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On Coulomb effects in N-particle scattering processes
Volume 82B, number 3,4
PHYSICS LETTERS
9 April 1979
ON COULOMB EFFECTS IN N-PARTICLE SCATTERING PROCESSES
Gy. BENCZE Central Research Institute for Physics, 1525 Budapest, Hungary and H. ZANKEL Institut fiir Theoretische Physik Universitiit Graz, A-8010 Graz, Austria Received 13 January 1979
An approximate treatment of Coulomb effects in N-particle scattering is proposed which is the generalisation of the channel distortion approximation developed earlier for nuclear three-body systems.
As is well known if Coulomb interactions are present in a nonrelativistic multiparticle system wave operators do not exist in the usual sense [ 1]. In order to be able to define wave and scattering operators for long range potentials the asymptotic condition has to be modified, i.e. generalised wave operators have to be introduced [2]. This procedure has been first applied by Dollard [3] who developed a complete time dependent theory for N-particle Coulomb scattering. However, despite the promising results obtained recently [4,5], as yet no exact and practical stationary scattering theory exists for Coulomb potentials. The nuclear multiparticle systems are characterised by strong short range nuclear interactions and repulsive Coulomb interactions. In such a situation it is physically reasonable to assume that the dynamics is dominated by the nuclear interactions and the main effect of the Coulomb forces is exhib~ited in the distortion of the relative motion of the fragments in the asymptotic reaction channels. This assumption is in accord with the modified asymptotic condition and is also supported by conventional nuclear reaction studies. It is therefore justified to seek for modifications of the existing time independent formalisms which allow for at least an approximate treatment of the dominant Coulomb effects. In the method developed by Noble [6] for the three-body problem the Coulomb interactions are in316
cluded into the unperturbed Hamiltonian. In this way the pure Coulomb contributions to the various processes can be separated off and the rest can be treated by modified Faddeev equations. For two charged particles this method is shown to be equivalent [7] to the more practical approach developed recently by Alt et al. [8]. In the present note we show that Noble's method can be generalised to N-particle systems in a straightforward way resulting in modified N-particle integral equations. If in these equations the "channel distortion approximation" (CDA) is introduced [9] a practical and systematic approximation scheme is obtained. Our starting point is the two-potential formula for N-particle scattering first derived by Cattapan and Vanzani [10,11 ]. This formula is a consequence of Kato's chain rule for wave operators [12] and holds even for Coulomb interactions and generalized wave operators [11-13]. Let us consider a system of N distinguishable particles with an arbitrary number of them being charged. The hamiltonian of the system is H=Ho+
~ aN-1
VaN_l+ ~
aN-1
UaN_I - H o + V + U , (1)
where VaN_I and UaN_I, respectively, denote the nuclear and Coulomb interaction acting between the pair of particles labelled by the partition aN_ 1, and
Volume 82B, number 3,4
PHYSICS LETTERS
H 0 is the kinetic energy operator. For an arbitrary partition a let us introduce the channel hamiltonian [10-131
G --no + v~ + G ,
(2)
HC~ = H 0 + V~ + U,
(3)
Hao(t) = Hc~t + Aa(t) ,
(4)
where A a ( t ) is Dollard's distortion operator [3]. In terms of the channel hamiltonians one can define the following wave operators ~2c(±) = s-lim exp [if/cat] exp [-iH~0(t)]
(5)
9 April 1979
H c=H a+U cm,
where Ucm denotes a two-body Coulomb interaction acting between the centres of masses which are assumed to carry the total charge of the fragments. With this choice the Coulomb waves (12) can be immediately constructed as a product of bound state wave functions and a two-body Coulomb scattering wave function describing the relative motion of the fragments. Moreover, Tc reduces to a two-body Coulomb T-matrix. On the other hand "polarisation forces" will necessarily appear in the Coulomb distorted transition operator. Thus instead of (10) one has the representation f;8~ = (V3 + Up~ol)(l + G(z) ( V ~ + U~ol)),
~2~ (+-) = s-lira exp [iHt] exp [ - i l l c t] ,
(6)
(13)
(14)
where
t--+± ~
~2(-+) = s-lira exp [iHt] exp [-iHe0(t)] .
(7)
t--+± ~
It follows then by the application of the chain rule [12] that the S-operator can be written in the form
s ~ -- ~ s
c - 2~i a~ (-)~ ~r~ ~c(+),
(s)
U~oI = U ~ - U cm .
(15)
It is important to emphasise that although the twopotential formula (11) is exact, it does not necessarily imply an exact description of the scattering process. Even if X(±) can be constructed, one still has to derive well-behaved equations for the transition operators T3~.
where a and 3 are arbitrary partitions of the N-particle system, S c denotes the Coulomb S-operator S c = f~c(-)? ac(+) ,
(9)
and the Coulomb distorted transition operator TSa can be represented by the expression [11,12] ? ~ = v~ + v ~ a ( z ) V ~ ,
(10)
where G(z) denotes the resolvent of the full hamiltonian (1). Eq. (8) can be rewritten for the on-shell Tmatrix elements in the more familiar form [12,14] q~t c
The basic advantage of Noble's method is that in the transition operator (10) only the nuclear interactions appear explicitly. The Coulomb interactions appear only in the resolvent of the full hamiltonian. Thus if one replaces the free propagator Go(Z ) = (z H 0 ) - I everywhere by the Coulomb propagator G~(z) = (z - H 0 U)-1 and defines connectivity with respect to nuclear interactions only, previous derivations of N-particle equations can be trivially repeated. In particular the modified BR-equations with minimal coupling can be immediately written down [17] 7"~
+ <×~-)I ~ ~ IG+)>,
(11)
= Vg + ~ K~ G~ ?'r~, 7~B
(16)
where T c denotes the pure Coulomb T-operator and the "distorted waves" X(~±) are defined as
where in complete analogy with [17] K~ is defined as the y-connected part (by the nuclear interactions) of the subsystem scattering operator
x(d ) = a c ( ± > G .
T ~ = V~ + VSG.~V 7 .
(12)
A special case of the two-potential formula (8) for two-cluster channels (partitions) has been rederived recently by the screening technique by Alt and Sandhas [15]. In ref. [15] instead o f ( 3 ) the following choice is made
(17)
For N = 3 eqs. (16) trivially reduce to the modified three-body equations of ref. [9], though there the AGS-form of the transition operators has been used. It is clear that the kernel of eq. (16) is fully connected after a single iteration. However, it is rather 317
Volume 82B, number 3,4
PHYSICS LETTERS
difficult to construct. Due to the presence of the Coulomb interactions in the resolvents G(~ and Gg, the construction of the kernel as well as the Coulomb Toperator T c and X(+-) requires the solution of several auxiliary N ' - b o d y problems, where N ' is the number of charged particles in the system. Thus even though eqs. (16) are exact, they cannot be solved unless some approximations are introduced. At this point it should be emphasised that contrary to the claims of refs. [15,16] only the case of two charged particles can be solved exactly at present. Indeed, even if only two-fragment channels are considered, so that ×(+-) can be readily obtained one still has to solve equations for the transition operators (14). Irrespectively of whatever set of equations determine them, i~t3~ can be used to trivially construct the resolvent G(z), which in turn uniquely determines its disconnected part G~(z), the Coulomb resolvent. Thus the exact solution of the problem implies the knowledge of the solution of the pure Coulomb problem. Let us now introduce the channel distortion approximation (CDA) into eqs. (16) in complete analogy with ref. [9]. The CDA is essentially equivalent to replacing the resolvent G(~ in each channel 3' by G~ ~ G~,), = (Z -- H 0 -- u ~ m ) - 1 ,
(lS)
which in turn implies Gg~(z-H
0-
U T - V7 - U ~ m ) -1
(19)
With the approximation (19) the subsystem scattering operators (17) can be constructed by convolution just as in ref. [9]. Actually the approximation (19) decouples the internal motion and the relative motion of the fragments. The CDA form of eqs. (16) can therefore be written as T~
= V~ + Z) KT-~a0~c frT~. -y¢13
(20)
The advantage of eqs. (20) is that they couple only two fragment channels. The CDA will of course also reduce Tac to a two-body Coulomb T-matrix just as X(+-) will also be two-fragment distorted waves just as in ref. [15]. Thus actually CDA is equivalent to neglecting polarisation forces in the treatment of ref. [15]. We have shown that Noble's method can be generalised to N-particles in a straightforward way. The in-
318
9 April 1979
troduction of the CDA greatly simplifies the otherwise intractable set of modified N-particle equations. As a result a practical approximation scheme is obtained. In particular it is easy to show that by a separable expansion of the subsystem T-operators the N-particle equations can be reduced to a set of effective two-body equations in the Coulomb wave basis. The details of the reduction process are exactly analogous to those of ref. [9], so that they will not be given here. In conclusion we want to point out that our aim in the present work was to obtain a practical approximation scheme to two cluster processes in a multiparticle scattering process. The approximate equations (20) are of course no substitute for an exact treatment. In any case we made it also clear that exact solution by the conventional stationary methods is not possible unless the pure Coulomb N-particle problem is solved first. The authors are grateful to M. Haftel for valuable comments. References
[1] W.O. Amrein et al., Scattering theory in quantum mechanics (Benjamin, Reading, MA, 1977). [2] W.O. Amrein et al., Helv. Phys. Acta 43 (1970) 313. [3] J. Dollard, J. Math. Phys. 5 (1964) 729; 14 (1973) 708. [4] C. Chandler and A.G. Gibson, J. Math. Phys. 15 (1974) 1366. [5] J. Zorbas, J. Math. Phys. 17 (1976) 498. [6] J.V. Noble, Phys. Rev. 161 (1967) 945. [7] P.U. Sauer, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 313. [8] E.O. Alt et al., Phys. Rev. C17 (1978) 1981. [9] Gy. Bencze, Nucl. Phys. A196 (1972) 135. [10] G. Cattapan and V. Vanzani, Lett. Nuovo Cim. 14 (1975) 465. [11] Gy. Bencze et al. Lett. Nuovo Cim. 20 (1977) 248. [12] Gy. Bencze, Lett. Nuovo Cim. 17 (1976) 91. [13] J. Zorbas, Rep. Math. Phys. 9 (1976) 309. [14] Z. Bajzer, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 365. [15] E.O. Alt and W. Sandhas, in: Few-body systems and nuclear forces (Springer, 1978) pp. 373-376. [16] E.O. Alt, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 271. [17] Gy. Bencze, Nucl. Phys. A210 (1973) 568; E.F. Redish, Nuel. Phys. A235 (1974) 16.
PHYSICS LETTERS
9 April 1979
ON COULOMB EFFECTS IN N-PARTICLE SCATTERING PROCESSES
Gy. BENCZE Central Research Institute for Physics, 1525 Budapest, Hungary and H. ZANKEL Institut fiir Theoretische Physik Universitiit Graz, A-8010 Graz, Austria Received 13 January 1979
An approximate treatment of Coulomb effects in N-particle scattering is proposed which is the generalisation of the channel distortion approximation developed earlier for nuclear three-body systems.
As is well known if Coulomb interactions are present in a nonrelativistic multiparticle system wave operators do not exist in the usual sense [ 1]. In order to be able to define wave and scattering operators for long range potentials the asymptotic condition has to be modified, i.e. generalised wave operators have to be introduced [2]. This procedure has been first applied by Dollard [3] who developed a complete time dependent theory for N-particle Coulomb scattering. However, despite the promising results obtained recently [4,5], as yet no exact and practical stationary scattering theory exists for Coulomb potentials. The nuclear multiparticle systems are characterised by strong short range nuclear interactions and repulsive Coulomb interactions. In such a situation it is physically reasonable to assume that the dynamics is dominated by the nuclear interactions and the main effect of the Coulomb forces is exhib~ited in the distortion of the relative motion of the fragments in the asymptotic reaction channels. This assumption is in accord with the modified asymptotic condition and is also supported by conventional nuclear reaction studies. It is therefore justified to seek for modifications of the existing time independent formalisms which allow for at least an approximate treatment of the dominant Coulomb effects. In the method developed by Noble [6] for the three-body problem the Coulomb interactions are in316
cluded into the unperturbed Hamiltonian. In this way the pure Coulomb contributions to the various processes can be separated off and the rest can be treated by modified Faddeev equations. For two charged particles this method is shown to be equivalent [7] to the more practical approach developed recently by Alt et al. [8]. In the present note we show that Noble's method can be generalised to N-particle systems in a straightforward way resulting in modified N-particle integral equations. If in these equations the "channel distortion approximation" (CDA) is introduced [9] a practical and systematic approximation scheme is obtained. Our starting point is the two-potential formula for N-particle scattering first derived by Cattapan and Vanzani [10,11 ]. This formula is a consequence of Kato's chain rule for wave operators [12] and holds even for Coulomb interactions and generalized wave operators [11-13]. Let us consider a system of N distinguishable particles with an arbitrary number of them being charged. The hamiltonian of the system is H=Ho+
~ aN-1
VaN_l+ ~
aN-1
UaN_I - H o + V + U , (1)
where VaN_I and UaN_I, respectively, denote the nuclear and Coulomb interaction acting between the pair of particles labelled by the partition aN_ 1, and
Volume 82B, number 3,4
PHYSICS LETTERS
H 0 is the kinetic energy operator. For an arbitrary partition a let us introduce the channel hamiltonian [10-131
G --no + v~ + G ,
(2)
HC~ = H 0 + V~ + U,
(3)
Hao(t) = Hc~t + Aa(t) ,
(4)
where A a ( t ) is Dollard's distortion operator [3]. In terms of the channel hamiltonians one can define the following wave operators ~2c(±) = s-lim exp [if/cat] exp [-iH~0(t)]
(5)
9 April 1979
H c=H a+U cm,
where Ucm denotes a two-body Coulomb interaction acting between the centres of masses which are assumed to carry the total charge of the fragments. With this choice the Coulomb waves (12) can be immediately constructed as a product of bound state wave functions and a two-body Coulomb scattering wave function describing the relative motion of the fragments. Moreover, Tc reduces to a two-body Coulomb T-matrix. On the other hand "polarisation forces" will necessarily appear in the Coulomb distorted transition operator. Thus instead of (10) one has the representation f;8~ = (V3 + Up~ol)(l + G(z) ( V ~ + U~ol)),
~2~ (+-) = s-lira exp [iHt] exp [ - i l l c t] ,
(6)
(13)
(14)
where
t--+± ~
~2(-+) = s-lira exp [iHt] exp [-iHe0(t)] .
(7)
t--+± ~
It follows then by the application of the chain rule [12] that the S-operator can be written in the form
s ~ -- ~ s
c - 2~i a~ (-)~ ~r~ ~c(+),
(s)
U~oI = U ~ - U cm .
(15)
It is important to emphasise that although the twopotential formula (11) is exact, it does not necessarily imply an exact description of the scattering process. Even if X(±) can be constructed, one still has to derive well-behaved equations for the transition operators T3~.
where a and 3 are arbitrary partitions of the N-particle system, S c denotes the Coulomb S-operator S c = f~c(-)? ac(+) ,
(9)
and the Coulomb distorted transition operator TSa can be represented by the expression [11,12] ? ~ = v~ + v ~ a ( z ) V ~ ,
(10)
where G(z) denotes the resolvent of the full hamiltonian (1). Eq. (8) can be rewritten for the on-shell Tmatrix elements in the more familiar form [12,14] q~t c
The basic advantage of Noble's method is that in the transition operator (10) only the nuclear interactions appear explicitly. The Coulomb interactions appear only in the resolvent of the full hamiltonian. Thus if one replaces the free propagator Go(Z ) = (z H 0 ) - I everywhere by the Coulomb propagator G~(z) = (z - H 0 U)-1 and defines connectivity with respect to nuclear interactions only, previous derivations of N-particle equations can be trivially repeated. In particular the modified BR-equations with minimal coupling can be immediately written down [17] 7"~
+ <×~-)I ~ ~ IG+)>,
(11)
= Vg + ~ K~ G~ ?'r~, 7~B
(16)
where T c denotes the pure Coulomb T-operator and the "distorted waves" X(~±) are defined as
where in complete analogy with [17] K~ is defined as the y-connected part (by the nuclear interactions) of the subsystem scattering operator
x(d ) = a c ( ± > G .
T ~ = V~ + VSG.~V 7 .
(12)
A special case of the two-potential formula (8) for two-cluster channels (partitions) has been rederived recently by the screening technique by Alt and Sandhas [15]. In ref. [15] instead o f ( 3 ) the following choice is made
(17)
For N = 3 eqs. (16) trivially reduce to the modified three-body equations of ref. [9], though there the AGS-form of the transition operators has been used. It is clear that the kernel of eq. (16) is fully connected after a single iteration. However, it is rather 317
Volume 82B, number 3,4
PHYSICS LETTERS
difficult to construct. Due to the presence of the Coulomb interactions in the resolvents G(~ and Gg, the construction of the kernel as well as the Coulomb Toperator T c and X(+-) requires the solution of several auxiliary N ' - b o d y problems, where N ' is the number of charged particles in the system. Thus even though eqs. (16) are exact, they cannot be solved unless some approximations are introduced. At this point it should be emphasised that contrary to the claims of refs. [15,16] only the case of two charged particles can be solved exactly at present. Indeed, even if only two-fragment channels are considered, so that ×(+-) can be readily obtained one still has to solve equations for the transition operators (14). Irrespectively of whatever set of equations determine them, i~t3~ can be used to trivially construct the resolvent G(z), which in turn uniquely determines its disconnected part G~(z), the Coulomb resolvent. Thus the exact solution of the problem implies the knowledge of the solution of the pure Coulomb problem. Let us now introduce the channel distortion approximation (CDA) into eqs. (16) in complete analogy with ref. [9]. The CDA is essentially equivalent to replacing the resolvent G(~ in each channel 3' by G~ ~ G~,), = (Z -- H 0 -- u ~ m ) - 1 ,
(lS)
which in turn implies Gg~(z-H
0-
U T - V7 - U ~ m ) -1
(19)
With the approximation (19) the subsystem scattering operators (17) can be constructed by convolution just as in ref. [9]. Actually the approximation (19) decouples the internal motion and the relative motion of the fragments. The CDA form of eqs. (16) can therefore be written as T~
= V~ + Z) KT-~a0~c frT~. -y¢13
(20)
The advantage of eqs. (20) is that they couple only two fragment channels. The CDA will of course also reduce Tac to a two-body Coulomb T-matrix just as X(+-) will also be two-fragment distorted waves just as in ref. [15]. Thus actually CDA is equivalent to neglecting polarisation forces in the treatment of ref. [15]. We have shown that Noble's method can be generalised to N-particles in a straightforward way. The in-
318
9 April 1979
troduction of the CDA greatly simplifies the otherwise intractable set of modified N-particle equations. As a result a practical approximation scheme is obtained. In particular it is easy to show that by a separable expansion of the subsystem T-operators the N-particle equations can be reduced to a set of effective two-body equations in the Coulomb wave basis. The details of the reduction process are exactly analogous to those of ref. [9], so that they will not be given here. In conclusion we want to point out that our aim in the present work was to obtain a practical approximation scheme to two cluster processes in a multiparticle scattering process. The approximate equations (20) are of course no substitute for an exact treatment. In any case we made it also clear that exact solution by the conventional stationary methods is not possible unless the pure Coulomb N-particle problem is solved first. The authors are grateful to M. Haftel for valuable comments. References
[1] W.O. Amrein et al., Scattering theory in quantum mechanics (Benjamin, Reading, MA, 1977). [2] W.O. Amrein et al., Helv. Phys. Acta 43 (1970) 313. [3] J. Dollard, J. Math. Phys. 5 (1964) 729; 14 (1973) 708. [4] C. Chandler and A.G. Gibson, J. Math. Phys. 15 (1974) 1366. [5] J. Zorbas, J. Math. Phys. 17 (1976) 498. [6] J.V. Noble, Phys. Rev. 161 (1967) 945. [7] P.U. Sauer, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 313. [8] E.O. Alt et al., Phys. Rev. C17 (1978) 1981. [9] Gy. Bencze, Nucl. Phys. A196 (1972) 135. [10] G. Cattapan and V. Vanzani, Lett. Nuovo Cim. 14 (1975) 465. [11] Gy. Bencze et al. Lett. Nuovo Cim. 20 (1977) 248. [12] Gy. Bencze, Lett. Nuovo Cim. 17 (1976) 91. [13] J. Zorbas, Rep. Math. Phys. 9 (1976) 309. [14] Z. Bajzer, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 365. [15] E.O. Alt and W. Sandhas, in: Few-body systems and nuclear forces (Springer, 1978) pp. 373-376. [16] E.O. Alt, in: Few-body nuclear physics (IAEA, Vienna, 1978) p. 271. [17] Gy. Bencze, Nucl. Phys. A210 (1973) 568; E.F. Redish, Nuel. Phys. A235 (1974) 16.