Subsections

• 4.1 Matrix Elements of Tensor Forces
  • 4.1.1 Spin-orbit Interactions
  • 4.1.2 Second-rank Tensor Forces
  
  
• 4.2 Inelastic Excitations
  • 4.2.1 Nuclear Rotational Model
  • 4.2.2 Coulomb Deformations
  • 4.2.3 Angular Momentum Coupling Coefficients
  
  
• 4.3 Single Particle Excitations
  • 4.3.1 Projectile Single-Particle Mechanisms
  • 4.3.2 Target Single-Particle Mechanisms
  
  
• 4.4 Particle Transfers
  • 4.4.1 Finite Range Transfers
  • 4.4.2 Zero Range Transfers
  • 4.4.3 Local Energy Approximation

4 Coupling Interactions

When two nuclei interact, a variety of kinds of elastic and inelastic potentials may be needed to describe their interaction. As well as the scalar nuclear attractions and scalar Coulomb repulsions, if either of the nuclei has spin $J \neq 0$, then there can be higher-order tensor interactions which couple together the spin and the orbital motion. If a nucleus has spin $J \geq\frac{1}{2}$, then there can be a spin-orbit component $V_{ls} (R) 2 {\bf l}\cdot {\bf J}$ in the Hamiltonian ${\cal H}$. and if its spin is one or greater ($J \geq 1$), there can be tensor forces of various kinds. The most commonly used tensor force is a T_r potential of the form $V_{Tr} (R) {\bf R}_2 ( {\bf R}, {\bf R}) {\bf .S}_2 (J,J)$. Similar tensor forces are also generated if the projectile and target spins coupled together can reach $J_p + J_t \geq 1$: such is the case with the tensor force between the proton and the neutron within the deuteron.

Inelastic potentials (4.2) arise when one or both of the nuclei have permanent deformations (as seen in their intrinsic frame), or are vibrationally deformable. The inelastic potentials which come from rotating a permanently deformed nucleus are described in the Hamiltonian by terms of the form

$${\bf V}_\lambda = \sum _ \mu {\sf V}_\lambda (R) D^\lambda_{\mu 0} Y^\mu_\lambda (\hat {\bf R})$$ (70)

where the form factors ${\sf V}_\lambda (R)$ have both nuclear and Coulomb components for angular momentum transfers $\lambda$. Their nuclear component is approximately proportional to the derivative of the scalar potential between the two reaction partners. Simultaneous excitations of both nuclei are also possible (see e.g. [17]), but have not been included in the present code. Vibrational excitations of a nucleus have more complicated form factors in general [1], but can still be expanded in the form of equation (70). For the more intricate level schemes of strongly-deformed nuclei, it will in general be necessary for each allowed transition to have its own transition rate specified independently of a particular rotational or vibrational model.

Inelastic potentials also arise when one of the nuclei can be decomposed into a `core' + `valence particle' structure 4.3), such that the opposing nucleus interacts with the two components with distinct potentials acting on distinct centres-of-mass. The valence particle can be a single nucleon, as in the case of ¹⁷O = ¹⁶O + n, or it can be a cluster of nucleons, as in ⁶Li =$\alpha$ + ²H, or ⁷Li =$\alpha$ + ³H. In all these cases, there arise inelastic potentials which can re-orient the ground state of the composite nucleus, or can excite the valence particle into higher-energy eigenstates.

Finally, transfer interactions (4.4) arise when the reaction brings about the transfer of a valence particle from one nucleus into a bound state around the other. As the incoming and outgoing projectiles have different centres-of-mass, with the targets likewise, the correct treatment of transfer interactions requires taking into account the effects of recoil and of the finite ranges of the binding potentials. These result in the coupling form factors becoming non-local, so that they must be specified by the two-dimensional kernel functions $V_{\alpha :\alpha'} (R_\kappa , R_{\kappa'} )$ in equation (30). They also require that the coupled equations be solved by iteration, as will be discussed in section 5. If the effects of recoil are neglected, the `no-recoil' (NR) approximation is obtained, but in general[72] this is inaccurate in ways which are difficult to predict. For that reason the NR approximation is not included in the present code. For many light-ion reactions, however, another `zero-range' approximation is available, and this does remove many of the finite-range requirements. Alternatively, a first-order correction for the finite-range effects may be estimated, to give the `local energy approximation'. These two special cases are discussed at the end of the section.

4.1 Matrix Elements of Tensor Forces

This section presents the matrix elements for spin-orbit forces and a variety of tensor interactions. The radial form factors ${\sf V}_Q (R)$ which multiply these matrix elements are not specified, since these are usually determined by a fitting procedure in an optical-model search code, and a wide variety of parameterised forms have been used.

We shall use the $\vert (LJ_p) J_1 ,J_t ;J_T\rangle$ representation for the order of coupling the spins, as in equation (24).

4.1.1 Spin-orbit Interactions

For the projectile spin-orbit force ${\bf L\cdot J}_p$

$$\left\langle (L J_p) J_1 , J_t; J_T\vert {\bf L\cdot J}_p \vert ( L'J_p) J'_1 , J_t; J_T\right\rangle$$

$$= \delta_{L L'} \delta_{ J_1 J'_1 } \frac{1}{2}[ J_1 ( J_1 +1) - L(L+1) - J_p( J_p+ 1) ]$$ (71)

This convention amounts to a $2 {\bf l}\cdot {\bf s}$ spin-orbit force, rather than one based on ${\bf l}\cdot {\bf\sigma}$. These are the same for nucleons and spin $\frac{1}{2}$ nuclei, but it means, for example, that the spin-orbit strengths for deuterons and ⁷Li will have to be decreased as they have ${\bf s}$ = 1 and 3/2 respectively.

For the target spin-orbit interaction ${\bf L\cdot J}_t$, we first transform

$$\vert (LJ_p) J_1 ,J_t ;J_T\rangle (-1)^{J_1 - L - J_p } \vert (J_p L) J_1 ,J_t ;J_T\rangle$$ =

$$(-1)^{J_1 - L - J_p } \sum _ {J_2} \vert J_p , (LJ_t ) J_2 ; J_T\rangle \hat J_1 \hat J_2 W( J_p L J_T J_t ; J_1 J_2 )$$ = (72)

so

$$\begin{eqnarray*}\nonumber &&\left\langle (L J_p ) J_1 , J_t ; J_T \vert {\bf L\... ...ta_{L L' }\frac{1}{2}[ J_2 ( J_2 +1) - L(L+1) - J_t ( J_t + 1) ] \end{eqnarray*}$$

4.1.2 Second-rank Tensor Forces

We use the notations of ref. [18]:

$$\langle S \Vert {\bf S}_2 \Vert S\rangle = {1 \over \sqrt 6} ~ { ... ...+1) \over \langle SK20 \vert SK\rangle} \mbox{ for any } \vert K \vert \leq S ,$$ (73)

and

$$\langle L' \Vert {\bf R}_2 \Vert L\rangle = \sqrt {2 \over 3} {\hat L \over \hat{L'}}\langle L~0~2~0~ \vert L' ~0~\rangle$$ (74)

for the reduced matrix elements of the second-rank spin and radial tensors respectively. With the projectile ${\bf T_r}$ tensor force ${\bf R}_2 \cdot {\bf S}_2 ( J_p J_p )$, the coupling interactions are

$$\left\langle (L J_p ) J_1 , J_t ; J_T \vert {\bf R}_2 \cdot {\bf S}_2 ( J_p J_p ) \vert ( L' J_p ) J'_1 , J_t ; J_T \right\rangle$$

$$~=~ \delta_{ J_1 J'_1 } \hat L \hat J_p (-1)^{ J_1 - L - J_p } W(... ... L \Vert {\bf R}_2 \Vert L'\rangle \langle J_p \Vert {\bf S}_2 \Vert J_p\rangle$$ (75)

For the target ${\bf T_r}$ tensor force ${\bf R}_2 \cdot {\bf S}_2 ( J_t J_t )$ the coupling interactions are

$$\left\langle (L J_p ) J_1 , J_t ; J_T \vert {\bf R}_2 \cdot {\bf S}_2 ( J_t J_t ) \vert ( L' J_p ) J'_1 , J_t ; J_T \right\rangle$$

$$~=~ (-1)^{J_1 - J_1' + L' - L} \sum _ {J_2} \hat J_1 \hat J'_1 (2 J_2 + 1) W( J_p L J_T J_t ; J_1 J_2 ) W( J_p L' J_T J_t ; J_1' J_2 )$$

$$\times \hat L \hat J_t (-1)^{ J_2 - L - J_t } W(L L' J_t J_t ; 2 ... ... L \Vert {\bf R}_2 \Vert L'\rangle \langle J_t \Vert {\bf S}_2 \Vert J_t\rangle$$ (76)

For the combined target-projectile ${\bf T_r}$ tensor force ${\bf R}_2 \cdot {\bf S}_2 ( J_p J_t )$ the coupling interactions are

$$\left\langle (L J_p ) J_1 , J_t ; J_T \vert {\bf R}_2 \cdot {\bf S}_2 ( J_p J_t ) \vert ( L' J_p ) J'_1 , J_t ; J_T \right\rangle$$

$$~=~ \sum_{S S' } \hat J_1 \hat J'_1 \hat S \hat S' W( L J_p J_T J_t ; J_1 S ) W( L' J_p J_T J_t ; J'_1 S' )$$

$$\times \hat L \hat S\ (-1)^{J_T - L - S' } W(L L' S S' ; 2 J_T ) ... ...Vert L'\rangle \langle ( J_p J_t )S \Vert {\bf S}_2 \Vert ( J_p J_t ) S'\rangle$$ (77)

where the second-rank reduced matrix element is

$$\langle ( J_p J_t )S \Vert {\bf S}_2 \Vert ( J_p J_t ) S'\rangle ... ... & 1& 1\end{array} \right ) ~ \sqrt { J_p ( J_p + 1) } \sqrt { J_t ( J_t + 1) }$$

4.2 Inelastic Excitations

4.2.1 Nuclear Rotational Model

Consider a deformed nucleus with deformation lengths $\delta_\lambda$. The effect of these deformations can be expressed as a change in the radius at which we evaluate the optical potentials, the change depending on the relative orientations of the radius vector to the intrinsic orientation of the nucleus. Deformation lengths are used to specify the these changes, rather than fractional deformations $\beta_\lambda$, to remove a dependence on the `average potential radius' R<sub>U</sub>. This is desirable because often the real and imaginary parts of the potential have different radii, and it is not clear which is to be used. It also removes a dependence on exactly how the `average radius' of a potential is to be defined.

When U(R) is the potential shape to be deformed, the coupling interaction is

$${\bf V}( {\bf\xi} , {\bf R}) = U(R - \delta( \hat{\bf R}, {\bf\xi} ))$$ (78)

where the `shift function' has the multipole expansion

$$\delta( \hat{{\bf R}'}) = \sum_{\lambda \neq 0} \delta_\lambda Y^0_\lambda (\hat{{\bf R}'})$$ (79)

($\hat {\bf R}'$ is the vector $\hat{\bf R}$ in the body-centred frame of coordinates defined by ${\bf\xi}$). Transforming to the space-fixed frame of reference, and projecting onto the spherical harmonics, the multipole expansion becomes

$${\bf V}( {\bf\xi} , {\bf R}) = \sum_{\lambda \mu} {\sf V}_\lambda (R) D^\lambda_{\mu 0} Y^\mu_\lambda (\hat{\bf R})$$ (80)

$$\mbox{where } {\sf V}_\lambda (R) = \frac{1}{2}\int _{-1}^{+1} U(~r(R,\cos \theta)~) Y_\lambda^\mu (\theta ,0) ~d(\cos \theta)$$ (81)

$$\mbox{and } r(R,u) = R - \sqrt{ {2 \lambda+1 \over 4 \pi} } P_\lambda (u) \delta_\lambda + \epsilon$$ (82)

$$\mbox{with } \epsilon = {\sum _ \lambda {\delta_\lambda}^2} / (4 \pi R_U )$$ (83)

The correction $\epsilon$ is designed ([45]) to ensure that the volume integral of the monopole potential ${\sf V}_0 (R)$ is the same as that of U(R), and is correct to second order in the $\{\delta_\lambda\}\}$.

When the $\{\delta_\lambda\}\}$ are small, the above multipole functions are simply the first derivatives of the U(R) function:

$${\sf V}_\lambda (R) = - {\delta_\lambda \over \sqrt {4 \pi}} ~{dU(R) \over dR} ,$$ (84)

with the same shape for all multipoles $\lambda> 0$.

4.2.2 Coulomb Deformations

The deformations of the Coulomb potential can also be defined by the $\delta_\lambda$, but unfortunately an average potential radius is again introduced. The dependence on models for average radii can be reduced by defining the Coulomb deformations in terms of a reduced matrix element such as that of Brink and Satchler [19], or that of Alder and Winther [20]. For the present purposes we adopt that of Alder and Winther, as it is hermitian upon interchanging the forward and reverse directions. We include, however, a simple phase factor to keep it real-valued. The new deformation parameter is called $M(E \lambda)$ and has units of $e . {\rm fm}^\lambda$. In terms of the Alder and Winther reduced matrix element it is

$$M(E \lambda) = i^{ I - I' + \vert I - I' \vert} \times \langle I' \Vert E \lambda \Vert I\rangle$$ (85)

and is directly related to the observable electro-magnetic transition rate without any model-dependent parameters entering (except a sign):

$$M(E \lambda) =\pm\sqrt { (2I+1) B(E \lambda , I \rightarrow I') }.$$ (86)

A model dependent radius parameter R_c only enters in the relation to the deformation lengths of the rotational model:

$$M(E \lambda , I \rightarrow I') = {3Z \delta_\lambda {R_c}^{\lamb... ... + \vert I - I' \vert} ~ \sqrt {2I+1} ~~\langle I K \lambda 0 \vert I' K\rangle$$ (87)

for transitions from a state of spin I to one of spin I' in a rotational band of projection K in a nucleus of charge Z. Within K=0 bands, $M(E \lambda , 0 \rightarrow I) = M(E \lambda , I \rightarrow 0)$ have the same sign as $\delta_\lambda$.

The only disadvantage of using reduced matrix elements as input parameters in this way is that the transitions in a rotational band do not all have the same matrix elements $M(E \lambda , I \rightarrow I')$, even when the deformation length is constant.

The radial form factors for Coulomb inelastic processes may be simply derived from the multipole expansion of $\vert {\bf r}- {\bf r}' \vert^{-1}$, giving

$${\sf V}_\lambda^c (R) = M(E \lambda ) ~ { \sqrt {4 \pi} e^2 \over... ...da+1} ~~(R \leq R_c)}\\ {1 / R^{\lambda+1} ~~ (R> R_c) } \end{array} \right. ~$$ (88)

remembering that a factor $\delta_\lambda {R_c}^{\lambda - 1} = \beta_\lambda {R_c}^\lambda$ is already included in the matrix element of equation (87) which appears in this form factor. This form factor is to be multiplied by the angular momentum coupling coefficients of the next section, and also by the charge of the opposing nucleus.

4.2.3 Angular Momentum Coupling Coefficients

The basic rotational coupling coefficient, with ${\bf V}_\lambda$ given by equation (70), is

$${\bf X}^{J \lambda}_{LI:L' I'} (R) = \langle LI; J \vert {\bf V}_\lambda \vert L' I' ; J\rangle$$ (89)

The Coulomb form factors ${\sf V}_\lambda^c (R)$ have coupling coefficients

$${\bf X}^{J \lambda}_{LI:L' I'} (R) = \hat L \hat {L'} (-1)^{J-I' ... ...' ; \lambda J) \langle L 0 L' 0 \vert \lambda 0\rangle ~~ {\sf V}_\lambda^c (R)$$ (90)

whereas the nuclear form factors ${\sf V}_\lambda (R)$ defined for a rotational band with projection K have coupling coefficients

$${\bf X}^{J \lambda}_{LI:L' I'} (R) \hat L \hat {L'} (-1)^{J-I' -L+L'} W(L L' I I' ; \lambda J) \langle L 0 L' 0 \vert \lambda 0\rangle ~~ {\sf V}_\lambda (R)$$ =

$$\hat{I' } \langle I' K \lambda 0 \vert I K\rangle .$$ (91)

For projectile inelastic excitation, this coupling coefficient may be used directly as

$$\langle (L J_p)J, J_t ; J_T \vert {\bf V}_\lambda \vert (L' J'_ p... ...delta_{J_t , J'_ t} ~\delta_{J , J'} ~ {\bf X}^{J \lambda}_{LJ_p :L' J'_ p} (R)$$ (92)

whereas for target excitations,

$$\langle (L J_p)J, J_t ; J_T \vert {\bf V}_\lambda \vert (L' J'_ p... ... J'_ t ; J_T\rangle = \delta_{J_p , J'_ p} ~ (-1)^{J-J' -L+L'} \hat J \hat {J'}$$

$$\times \sum_{J_2} (2J_2 + 1) W(J_p L J_T J _t ; J J_2) W(J_p L' J_T J'_ t ; J' J_2) {\bf X}^{J_2 \lambda}_{LJ_t :L' J'_ t} (R)$$ (93)

4.3 Single Particle Excitations

When a nucleus consists of a single particle outside a core, the state of the particle can be disturbed by the interaction with1 another nucleus, as the force of that nucleus can act differentially on the particle and the core. If $V_{cc} ({\bf R}_c)$ and $V_p ({\bf r}')$ are the interactions of the second nucleus with the core and particle respectively, then the excitation coupling from state $\vert (\ell' L') \lambda\rangle$ to state $\vert (\ell L) \Lambda\rangle$ is given by the single-folding expression

$${\bf X}^\Lambda_{\ell L: \ell' L'} (R) = \langle (\ell L) \Lambda... ... R}_c) + V_p ({\bf r}') - U_{\rm opt} ({\bf R}) \vert (\ell' L') \Lambda\rangle$$ (94)

where $U_{\rm opt} ({\bf R})$ is the optical potential already defined for these channels. This optical potential is subtracted to avoid double counting of either the Coulomb or the nuclear potentials, rather than disabling the potentials which have already been defined. This means that the `monopole' potential ${\sf V}_0 (R,r)$ (to be constructed) will have no long-range Coulomb component, and will not disturb the matching of the wave functions to the asymptotic Coulomb functions. It also means that if a nuclear well has already been defined, the new monopole form factor will be simply the difference between this well and that desired well calculated from the folding procedure.

If the potentials $V_{cc} ({\bf R}_c)$and $U_{\rm opt} ({\bf R})$ contain only scalar components, then the R- and r- dependent Legendre multipole potentials can be formed as

$${\sf V}_K (R,r) =\frac{1}{2}\int_{-1}^{+1} \left [ V_{cc} ({\bf R}_c) + V_p ({\bf r}') - U_{\rm opt} ({\bf R}) \right ] . P_K (u) du$$ (95)

where

$$\mbox{ the multipole moment,} \nonumber$$ K =

$$\hat{\bf r}\cdot \hat{\bf R} \mbox{ is the cosine of the angle between } {\bf r}\mbox{ and } {\bf R},$$ u =

$${\bf r} a {\bf R}+ b {\bf r}\mbox{ is the particle-core vector,}$$ =

$$\mbox{and } {\bf R}_c p {\bf R}+ q {\bf r}\mbox{ is the core-nucleus vector.}$$ =

The coupling form factor between states $u_{\ell'} (r)$ and $u_\ell (r)$ is then

$${\bf X}^\Lambda_{\ell L: \ell' L'} (R) \frac{1}{2}\sum _ K { \int_ 0^ {R_m} u_\ell (r)^* {\sf V}_K (R,r) u_{\ell'} (r) dr} (-1)^{\Lambda+K} \hat\ell \hat L \hat {\ell'} \hat {L'}$$ =

$$(2K+1) W(\ell \ell' L L' ;K \Lambda) \left ( \begin{array}{ccc}K ... ...{array} \right ) \left ( \begin{array}{ccc}K&L&L'\\ 0&0&0 \end{array} \right )$$ x (96)

4.3.1 Projectile Single-Particle Mechanisms

If the projectile has the particle - core composition, then the coupling interaction is

$$V^{J_T}_{\alpha :\alpha'} (R) = \langle (L J _p)J, J_t ;J_T \vert {\bf V} \vert (L' J'_ p)J, J_t ;J_T\rangle$$ (97)

where the initial (primed) and final (unprimed) states are

$$\phi_{J'_ p} (\xi_p , r) = \sum_{\ell' sj'} A_{\ell ' sj'}^{j' I_... ... = \sum_{\ell sj} A_{\ell sj}^{jI_p J_p} \vert ( \ell s) j , I_p ; J_p\rangle ,$$ (98)

respectively, and I_p is the (fixed) spin of the core. Then

$$V^{J_T}_{\alpha :\alpha'} (R) \sum_{\begin{array}{c}F \Lambda I_p\\ j j' \ell \ell'\end{array}... ...hat {J'_ p} (2 \Lambda+1) W(\ell s J_p I_p ; j F) W(\ell' s J'_ p I_p ; j' F) }$$ =

$$\times A_{\ell ' sj'}^{j' I_p J'_ p} ~ A_{\ell sj}^{jI_p J_p} ~ W... ...p) W(L' \ell' J F; \Lambda J'_ p) \times {\bf X}^\Lambda_{\ell L: \ell' L'} (R)$$ (99)

4.3.2 Target Single-Particle Mechanisms

If the target has the particle - core composition, then the coupling interaction is

$$V^{J_T}_{\alpha :\alpha'} (R) = \langle(L J_p)J, J'_ t ;J_T \vert {\bf V} \vert (L' J_p)J, J _t ;J_T\rangle$$ (100)

where the initial (primed) and final (unprimed) states are

$$\phi_{J'_ t} (\xi_t , r) = \sum_{\ell' sj'} A_{\ell ' sj'}^{j' I_... ... = \sum_{\ell sj} A_{\ell sj}^{jI_t J_t} \vert ( \ell s) j , I_t ; J_t\rangle ,$$ (101)

respectively, and I_t is the (fixed) spin of the core in the target. Then

$$V^{J_T}_{\alpha :\alpha'} (R) \sum_{I_t j j' \ell \ell'} A_{\ell ' sj'}^{j' I_t J'_ t} ~ A_{\ell sj}^{jI_t J_t}$$ =

$$\times \sum_{J_a} (2J_a + 1) \hat{J'_ t} \hat {J_t} ~ W(J j J_T I_t ; J_a J_t) ~ W(J' j' J_T I_t ; J_a J'_ t)$$

$$\times {\sum_{\Lambda s_a } } \left \{ \begin{array}{ccc} L&\ell&... ...p&s&s_a\\ J'&j'&J_a\end{array}\right \} {\bf X}^\Lambda_{\ell L: \ell' L'} (R)$$ (102)

4.4 Particle Transfers

4.4.1 Finite Range Transfers

To calculate the coupling term that arises when a particle is transferred, for example from a target bound state to being bound in the projectile, we need to evaluate source terms of the form

$$S_\alpha (R) = \int_ 0 ^ \infty \langle (LJ_p)J,J_t ;J_T \vert {\... ...ert (L' J'_ p)J' ,J'_ t ;J_T\rangle ~ f^{J_T} _{ (L' J'_ p)J' ,J'_ t} (R' ) dR'$$ (103)

where the initial (primed) state has a composite target with internal coordinates $\xi'_ t \equiv \{ \xi_t , {\bf r}' \}:$ $\phi_{J'_ t} (\xi_t , {\bf r}' ) = \vert ( \ell' s) j' , J_t ; J'_ t\rangle$ and the final (unprimed) state has a composite projectile with internal coordinates $\xi_p \equiv \{ \xi_{p'} , {\bf r}\}:$ $\phi_{J_p} (\xi'_ p , {\bf r}) = \vert ( \ell s) j , J'_ p ; J_p\rangle .$

The ${\bf V}$ is the interaction potential, of which the prior form is

$${\bf V} = V_{\ell sj} ({\bf r}) + U_{cc} (R_c) - U_{\alpha'} ({\bf R}')$$ (104)

and the post form is

$${\bf V} = V_{\ell' sj'} ({\bf r}') + U_{cc} (R_c) - U _\alpha ({\bf R})$$ (105)

where $V_\beta ({\bf r})$ is the potential which binds $\varphi_\beta ({\bf r}),$ $U _\alpha ({\bf R})$ are the optical potentials, and $U_{cc} ({\bf R}_c)$ is the `core-core' potential, here between the p' and the t nuclei. The $V_\beta$ will be real, but the $U _\alpha$ and U_cc will typically have both real and imaginary components.

This source function $S_\alpha (R)$ evaluates a non-local integral operator, as it operates on the function $f_{\alpha'} (R' )$ to produce a function of R. This section therefore derives the non-local kernel $V_{\alpha ,\alpha'} (R,R' )$ so that the source term, which initially involves a five dimensional integral over ${\bf r}$ and $\hat{\bf R}$, may be calculated by means of a one-dimensional integral over R':

$$S_\alpha (R) = \int_ 0 ^ {R_m} V_{\alpha ,\alpha'} (R,R' ) f_{\alpha'} (R' ) dR' .$$ (106)

Note that when the initial and final single-particle states are real, then the kernel function is symmetric

$$V_{\alpha ,\alpha'} (R,R' ) = V_{\alpha' ,\alpha} (R' ,R),$$ (107)

whereas if the states are unbound and complex-valued, then the kernel function is hermitian provided the interaction potential ${\bf V}$ is real. If the particle states and the interaction potential are complex, then both the forward and reverse kernels must be each calculated independently.

When the potential V contains only scalar potentials, the kernel calculation can be reduced to the problem of finding ${\bf X}^\Lambda_{\ell L: \ell' L'} (R,R' )$ such that, given

$$\langle (LJ_p)J,J_t ;J_T \vert {\bf V}\vert(L' J'_ p)J' ,J'_ t ;J... ...\begin{array}{ccc}L'&J'_p&J'\\ \ell'&s'&j'\\ \Lambda&F&J\end{array} \right \}$$

$$\times W(J_t j' J_T J' ;~ J'_ t J) W(ls J_p J'_ p ;~jF) W(L \ell ... ...a J_p ) \langle \ell L; \Lambda \vert {\bf V} \vert \ell' L' ; \Lambda\rangle ,$$ (108)

the integral operator $\langle \ell L; \Lambda \vert {\bf V} \vert \ell' L' ; \Lambda\rangle \mbox{ has the kernel function } {\bf X}^\Lambda_{\ell L: \ell' L'} (R,R' ).$ Note that the F summation may be performed in an inner loop that does not evaluate the kernel function.

Now the ${\bf r}$ and ${\bf r}'$ are linear combinations of the channel vectors ${\bf R}$ and ${\bf R}'$: ${\bf r}= a {\bf R}+ b {\bf R}' \mbox{ and } {\bf r}' = a' {\bf R}+ b' {\bf R}'$ where, when $\varphi_\ell ({\bf r})$ is the projectile bound state,

$$a = \nu_t \omega , ~~~ b = - \omega , ~~~ a' = \omega , ~~~ b' = - \nu_p \omega ,$$ (109)

with $\nu_p \equiv A_{\kappa' p} / A_{\kappa p}$ , $\nu_t \equiv A_{\kappa t} / A_{\kappa' t}$ , and $\omega = (1 - \nu_p \nu_t ) ^ {-1}$ . When $\varphi_\ell ({\bf r})$ is the target bound state

$$a = - \nu_p \omega, ~~~ b = \omega , ~~~ a' = - \omega , b' = \nu_t \omega , ~~~$$ (110)

with $\nu_p \equiv A_{\kappa p} / A_{\kappa' p}$ , $\nu_t \equiv A_{\kappa' t} / A_{\kappa t}$ , and $\omega = (1 - \nu_p \nu_t ) ^ {-1}$ . The `core-core' vector is always ${\bf R}_c = {\bf r}' - {\bf r} = (a' - a) {\bf R}+ (b' - b) {\bf R}' .$

Thus the spherical harmonics $Y_\ell ( \hat {\bf r})$ and $Y_{\ell'} ( \hat{\bf r}' )$ can be given in terms of the spherical harmonics $Y_n ( \hat{\bf R})$ and $Y_{n'} ( \hat{\bf R}' )$ by means of the Moshinsky [71] solid-harmonic expansion (see also refs. [21] and [46]

$$Y_\ell^m ( \hat {\bf r}) = \sqrt {4 \pi} \sum_{n \lambda} c( \ell... ...mbda ( \hat{\bf R}') \langle \ell - n m - \lambda n \lambda \vert \ell m\rangle$$ (111)

where

$$c( \ell ,n) = \sqrt { {1 \over 2n+1} ~~ \left ( \begin{array}{c}2 \ell + 1\\ 2n\end{array} \right ) } ,$$

with $\left ( \begin{array}{c}x\\ y\end{array} \right )$ the binomial coefficient (Appendix A).

We now perform the Legendre expansion

$${\bf V} {u_{\ell sj} (r) \over r^{\ell +1}} ~ {u_{\ell' sj'} (r')... ...r {r'}^{\ell' +1} }~ = \sum _ T (2T+1) {\bf q}^T_ {\ell , \ell'} (R,R') P_T (u)$$ (112)

where the Legendre polynomials P_T (u) are functions of u, the cosine of the angle between ${\bf R}$ and ${\bf R}' ,$ by using r = (a² R² +b² R'² + 2abRR' u)^1/2 (with r' analogously) in the numerical quadrature of the integral

$${\bf q}^T_ {\ell , \ell'} (R,R') = \frac{1}{2}\int_ {-1} ^ {+1} {... ...(r)\over r^{\ell + 1}} ~ {u_{\ell' sj'} (r')\over {r'}^{\ell' +1}} ~ P_T (u) du$$ (113)

The quadrature methods used here, and the accuracy attained, are discussed in section 5.3.

Using the Legendre expansion, the radial kernel function

$${\bf X}^\Lambda_{\ell L: \ell' L'} (R,R' ) = {\vert b \vert^3 \ov... ...ll' n') R R' (aR)^{\ell - n} (bR')^n (a' R)^{\ell' - n'} (b' R')^{n'} \nonumber$$

$$\times \sum_T {\bf q}^T_ {\ell , \ell'} (R,R') (2T+1) (-1)^{\Lamb... ...hat{(\ell - n)}~ \hat{(\ell' - n')} \hat n \hat {n'} \hat L \hat {L'} \nonumber$$

$$\times \sum_{K K'} (2K+1)(2K' + 1) \left ( \begin{array}{ccc}\ell... ...ght ) \left ( \begin{array}{ccc}K'&L'&T\\ 0&0&0 \end{array} \right ) \nonumber$$

$$\times \sum_{Q} (2Q+1) W(\ell L \ell' L' ; \Lambda Q) W(K L K' L'... ...array}{ccc} \ell'&Q&\ell\\ n'&K&\ell-n\\ \ell' - n'&K'&n \end{array} \right )$$ (114)

These formulae can also be used with ${\bf V} \equiv 1$ to calculate the kernel functions $K^\Lambda_{\ell L: \ell' L'} (R,R' )$ for the wave function overlap operator $K_{ij} \equiv\langle\phi_i \vert\phi_j\rangle$ needed in evaluating the non-orthogonality terms of section 2.3.

One disadvantage of this method of calculating the two-dimensional radial kernels ${\bf X}^\Lambda_{\ell L: \ell' L'} (R,R' )$ is that in the process of transforming the solid harmonics of ${\bf r}$ and ${\bf r}'$ into those of ${\bf R}$ and ${\bf R}'$, there appears summations containing high powers of the coefficients a, b, a' and b' These products will become larger than unity by several orders of magnitude, will the summed result is typically of the order of unity. This means that the summations involve large cancellations, and as the degree of cancellation gets worse for large $\ell$ and $\ell' ,$ the cancellation places a limit on the maximum value $\ell + \ell'$ of the transferred angular momentum.

One way of circumventing this loss of accuracy is that proposed by Tamura and Udagawa [47], whereby solid harmonics are avoided in favour of a suitable choice of axes to render it practical to calculate m-dependent form factors directly. If the ${\bf z}$ axis is not (as usual) parallel to the incident momentum, but set parallel to ${\bf R}$, and the ${\bf x}'$ axis set in the plane determined by ${\bf R}$ and ${\bf R}' ,$ then the ${\bf r}$ and ${\bf r}'$ vectors are also in this plane. The radial kernels may then be calculated as a sum of m-dependent integrals over $\cos \theta = \hat {\bf R}\cdot \hat{{\bf R}'}$, as before the cosine of the angle between ${\bf R}$ and ${\bf R}'$. Although there are hence a larger number of radial integrals to be performed, there are no large cancellations between the separate terms, and there is no limit on the size of the transferred angular momentum.

A third method [22] of calculating the transfer form factors is that involving expanding the initial and final channel wave functions in terms of spherical Bessel functions:

$$f _\alpha (R) = \sum_{n=1}^ {N(L)} {a _\alpha (K_n) R ~ j_L (K_n R) } .$$ (115)

Using then the Fourier transform of the bound state wave functions $u_\ell (r)$ and $u_{\ell'} (r')$, a transfer T-matrix element may be written as a sum of a set of one-dimensional integrals over a momentum variable. Efficient codes [23] have been written for CCBA calculations of transfers induced by light ions (up to masses $\sim$ 10 to 15 amu).

This plane-wave expansion method has however several disadvantages when it comes to solving problems with coupled reaction channels. If transfers are to be calculated at each iteration of the coupled equations, then the expansion (115) has to be recalculated at each step. Another difficulty is that the method is not suited to heavy-ion induced transfers, as the large degree of absorption inside the nuclei in these cases requires a large number of momentum basis states K_n to be represented accurately. The plane-wave expansion becomes uneconomical, and sometimes the determination of the $a _\alpha (K_n)$ coefficients becomes numerically ill-conditioned

We will see in section 5.3.1, however, that if the cancellation which occurs in the first method is monitored, and steps taken to keep it to a minimum, a workable code [34] results which can produce accurate results for L-transfers up to around 6.

4.4.2 Zero Range Transfers

When the projectile wave functions $\varphi_\ell ({\bf r})$ are all s-states ($\ell=0$ and the interaction potential is of zero-range $({\bf V} \varphi ({\bf r}) \sim D_0 \delta ( {\bf r})~) ,$ then the form factor ${\bf X}^\Lambda_{\ell L: \ell' L'} (R,R' )$ of equation (114) can be simplified to

$${\bf X}^L_{0L: \ell' L'} (R,R') = D_0 ~ {(-1)^{L' - \ell'} \over ... ...ay} \right ) ~ {1 \over R} u_{\ell' sj'} (R ) ~ {b^2 \over a} \delta (aR+bR') .$$ (116)

This can be made local by defining a new step size $h' = -ah/b \equiv \nu_t h$ in the stripping channel $\alpha'$.

4.4.3 Local Energy Approximation

If the interaction potential is of small range, though not zero, and the projectile still contains only s-states, then a first-order correction may be made to the above form factor. This correction will depend on the rate of oscillation of the source wave function fJT (L' J'p),J' ,J't (R' ) within a `finite-range effective radius' $\rho$. The rate of oscillation is estimated from the local energy in the entrance and exit channels, and the result [24] is to replace $u_{\ell' sj'} (R)$ in the previous section by

$$u_{\ell' sj'} (R) \rightarrow u_{\ell' sj'} (R) \left [ 1 + \rho^... ...'} ( R) + V_{\ell' sj'} (R) -U _\alpha (R) + \epsilon _\alpha \right ) \right ]$$ (117)

where the U(R) are the optical potentials, with $V_{\ell' sj'} (r)$ the single-particle binding potential in the target. The $\mu _\alpha^{(p)}$ is the reduced mass of the particle in the projectile, and $\epsilon _\alpha$ its binding energy.

At sub-Coulomb incident energies [25], the details of the nuclear potentials in equation (117) become invisible, and as the longer-ranged Coulomb potentials cancel by charge conservation, the form factor can be simplified to

$$D_0 u_{\ell' sj'} (R) \rightarrow u_{\ell' sj'} (R) D_0 \left [ 1 + \rho^2 {2 \mu _\alpha^{(p)}\over \hbar^2} \epsilon _\alpha \right ] u_{\ell' sj'} (R) D$$ = (118)

where

$$D = D_0 \left [ 1 + \left ( \rho k _\alpha^{(p)} \right ) ^2 \right ]$$ (119)

is the effective zero-range coupling constant for sub-Coulomb transfers.

The parameters  D₀ and D can be derived from the details of the projectile bound state $\varphi_{0ss} ( {\bf r})$. The zero-range constant D₀ may be defined as

$$D_0 = \sqrt {4 \pi} \int_ 0^\infty r V(r) u_{0ss} (r) dr.$$ (120)

The parameter D, on the other hand, reflects the asymptotic strength of the wave function u_0ss (r) as $r\rightarrow \infty$, as it is the magnitude of this tail which is important in sub-Coulomb reactions:

$$u_{0ss} (r) = _ {r \rightarrow \infty} {2 \mu _\alpha^{(p)}\over \hbar^2} {1 \over \sqrt {4 \pi}} ~ D e^{-k _\alpha^{(p)} r} .$$ (121)

It may be also found, using Schrödinger's equation, from the integral

$$D = \sqrt {4 \pi} \int_ 0 ^\infty {\sinh (k^{(p)} _\alpha r)\over k^{(p)} _\alpha} ~ V(r) u_{0ss} (r) dr.$$ (122)

From this equation we can see that as the range of the potential becomes smaller, D approaches D₀. The `finite-range effective radius' $\rho$ of equation (119) is thus some measure of the mean radius of the potential V(r).