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Inelastic sum rule for high-energy hadron scattering and nucleon-nucleon correlations
Volume 43B, number 5
INELASTIC
PHYSICS LETTERS
5 March 1973
SUM RULE FOR HIGH-ENERGY HADRON SCATTERING AND NUCLEON-NUCLEON CORRELATIONS E. BLESZYlqSKA
Uniwersytet Jagiellohski, Instytut FizyM, Krak6w, Poland M. BLESZYIqSKI, A. MALECKI
Instytut Fizyki Jadrowej, Krak6w 23, Poland and P. PICCHI
Laboratori Nazionali del CNEN, Frascati, Italy Received 28 January 1973 We have considered the influence of short-range dynamical and statistical correlations on the elastic and summed inelastic scattering of high-energy hadrons from light nuclei. Our calculations are based on the complete multiple scattering series of Glauber. The correlation effects are discussed on the example of the inelastic sum rule for p-160 scattering.
The use of sum rules for inelastic scattering has the attractive feature that the information obtained pertains only to the ground state while the final states of the target need not be identified. The subject has been widely discussed for the electron-nucleus scattering [ 1], the sum rule being in this case intimately related to the nucleon pair correlation function of the target. It is very little known, however, what information concerning nuclear structure might be deduced from the study of sum rules with strongly interacting probes. Challenged by this problem, we have investigated the role of nucleon-nucleon correlations in the inelastic scattering of high-energy hadrons from nuclei. This letter contains the first results of our work. We consider the total inelastic cross-section defined by subtracting from the sum of all purely nuclear scatterings (denoted desum/d~2 and called the summed cross-section) the elastic contribution: dOinel/d~2 = dOsum/d~2 - dOel/d~2
(1)
Let us assume a small angle scattering of hadrons at high energies. Since the m o m e n t u m transfer is then nearly perpendicular to the incident m o m e n t u m (qLp) it is convenient to use the impact parameter representation of the transition amplitude. For elastic scattering we write: deel/dn = [(ip/21r)fd2b exp ( ~ . b)<¢lF(b)l~) 2®CM
(2)
where ~ is the ground state wave function and P(b) is called nuclear profile [2], OCM being a correction due to the fact that in general ~ depends on the coordinates of all A nucleons rather than on the intrinsic coordinates. We will discuss the sum rule at fixed m o m e n t u m transfer q. Constant q makes possible the use of closure and the elimination of final nuclear states. The summed cross-section may be expressed as follows: dOsum/d~2 = (p/27r)2fd2b 1 d2b2 exp { iq "(b 1 - b2) } ( ~[F+(bl ) l-'(b2)l ~ )"
(3)
The nuclear profile F(b) depends on the positions of the nucleons S~ in the plane perpendicular to p. Following Glauber, we assume [2]:
355
Volume 43B, number 5
PHYSICS LETTERS
A F(b, s 1 .... SA) = 1 -- I-I [1 /=1
5 March 1973
7/(b-s])]
(4)
where the profiles of the nucleons 3'] are to be expressed through hadron-nucleon elastic scattering amplitudes. Let us now turn to the nuclear wave function. For the ground state density we have used the pair correlation model [3]:
1412 = l--I o(9) 1 + ~ (2tl!) -1 /=1
1=1
~
Jl :/:kl ~...~Jl@kl
[A(/1, kl)
... A(jl, kl) ] ,
A(j, k) = G2(r/., rk) -- 1
(5)
which is capable of accounting for the short-range interactions between particles. The two inputs of the model are the single particle density p(r) and the correlation operator G(rj, rk) that serves to determine the type of the interaction. The successive terms of (5) correspond to the expansion in numbers of correlated pairs: independent particles, one correlated pair, etc. Using (4) and (5) one obtains for the expectation values of the nuclear profiles: (q:lr(b)14,> =
A/2 1- ~ 1=1
A! SA_21(b)Tl(b ) 2ll!(A- 21)! A/2
( ~lF+(bi ) F(b2)l ~) = 1 - ~ t=0 where the functions
S(b) = fd3rp(r) T(b) = f d 3 r l
A! 2tI!(A - 2/)!
(6a)
[sA-21(bl ) T/(bl ) + s*A-21(b 2) T*/(b2 ) - U A-2l(b l, b2) wl(bl , b2)] (6b)
S, T, U, W are defined as follows:
[1 - 3'(b- s)]
d3r2P(rl)P(r2 ) A(1, 2)[I - 3'(b-Sl) ] [1 - 3,(b- Sz)] (7)
U(b 1, b2) = fd3rp(r)[l - 3'(b 1 - S ) ] [1 - 3'*(b 2 -,7)] W(bl, b2) = fd3r t d3r2P(r 1) P(r2) A(1, 2) [ 1 -3'(b 1- S l ) ] [1 - 7(b 1- $2) ] [1 - 7*(b 2 - Sl) ] [ 1 - 3'*(b 2 - s2) ] We have performed the calculations for the p-shell nuclei. Let us discuss in detail the case of 160. The singleparticle density was assumed in the harmonic oscillator form:
p(r) = p(0)[1 + 2r2/R 2] exp (-r2/R2),
(8)
R being the oscillator size parameter. The c.m. correction is then OCM = exp (q2R2/32). As for the correlation operator, we considered the following cases: (a) independent particle model (IPM) : G 2 - 1, (b) dynamical correlations cdrresponding to a repulsive interaction of short range: G2yn
=g2/N '
g2 = 1 - e x p
{-X(rl-r2)2/R2},
(9)
the coefficient N being determined by the normalization, (c) statistical correlations: G stat 2 = [O(rl) P(r2)] -1 [ ~ P(rl ) P(r2 ) _ i~sp2(0)(1 + 2r 1 .r2/R 2)2 exp {-(r~ 356
+r~)/R 2 }]
(10)
Volume 43B, number 5
PHYSICS LETTERS
i
INELASTICHADRONSC,~TTERINGSUM'RULE
3ooL
5 March 1973
~INELASnCH~,DRONSCATIE ' RrNOSUMRULtE
STATISTICAL+ DYNAMICAL CORRELATIONS
i~0
'60 •
~
DYNAhlCAL CORRELATIONS
. ~ ' .~. / ~../. / .. . / ~ ,,..~
300 . '..... DYNAMICAL
CO,RELAT,ONSI --..: /
/
/
,•
/
/
/'
// ."
"".x"
~t% "'..... -'~,~.
[ "-.~"'- ..... " l
\ \This calculation
" ~ , . \ "--k. /""-../ ............. ""-~ "'-.. ....... ......... .....
"-. .........
2oo
../
...... ..... _
...... "¢'",,I / /..."/./ ..'CO CO,ELA, ,ELA, SS '...
/
200.
I
[
t
.tT~s\
t:
i..~.,o.'..~
I .1
;r
"
100
I/
I ......
INDEPENDENT PARTICLES
.
"'-..
......
, ~
~..
\
....... ~. "'".. ~
,
,
200
300
4
"4
J I
'..;...
.... -
~'J "1
100
~)0
i
200
3(X)
i
q(MeV)
400 q(MeV)
400
Fig. 1. Inelastic cross-section for p r o t o n ) 6 0 scattering in the function of momentum transfer. The various forms of nuclear density and their parameters are discussed in the text.
Fig. 2. Inelastic proton-160 scattering sum rule computed on the basis of complete (this calculation) and
approximate (ref. [6]) multiple scattering series. The results are compared for the two types of nuclear density: independent particles and dynamical correlations.
The term in brackets is the two-body density calculated with the Slater determinant of the oscillator orbitals. The pair correlation description given by (5) and (10) was compared with the exact treatment (by means of the Slater determinant) of statistical correlations [4] ; the differences are very small (2%). (d) dynamical and statistical correlations: G2 = c : 2
~stat
.G 2
dyn "
(11)
This assumption corresponds to the independent pair treatment of short range correlations. The nuclear parameters in our computations were: R = 1.71 fro, 2~= 3.4 which gives the correlation length - 0 . 8 2 fm. The nucleon profiles were assumed equal each other as ~(b) = (1 -ic0(41ra) -10NN exp (-b2/2a)
(12)
which corresponds to a Gaussian dependence of the nucleon-nucleon scattering amplitude at small momentum transfer. We used the parameters: a = -0.33, a = 5 GeV -2, aNN = 44 mb, reasonable for protons of ~ 1 GeV energY. Under assumptions (8)-(12) all the functions defined in (7) may be'calculated analytically. The remaining integrations (single for elastic, triple for summed cross-section) were performed on the computer. We would like to stress that our calculations are exact in the sense that, starting from eqs. (2), (3), (4), no approximations concerning the transition operator l-'(b, s 1, ... SA) were made. We did not profit from any exponentiation or truncation in the multiple scattering series, as is sometimes practised [5, 6]. On the contrary, in performing the integrations we included all powers of the elementary profiles. Thus in our calculations the only approximations concern the form of the nuclear density - eqs. (5, 8-11). From fig. 1, where our results are presented, one can see that the correlation effects strongly depend on the momentum transfer q. For small q ( < 80 MeV) the statistical and dynamical correlations add constructively, lowering the inelastic cross-section predicted by IPM by 40%. At medium momentum transfers (80 < q < 250 MeV) the two types of correlations act in opposite directionst'. The repulsive character of the dynamical correlations causes the cross-section to rise. The statistical correlations, though also repulsive at short distances, contain an im"fA similar counteraction of the statistical and dynamical correlations was found in ref. [7], where the sum rule for Coulomb scattering of electrons is discussed.
357
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
portant non-central part which results in the reduction of the cross-section. The net effect is the lowering of the IPM curve by 20-30%. For large momentum transfer both the correlations again add constructively, raising the cross-section. In fig. 2 we compare our results, based on the complete multiple scattering series with the curves computed on the basis of ref. [6]. This comparison constitutes a rather negative test of the various approximations applied in [6]. Probably these approximations work better for large nuclei, though such a large difference for 160 allows us to question their validity. In fact, for elastic hadron scattering the so-called optical limit (large A) works pretty well for this nucleus [8]. In conclusion, let us say a few words about the experimental feasibility of the discussed sum rule. For large momentum transfer (q > 250 MeV) where the elastic contribution is small, the inelastic cross-section practically coincides with the summed cross-section. Note that for large enough projectile energy the momentum transfer is almost insensitive to the excitation energy of the target. In consequence the constant q sum rule may be measured in the experiment with poor energy resolution ( A E -~ 100 MeV) at fixed scattering angle. Such measurements have already been performed for a number of nuclei [9]. For small momentum transfers the situation is much more difficult. The inelastic sum rule appears here as a small difference of the two cross-sections, which are much greater by one or even two orders of magnitude than itself. This poses the requirement of a very high precision in the measurements. Since for small q the variation of momentum transfer with energy loss is quite important, the sum rule is no longer accessible as a poor energy resolution cross-section; the scattered hadron spectrum has first to be resolved and then integrated. We are grateful to Professor W. Czy~t for his encouragement and valuable discussions.
References [1] K.W. McVoy and L. Van Hove, Phys. Rev. 125 (1962) 1034; W. Czy~t,L. Le~niak and A. Matecki, Ann. of Phys. 42 (1967) 119. [2]' R.J. Glauber, High-energy physics and nuclear structure, ed. S. Devons (Plenum Press, New York 1970) p. 207. [3] A. Matecki and P. Picchi, Riv. Nuovo Cim. 2 (1970) 119. [4] W. Czy$, L. Legniak and H. Wotek, Nucl. Phys. B19 (1970) 125;B52 (1971) 638. [5] R.J. Glauber and G. Matthiae, Nucl. Phys. B21 (1970) 135. [6] G. Von Bochman, B. Margolis and C.L. Tang, Phys. Lett. 30B (1969) 254. [7] A. Malecki and P. Picchi, Institute of Nuclear Physics, Cracow, INP No. 811/PH/PL (1972). [8] W. Czy~ and L.C. Maximon, Ann. of Phys. 52 (1969) 59. [9] G. Bellettini et al., Nucl. Phys. 79 (1966) 609.
358
INELASTIC
PHYSICS LETTERS
5 March 1973
SUM RULE FOR HIGH-ENERGY HADRON SCATTERING AND NUCLEON-NUCLEON CORRELATIONS E. BLESZYlqSKA
Uniwersytet Jagiellohski, Instytut FizyM, Krak6w, Poland M. BLESZYIqSKI, A. MALECKI
Instytut Fizyki Jadrowej, Krak6w 23, Poland and P. PICCHI
Laboratori Nazionali del CNEN, Frascati, Italy Received 28 January 1973 We have considered the influence of short-range dynamical and statistical correlations on the elastic and summed inelastic scattering of high-energy hadrons from light nuclei. Our calculations are based on the complete multiple scattering series of Glauber. The correlation effects are discussed on the example of the inelastic sum rule for p-160 scattering.
The use of sum rules for inelastic scattering has the attractive feature that the information obtained pertains only to the ground state while the final states of the target need not be identified. The subject has been widely discussed for the electron-nucleus scattering [ 1], the sum rule being in this case intimately related to the nucleon pair correlation function of the target. It is very little known, however, what information concerning nuclear structure might be deduced from the study of sum rules with strongly interacting probes. Challenged by this problem, we have investigated the role of nucleon-nucleon correlations in the inelastic scattering of high-energy hadrons from nuclei. This letter contains the first results of our work. We consider the total inelastic cross-section defined by subtracting from the sum of all purely nuclear scatterings (denoted desum/d~2 and called the summed cross-section) the elastic contribution: dOinel/d~2 = dOsum/d~2 - dOel/d~2
(1)
Let us assume a small angle scattering of hadrons at high energies. Since the m o m e n t u m transfer is then nearly perpendicular to the incident m o m e n t u m (qLp) it is convenient to use the impact parameter representation of the transition amplitude. For elastic scattering we write: deel/dn = [(ip/21r)fd2b exp ( ~ . b)<¢lF(b)l~) 2®CM
(2)
where ~ is the ground state wave function and P(b) is called nuclear profile [2], OCM being a correction due to the fact that in general ~ depends on the coordinates of all A nucleons rather than on the intrinsic coordinates. We will discuss the sum rule at fixed m o m e n t u m transfer q. Constant q makes possible the use of closure and the elimination of final nuclear states. The summed cross-section may be expressed as follows: dOsum/d~2 = (p/27r)2fd2b 1 d2b2 exp { iq "(b 1 - b2) } ( ~[F+(bl ) l-'(b2)l ~ )"
(3)
The nuclear profile F(b) depends on the positions of the nucleons S~ in the plane perpendicular to p. Following Glauber, we assume [2]:
355
Volume 43B, number 5
PHYSICS LETTERS
A F(b, s 1 .... SA) = 1 -- I-I [1 /=1
5 March 1973
7/(b-s])]
(4)
where the profiles of the nucleons 3'] are to be expressed through hadron-nucleon elastic scattering amplitudes. Let us now turn to the nuclear wave function. For the ground state density we have used the pair correlation model [3]:
1412 = l--I o(9) 1 + ~ (2tl!) -1 /=1
1=1
~
Jl :/:kl ~...~Jl@kl
[A(/1, kl)
... A(jl, kl) ] ,
A(j, k) = G2(r/., rk) -- 1
(5)
which is capable of accounting for the short-range interactions between particles. The two inputs of the model are the single particle density p(r) and the correlation operator G(rj, rk) that serves to determine the type of the interaction. The successive terms of (5) correspond to the expansion in numbers of correlated pairs: independent particles, one correlated pair, etc. Using (4) and (5) one obtains for the expectation values of the nuclear profiles: (q:lr(b)14,> =
A/2 1- ~ 1=1
A! SA_21(b)Tl(b ) 2ll!(A- 21)! A/2
( ~lF+(bi ) F(b2)l ~) = 1 - ~ t=0 where the functions
S(b) = fd3rp(r) T(b) = f d 3 r l
A! 2tI!(A - 2/)!
(6a)
[sA-21(bl ) T/(bl ) + s*A-21(b 2) T*/(b2 ) - U A-2l(b l, b2) wl(bl , b2)] (6b)
S, T, U, W are defined as follows:
[1 - 3'(b- s)]
d3r2P(rl)P(r2 ) A(1, 2)[I - 3'(b-Sl) ] [1 - 3,(b- Sz)] (7)
U(b 1, b2) = fd3rp(r)[l - 3'(b 1 - S ) ] [1 - 3'*(b 2 -,7)] W(bl, b2) = fd3r t d3r2P(r 1) P(r2) A(1, 2) [ 1 -3'(b 1- S l ) ] [1 - 7(b 1- $2) ] [1 - 7*(b 2 - Sl) ] [ 1 - 3'*(b 2 - s2) ] We have performed the calculations for the p-shell nuclei. Let us discuss in detail the case of 160. The singleparticle density was assumed in the harmonic oscillator form:
p(r) = p(0)[1 + 2r2/R 2] exp (-r2/R2),
(8)
R being the oscillator size parameter. The c.m. correction is then OCM = exp (q2R2/32). As for the correlation operator, we considered the following cases: (a) independent particle model (IPM) : G 2 - 1, (b) dynamical correlations cdrresponding to a repulsive interaction of short range: G2yn
=g2/N '
g2 = 1 - e x p
{-X(rl-r2)2/R2},
(9)
the coefficient N being determined by the normalization, (c) statistical correlations: G stat 2 = [O(rl) P(r2)] -1 [ ~ P(rl ) P(r2 ) _ i~sp2(0)(1 + 2r 1 .r2/R 2)2 exp {-(r~ 356
+r~)/R 2 }]
(10)
Volume 43B, number 5
PHYSICS LETTERS
i
INELASTICHADRONSC,~TTERINGSUM'RULE
3ooL
5 March 1973
~INELASnCH~,DRONSCATIE ' RrNOSUMRULtE
STATISTICAL+ DYNAMICAL CORRELATIONS
i~0
'60 •
~
DYNAhlCAL CORRELATIONS
. ~ ' .~. / ~../. / .. . / ~ ,,..~
300 . '..... DYNAMICAL
CO,RELAT,ONSI --..: /
/
/
,•
/
/
/'
// ."
"".x"
~t% "'..... -'~,~.
[ "-.~"'- ..... " l
\ \This calculation
" ~ , . \ "--k. /""-../ ............. ""-~ "'-.. ....... ......... .....
"-. .........
2oo
../
...... ..... _
...... "¢'",,I / /..."/./ ..'CO CO,ELA, ,ELA, SS '...
/
200.
I
[
t
.tT~s\
t:
i..~.,o.'..~
I .1
;r
"
100
I/
I ......
INDEPENDENT PARTICLES
.
"'-..
......
, ~
~..
\
....... ~. "'".. ~
,
,
200
300
4
"4
J I
'..;...
.... -
~'J "1
100
~)0
i
200
3(X)
i
q(MeV)
400 q(MeV)
400
Fig. 1. Inelastic cross-section for p r o t o n ) 6 0 scattering in the function of momentum transfer. The various forms of nuclear density and their parameters are discussed in the text.
Fig. 2. Inelastic proton-160 scattering sum rule computed on the basis of complete (this calculation) and
approximate (ref. [6]) multiple scattering series. The results are compared for the two types of nuclear density: independent particles and dynamical correlations.
The term in brackets is the two-body density calculated with the Slater determinant of the oscillator orbitals. The pair correlation description given by (5) and (10) was compared with the exact treatment (by means of the Slater determinant) of statistical correlations [4] ; the differences are very small (2%). (d) dynamical and statistical correlations: G2 = c : 2
~stat
.G 2
dyn "
(11)
This assumption corresponds to the independent pair treatment of short range correlations. The nuclear parameters in our computations were: R = 1.71 fro, 2~= 3.4 which gives the correlation length - 0 . 8 2 fm. The nucleon profiles were assumed equal each other as ~(b) = (1 -ic0(41ra) -10NN exp (-b2/2a)
(12)
which corresponds to a Gaussian dependence of the nucleon-nucleon scattering amplitude at small momentum transfer. We used the parameters: a = -0.33, a = 5 GeV -2, aNN = 44 mb, reasonable for protons of ~ 1 GeV energY. Under assumptions (8)-(12) all the functions defined in (7) may be'calculated analytically. The remaining integrations (single for elastic, triple for summed cross-section) were performed on the computer. We would like to stress that our calculations are exact in the sense that, starting from eqs. (2), (3), (4), no approximations concerning the transition operator l-'(b, s 1, ... SA) were made. We did not profit from any exponentiation or truncation in the multiple scattering series, as is sometimes practised [5, 6]. On the contrary, in performing the integrations we included all powers of the elementary profiles. Thus in our calculations the only approximations concern the form of the nuclear density - eqs. (5, 8-11). From fig. 1, where our results are presented, one can see that the correlation effects strongly depend on the momentum transfer q. For small q ( < 80 MeV) the statistical and dynamical correlations add constructively, lowering the inelastic cross-section predicted by IPM by 40%. At medium momentum transfers (80 < q < 250 MeV) the two types of correlations act in opposite directionst'. The repulsive character of the dynamical correlations causes the cross-section to rise. The statistical correlations, though also repulsive at short distances, contain an im"fA similar counteraction of the statistical and dynamical correlations was found in ref. [7], where the sum rule for Coulomb scattering of electrons is discussed.
357
Volume 43B, number 5
PHYSICS LETTERS
5 March 1973
portant non-central part which results in the reduction of the cross-section. The net effect is the lowering of the IPM curve by 20-30%. For large momentum transfer both the correlations again add constructively, raising the cross-section. In fig. 2 we compare our results, based on the complete multiple scattering series with the curves computed on the basis of ref. [6]. This comparison constitutes a rather negative test of the various approximations applied in [6]. Probably these approximations work better for large nuclei, though such a large difference for 160 allows us to question their validity. In fact, for elastic hadron scattering the so-called optical limit (large A) works pretty well for this nucleus [8]. In conclusion, let us say a few words about the experimental feasibility of the discussed sum rule. For large momentum transfer (q > 250 MeV) where the elastic contribution is small, the inelastic cross-section practically coincides with the summed cross-section. Note that for large enough projectile energy the momentum transfer is almost insensitive to the excitation energy of the target. In consequence the constant q sum rule may be measured in the experiment with poor energy resolution ( A E -~ 100 MeV) at fixed scattering angle. Such measurements have already been performed for a number of nuclei [9]. For small momentum transfers the situation is much more difficult. The inelastic sum rule appears here as a small difference of the two cross-sections, which are much greater by one or even two orders of magnitude than itself. This poses the requirement of a very high precision in the measurements. Since for small q the variation of momentum transfer with energy loss is quite important, the sum rule is no longer accessible as a poor energy resolution cross-section; the scattered hadron spectrum has first to be resolved and then integrated. We are grateful to Professor W. Czy~t for his encouragement and valuable discussions.
References [1] K.W. McVoy and L. Van Hove, Phys. Rev. 125 (1962) 1034; W. Czy~t,L. Le~niak and A. Matecki, Ann. of Phys. 42 (1967) 119. [2]' R.J. Glauber, High-energy physics and nuclear structure, ed. S. Devons (Plenum Press, New York 1970) p. 207. [3] A. Matecki and P. Picchi, Riv. Nuovo Cim. 2 (1970) 119. [4] W. Czy$, L. Legniak and H. Wotek, Nucl. Phys. B19 (1970) 125;B52 (1971) 638. [5] R.J. Glauber and G. Matthiae, Nucl. Phys. B21 (1970) 135. [6] G. Von Bochman, B. Margolis and C.L. Tang, Phys. Lett. 30B (1969) 254. [7] A. Malecki and P. Picchi, Institute of Nuclear Physics, Cracow, INP No. 811/PH/PL (1972). [8] W. Czy~ and L.C. Maximon, Ann. of Phys. 52 (1969) 59. [9] G. Bellettini et al., Nucl. Phys. 79 (1966) 609.
358