170(lds/2 )
5.3
0.77
0.25 + 0.10
l
J
0.26 ± 0.10
---
a For the excited states, the experimental results are the sums of the excitation functions for states in the region of the first excited state. We quote here the results of a •reanalysis of ref. [1], quoted in ref. [5], which yields values smaller by about a factor of 3 than those originally presented.
(Esep ~ 1.5 MeV), although in fact n o bound state o f 5He exists, the Born cross section is 23.04/ab/sr. : Distortions reduce this to about 0.12/ab/sr, while the experimental value [2] is 26 +- 3 tab/s.r. Presumably this lack o f agreement reflects either the use of bound wave functions or the non resolution o f 1 P3/2 from 1 Pl/2 states or both.) In general the DWBA results o f table 3 yield better agreement with experiment than that o f the previously
22
8 January 1973
We gratefully acknowledge grants for computer work from the Center for Advanced Studies and from the Computer Science Center at the University o f Virginia.
References [1] [2] [3] [4]
J.J. Domingo et al., Phys. Lett. 32B (1970) 309. K. Gabathuler et al., Nucl. Phys. B40 (1972) 32. C.H.Q. Ingram et al., Nucl. Phys. B31 (1971) 331. J. LeTourneux and J.M. Eisenberg, Nucl. Phys. 87 (1966) 331. [5] A. Reitan, Nucl. Phys. B29 (1971) 525; preprint. [6] J. Rohlin et al., Phys. Lett. 40B (1972) 539. [71 W.B. Jones and J.M. Eisenberg, Nucl. Phys. A154 (1970) 49; M.V. BarnhiU, Nucl. Phys. A131 (1969) 106. [ 8] R.J. Glanber, Lectures in theoretical physics, ed. W.E. Brittin and L.G. Dunham (Interscience, New York, 1959). [9] D.S. Koltun, Adv. NucL Phys. 3 (1969) 71. [10] R.H. Bassel and C. Wilkin, Phys. Rev. 174 (1968) 1179 [ 11 ] J.M. Eisenberg and W. Greiner, Excitation mechanisms of the nucleus (North-Holland, Amsterdam, 1970). [12] D.V. Bugg et aL, Phys. Rev. 146 (1966) 980. [13] G. HShler, G. Ebel and J. Giesecke, Z. Physik 180 (1964) 430. [14] ].V. Noble, Phys. Rev. C1 (1970) 1900, and references contained therein.