- Email: [email protected]
The transfer angular momentum scheme in nuclear reactions
&*H__ we
Nuclear Instruments and Methodsin Physics ResearchB 99 (1995) 301-304
Bum Interactions with Materials & Atoms
!!!Z EISEVIER
The transfer angular momentum scheme in nuclear reactions * D. Robson
*
Physics Department, Florida State Uniuersity, Tallahassee, FL 32306-3016, USA
Abstract The use of transferred angular momentum (TROISAM) quantum numbers to describe nuclear reactions is discussed without assuming a particular reaction mechanism. The specific reaction ‘*@Li, o)14N* with polarized lithium is discussed as an example of the formalism.
of the initial and final channel spins denoted by s and s’ respectively.
1. Introduction In the literature most nuclear reactions at energies significantly above the relevant Coulomb barriers are analyzed in terms of direct or few-step reaction theories. In many cases the concept of angular momentum transfer between initial and final states is used to characterize the reaction amplitudes, e.g., in stripping reactions like A + d + (A + n) * f p the angular distributions are characterized by the “neutron transfer” quantum numbers I and j = (1+ i) for the configuration (A + r-r)*. Usually the approach uses models and approximations and it is advantageous to know if the transferred quantum numbers can be derived in an exact formulation of the reaction theory. The fact is that the transferred angular momentum (TRANSAM) scheme is just as general as the more conventional general partial wave schemes typified by the channel spin and total angular momentum representation. The advantage of the transfer representation lies not only in the fact that there is a finite number of amplitudes allowed by conservation of angular momentum and parity, but also in their properties under coordinate rotations. In particular each amplitude is characterized by a quantum number L and its (2L + 1) projections M which transform under rotations as an irreducible tensor of rank L. Physically the L quantum number is the angular momentum difference between the final orbital angular momentum 1’ and the initial orbital angular momentum 1, i.e., L = f’ - 1. Conservation of angular momentum in a reaction of the form A + a + B + b leads to the requirement that L = s - s’, so that L is constrained by the allowed finite values
2. Formalism The derivation of explicit formulas can only be outlined here. The derivation given here begins from the T-matrix in the uncoupled representation: (k’, sb/.+,s,~nITlk, = 2aikK’
s~cLA/+,)
c (c’ 1tl c)Y,;(i)Y,,(&‘), ll’mm’
(1)
in which c, c’ are collective channel labels for the uncoupled basis states 1s, pasA pAlm), I sbpbsB p,l’m’) respectively and %, p are the unit wavevectors for the directions of the initial and final relative motions in the center of momentum frame. The spins s, and their projections pi are self explanatory. Following the usual formalism for a single system density matrix written as a series of spin tensors of rank sir [l] where: (%/$I =
PIsiP?)
C
(“isfPiJ
-Pf
I Sif~if)(-)Sf-fLfPS,Sf~i~~,~t
C2>
sir Pif
we generalize to a system with two subsystems and write our T-matrix as: (k’, st,w,pBITlk
=
s,I~,sAP~)
c (SaSbCL,, -Pblw,b)(-_)Sb-Pb %b Cd X(wl3CLAt -/+I sABl*&( -p-ILB X(s,bsABCL.b~~ILM)T~~~B(k’, k).
* Work supported in part by the US DOE Grant DE-FGOS86ER-40273. * Tel. + 1 904 644 [email protected].
1767,
fax
+ 1 904 644 8630,
e-mail:
0168-583X/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(94)00639-3
(3)
this expansion the coupled spin tensors TlbyA,(k’, k) are irreducible tensors of rank L under rotations. Inverting this
In
V. NUCLEAR
PHYSICS
302
D. Robson /Nucl. Instr. and Meth. in Phys. Res. B 99 (1995) 301-304
expansion and expressing (c’ ) t ) c) in Eq. (1) as a series of coupled t-matrix elements in the channel spin and total angular momentum representation (S = s, + s,, J = s + 1, etc.) yields after some tedious algebra the result:
=2nK’
~tjf~SABL(il’mm’
It is also easy to relate the tensors in the transverse frame to the Madison by rotating through the Euler angles (Y= -7r/2, /3= -rr/2and y=O, i.e.,
ILM)Y,,(i)Y,c,(P),
(7)
II’ (4)
in which the dynamics are contained in the tiLrABLgiven by
%
t[r %lJ,& = c (25 + l)W(sls’l’; SS’J
’
[
sb
%b
sA
‘B
SAB
s
s’
L
X(s’l’J
JL)
1
ws+L-J-r
II t II SW),
TLM
(5)
with the W being a Racah coefficient and the square array being the standard LS - jj recoupling coefficient. Bqs. (4) and (5) provide the connection between the and the conventional partial wave series. It tensors TSfbyAB is not always appropriate to use this connection because of the sheer number of partial waves which occur and we included the relationship here simply to show the generality of the transfer representation. Other models can be used to calculate TkyAB such as approximate amplitudes from DWBA, CCBA or other appropriate direct reaction theories.
3. Frame dependence As might be anticipated the tensors TX”,”are not all independent of each other. The particular relationships which occur are frame dependent. Here we indicate special results for two commonly used coordinate frames, i.e., the Madison convention and the “transverse frame”. In the Madison frame the z-axis is parallel to fi and the y-axis is parallel to k X k. In contrast the transverse frame has the z-axis parallel to i X I? and the x-axis parallel to fi. Defining the intrinsic parities of each subsystem by ni then the product ~~n~rrb~n represents the parity transfer rrr which by conservation of parity is also equal to (_)[+I’_ Using the explicit Eq. (4) above for TX”,”and the properties of vector addition coefficients yields the relations: TLM = q( -XY
-)‘+“Tx:-M
(Madison),
where we introduce the labels T and M to denote the particular frame for each tensor. Of course the number of independent amplitudes is frame independent and a particular choice is either a matter of convention or convenience. The special case of elastic scattering of spin-l particles by spin zero targets has been considered [2] using a spin tensor expansion and the results above in Eq. (6a) reduced to (with x = sab =L, y=ss,=O), =
(_
)L+“~TL,-M
(8)
for ,.any z-axis in the plane containing k and k’ and y 11k X k. In the particular case where the z-axis is chosen along the direction of k + k’ the additional symmetry from time reversal yields zero values for TLM when L f M = odd. This frame choice appears to be useful only for elastic scattering and is denoted by a bar over the tensor.
4. Special cases The special cases which are of interest are those where at least two of the particles a, A, b, B have spins which are zero. In these cases there are strong limitations on the transferred angular momentum quantum numbers sab? sAB, and L. If only one spin, say si, is non-zero, then one of sab and sA8 is zero and the other is equal to si. Also in this case L = si only. For all of these four special cases the T-matrix in Eq. (3) reduces to: (k’, SbP&rCLnITlk =TSipi(k’,
S~CLAPA)
k), if si=s,
= ( _)SI-PiTS,, -‘+(k’,
or s,, k), if si = sb or sB.
(9)
with the VdUeS of s,b, Sm being fixed as indicated above. The special cases when only two spins, say si, sj are non-zero lead to six possibilities all having the constraints that si and sj must both be integers or both be half-integers. Moreover the allowed values of L are given by the vector addition rules, i.e., L = I si - sj I, ( si - sj ( + 1 . . . si + sj, so that the T-matrix in Eq. (3) reduces to a single sum over L of the form:
(ha) L
and
=‘i+?TLM(,f,
k),
x(-)
T*“,”= 0 for ( - ) M + rrr
(transverse).
(6b)
(10)
303
D. Robson / Nucl. Ins&. and Meth. in Phys. Res. B 99 (1995) 301-304 wherein yi, -yj are equal to + 1 or - 1 corresponding to i, j being initial particles or final particles respectively. Similarly oi = 0 or si - pi, ‘Ye= 0 or sj - kj for i, j being initial or final particles.
5. Example As an example of the TRANSAM representation we consider the special reaction “C(%, a)14N * with a polarized 6Li beam. Since two of the spins and parities are Of the reaction matrix is reduced to a special case of Eq. (10). If we suppress the labels sA, s&= 0) we have with s,= 1, (k’, .QrP*ITlk,
involved in the TRANSAM scheme. However, the analyzing powers characterized by Tkq = E&/E,& with k = 1 (vector) and k = 2 (tensor) will in general have interference effects between the different values of the transfer quantum numbers. For the example here when sn = O- one obtains the five entities in the Madison frame in terms of AM = MT’“(k’, k): iT,, = GAY = - fi2 T,,=&[~A’I”T,, = - 62
IA’f]/N, Re( A’( A’)* )/N,
Tz2= -&21A’12/N,
IPa>
N=2jA’12+
x T$;a-“b(
k’, k).
(11)
For a polarized beam and no spin sensitive detectors observables are given in terms of efficiency tensors which are defined by:
the Ed;
where the density matrix for this situation is given by: C(k’, 6%
Sn~aITlk,
l/Q*
X(k’,
sBpB17’lk.
I/L,).
(13)
Substituting Eq. (11) into Eqs. (12) and (13) and performing the magnetic summations yields: $++ = Lg
( _)+‘+sB-L’(2L
x (2k + 1)“2w(L1L’1; X(LkM,
+ I)‘/2
Snk)
-sIL’M-q)T~~~(T~*~-q)*,
jA”j2.
(16)
which obeys the general quadratic relationship reactions: (i”T,,)’
+ i”T;o + “T& +“T&
= 1
[3] for such
(17)
and the relations (12)
P&,=
Im( A’( A’)* )/N,
(14)
with the k = 0 = q tensor being: (15) This special tensor is proportional to the differential cross section for an unpolarized beam. It is worth noting that in the most general case (none of the four particles having zero spin) the differential cross section for unpolarized beam and target and spin insensitive detectors involves an incoherent sum over all the transfer quantum numbers sat.,, s,, L and M. This follows from the usual definition of the unpolarized cross section which involves an incoherent sum over the four spin projections of sa, s,, sbr and sn and the orthogonality of the Clebsch-Gordan coefftcients
(18) Clearly the 5 observables involve only 3 variables: 1A’ I, ( A0 ( and the relative phase tVol of these two amplitudes. An interesting usage of these relationships has been made recently to deduce iT,, for our reaction example from measurements of tensor analyzing powers [4]. Still more recently [5] measurements of i”T,,, TT20, MT and MT22 have been made for several excited states T$ . N m the <%, cx) reaction at E, = 33 MeV. The of measurements show that TT20 tends to have a definite sign which is positive for unnatural parity states (S, = O-, l+, 2-, 3+) and negative for natural parity states (S, = l-, 2+, 3- . . . ). The TRANSAM scheme for this quantity does indeed predict an alternating sign for a constant leading term in TT20, at least for S, = O+, O-, If, lwhere explicit results have been calculated. The present formalism can also be used to analyze reactions involving other polarized beams such as deuterium, ‘Li and ‘“Na. It is also of some interest when the beam is unpolarized and the final state nucleus B decays into two components which are detected. In this latter case the spin tensor for the nucleus B is characterized by the formation amplitudes in the TRANSAM representation and the measurements given in terms of the general angular correlation function for the sequential reaction a + A + b + B * + b + (c + d). The ability to make measurements of such reactions and the marked improvements in polarized beam technology suggest that there will be several applications of the TRANSAM representation in the near future.
V. NUCLEAR PHYSICS
304
D. Robson / Nucl. Instr. and Meth. in Phys. Res. B 99 (1995) 301-304
References [I] D.M. Brink and G.R. Satchler, Angular Momentum (Clarendon, Oxford, 1968) p. 109. [2] D.J. Hooton and R.C. Johnson, Nucl. Phys. A 175 (1971) 583; H. Nishioka and R.C. Johnson, Nucl. Phys. A 440 (1985) 557.
[3] K. Stephenson, L.D. Knutson and W. Haeberli, Nucl. Phys. A 277 (1977) 365; P. Zupranski et al., Nucl. Instr. and Meth. 167 (1979) 193. [4] A.J. Mendez et al., Nucl. Pbys. A 567 (1994) 655. [5] A.J. Mendez et al., submitted to Phys. Rev. C.
Nuclear Instruments and Methodsin Physics ResearchB 99 (1995) 301-304
Bum Interactions with Materials & Atoms
!!!Z EISEVIER
The transfer angular momentum scheme in nuclear reactions * D. Robson
*
Physics Department, Florida State Uniuersity, Tallahassee, FL 32306-3016, USA
Abstract The use of transferred angular momentum (TROISAM) quantum numbers to describe nuclear reactions is discussed without assuming a particular reaction mechanism. The specific reaction ‘*@Li, o)14N* with polarized lithium is discussed as an example of the formalism.
of the initial and final channel spins denoted by s and s’ respectively.
1. Introduction In the literature most nuclear reactions at energies significantly above the relevant Coulomb barriers are analyzed in terms of direct or few-step reaction theories. In many cases the concept of angular momentum transfer between initial and final states is used to characterize the reaction amplitudes, e.g., in stripping reactions like A + d + (A + n) * f p the angular distributions are characterized by the “neutron transfer” quantum numbers I and j = (1+ i) for the configuration (A + r-r)*. Usually the approach uses models and approximations and it is advantageous to know if the transferred quantum numbers can be derived in an exact formulation of the reaction theory. The fact is that the transferred angular momentum (TRANSAM) scheme is just as general as the more conventional general partial wave schemes typified by the channel spin and total angular momentum representation. The advantage of the transfer representation lies not only in the fact that there is a finite number of amplitudes allowed by conservation of angular momentum and parity, but also in their properties under coordinate rotations. In particular each amplitude is characterized by a quantum number L and its (2L + 1) projections M which transform under rotations as an irreducible tensor of rank L. Physically the L quantum number is the angular momentum difference between the final orbital angular momentum 1’ and the initial orbital angular momentum 1, i.e., L = f’ - 1. Conservation of angular momentum in a reaction of the form A + a + B + b leads to the requirement that L = s - s’, so that L is constrained by the allowed finite values
2. Formalism The derivation of explicit formulas can only be outlined here. The derivation given here begins from the T-matrix in the uncoupled representation: (k’, sb/.+,s,~nITlk, = 2aikK’
s~cLA/+,)
c (c’ 1tl c)Y,;(i)Y,,(&‘), ll’mm’
(1)
in which c, c’ are collective channel labels for the uncoupled basis states 1s, pasA pAlm), I sbpbsB p,l’m’) respectively and %, p are the unit wavevectors for the directions of the initial and final relative motions in the center of momentum frame. The spins s, and their projections pi are self explanatory. Following the usual formalism for a single system density matrix written as a series of spin tensors of rank sir [l] where: (%/$I =
PIsiP?)
C
(“isfPiJ
-Pf
I Sif~if)(-)Sf-fLfPS,Sf~i~~,~t
C2>
sir Pif
we generalize to a system with two subsystems and write our T-matrix as: (k’, st,w,pBITlk
=
s,I~,sAP~)
c (SaSbCL,, -Pblw,b)(-_)Sb-Pb %b Cd X(wl3CLAt -/+I sABl*&( -p-ILB X(s,bsABCL.b~~ILM)T~~~B(k’, k).
* Work supported in part by the US DOE Grant DE-FGOS86ER-40273. * Tel. + 1 904 644 [email protected].
1767,
fax
+ 1 904 644 8630,
e-mail:
0168-583X/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-583X(94)00639-3
(3)
this expansion the coupled spin tensors TlbyA,(k’, k) are irreducible tensors of rank L under rotations. Inverting this
In
V. NUCLEAR
PHYSICS
302
D. Robson /Nucl. Instr. and Meth. in Phys. Res. B 99 (1995) 301-304
expansion and expressing (c’ ) t ) c) in Eq. (1) as a series of coupled t-matrix elements in the channel spin and total angular momentum representation (S = s, + s,, J = s + 1, etc.) yields after some tedious algebra the result:
=2nK’
~tjf~SABL(il’mm’
It is also easy to relate the tensors in the transverse frame to the Madison by rotating through the Euler angles (Y= -7r/2, /3= -rr/2and y=O, i.e.,
ILM)Y,,(i)Y,c,(P),
(7)
II’ (4)
in which the dynamics are contained in the tiLrABLgiven by
%
t[r %lJ,& = c (25 + l)W(sls’l’; SS’J
’
[
sb
%b
sA
‘B
SAB
s
s’
L
X(s’l’J
JL)
1
ws+L-J-r
II t II SW),
TLM
(5)
with the W being a Racah coefficient and the square array being the standard LS - jj recoupling coefficient. Bqs. (4) and (5) provide the connection between the and the conventional partial wave series. It tensors TSfbyAB is not always appropriate to use this connection because of the sheer number of partial waves which occur and we included the relationship here simply to show the generality of the transfer representation. Other models can be used to calculate TkyAB such as approximate amplitudes from DWBA, CCBA or other appropriate direct reaction theories.
3. Frame dependence As might be anticipated the tensors TX”,”are not all independent of each other. The particular relationships which occur are frame dependent. Here we indicate special results for two commonly used coordinate frames, i.e., the Madison convention and the “transverse frame”. In the Madison frame the z-axis is parallel to fi and the y-axis is parallel to k X k. In contrast the transverse frame has the z-axis parallel to i X I? and the x-axis parallel to fi. Defining the intrinsic parities of each subsystem by ni then the product ~~n~rrb~n represents the parity transfer rrr which by conservation of parity is also equal to (_)[+I’_ Using the explicit Eq. (4) above for TX”,”and the properties of vector addition coefficients yields the relations: TLM = q( -XY
-)‘+“Tx:-M
(Madison),
where we introduce the labels T and M to denote the particular frame for each tensor. Of course the number of independent amplitudes is frame independent and a particular choice is either a matter of convention or convenience. The special case of elastic scattering of spin-l particles by spin zero targets has been considered [2] using a spin tensor expansion and the results above in Eq. (6a) reduced to (with x = sab =L, y=ss,=O), =
(_
)L+“~TL,-M
(8)
for ,.any z-axis in the plane containing k and k’ and y 11k X k. In the particular case where the z-axis is chosen along the direction of k + k’ the additional symmetry from time reversal yields zero values for TLM when L f M = odd. This frame choice appears to be useful only for elastic scattering and is denoted by a bar over the tensor.
4. Special cases The special cases which are of interest are those where at least two of the particles a, A, b, B have spins which are zero. In these cases there are strong limitations on the transferred angular momentum quantum numbers sab? sAB, and L. If only one spin, say si, is non-zero, then one of sab and sA8 is zero and the other is equal to si. Also in this case L = si only. For all of these four special cases the T-matrix in Eq. (3) reduces to: (k’, SbP&rCLnITlk =TSipi(k’,
S~CLAPA)
k), if si=s,
= ( _)SI-PiTS,, -‘+(k’,
or s,, k), if si = sb or sB.
(9)
with the VdUeS of s,b, Sm being fixed as indicated above. The special cases when only two spins, say si, sj are non-zero lead to six possibilities all having the constraints that si and sj must both be integers or both be half-integers. Moreover the allowed values of L are given by the vector addition rules, i.e., L = I si - sj I, ( si - sj ( + 1 . . . si + sj, so that the T-matrix in Eq. (3) reduces to a single sum over L of the form:
(ha) L
and
=‘i+?TLM(,f,
k),
x(-)
T*“,”= 0 for ( - ) M + rrr
(transverse).
(6b)
(10)
303
D. Robson / Nucl. Ins&. and Meth. in Phys. Res. B 99 (1995) 301-304 wherein yi, -yj are equal to + 1 or - 1 corresponding to i, j being initial particles or final particles respectively. Similarly oi = 0 or si - pi, ‘Ye= 0 or sj - kj for i, j being initial or final particles.
5. Example As an example of the TRANSAM representation we consider the special reaction “C(%, a)14N * with a polarized 6Li beam. Since two of the spins and parities are Of the reaction matrix is reduced to a special case of Eq. (10). If we suppress the labels sA, s&= 0) we have with s,= 1, (k’, .QrP*ITlk,
involved in the TRANSAM scheme. However, the analyzing powers characterized by Tkq = E&/E,& with k = 1 (vector) and k = 2 (tensor) will in general have interference effects between the different values of the transfer quantum numbers. For the example here when sn = O- one obtains the five entities in the Madison frame in terms of AM = MT’“(k’, k): iT,, = GAY = - fi2 T,,=&[~A’I”T,, = - 62
IA’f]/N, Re( A’( A’)* )/N,
Tz2= -&21A’12/N,
IPa>
N=2jA’12+
x T$;a-“b(
k’, k).
(11)
For a polarized beam and no spin sensitive detectors observables are given in terms of efficiency tensors which are defined by:
the Ed;
where the density matrix for this situation is given by: C(k’, 6%
Sn~aITlk,
l/Q*
X(k’,
sBpB17’lk.
I/L,).
(13)
Substituting Eq. (11) into Eqs. (12) and (13) and performing the magnetic summations yields: $++ = Lg
( _)+‘+sB-L’(2L
x (2k + 1)“2w(L1L’1; X(LkM,
+ I)‘/2
Snk)
-sIL’M-q)T~~~(T~*~-q)*,
jA”j2.
(16)
which obeys the general quadratic relationship reactions: (i”T,,)’
+ i”T;o + “T& +“T&
= 1
[3] for such
(17)
and the relations (12)
P&,=
Im( A’( A’)* )/N,
(14)
with the k = 0 = q tensor being: (15) This special tensor is proportional to the differential cross section for an unpolarized beam. It is worth noting that in the most general case (none of the four particles having zero spin) the differential cross section for unpolarized beam and target and spin insensitive detectors involves an incoherent sum over all the transfer quantum numbers sat.,, s,, L and M. This follows from the usual definition of the unpolarized cross section which involves an incoherent sum over the four spin projections of sa, s,, sbr and sn and the orthogonality of the Clebsch-Gordan coefftcients
(18) Clearly the 5 observables involve only 3 variables: 1A’ I, ( A0 ( and the relative phase tVol of these two amplitudes. An interesting usage of these relationships has been made recently to deduce iT,, for our reaction example from measurements of tensor analyzing powers [4]. Still more recently [5] measurements of i”T,,, TT20, MT and MT22 have been made for several excited states T$ . N m the <%, cx) reaction at E, = 33 MeV. The of measurements show that TT20 tends to have a definite sign which is positive for unnatural parity states (S, = O-, l+, 2-, 3+) and negative for natural parity states (S, = l-, 2+, 3- . . . ). The TRANSAM scheme for this quantity does indeed predict an alternating sign for a constant leading term in TT20, at least for S, = O+, O-, If, lwhere explicit results have been calculated. The present formalism can also be used to analyze reactions involving other polarized beams such as deuterium, ‘Li and ‘“Na. It is also of some interest when the beam is unpolarized and the final state nucleus B decays into two components which are detected. In this latter case the spin tensor for the nucleus B is characterized by the formation amplitudes in the TRANSAM representation and the measurements given in terms of the general angular correlation function for the sequential reaction a + A + b + B * + b + (c + d). The ability to make measurements of such reactions and the marked improvements in polarized beam technology suggest that there will be several applications of the TRANSAM representation in the near future.
V. NUCLEAR PHYSICS
304
D. Robson / Nucl. Instr. and Meth. in Phys. Res. B 99 (1995) 301-304
References [I] D.M. Brink and G.R. Satchler, Angular Momentum (Clarendon, Oxford, 1968) p. 109. [2] D.J. Hooton and R.C. Johnson, Nucl. Phys. A 175 (1971) 583; H. Nishioka and R.C. Johnson, Nucl. Phys. A 440 (1985) 557.
[3] K. Stephenson, L.D. Knutson and W. Haeberli, Nucl. Phys. A 277 (1977) 365; P. Zupranski et al., Nucl. Instr. and Meth. 167 (1979) 193. [4] A.J. Mendez et al., Nucl. Pbys. A 567 (1994) 655. [5] A.J. Mendez et al., submitted to Phys. Rev. C.