Abstract
In this chapter we introduce the basic concepts of multichannel collision theory and we apply this theory for illustrative purposes to non-relativistic electron collisions with multi-electron atoms and atomic ions. This chapter thus provides an introduction to our discussion of resonances and threshold behaviour presented in Chap. 3 and to R-matrix theory and applications, presented in Chap. 4 and later chapters in this monograph. We will be mainly concerned in this chapter with low-energy elastic scattering and excitation processes. However, we will show in Chap. 6 that the theory and methods developed in this chapter are the basis of R-matrix methods which enable accurate excitation and ionization processes to be calculated at intermediate energies.
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Keywords
- Differential Cross Section
- Target State
- Partial Wave Analysis
- Close Coupling
- Asymptotic Boundary Condition
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In this chapter we introduce the basic concepts of multichannel collision theory and we apply this theory for illustrative purposes to non-relativistic electron collisions with multi-electron atoms and atomic ions. This chapter thus provides an introduction to our discussion of resonances and threshold behaviour presented in Chap. 3 and to R-matrix theory and applications, presented in Chap. 4 and later chapters in this monograph. We will be mainly concerned in this chapter with low-energy elastic scattering and excitation processes. However, we will show in Chap. 6 that the theory and methods developed in this chapter are the basis of R-matrix methods which enable accurate excitation and ionization processes to be calculated at intermediate energies.
We commence our discussion of multichannel collision theory in Sect. 2.1 by considering the solution of the time-independent Schrödinger equation describing low-energy electron collisions with multi-electron atoms and atomic ions which contain N electrons and have nuclear charge number Z. We define the scattering amplitude in terms of the asymptotic form of the solution of the Schrödinger equation. The differential and total cross sections are then defined in terms of this scattering amplitude. In Sect. 2.2 we consider the atomic or ionic target eigenstates which take part in the collision process. In order to obtain accurate scattering amplitudes and cross sections it is necessary to represent the target by accurate wave functions. We therefore give a brief overview in this section of representations of the target eigenstates, used in most practical applications, where electron exchange and correlation effects are both accurately represented. We also introduce the concept of pseudostates, which enable long-range polarization effects to be accurately included in low-energy electron–atom collisions. A discussion of the further role of pseudostates in representing inelastic effects due to excitation of high-lying bound states and continuum states at incident electron energies close to and above the ionization threshold is reserved for Chap. 6.
In Sect. 2.3 we turn our attention to the derivation of the close coupling equations that can yield accurate low- and intermediate-energy solutions of the time-independent Schrödinger equation describing electron–atom and electron–ion collisions. We commence by showing that the wave function can be expanded in terms of an antisymmetrized sum over target eigenstates and pseudostates multiplied by functions representing the motion of the scattered electron. This close coupling expansion is then substituted into the Schrödinger equation leading to the close coupling equations which are a set of coupled second-order integrodifferential equations satisfied by functions representing the radial motion of the scattered electron. Finally in this section we examine the form of the local and non-local potentials that occur in these close coupling equations. In Sect. 2.4 we examine the asymptotic form of the solution of the close coupling equations which enables us to define the K-matrix which is a generalization of the expression for this quantity in potential scattering given in Chap. 1. We show that the solution of the close coupling equations satisfies the Kohn variational principle, and hence the corresponding K-matrix is correct to second order in the error in the collision wave function. We also show from general considerations that the K-matrix is real and symmetric. Finally, in Sect. 2.5 we define the multichannel S- and T-matrices in terms of the K-matrix, which in turn leads to the derivation of expressions for the differential and total cross sections. In this section we also summarize the angular momentum transfer formalism, which enables several qualitative features of angular distributions to be simply understood, and we define the collision strength and the effective collision strength which have been widely used in plasma physics and astrophysics applications.
1 Wave Equation and Cross Section
We illustrate multichannel collision theory in this chapter by considering non-relativistic low-energy elastic and inelastic electron collisions with multi-electron atoms and atomic ions represented by the equation
where A i and A j are the initial and final bound states of the target. The time-independent Schrödinger equation satisfied by the wave function Ψ describing process (2.1) is
where \(H_{N+1}\) is the non-relativistic Hamiltonian defined in atomic units by
and E is the total energy. It then follows that (2.2) and (2.3) describe the collision of an electron with an atom or atomic ion containing N electrons and with nuclear charge number Z, where we limit ourselves in this chapter to low Z atomic targets so that relativistic effects are negligible. In (2.3) we have taken the origin of coordinates to be the target nucleus, which we assume has infinite mass. Also ∇ i 2 is the Laplacian operator defined in spherical polar coordinates in Appendix B.3 and we have written \(r_{ij}=\left|\textbf{r}_i-\textbf{r}_j\right|\) where r i and r j are the vector coordinates of the ith and jth electrons.
In order to define the scattering amplitude and cross sections we first rewrite \(H_{N+1}\) in terms of the target Hamiltonian H N as follows:
where H N is defined by (2.3) with \(N+1\) replaced by N. We next introduce a set of target eigenstates, and possibly pseudostates, Φ i , and their corresponding energies e i which satisfy the equation
where the integration in this equation is carried out over the space and spin coordinates of the N target electrons. We then look for the solution of (2.2) corresponding to the process represented by (2.1), where an electron in spin state \(\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2}m_i}\) collides with a target atom or ion in state Φ i and is scattered into spin state \(\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2}m_j}\), leaving the atom or ion in state Φ j allowed by the conservation relations, where the z-axis is chosen to lie along the incident beam direction. The asymptotic form of the wave function in the case of a neutral target where \(N=Z\) is then
where r, θ and φ are the radial and spherical polar coordinates of the scattered electron and where \(f_{ji}(\theta,\phi)\) is the scattering amplitude for a transition from state \(\Phi_i\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2}m_i}\) to state \(\Phi_j\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2}m_j}\) corresponding to the scattering angles θ, φ. The direction of spin quantization is usually taken to be the incident beam direction and the wave numbers k i and k j , for the incident and scattered electrons, are related to the total energy E of the system and to the target eigenenergies e i and e j by the equation
The outgoing wave term in (2.6) contains contributions from all target states that are energetically allowed, that is for which \(k_j^2\geq 0\). The remaining states, for which \(k_j^2< 0\), can only occur virtually during the collision process. These virtual states play an important role when the scattered electron lies within the target charge cloud and we will see that they can give rise to resonances in the collision process.
If the incident electron energy is high enough then continuum states of the target can be excited and contribute to the asymptotic form in (2.6). These terms correspond to ionizing collisions. We will consider this possibility in Sect. 3.3.5 when we discuss the threshold behaviour of ionization and in Chap. 6 when we discuss intermediate-energy electron–atom collisions.
We also note that when the target is an atomic ion, logarithmic phase factors must be included in the exponentials in (2.6) to allow for the long-range distortion caused by the Coulomb potential. This introduces no essential complications so we will not consider these factors further here, but will return to consider their effect on the cross section in Sect. 2.5.
The differential cross section for a transition from an initial atomic state Φ i to a final atomic state Φ j , with the scattered electron spin magnetic quantum number changing from m i to m j and its wave number changing from k i to k j , can be obtained by calculating the incident and scattered fluxes in (2.6). We obtain
in units of a0 2/steradian, where a0 is the Bohr radius of the hydrogen atom in its ground state. The total cross section is then obtained by averaging over the initial spin states, summing over the final spin states and integrating over all scattering angles.
2 Target Eigenstates and Pseudostates
In order to calculate the wave function Ψ in (2.1) describing the collision process and hence the scattering amplitude and cross sections we must first consider how the target eigenstates, and possibly pseudostates, Φ j , are represented in the theory. In this section we give a brief overview of the representations that are adopted for these target states in non-relativistic electron–atom and electron–ion collision calculations.
2.1 Target Eigenstates
For multi-electron atoms and ions, the target eigenstates are not known exactly. Hence in most electron collision calculations they are written as configuration interaction expansions in terms of sums over an orthonormal set of target basis configurations φ i in the form
as discussed by Hartree [445], Froese Fischer et al. [346, 349], Hibbert [465] and Cowan [233]. In this equation \(\textbf{X}_N\equiv \textbf{x}_1,\dots,\textbf{x}_N\), where \(\textbf{x}_i\equiv\textbf{r}_i\sigma_i,\; i=1,\dots,N\), represent the space and spin coordinates of the N target electrons and the expansion coefficients c ij are obtained by diagonalizing the target Hamiltonian in (2.5) in this basis. These calculations can be carried out using one of a number of atomic multiconfiguration atomic structure programs, which we refer to in Sect. 5.1.1. We assume in the following discussion that the atomic orbitals for each orbital angular momentum are constrained to be orthogonal, corresponding to most atomic structure and collision programs. However, we observe that non-orthogonal orbitals are finding increasing use in atomic structure and collision calculations considered in later chapters.
The basis configurations φ i in (2.9) are constructed from N one-electron orbital and spin functions which have the form
where the reduced radial orbitals \(P_{n\ell}(r)\) satisfy the orthonormality relations
for each orbital angular momentum ℓ. Also \(Y_{\ell m_\ell}(\theta,\phi)\) are spherical harmonics, which are defined and discussed in Appendices B.3 and B.4, and \(\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2} m_i}(\sigma)\) are electron spin eigenfunctions. In the absence of relativistic terms in the Hamiltonian, the orbital and spin angular momenta of the one-electron functions are coupled together to yield completely antisymmetrized configurations, which are eigenfunctions of the square of the total N-electron target orbital and spin angular momentum operators L 2 and S 2 and their z-components L z and S z as well as the total target parity operator π. We can write these basis configurations more explicitly as
where the N ji are the occupation numbers of the target shells, which satisfy
Also in (2.12), β i denotes the coupling of the target shells, L i and S i are the total target orbital and spin angular momentum quantum numbers, \(M_{L_{i}}\) and \(M_{S_{i}}\) are the corresponding magnetic quantum numbers in some preferred direction and π i is the total target parity quantum number. Each target eigenstate Φ j involves a summation over basis configurations φ i that have the same total orbital, spin and parity quantum numbers but differ in the occupation numbers or the coupling. We can thus write these target eigenstates more explicitly as
where the quantity α j serves to distinguish different target states with the same total target orbital, spin and parity quantum numbers.
The basis configurations in (2.9) usually include the Hartree–Fock configuration of the target ground state or a low-lying excited state. Hence the reduced radial orbitals \(P_{n\ell}(r)\) include the self-consistent field (SCF) orbitals. Additional “physical orbitals” are then included to represent the other target states of interest in the calculation and possibly further “pseudo-orbitals” are included to represent additional correlation and polarization effects. These orbitals are either expressed in analytical form as a sum of Slater-type orbitals (STOs) defined by
where \(k_j\ge \ell+1\) and the coefficients b j , k j and ξ j depend on n and ℓ, or the orbitals are tabulated at a grid of points.
As an example, we consider the target eigenstates that have been adopted in several studies of low-energy electron collisions with Be-like ions C2+ and O4+. In this case electron collisional excitation cross sections between the following six target eigenstates are important in many applications (see, for example, [97])
Accurate low-energy excitation cross sections can then be obtained using the following physical orbitals and pseudo-orbitals in the representation of the target eigenstates
where we distinguish the \(\overline{3\textrm{s}}, \overline{\textrm{3p}}\) and \(\overline{\textrm{3d}}\) pseudo-orbitals from the 1s, 2s and 2p physical orbitals by placing a bar over the pseudo-orbitals.
The target eigenstates are constructed by diagonalizing the target Hamiltonian matrix, defined in (2.5), in the basis of configurations defined by (2.9). These configurations are constructed from the physical and pseudo-orbitals assuming that the 1s orbital remains doubly occupied. Configurations where one or two electrons are excited out of the 1s orbital correspond to high-energy excitations which are not important in low-energy electron collisions. A list of configurations that can be constructed from the orbital basis defined by (2.17) for each target eigenstate is given in Table 2.1, where we find it convenient to put these configurations into categories depending on whether zero, one or two electrons are excited from physical to pseudo-orbitals.
The choice of the physical and pseudo-orbitals is not unique and care must be taken in choosing them. One appropriate choice is to take the 1s and 2s orbitals to be the Hartree–Fock orbitals from the \(\textrm{1s}^22\textrm{s}^2\;^1\textrm{S}^{\textrm{e}}\) ground state and the 2p orbital to be the Hartree–Fock orbital from the \(\textrm{1s}^22\textrm{s}2\textrm{p}\;^3\textrm{P}^{\textrm{o}}\) first excited state. The \(\overline{\textrm{3s}}\), \(\overline{\textrm{3p}}\) and \(\overline{\textrm{3d}}\) pseudo-orbitals, which are orthogonal to the physical orbitals with the same angular symmetry, can then be chosen to optimize the energies of the remaining four excited states in (2.16). If we assume that the target eigenstates Φ j in (2.9) are expanded in terms of zero-electron and one-electron excitation configurations from Table 2.1, then the six target eigenstates in (2.16) are expanded as follows:
where the dominant configuration is the first configuration in the list in each case. The coefficients defining the pseudo-orbitals in (2.15) can be determined by minimizing the energies of the excited states defined by (2.18) using an atomic structure program (e.g. [464], or an equivalent program). For example, the \(\overline{\textrm{3s}}\) pseudo-orbital could be chosen to allow for the difference of the 2s orbital in the \(\textrm {1s}^22\textrm{s}^2\; ^1\textrm{S}^{\textrm{e}}\) ground state and in the \(\textrm{1s}^22\textrm{s}2\textrm{p}\; ^3\textrm{P}^{\textrm{o}}\) and \(\textrm{1s}^22\textrm{s}2\textrm{p}\; ^1\textrm{P}^{\textrm{o}}\) excited states. Thus the \(\overline{\textrm{3s}}\) pseudo-orbital coefficients could be optimized on a linear combination of the \(\textrm{1s}^22\textrm{s}2\textrm{p}\; ^3\textrm{P}^{\textrm{o}}\) and \(\textrm{1s}^22\textrm{s}2\textrm{p}\; ^1\textrm{P}^{\textrm{o}}\) excited state energies. Also the \(\overline{\textrm{3p}}\) pseudo-orbital could be chosen to allow for the difference of the 2p orbital in the \(\textrm {1s}^22\textrm{s}2\textrm{p}\; ^3\textrm{P}^{\textrm{o}}\) first excited state and in the \(\textrm{1s}^22\textrm{p}^2\; ^3\textrm{P}^{\textrm{e}}\) and \(\textrm{1s}^22\textrm{p}^2\; ^1\textrm{S}^{\textrm{e}}\) excited states. Thus the \(\overline{\textrm{3p}}\) pseudo-orbital coefficients could be optimized on a linear combination of the \(\textrm{1s}^22\textrm{p}^2\; ^3\textrm{P}^{\textrm{e}}\) and \(\textrm{1s}^22\textrm{p}^2\; ^1\textrm{S}^{\textrm{e}}\) excited state energies. Finally, the \(\overline{\textrm{3d}}\) pseudo-orbital coefficients could be optimized on the \(\textrm{1s}^22\textrm{p}^2\; ^1\textrm{D}^{\textrm{e}}\) excited state energy.
We see from Table 2.1 that in addition to the zero-electron and one-electron excitation configurations, which we included in expansions (2.18) of the target eigenstates, we could also include two-electron excitation configurations. This would improve the target eigenstates by including additional electron–electron correlation effects. However, it is important to ensure that the correlation effects included in the target states balance those included in the collision wave function in order to obtain accurate collision results. We will see in Chap. 6, where we discuss electron collisions at intermediate energies, that the inclusion of two-electron excitation configurations in the collision wave function can give rise to unphysical or pseudo-resonances at these energies. We will therefore defer further discussion of this point until that chapter.
2.2 Target Pseudostates
In certain circumstances determination of target states which are not eigenstates of the target Hamiltonian is required to obtain accurate electron–atom collision cross sections. These states, which are usually called pseudostates, are found to be particularly useful in low-energy electron–atom and electron–molecule collisions, where the long-range polarization potential gives an important contribution to the cross section. We will see in Chaps. 6 and 11 that target pseudostates can also be used to represent the ionization continuum in electron–atom and electron–molecule collisions at intermediate energies.
For an atom in a non-degenerate S-state, the long-range polarization potential has the asymptotic form
where the quantity α which appears in this equation is the dipole polarizability. This is defined by the expression (see [243])
where the summation and integration in this equation are taken over all target eigenstates Φ k , including the continuum, which are coupled to the ground state Φ 0 by the dipole operator
and where the eigenenergies e k are defined by
It was shown by Castillejo et al. [205] that in the case of electron collisions with atomic hydrogen in its ground state, 65.8% of the dipole polarizability comes from including the 2p state in expansion (2.20) while 81.4% of the dipole polarizability comes from the sum over all bound states, the remaining 18.6% coming from the continuum terms in the expansion. In this case, Damburg and Karule [245] showed that a p-wave pseudostate denoted by \(\overline{\textrm{2p}}\) enables expansion (2.20) to be replaced by a single term. This polarized pseudostate has the same range as the 1s ground state orbital and has the reduced radial form
The corresponding pseudostate energy e p, defined by
has the value −7/86 a.u., where Φ p is the pseudostate wave function. Clearly this energy is not an eigenenergy of the target Hamiltonian. However, if this polarized pseudostate, as well as the ground target eigenstate, is included in the close coupling expansion of the collision wave function, as discussed in Sect. 2.3 and Chaps. 5 and 6, then the full long-range part of the polarization potential given by (2.19) is represented in the collision process. Elastic e−–H collision calculations including this pseudostate were first carried out at energies below the 2s and 2p excitation threshold by Burke et al. [177].
In the case of multi-electron atoms and atomic ions, the polarized pseudostates, like the target eigenstates, cannot be written down exactly. In this case a variational principle [166, 167, 941] can be used to calculate these pseudostates. We consider the following inhomogeneous equation for the unnormalized pseudostate \(\widetilde{\Phi}^{p}\)
This equation has the formal solution
where the spectral representation of the Green’s function \((H_N-e_0)^{-1}\) has been used, which involves a summation over the discrete spectrum and an integration over the continuum spectrum of H N . Substituting (2.26) into (2.20) then gives
In order to write this equation in the form of (2.20) containing a single term we normalize \(\widetilde{\Phi}^{p}\) by introducing the constant n p defined by
The normalized pseudostate Φ p is then given by
Substituting this result into (2.27) then gives
The final step is to eliminate \(n_{p}^{1/2}\) from this equation. To do this we project (2.25) onto \(\widetilde{\Phi}^{p}\) yielding
which gives, after using (2.29)
Substituting this result for \(n_{p}^{1/2}\) into (2.30) then gives
where the energy e p of the polarized pseudostate is given by (2.24). If we include the ground state Φ 0 and the polarized pseudostate Φ p in the close coupling expansion of the wave function describing electron collisions with atomic hydrogen, as discussed in Sect. 2.3 and Chaps. 5 and 6, then the full long-range part of the dipole polarization potential given by (2.19) is represented in the collision process. In this way we have replaced the summation and integration in the expression for the dipole polarizability given by (2.20) by a single pole term given by (2.33).
The problem of calculating polarized pseudostates for complex targets reduces to solving the inhomogeneous equation (2.25) to obtain \(\widetilde{\Phi}^{p}\) and then normalizing this solution using (2.29) to give the required polarized pseudostate Φ p. We can solve (2.25) by introducing a trial function \(\widetilde{\Phi}_\textrm{t}^{{p}}\) and considering the variational functional
The first-order variation δ J of the functional J with respect to small variations \(\delta\widetilde{\Phi}_\textrm{t}^{p}\) in the trial function \(\widetilde{\Phi}_\textrm{t}^{p}\) is
which is zero when \(\widetilde{\Phi}_\textrm{t}^{p}\) is an exact solution of (2.25). We construct a trial function in analogy with the target eigenstates given by (2.9), by expanding \(\widetilde{\Phi}_\textrm{t}^{p}\) in terms of a sum of orthonormal basis configurations \(\widetilde{\phi}_j\) with the appropriate symmetry as follows:
Substituting this expansion into (2.34) and varying the coefficients b j leads to the system of m linear simultaneous equations
which can be solved to yield the coefficients b j , and hence \(\widetilde{\Phi}_\textrm{t}^{p}\) and the normalized polarized pseudostate Φ p can be constructed.
Also we see from (2.34) that the second-order variation \(\delta^2 J[\widetilde{\Phi}_\textrm{t}^{p}]\) satisfies
since e 0 is the lowest eigenvalue of H N so that \(H_N-e_0\) is a positive definite operator. Hence the minimum value of J is obtained when \(\widetilde{\Phi}_\textrm{t}^{p}\) is the exact solution of (2.25). Further at the minimum \(J_{\textrm{min}}\) of J we have
which follows from (2.25), (2.27) and (2.34). Hence in constructing the polarized pseudostate it is possible to improve this state by varying the radial orbitals used in the definition of the basis configurations in (2.36) to minimize J or to maximize α.
As an example of the above theory we consider the calculation of the polarized pseudostate required to represent low-energy elastic electron collisions with neon. In this case a reasonably good approximation for the elastic collision process is obtained by representing the neon ground state by the Hartree–Fock \(\textrm{1s}^22\textrm{s}^22\textrm{p}^6\;^1\textrm{S}^{\textrm{e}}\) configuration and the polarized pseudostate, which has \(^\textrm{1}\textrm{P}^{\textrm{o}}\) symmetry, by a linear combination of the following basis configurations:
The additional polarized pseudo-orbitals \(\overline{j\hbox{s}}\), \(\overline{j\hbox{p}}\) and \(\overline{j\hbox{d}}\) must satisfy the usual orthonormality relations given by (2.11) but are not physical. Indeed, like the \(\overline{2\textrm{p}}\) orbital representing the polarized pseudostate in atomic hydrogen given by (2.23), their range is determined by the range of the ground state of the target atom whose polarizability they are representing rather than by the range of the excited states. In a study carried out by Burke and Mitchell [166], the \(\overline{j\textrm{s}}, \overline{j\textrm{p}}\) and \(\overline{j\textrm{d}}\) polarized pseudo-orbitals were expanded in terms of basis orbitals with the following reduced radial form:
In this equation, β is a range parameter and the coefficients \(a_{ij\ell}\) were chosen so that these pseudo-orbitals were orthogonal to the 1s, 2s and 2p Hartree–Fock orbitals of the same angular symmetry and orthonormal to each other. Equation (2.37) was then solved for a series of values of this range parameter. Figure 2.1 shows the variation of the dipole polarizability with the range parameter β for three expansions including all configurations in (2.40) with \(n= \) 4, 5 and 6, respectively. For example, when \(n= 4\) six basis configurations corresponding to the pseudo-orbitals \(\overline{3\textrm{s}}, \overline{4\textrm{s}}\), \(\overline{3\textrm{p}}, \overline{4\textrm{p}}\), \(\overline{3\textrm{d}}\) and \(\overline{4\textrm{d}}\) are retained in expansion (2.36). Also shown in Fig. 2.1 is the experimental value of the dipole polarizability determined from experimental data by Dalgarno and Kingston [242]. As the number of terms in the basis increases the curves become flatter and converge towards the experimental value. However, this calculation does not give a rigorous lower bound on the exact dipole polarizability since an exact target ground state was not used. Polarized pseudostates of this type have been used in electron–neon elastic scattering calculations by Blum and Burke [120] and Fon and Berrington [327].
The above theory can be modified in a straightforward way so that pseudostates can be calculated which represent higher multipole polarizabilities of the target and also which represent the dipole polarizabilities of excited states of the target. In addition, pseudostates can be chosen which allow in an average way for the loss of flux into the infinite number of high-lying Rydberg states and continuum states of the target, thus representing ionization in electron–atom collisions. We will discuss the construction and application of such pseudostates when we consider electron–atom collisions at intermediate energies in Chap. 6.
3 Close Coupling Equations
In the previous section we showed how accurate wave functions can be obtained for the target eigenstates and pseudostates which occur in electron collisions with multi-electron atoms or atomic ions. We turn our attention in this section to the determination of the electron–atom or electron–ion collision wave function Ψ that satisfies the non-relativistic Schrödinger equation (2.2). In Sect. 2.3.1 we review the foundations of the method which involves the solution of a set of “close coupling equations” also known as “coupled ID equations” which enables accurate excitation and ionization cross sections to be determined at low and intermediate energies. Then in Sect. 2.3.2 we describe the explicit form of the close coupling equations which must be solved in practical calculations.
3.1 Foundations of the Method
The foundations of methods for solving the Schrödinger equation (2.2) to obtain accurate elastic scattering, excitation and ionization cross sections for low- and intermediate-energy electron–atom and electron–ion collisions were laid by Massey and Mohr [642, 643] and Mott and Massey [665]. They introduced the following “close coupling” expansion of the total wave function describing electron collisions with an N-electron atom or atomic ion
where \(\textbf{X}_{N+1}\equiv \textbf{x}_1,\dots,\textbf{x}_{N+1}\) and where \(\textbf{x}_i\equiv \textbf{r}_i\sigma_i,\;i=1,\dots,N+1,\) represent the space and spin coordinates of the \(N+1\) electrons. The summation in (2.42) goes over the bound target eigenstates and the integration goes over the continuum target eigenstates, which are described by \(\Phi_i(\textbf{X}_N)\), and the functions \(F_i(\textbf{ x}_{N+1})\) describe the corresponding motion of the scattered electron. We now substitute expansion (2.42) into the Schrödinger equation (2.2) and project onto the target eigenstates \(\Phi_i(\textbf{X}_N)\) to yield the following infinite set of coupled second-order partial differential equations satisfied by the functions \(F_i(\textbf{x}_{N+1})\)
Here k i 2 is defined by (2.5) and (2.7) and the potential matrix \(V_{ij}(\textbf{x}_{N+1})\) is defined by
where the integration in this matrix element goes over the space and spin coordinates of the N target electrons.
Although the solution of Schrödinger’s equation (2.2) given by (2.42) and (2.43) in principle gives an accurate description of the collision, one question which arises is how can electron exchange, which is implicit in the theory, be calculated. The importance of exchange is well known in many applications, for example in “forbidden transitions” between the \(\textrm{1s}^22\textrm{s}^22\textrm{p}^2\:^3\textrm{P}^{\textrm{e}}, ^1\textrm{D}^{\textrm{e}}\) and \(^1\textrm{S}^{\textrm{e}}\) terms of O III (O2+) which give rise to prominent lines in the spectra of many gaseous nebulae and active galactic nuclei [709]. On examining expansion (2.42) we see that electron exchange arises from the continuum terms in the expansion. In this case the incident electron labelled \(N+1\) is captured into a bound eigenstate and one of the target electrons labelled \(1,\dots,N\) is ejected into a continuum state. While this process can be calculated using perturbation theory the resultant cross section can be significantly in error (e.g. [77]). On the other hand, the corresponding solution of the coupled equations (2.43) gives rise to difficulties owing to singularities which occur in integration over the continuum terms in the expansion corresponding to electron exchange. In nuclear structure and collisions these singularities were avoided in early work by Wheeler [960, 961], by expanding the total wave function in antisymmetric resonating groups of nucleons. We now discuss how these difficulties are resolved in electron–atom multichannel collision theory.
In electron–atom collisions the difficulties owing to singularities arising in the continuum due to electron exchange were overcome in a fundamental paper by Seaton [848], who extended the Hartree–Fock equations for bound states, in which electron exchange is treated using explicitly antisymmetric wave functions, to the treatment of continuum states. In this paper expansion (2.42) is replaced by the close coupling expansion
where the summation in this equation is now restricted to a finite number of bound antisymmetric target eigenstates \(\Phi_i(\textbf{X}_N)\) which satisfy (2.5). In addition \({\cal A}\) is the antisymmetrization operator which ensures that each term in expansion (2.45) is antisymmetric with respect to interchange of the space and spin coordinates of any pair of the \(N+1\) electrons. We find that \({\cal A}\) is defined by
where \(P_{iN+1}\) is the operator which interchanges the space and spin coordinates of electrons labelled i and \(N+1\). It follows that the total wave function \(\Psi(\textbf{X}_{N+1})\) defined by (2.45) is antisymmetric with respect to interchange of the space and spin coordinates of any pair of the \(N+1\) electrons, in accordance with the Pauli exclusion principle.
Following Burke and Seaton [164], we consider first the uniqueness of the solution defined by (2.45) and (2.46). In the case where \(N=1\), corresponding to electron collisions with hydrogenic targets, (2.45) becomes, after using (2.46),
We now write
where the summation in this equation goes over the same set of bound target eigenstates \(\Phi_i(\textbf{x})\) which are retained in (2.47). In this way we have defined a new function \(\overline{F}_i(\textbf{x})\) for any given set of coefficients b ij . Substituting (2.48) into the right-hand side of (2.47) then gives
We see that if
then the second summation on the right-hand side of (2.49) vanishes and hence the wave function \(\Psi(\textbf{x}_1,\textbf{x}_2)\) is unaltered by the transformation defined by (2.48). It follows that the functions \(F_i(\textbf{x})\) defined by (2.48) are not unique and that different functions defined by this equation will yield the same wave function \(\Psi(\textbf{x}_1,\textbf{x}_2)\) provided that the coefficients b ij satisfy the symmetry relations given by (2.50).
On the other hand, the asymptotic form of the functions \(F_i(\textbf{x})\) is unique since the bound target eigenstates \(\Phi_i(\textbf{x})\) retained in (2.47) vanish asymptotically. Hence the scattering amplitudes and cross sections which are determined from the asymptotic form of the functions \(F_i(\textbf{x})\) are not modified by transformation (2.48).
We now consider three different procedures for choosing the coefficients b ij in (2.48):
-
i.
We may use expansion (2.47) without introducing explicit conditions which suffice to define \(F_i(\textbf{x})\) uniquely. This may lead to loss of accuracy in the numerical solution of the coupled integrodifferential equations, which we will see below are satisfied by the functions F i (x).
-
ii.
We may introduce conditions which are sufficient to define the functions \(F_i(\textbf{x})\) uniquely, but which do not change the form of (2.47). Thus, for example, we could impose the orthogonality conditions
$$\langle \Phi_i|F_j\rangle=0,\quad i\leq j,$$((2.51))where we list the states in some definite order. This procedure has been widely discussed [848, 161, 164, 289] and was shown by Norcross [692] to improve the accuracy of the numerical integrations.
-
iii.
We may impose the orthogonality conditions
$$\langle \Phi_i|F_j\rangle=0,\quad \hbox{all } i,\;j,$$((2.52))and replace (2.47) by
$$\begin{array}{rcl}\Psi(\textbf{x}_1,\textbf{x}_2)&&=\frac{1}{\sqrt{2}} \sum_i\left[\Phi_i(\textbf{x}_1)F_i(\textbf{x}_2)- \Phi_i(\textbf{x}_2)F_i(\textbf{x}_1)\right]\\ &&\quad+\frac{1}{\sqrt{2}}\sum_{i\leq j}\left[\Phi_i(\textbf{ x}_1)\Phi_j(\textbf{x}_2) -\Phi_i(\textbf{x}_2)\Phi_j(\textbf{ x}_1)\right]c_{ij}. \end{array}$$((2.53))We then have to solve for the functions \(F_i(\textbf{x})\) and for the coefficients c ij , subject to the orthogonality conditions (2.52). This method has the advantage of being easy to generalize to the case of electron collisions with atoms and ions containing many electrons and has been adopted in many recent theoretical developments, which we discuss in Sect. 2.3.2 and in Chap. 5.
Returning to (2.45), we now substitute this expansion into the Schrödinger equation (2.2) and project onto the target eigenstates Φ i to yield the following set of coupled second-order integrodifferential equations, satisfied by the functions \(F_i(\textbf{x})\)
In this equation the potential matrix elements \(V_{ij}(\textbf{x})\) coupling the target states are the same as in (2.43) and the new exchange kernel \(K_{ij}(\textbf{x},\textbf{x}^{\prime})\) arises from the operator \(P_{iN+1}\) in (2.46) and gives rise to electron exchange in the collision. The solution of (2.54) now yields both the direct and exchange scattering amplitudes for transitions between the target states Φ i retained in the original expansion (2.45).
However, we observe that an exact solution of these coupled equations will not yield an exact solution of the original Schrödinger equation (2.2) because of the truncation of expansion (2.45) to a finite number of bound target eigenstates. In many cases of interest, involving transitions between strongly coupled low-energy eigenstates, the resultant solution will be accurate. However, the omission from the expansion of an infinite number of bound target eigenstates lying close to the ionization threshold, as well as all the continuum target eigenstates can lead to substantial errors for some transitions, particularly for incident-electron energies close to and above the ionization threshold, often referred to as “intermediate energies”. In addition, since the continuum eigenstates are omitted from expansion (2.45), the possibility of determining ionization resulting from the excitation of these continuum eigenstates is not included in the calculation.
We now consider a straightforward extension of the close coupling expansion (2.45) which has enabled accurate ionization cross sections as well as excitation cross sections to be determined at intermediate energies. We observed in Sect. 2.2.2 that an effective way of representing the long-range polarization potential, where a substantial contribution to this potential comes from intermediate target eigenstates lying in the continuum, is to introduce a quadratically integrable polarized pseudostate which has a substantial overlap with the continuum. This pseudostate replaces the usual integral expression for the dipole polarizability, given by (2.20), by a single pole term, given by (2.33). In an analogous way, including a finite number of discrete quadratically integrable target pseudostates in the expansion has been found to be an effective way of representing the continuum in electron collisions. In this approach the eigenstate close coupling expansion (2.45) is replaced by the following “close coupling with pseudostates” expansion suggested by Burke and Schey [160]
The first summation in this equation goes over a finite number of bound target eigenstates \(\Phi_i(\textbf{X}_N)\), as in (2.45), and the second summation goes over a finite number of suitably chosen quadratically integrable target pseudostates \(\Phi_i^{p}(\textbf{X}_N)\) representing the highly excited and continuum target eigenstates. The functions \(F_i(\textbf{x})\) and \(G_i(\textbf{x})\) represent the corresponding motion of the scattered electron. The pseudostates are chosen to be orthogonal to the bound target eigenstates retained in the first expansion in (2.55) and to diagonalize the target Hamiltonian H N as follows:
where the pseudostate energies e i p partially span the energy range, including the continuum, which is omitted from the first expansion. Substituting expansion (2.55) into the Schrödinger equation (2.2) and projecting onto the target eigenstates \(\Phi_i(\textbf{X}_N)\) and onto the target pseudostates \(\Phi_i^{p}(\textbf{X}_N)\) then yields a set of coupled second-order integrodifferential equations satisfied by the functions \(F_i(\textbf{x})\) and \(G_i(\textbf{x})\) which have the same form as (2.54).
We will consider in detail the choice and role of pseudostates in the close coupling expansion when we discuss electron collisions at low and intermediate energies in Chaps. 5 and 6, respectively. We will see in these chapters that approaches based on the close coupling with pseudostates expansion, including the R-Matrix with PseudoStates (RMPS) method, introduced by Bartschat et al. [70, 71] and discussed in Sect. 6.2, and the convergent close coupling (CCC) method, introduced by Bray and Stelbovics [126–128] and reviewed in Sect. 6.1, yield accurate cross sections over a wide range of electron collision energies.
3.2 Derivation of the Close Coupling Equations
We now turn our attention to determine the explicit form of the close coupling equations which must be solved in practical calculations. We first observe that in order to minimize the computational effort we must use the symmetry of the Hamiltonian to separate these equations into uncoupled blocks, corresponding to the conserved quantum numbers, which can be solved independently. In addition, in order to make the solution of these coupled equations tractable for electron collisions with multi-electron atoms and ions, a partial wave analysis must also be carried out. In this way we obtain sets of coupled second-order integrodifferential equations which are satisfied by the wave functions representing the radial motion of the scattered electron. We will then examine the detailed form of these close coupling equations including the local direct and the non-local exchange and correlation potentials that arise. In this way we provide the basis of the R-matrix theory approach for solving these equations which we will discuss in Chap. 5.
Following Burke [159] the required close coupling with pseudostates expansion, which replaces (2.55), has the following form for each set of conserved quantum numbers represented by Γ
where j labels the linearly independent solutions of the Schrödinger equation (2.2), which we will discuss in detail in Sect. 2.4 when we consider the asymptotic boundary conditions satisfied by the functions \(F_{ij}^\Gamma(r_{N+1})\). The conserved quantum numbers represented by Γ in (2.57) correspond to the eigenvalues of the complete set of operators which commute with the Hamiltonian. In the case of the non-relativistic Hamiltonian defined by (2.3) these conserved quantum numbers are given by
where L and S are the total orbital and spin angular momentum quantum numbers, M L and M S are the corresponding magnetic quantum numbers in some preferred direction z, π is the total parity quantum number and α represents any further quantum numbers which are conserved in the collision. Also, the channel functions \(\overline{\Phi}_i^\Gamma(\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\) in (2.57) are obtained by coupling the target eigenstates and pseudostates retained in the expansion with the spin–angle functions of the scattered electron to form eigenfunctions of the square of the total orbital and spin angular momentum operators L 2 and S 2 and their z-components as well as the parity operator π. Hence the channel functions can be written as follows:
where \(\Phi_i(\textbf{X}_N)\) are the antisymmetric target eigenstates and pseudostates, discussed above, \(Y_{\ell_im_{\ell_i}}(\theta_{N+1},\phi_{N+1})\) are spherical harmonics, defined in Appendix B.3, which describe the angular motion of the scattered electron, \(\hbox{\raisebox{.4ex}{$\chi$}}_{\frac{1}{2} m_i}(\sigma_{N+1})\) are electron spin functions which describe the spin motion of the scattered electron and \((abcd|ef)\) are Clebsch–Gordan coefficients defined in Appendix A.1. Returning to (2.57), the reduced radial functions \(F_{ij}^\Gamma(r_{N+1})\) describe the radial motion of the scattered electron in the ith channel and the \(\hbox{\raisebox{.4ex}{$\chi$}}_i^\Gamma(\textbf{X}_{N+1})\) are quadratically integrable functions which vanish at large distances from the nucleus. These quadratically integrable functions are usually constructed from the same set of physical and pseudo-orbitals used to construct the target eigenstates and pseudostates \(\Phi_i(\textbf{X}_N)\) and are antisymmetric with respect to interchange of the space and spin coordinates of any pair of the \(N+1\) electrons. We discuss the reasons for the inclusion of these quadratically integrable functions in the expansion of the wave function below. Finally, the antisymmetrization operator \({\cal A}\) defined by (2.46) ensures that the total wave function is explicitly antisymmetric with respect to interchange of the space and spin coordinates of any pair of the \(N+1\) electrons, in accordance with the Pauli exclusion principle.
We observe that in order to obtain accurate scattering amplitudes we must include in the first expansion on the right-hand side of (2.57) all the target states of physical interest. By this we mean that we must include both the initial and final target eigenstates corresponding to the scattering amplitude of interest, as well as all other target eigenstates that are expected to play an important role as intermediate states in the transitions of interest. In particular, if one term of a target configuration is included in this expansion then all other terms corresponding to this configuration are normally strongly coupled to this configuration and should also be included. For example, if we are considering electron collisions with atomic oxygen in its \(\textrm{1s}^22\textrm{s}^22\textrm{p}^4\;^3\textrm{P}^{\textrm{e}}\) ground state, then the other \(\textrm{1s}^22\textrm{s}^22\textrm{p}^4\;^1\textrm{D}^{\textrm{e}}\) and \(\textrm{1s}^22\textrm{s}^22\textrm{p}^4\;^1\textrm{S}^{\textrm{e}}\) terms corresponding to this ground state configuration will be strongly coupled and must also be included to obtain accurate results. We may also need to include pseudostates in this expansion, either to accurately represent polarization effects at low energies, as discussed in Sect. 2.2.2, or to represent the highly excited and continuum states of the target in order to obtain accurate excitation and ionization cross sections at intermediate energies, as discussed later in this section and in Chap. 6.
The second expansion on the right-hand side of (2.57), over the quadratically integrable functions \(\hbox{\raisebox{.4ex}{$\chi$}}_i^\Gamma(\textbf{X}_{N+1})\), is included for two reasons. First, as discussed in Sect. 2.3.1, the reduced radial functions \(F_{ij}^\Gamma(r)\) are in many calculations constrained to be orthogonal to the physical orbitals and pseudo-orbitals with the same angular symmetry which are used in the construction of the target states \(\Phi_i(\textbf{X}_N)\). For example, in electron collisions with atomic oxygen the p-wave reduced radial function is constrained to be orthogonal to the 2p orbital in the target. However, this constraint means that the \(\textrm{ 1s}^22\textrm{s}^22\textrm{p}^5\;^2\textrm{P}^{\textrm{o}}\) configuration, which plays an important role as an intermediate state in 2Po electron collisions with the \(\textrm{1s}^22\textrm{s}^22\textrm{p}^4\;^3\textrm{P}^{\textrm{e}}, ^1\textrm{D}^{\textrm{e}}\) and \(^1\textrm{S}^{\textrm{e}}\) target states, is not represented in the first expansion. This configuration must therefore be included in the second expansion for completeness, to ensure that the 2Po collision wave function represents this possibility. This example also re-emphasizes the importance of including all three target state terms belonging to the \(\textrm{1s}^22\textrm{s}^22\textrm{p}^{4}\) configuration in the first expansion in (2.57), since they are strongly coupled through the \(\textrm{ 1s}^22\textrm{s}^22\textrm{p}^5\;^2\textrm{P}^{\textrm{o}}\) intermediate quadratically integrable function and their omission would lead to inconsistencies, including the appearance of low-energy pseudoresonances in the cross sections, as pointed out by Gorczyca et al. [398].
The second reason for including quadratically integrable functions in the second expansion in (2.57) is to represent short-range electron–electron correlation effects, which may be difficult to represent accurately by including a finite expansion over target states and pseudostates in the first expansion in (2.57). In the case of one-electron targets, such as H and He+, highly accurate electron collision phase shifts and cross sections have been obtained at low energies by Schwartz [838, 839], Burke and Taylor [162] and Armstead [24], by taking the terms in the second expansion to be Hylleraas-type functions. In the case of multi-electron targets these correlation effects are usually included by the introduction of additional contracted pseudo-orbitals with approximately the same range as the Hartree–Fock orbitals used to construct the target wave functions, but with more nodes. Additional (\(N+1\))-electron quadratically integrable functions, constructed from the physical orbitals and the pseudo-orbitals, must then be included in the second expansion in (2.57) for consistency. Finally, we note that the inclusion of quadratically integrable functions in the second expansion in (2.57), to represent short-range electron–electron correlation effects, gives rise to unphysical pseudoresonances at intermediate energies. We discuss the role of these pseudoresonances later in this section and in Chap. 6.
We now derive coupled second-order integrodifferential equations satisfied by the reduced radial functions \(F_{ij}^\Gamma(r_{N+1})\) in (2.57). These equations are obtained by substituting the representation for \(\Psi_{jE}^\Gamma(\textbf{X}_{N+1})\) given by (2.57) into the Schrödinger equation (2.2) and projecting onto the channel functions \(\overline{\Phi}_i^\Gamma\), defined by (2.59) and onto the quadratically integrable functions \(\hbox{\raisebox{.4ex}{$\chi$}}_i^\Gamma(\textbf{X}_{N+1})\). In this way we obtain the following set of \(n+m\) coupled equations
and
subject to the orthogonality constraints
The prime on the Dirac bracket in (2.60) and later equations means that the integration is carried out over the space and spin coordinates of all \(N+1\) electrons except the radial coordinate \(r_{N+1}\) of the scattered electron. In (2.61) the integration is carried out over the space and spin coordinates of all \(N+1\) electrons. Finally, the orthogonality constraints (2.62) are required to ensure that the reduced radial functions \(F_{ij}^\Gamma(r)\) are orthogonal to all the physical and pseudo-orbitals \(P_{n_s\ell_i}(r)\) of the same angular symmetry ℓ i , which are used to construct the target states retained in expansion (2.57), as discussed above.
We now eliminate the expansion coefficients \(c_{ij}^\Gamma\) in (2.57) between (2.60) and (2.61), by substituting the expression for these coefficients obtained from (2.61) into (2.60). After writing the Hamiltonian \(H_{N+1}\) in terms of H N using (2.4), we find that the reduced radial functions \(F_{ij}^\Gamma(r)\) satisfy the following set of n coupled second-order integrodifferential equations which are called the “close coupling equations” or “coupled ID equations”
In (2.63) ℓ i is the orbital angular momentum of the scattered electron, k i 2 is the wave number squared of the scattered electron, \(V_{ii^{\prime}}^\Gamma(r)\), \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) are the local direct, non-local exchange and non-local correlation potentials, respectively, \(\lambda_{in_{sj}}\) are Lagrange multipliers which are chosen so that the orthogonality constraints defined by (2.62) are satisfied and j labels the linearly independent solutions of these equations.
After writing \(H_{N+1}\) in terms of H N using (2.4), we find that the channel wave numbers squared k i 2 in (2.63) are given by
where the channel energies \(\overline{e}_i^\Gamma\) are defined by
where the channel functions \(\overline{\Phi}_i^\Gamma\) are defined by (2.59). The local direct potential \(V_{ii^{\prime}}^\Gamma(r)\) in (2.63), which arises from the direct terms in the first expansion in (2.57), has the explicit form
where the term \(-N/r_{N+1}\) is included in this definition so that the long-range Coulomb potential in electron collisions with atomic ions is included on the left-hand side of (2.63). The non-local exchange potential \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in (2.63), which arises from the exchange terms in the first expansion in (2.57) has the explicit form:
where the double prime on the Dirac bracket means that the integration is carried out over the space and spin coordinates of all \(N+1\) electrons except the radial coordinates \(r_{N+1}\) and r N of the incident and scattered electrons. Finally, the non-local correlation potential \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in (2.63) arises from the quadratically integrable functions \(\hbox{\raisebox{.4ex}{$\chi$}}_i^\Gamma(\textbf{X}_{N+1})\) in expansion (2.57). We can choose these functions without approximation to diagonalize \(H_{N+1}\) as follows:
We then define the radial functions
The elimination of expansion coefficients \(c_{ij}^\Gamma\) in (2.57) between (2.60) and (2.61) then yields the following expression for the correlation potential:
Explicit forms for the potentials \(V_{ii^{\prime}}^\Gamma(r)\), \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) have been given in a few simple cases, for example, for e− – H collisions by Percival and Seaton [726] and for electron collisions with atoms or ions with open 2pq and 3pq shells by Henry et al. [455]. However, it is not feasible or necessary to write down the explicit form of these potentials for electron collisions with general atoms and atomic ions. Instead, they are constructed as part of R-matrix computer programs for solving the close coupling equations (2.63) that we refer to in Sect. 5.1.1. Numerical methods for solving (2.63) have been discussed by Burke and Seaton [164] and the R-matrix approach for their solution will be presented in Chap. 5.
Nevertheless, certain general statements can be made about the form of potentials \(V_{ii^{\prime}}^\Gamma(r), W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) which are important in understanding the physical properties of the solution of the close coupling equations (2.63). First, we observe that the phases of the angular integrals in (2.66), (2.67) and (2.69) can be chosen so that these potentials are real. Also, it follows from these equations that the potentials satisfy the following symmetry relations:
These reality and symmetry conditions follow from the time-reversal invariance and hermiticity of the Hamiltonian which we will see in Sects. 2.4 and 2.5 lead to the reality and symmetry of the K-matrix and to the unitarity and symmetry of the S-matrix.
We also mention here some further properties of the potentials \(V_{ii^{\prime}}^\Gamma(r), W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\). We will see when we discuss Feshbach projection operator theory of resonances in Sect. 3.2.5 that we can divide Hilbert space spanned by the eigensolutions of the Schrödinger equation into two mutually orthogonal spaces by two projection operators P and Q, where the “optical potential” corresponding to scattering in Q-space is represented by the sum of pole terms in the expression defined by (3.108). We can relate Feshbach theory directly to the close coupling with pseudostates expansion (2.57), where P-space corresponds to the space spanned by those target states in this expansion which give rise to channels which are open at the energy under consideration, while Q-space corresponds to the remaining target states and pseudostates in this expansion, together with the quadratically integrable functions representing short-range electron–electron correlation effects included in the second expansion in (2.57). It is important to emphasize that the quadratically integrable functions included to remove the orthogonality constraints imposed on the radial functions \(F_{ij}^\Gamma(r_{N+1})\) in (2.57) must be regarded as part of P-space and must therefore be carefully chosen for this purpose, as discussed by Gorczyca et al. [398]. As a result of this choice of P- and Q-spaces, we will show in our discussion of Feshbach projection operator theory that the poles in the optical potential give rise to resonances. Also, since the resonances arise from pseudostates and quadratically integrable functions representing the effect of physical states not explicitly included in the first expansion, they are unphysical pseudoresonances and the corresponding T-matrix must be averaged over energy to obtain physically meaningful results at intermediate energies. We will consider this averaging procedure in the context of intermediate energy collisions in Chap. 6.
We consider next the asymptotic form of the direct potential \(V_{ii^{\prime}}^\Gamma(r)\) defined by (2.66). This potential can be simplified at large distances using (B.49) which can be written as
where \(\theta_{kN+1}\) is the angle between the unit vectors \(\hat{\textbf{r}}_\textbf{k}\) and \(\hat{\textbf{r}}_\textbf{N+1}, P_\lambda(x)\) is a Legendre polynomial and r < and r > are the smaller and larger of the two scalar distances r k and \(r_{N+1}\). We now observe that the integral over \(r_k,\;k=1,\dots,N\) in (2.66) involves the target states and pseudostates \(\Phi_i(\textbf{X}_N)\) and \(\Phi_{i^{\prime}}(\textbf{X}_N)\), retained in the original expansion (2.57) which vanish exponentially at large r k . Hence we can choose a value of the radial distance, say a 0, beyond which all the target states and pseudostates are effectively zero. The corresponding contributions to the integrals over \(r_k > a_0,\;k=1,\dots,N\) in (2.66) are then zero. It follows that when the scattered electron coordinate \(r_{N+1}\geq a_0\), then \(r_<^\lambda/r_>^{\lambda+1}\) in (2.72) can be replaced by \(r_k^\lambda/r_{N+1}^{\lambda+1}\). Hence (2.66) can be rewritten as
where the long-range potential coefficients \(\alpha_{ii^{\prime}\lambda}^\Gamma\) are defined by
We derive an explicit expression for these coefficients in Appendix D.1; see (D.12) and (D.21).
The leading term in the expansion of the potential matrix \(V_{ii^{\prime}}^\Gamma(r)\) in inverse powers of r behaves as r −2 since we remember that we have included the Coulomb potential \((Z-N)/r\) between the scattered electron and the target on the left-hand side of (2.63). The upper limit \(\lambda_{\textrm{max}}\) in the summation over λ in (2.73) results from the triangular relations satisfied by the angular momentum quantum numbers which arise in the integral in (2.74). We will see in Chaps. 3 and 5 that the long-range potentials given by (2.73) play a crucial role in many low-energy electron–atom collision cross sections. In particular, the leading dipole potential terms behaving as r −2, which from (2.74) can be seen to couple target states between which optically allowed transitions occur, give rise in second order to the long-range polarization potential defined by (2.19) and (2.20). Also, we will see in Sect. 3.3.2 that the long-range dipole potential which couples degenerate or almost degenerate target states of neutral atoms gives rise to resonances which lie below the thresholds for exciting these degenerate states. It follows that these long-range potentials must be accurately represented in any computational approach which is adopted in the low-energy electron collision region.
Finally we consider the asymptotic forms of the non-local exchange and correlation potentials \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in (2.63). We see from the definition of \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\), given by (2.67), that its behaviour as \(r\rightarrow\infty\) and \(r^{\prime}\rightarrow\infty\) is determined by the asymptotic behaviour of the channel functions \(\overline{\Phi}_i^\Gamma(\textbf{ x}_1,\dots,\textbf{x}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\) and \(\overline{\Phi}_{i^{\prime}}^\Gamma(\textbf{x}_1,\dots,\textbf{ x}_{N-1},\textbf{x}_{N+1};\hat{\textbf{r}}_N\sigma_N),\) respectively. It then follows from (2.59), defining these functions in terms of the target states and pseudostates, that the first channel function vanishes exponentially as \(r_N\rightarrow\infty\) and the second as \(r_{N+1}\rightarrow\infty\). Hence \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) vanishes exponentially as \(r\rightarrow\infty\) or as \(r^{\prime}\rightarrow\infty\). In a similar way we can see from the definition of \(X_{ii^{\prime}}^\Gamma(r_{N+1},r_N)\), given by (2.70), that its behaviour as \(r_N\rightarrow\infty\) and \(r_{N+1}\rightarrow\infty\) is determined by the asymptotic behaviour of the radial functions \(U_{i^{\prime}k}^\Gamma (r_N)\) and \(U_{ik}^\Gamma (r_{N+1}),\) respectively. It then follows from (2.69), defining these radial functions, that the first radial function vanishes exponentially as \(r_N\rightarrow\infty\) and the second as \(r_{N+1}\rightarrow\infty\). Hence \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) vanishes exponentially as \(r\rightarrow\infty\) or as \(r^{\prime}\rightarrow\infty\). We then find that the radius a 0 where \(V_{ii^{\prime}}^\Gamma(r)\) achieves its asymptotic form given by (2.73) can also be chosen so that
The close coupling equations (2.63) then reduce to coupled second-order differential equations given by
which can be solved in a straightforward way in the region \(r\ge a_0\) to yield the K-matrix, S-matrix and cross sections as discussed in Sects. 2.4 and 2.5. This result is of crucial importance in the development of the R-matrix approach for solving the coupled integrodifferential equations (2.63) which we discuss in Chap. 5.
We conclude this section by considering, as an example, the number of coupled channels and quadratically integrable functions which arise in the close coupling expansion (2.57) for electron collisions with Be-like ions. We consider the example where the six target eigenstates given by (2.16) are retained in the close coupling expansion (2.57), where these target eigenstates are represented by (2.18) in terms of three physical orbitals and three pseudo-orbitals 1s, 2s, 2p, \(\overline{\textrm{3s}}, \overline{\textrm{3p}}\) and \(\overline{\textrm{3d}}\).
We give in Table 2.2 the orbital angular momenta ℓ i of the scattered electron which are coupled to each of these six target states for total spin \(S=1/2\) and for each L and π combination with \(L\leq 4\). We also give the total number of channels which are coupled to these target states for each L and π combination. We see that the number of coupled channels for \(L\ge 2\) equals 10 when \(\pi=(-1)^L\) and equals 6 when \(\pi=(-1)^{L+1}\). We also note that if the total spin \(S=3/2\), then only the two triplet target states \(\textrm{1s}^22\textrm{s}2\textrm{p}\; ^3\textrm{P}^{\textrm{o}}\) and \(\textrm{1s}^22\textrm{p}^2\; ^3\textrm{P}^{\textrm{e}}\) are coupled where the orbital angular momenta l i of the coupled channels are given by the corresponding rows in Table 2.2.
We next consider the quadratically integrable functions which must be included in expansion (2.57) for electron collisions with Be-like ions. We give in Table 2.3 the number of quadratically integrable functions included for total spin \(S=1/2\) and for each L and π combination with \(L\leq 4\), where we assume that the 1s orbital remains doubly occupied and a maximum of one electron is excited to one of the \(\overline{3\textrm{s}}, \overline{3\textrm{p}}\) or \(\overline{3\textrm{d}}\) pseudo-orbitals. These quadratically integrable functions are those that must be retained in expansion (2.57) to ensure that the orthogonality constraints defined by (2.62), which we assume are applied, do not lead to incompleteness in the target orbital basis. As an example of one entry in Table 2.3, we observe that the \(\textrm{1s}^22\textrm{p}^2\overline{3\textrm{p}}\) configuration gives rise to the following three quadratically integrable functions
which must be included in expansion (2.57) for \(L=1, S=1/2\) and \(\pi=-1\).
In our discussion of the target eigenstates in Sect. 2.2.1, we observed that in addition to zero-electron and one-electron excitation configurations we could also include two-electron excitation configurations which would improve the target eigenstates. However, in order to balance the correlation effects in the target and the collision wave functions we would also have to include two-electron excitation configurations in the collision wave function which would give rise to quadratically integrable functions with two electrons in pseudo-orbitals. While the inclusion of these additional configurations usually give improved collision results at low energies they will also give rise to unphysical pseudo-resonances at intermediate energies, close to and above the ionization threshold, which have to be energy averaged to give reliable scattering amplitudes and cross sections. We will return to this question in Chap. 6 when we discuss electron collisions with multi-electron atoms and ions at intermediate energies.
Finally, we observe that the quadratically integrable configurations included in Table 2.3 do not contribute to L, S and π combinations with \(L\ge 5\). This is an example of a general result that quadratically integrable configurations, which are defined in terms of target physical and pseudo-orbitals, only contribute to low L collisions. Hence the correlation potential \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in (2.63) does not contribute at high L. Also, while the non-local exchange potential \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in (2.63) is in principle non-zero for all L, its contribution to the scattering amplitude and cross section becomes negligible compared with the contribution from the local direct potential \(V_{ii^{\prime}}^\Gamma(r)\) for large L. This is because for large L the repulsive angular momentum term \(-\ell_i(\ell_i+1)/r^2\) in (2.63) ensures that the scattered electron does not appreciably penetrate the internal region, and hence it only experiences the long-range contribution from the direct potential on the right-hand side of (2.76). This result also has implications for methods of solution of the close coupling equations (2.63) for large L, where the repulsive angular momentum term causes the contribution from the direct potential to become small and hence the Born approximation for the scattering amplitude and cross section becomes applicable, as discussed in Sect. 2.4.
4 K-Matrix and Kohn Variational Principle
In this section we consider the asymptotic form of the solution of the close coupling equations (2.63) as \(r\rightarrow\infty\). In this limit, we have seen that the local direct potential \(V_{ii^{\prime}}^\Gamma(r)\) has the asymptotic form given by (2.73) and the non-local exchange and correlation potentials \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) vanish exponentially so that (2.63) reduces to (2.76). We first generalize the expression for the K-matrix given in Sect. 1.1 for potential scattering to multichannel collisions considered in this and later chapters. We then show that the exact solution of (2.63) satisfies the Kohn variational principle [542] and we derive a Born series expansion for the K-matrix. Finally, we show that the Kohn variational principle enables a corrected K-matrix to be obtained from an approximate solution of (2.63) where this K-matrix is correct to second order in the error in the collision wave function.
We commence our discussion by ordering the target eigenstates and pseudostates retained in expansion (2.57), so that their energies e i defined by (2.5) are in increasing order. It then follows from (2.64) and (2.65) that when the total energy E is real the square of the channel wave numbers k i are real and satisfy
The equalities in this expression arise either because some of the target states included in expansion (2.57) are degenerate or because more than one channel function \(\overline{\Phi}_i^\Gamma\) in expansion (2.57) corresponds to a given target state, as is the case when the target orbital angular momentum L i in (2.59) is non-zero (see, for example, Table 2.2). We now assume that at the total energy E of interest, the first n a channels are open (i.e. have \(k_i^2\ge 0\)) so that the corresponding reduced radial functions \(F_{ij}^\Gamma(r)\) in (2.57) and (2.63) are oscillatory or linear as \(r\rightarrow \infty\) and the last n b channels are closed (i.e. have \(k_i^2< 0\)) so that the corresponding reduced radial functions \(F_{ij}^\Gamma(r)\) in (2.57) and (2.63) vanish as \(r\rightarrow \infty\). Hence
where the quantities n a and n b depend on the total energy E. We see that when E is greater than all the energies \(\overline{e}_i^\Gamma\) defined by (2.65), the channels are all open so that \(n_a=n\) and \(n_b =0\). On the other hand, when E is less than all the energies \(\overline{e}_i^\Gamma\), the channels are all closed, so that \(n_a=0\) and \(n_b =n\), corresponding to a bound state of the electron–atom or electron–ion system.
In order to define the asymptotic boundary conditions satisfied by the reduced radial wave functions \(F_{ij}^\Gamma(r)\) we must consider the second index j on these functions. As we have already mentioned in Sect. 2.3.2, this second index labels the linearly independent solutions of the n close coupling equations (2.63) satisfied by the functions \(F_{ij}^\Gamma(r)\). It follows from the general theory of linear coupled second-order differential equations that n coupled equations have in general 2n linearly independent solutions. However, the requirement that the total wave function must be normalizable near the origin implies that
where n ij are normalization factors. Hence the reduced radial functions vanish at the origin. The n conditions (2.80) reduce the number of physical linearly independent solutions from 2n to n. We will see below that when some of the channels are closed, so that \(n_b>0\), the number of linearly independent solutions is further reduced to \(n-n_b=n_a\). The second index j is thus required to label these n a linearly independent solutions.
We consider first the situation where all channels are open, so that \(n_a=n\). The asymptotic boundary conditions satisfied by the n linearly independent solutions of (2.63), which reduce to (2.76) asymptotically, can be written in analogy with (1.71) in potential scattering in the form
The quantity \(\theta_i(r)\) in (2.81) is defined in analogy with (1.58) and (1.59) by
where
and
for electron collisions with atoms or ions with N electrons and nuclear charge number Z. The factor \(k_i^{-1/2}\) in (2.81), normalizes the ingoing spherical wave to unit flux which we will see below means that the \(n\times n\)-dimensional K-matrix K ij Γ, defined by the asymptotic boundary conditions (2.81), is symmetric.
When n b channels are closed then the corresponding terms \(\sin \theta_i(r)\) and \(\cos \theta_i(r),\;i=n_a+1,\dots,n\) in (2.81) diverge exponentially asymptotically. This follows from (2.64) since \(k_i^2 <0,\;i=n_a+1,\dots,n\), and hence k i is pure imaginary. Such divergent solutions are physically inadmissible since they are not normalizable. Hence they must be eliminated by combining together the n linearly independent solutions \(F_{ij}^\Gamma (r),\;j=1,\dots,n\) in (2.81). Since there are n b divergent terms to be eliminated we are left with \(n_a=n-n_b\) linearly independent physical solutions which are finite at infinity. We choose these n a solutions of (2.63) to satisfy the asymptotic boundary conditions
Equations (2.85) define a reduced \(n_a\times n_a\)-dimensional K-matrix K ij Γ which connects the n a open channels.
Also, since the potentials \(V_{ii^{\prime}}^\Gamma(r)\), \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) in the close coupling equations (2.63) are real and the normalization factors n ij in (2.80) can be chosen to be real, then the solutions \(F_{ij}^\Gamma(r)\) are real. It follows that all the quantities in the asymptotic boundary conditions (2.81) or (2.85) are real and hence the K-matrix must be real. We will show below that the \(n_a\times n_a\)-dimensional K-matrix is also symmetric.
We now derive the multichannel Kohn variational principle for the K-matrix satisfied by the solutions of (2.63). We consider the following integral taken over the space and spin coordinates of all \(N+1\) electrons
where the solutions Ψ jE Γ and \(\Psi_{j^{\prime}E}^\Gamma\) are defined by (2.57). We find that
where we have used the same procedure that we adopted to reduce (2.60), (2.61), (2.62) and (2.63). In (2.87) the subscripts j and j ′ label the linearly independent solutions, and in the following discussion we assume there are n a open channels so that the K-matrix has dimensions \(n_a\times n_a\). Also in (2.87) the local direct potential \(V_{ii^{\prime}}^\Gamma(r)\) and the non-local exchange and correlation potentials \(W_{ii^{\prime}}^\Gamma(r,r^{\prime})\) and \(X_{ii^{\prime}}^\Gamma(r,r^{\prime})\) are defined by (2.66), (2.67) and (2.70), respectively. In (2.87), and in the following analysis, we find it convenient not to impose the orthogonality constraints, defined by (2.62). This means that the Lagrange multiplier terms in (2.63) and the additional quadratically integrable functions, which would otherwise need to be included in expansion (2.57) for completeness, are no longer required although the K-matrix, which is defined by the asymptotic form of the wave function, will be unaltered.
It is convenient to rewrite the integral, defined by (2.87), using Dirac bracket notation as follows:
so that the close coupling equations (2.63) can be written in the following matrix form
Hence \(\textbf{I}^\Gamma =0\) when \(\textbf{F}^\Gamma(r)\) is an exact solution of (2.89). It follows that I Γ is an \(n_a\times n_a\)-dimensional matrix, L Γ is an \(n\times n\)-dimensional integrodifferential matrix operator and \(\textbf{F}^\Gamma(r)\) is an \(n\times n_a\)-dimensional solution matrix satisfying the boundary conditions
corresponding to (2.80) and (2.85). Also in (2.90), we have only considered the non-vanishing asymptotic components of \(\textbf{F}^\Gamma(r)\) so that \(\textbf{k}, \boldsymbol{\theta}(r)\) and K Γ are \(n_a\times n_a\)-dimensional matrices where both k and \(\boldsymbol{\theta}(r)\) are diagonal.
We now consider variations in I Γ due to arbitrary small variations \(\delta \textbf{F}^\Gamma(r)\) about the exact solution of (2.89) satisfying the boundary conditions (2.90) where the variations satisfy the boundary conditions
The corresponding first-order variation in I Γ is then
which after using (2.89) becomes
We can evaluate the right-hand side of (2.93) by rewriting L Γ in the form
where D Γ is the second-order differential operator term \(-\frac{1}{2}\textbf{I}d^2/dr^2\) in L Γ and O Γ represents the remaining terms in L Γ. It follows from the reality and symmetry relations (2.71) satisfied by the potentials, that s is real and symmetric so that
Hence using (2.89), (2.94) and (2.95), we find that (2.93) reduces to
Integrating the terms on the right-hand side of (2.96) by parts then yields
where the superscript T means transpose. The surface terms in (2.97) can be evaluated using the boundary conditions (2.90) and (2.91) satisfied by \(\textbf{F}^\Gamma(r)\) and \(\delta\textbf{F}^\Gamma(r)\) at r = 0 and ∞ giving
Hence we obtain the Kohn variational principle for the K-matrix [542]
which is satisfied by the exact solution of (2.89). This equation is the multichannel generalization of the result obtained in potential scattering given by (1.212).
It follows from the above analysis that since the potential operator O Γ in (2.94) is real and symmetric then K Γ is a real symmetric \(n_a\times n_a\)-dimensional matrix. The K-matrix thus depends on \(n_a(n_a+1)/2\) real parameters, where n a is the number of open channels.
We can extend this result to obtain an integral expression for the K-matrix. We consider the variation of the solution \(\delta\textbf{F}^\Gamma(r)\) corresponding to a variation in the operator δ L Γ. Equation (2.89) then becomes
After using (2.89), (2.100) formally reduces to
where δ M Γ is obtained by expanding \(\left(\textbf{L}^\Gamma +\delta\textbf{L}^\Gamma\right)^{-1}\) yielding
Hence we obtain from (2.101)
Substituting for the left-hand side of (2.103) from (2.93) and (2.98) we then obtain
It follows from (2.102) that (2.104) is an exact integral expression relating the variation δ K Γ in the K-matrix K Γ to the variation δ L Γ in the integrodifferential operator L Γ.
We can choose the variation δ L Γ to correspond to the sum of the direct, exchange and correlation potentials in (2.63), that is
Equation (2.100) can then be rewritten as
where L 0 Γ is the diagonal differential operator on the left-hand side of (2.63) corresponding to pure Coulomb scattering in the absence of the potential \(\textbf{U}^\Gamma, \textbf{F}_0^\Gamma(r)\) is the corresponding diagonal Coulomb solution defined by the boundary conditions (2.90) with \(\textbf{K}^\Gamma=0\) and \(\delta\textbf{F}^\Gamma(r)\) is the variation in the solution caused by the potential U Γ. It follows from (2.104) that the K-matrix corresponding to the operator \((\textbf{L}_0^\Gamma +\textbf{U}^\Gamma)\) is
where M Γ is formally defined by the expression
Equations (2.107) and (2.108) correspond to the Born series expansion of the K-matrix. This expansion converges if the incident electron is fast or if the potential interaction U Γ is sufficiently weak, which occurs, for example, when the total orbital angular momentum L becomes sufficiently large. In these cases, the first-order Born term in the expansion of M Γ often yields an accurate estimate for the K-matrix. However, when these conditions do not apply, higher order terms in the Born series must be included to obtain accurate results and, even if the Born series converges, these higher order terms are difficult to evaluate for electron collisions with multi-electron atoms and ions. It is then usually preferable to solve the close coupling equations (2.89) directly to obtain the K-matrix. We will consider an accurate solution of these equations using the R-matrix method in Chap. 5. We will also consider how the Born series approach can be combined with the R-matrix method to obtain accurate results at intermediate energies in Chap. 6. Further discussion of the convergence properties of the Born series expansion has been given by Goldberger and Watson [387] and by Joachain [503].
The Kohn variational principle can also be used to improve an approximate solution of (2.89). Thus if \(\textbf{F}_\textrm{t}^\Gamma(r)\) is an approximate trial solution of (2.89) and K t Γ is the corresponding approximate K-matrix, then it follows from (2.99) that an improved K-matrix, correct to second order in the error in the collision wave function, is given by the Kohn-corrected K-matrix
where I t Γ is calculated from (2.88) using the trial solution \(\textbf{F}_\textrm{t}^\Gamma(r)\). We note that if an accurate solution of (2.89) is obtained, for example, by using the R-matrix method, then the correction I t Γ to the corresponding K-matrix, given by (2.109), vanishes.
The variational principle (2.99) clearly depends on the asymptotic boundary condition, defined by (2.90), chosen for the reduced radial functions \(\textbf{F}^\Gamma(r)\). However, this asymptotic form is not unique and as we have already shown in Sect. 1.5 different asymptotic forms can lead to different variational principles in potential scattering. In the present multichannel collision situation an infinity of different variational principles can be constructed by taking different linear combinations of the n a linearly independent solutions defined by (2.90). As an example, in Sect. 2.5 we will form n a solutions satisfying S-matrix asymptotic boundary conditions. The corresponding Kohn variational principle for the S-matrix has been applied to reactive scattering and electron–molecule collisions, for example, by Miller [651] and by McCurdy and Rescigno [614] where it has been shown to have computational advantages over the Kohn variational principle for the K-matrix. However, all of these variational principles are satisfied by an accurate solution of the close coupling equations (2.89).
5 S-Matrix, T-Matrix and Cross Sections
In this section we define the S-matrix and T-matrix in terms of the K-matrix and hence obtain expressions for the total and differential cross sections for electron–atom collisions. In order to determine the S-matrix it is necessary to express the asymptotic solutions of the close coupling equations (2.63) in terms of ingoing and outgoing waves rather than in terms of sine and cosine waves as in (2.85). When n a channels are open the required solutions are defined by the asymptotic boundary conditions
which are linear combinations of the solutions defined by (2.85). The relationship between these solutions is given by the matrix equation
where \(\textbf{F}^\Gamma(r)\) satisfies the asymptotic boundary conditions (2.85) and \(\textbf{G}^\Gamma(r)\) satisfies the asymptotic boundary conditions (2.110). The \(n_a\times n_a\)-dimensional open channel S-matrix S Γ, defined by (2.110), is related to the \(n_a\times n_a\)-dimensional K-matrix K Γ, defined by (2.85), by the matrix equation
Since K Γ is real and symmetric it follows from (2.112) that S Γ is unitary and symmetric. Hence S Γ can be diagonalized by the real orthogonal transformation which also diagonalizes K Γ. Hence we can write
where A Γ is a real orthogonal matrix and \((\textbf{A}^\Gamma)^\textrm{T}\) is the transpose of A Γ. The diagonal matrix \(\exp(2\textrm{i} \boldsymbol\Delta)\) can be written explicitly as
where \(\delta_1,\;\delta_2,\dots,\delta_{n_a}\) are n a real eigenphases. In the situation when only one channel is open \((n_a=1), \delta_1\) can be identified with the potential scattering phase shift defined by (1.9) and (1.71). However, when \(n_a>1\), a further \(n_a(n_a-1)/2\) real mixing parameters are necessary to completely specify the S-matrix. These parameters define the real orthogonal matrix A Γ in (2.113) and are related to the independent rotations possible in n a dimensions (i.e. to the three Euler rotation angles discussed in Appendix B.5, when \(n_a=3\)). Hence the S-matrix as well as the K-matrix corresponding to n a open channels are specified by \(n_a(n_a+1)/2\) real parameters.
As an example, when \(n_a=2, \exp(2\textrm{i} \boldsymbol\Delta)\) is represented by two eigenphases δ 1 and δ 2 and the corresponding orthogonal matrix A Γ can be expressed in terms of an additional mixing parameter as follows:
where ε is called the mixing angle. The corresponding S-matrix defined by (2.113) is then given by
This equation expresses the S-matrix for two open channels explicitly in terms of the three real parameters \(\delta_1, \delta_2\) and ε.
It is also useful to define solutions of (2.63) satisfying outgoing wave T-matrix asymptotic boundary conditions as follows
These solutions can be related to those satisfying S-matrix asymptotic boundary conditions given by (2.110) by the matrix equation
It follows that the \(n_a\times n_a\)-dimensional T-matrix is related to the \(n_a\times n_a\)-dimensional S-matrix by the matrix equation
We will see that the T-matrix occurs in the expressions for the cross sections given below. Finally, we can define solutions of (2.63) satisfying ingoing wave asymptotic boundary conditions by
where \(T_{ij}^{\Gamma^{\star}}\) is the complex conjugate of T ij Γ. These solutions are related to those satisfying outgoing wave boundary conditions by
and are required in the calculation of transition amplitudes, for example, the atomic photoionization amplitude discussed in Sect. 8.1.
Having obtained expressions for the multichannel S-matrix and T-matrix we can now derive formulae for the total and differential cross sections for transitions between the target states retained in expansion (2.57). We consider first electron collisions with neutral atomic targets and we then generalize our results to electron collisions with atomic ions.
Our basic problem is to relate the scattering amplitude \(f_{ji}(\theta,\phi)\), defined by (2.6), to the T-matrix T ij Γ, defined by (2.117). In order to derive this relation we first rewrite the wave function Ψ i in (2.6) in terms of incident and scattered wave functions as
After introducing the space and spin coordinates of the N target electrons X N and the scattered electron \(\textbf{x}_{N+1}\) we can write the asymptotic forms of these wave functions for neutral atomic targets as
and
The scattering amplitude \(f_{ji}(\theta_{N+1},\phi_{N+1})\) in (2.124) thus describes a transition from the target state and incident electron spin state denoted by the quantum numbers \(i\equiv\alpha_iL_iS_iM_{L_i}M_{S_i}\pi_i m_i\) to the target state and scattered electron spin state denoted by the quantum numbers \(j\equiv\alpha_jL_jS_jM_{L_j}M_{S_j}\pi_j m_j\), where we have used the notation of (2.14) in describing the target states.
Following the procedure which we have used for potential scattering in Sect. 1.1, we expand the plane wave term in \(\Psi_i^{\textrm{inc}}\) in partial waves using (1.27). We also introduce the channel functions \(\overline{\Phi}_i^\Gamma(\textbf{ X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\) which are defined in terms of the target states \(\Phi_i(\textbf{X}_N)\) by (2.59). After inverting (2.59), using the orthogonality conditions satisfied by the Clebsch–Gordan coefficients given in Appendix A.1, we obtain
where \(\theta_i(r_{N+1})=k_ir_{N+1}-\frac{1}{2}\ell_i\pi\) for neutral atomic targets. We now carry out a partial wave decomposition of Ψ i in (2.122) by writing
where the functions \(\Psi_i^\Gamma(\textbf{X}_{N+1}) \) have the asymptotic form
which follows from expansion (2.57) since the exchange terms and the quadratically integrable functions vanish in this limit. Also in (2.127) the reduced radial wave functions \(G_{ji}^\Gamma(r)\) are chosen to satisfy the S-matrix asymptotic boundary conditions (2.110). The coefficients \(B_i^\Gamma(E)\) in (2.126) are then chosen so that the ingoing wave terms in Ψ i and \(\Psi_i^{\textrm{inc}}\) are the same. This yields
Substituting this result into (2.126) and using (2.122) and (2.124) then gives
The scattering amplitude is obtained by expanding the channel functions \(\overline{\Phi}_i^\Gamma (\textbf{X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\) in (2.129) in terms of the target states \(\Phi_i(\textbf{X}_N)\) using (2.59) and comparing with (2.124). This gives
where the T-matrix elements T ji Γ in this equation are defined in terms of the S-matrix elements S ji Γ by (2.119).
The differential cross section for a transition from a state represented by the quantum numbers \(\alpha_i L_i S_i \pi_i\) to a state represented by the quantum numbers \(\alpha_j L_j S_j \pi_j\) is obtained by substituting (2.130) into (2.8) resulting in an expression for the differential cross section, given, for example, by Blatt and Biedenharn [116]. The total cross section is then obtained by averaging this expression for the differential cross section over the initial quantum numbers of the target atomic state and incident electron, summing over their final quantum numbers and integrating over the scattering angles of the outgoing electron. We obtain the following result for the total cross section:
where the partial wave cross sections \(\sigma^{LS\pi}(i\rightarrow j)\) corresponding to the conserved quantum numbers LSπ are given in units of \(\pi\textrm{a}_0^2\) by
We note that the slow convergence of expansion (2.132) for optically allowed transitions can be overcome by using a method proposed by Burke and Seaton [190] using the Burgess sum rule [149].
The expression for the differential cross section obtained by substituting (2.130) into (2.8) can be simplified using the angular momentum transfer formalism introduced by Fano and Dill [310]. We define the angular momentum transferred from the scattered electron to the target during the collision by ℓ t where
The relationship between these vectors and the total orbital angular momentum vector L is illustrated in Fig. 2.2. We now introduce a transformed T-matrix \(\widetilde{T}_{ji}^{tS}\) by the equation
where \(W(abcd;ef)\) are Racah coefficients defined in Appendix A.2. We can then show that (2.130) for the scattering amplitude can be rewritten as
Relationship between the angular momentum transfer vector ℓ t and the vectors \(\textbf{L}_i, \textbf{L}_j, \boldmath{\ell}_i, \boldmath{\ell}_j\) and L
The differential cross section, obtained by averaging (2.8) over the initial magnetic quantum numbers and summing over the final magnetic quantum numbers, can be written as
where
The subscripts i, j, i ′ and j ′ on \(\widetilde{T}_{ji}^{tS}\) and \(\widetilde{T}_{j^{\prime} i^{\prime}}^{tS}\) denote the channel quantum numbers \(\alpha_i L_i S_i \ell_i\pi_i\), \(\alpha_j L_j S_j \ell_j\pi_j, \alpha_i L_i S_i \ell_{i^{\prime}}\pi_{i^{\prime}}\) and \(\alpha_j L_j S_j \ell_{j^{\prime}}\pi_{j^{\prime}},\) respectively.
The introduction of the angular momentum transfer ℓ t in the expression for the differential cross section given by (2.136) and (2.137) replaces the double summation over L and L ′ in the earlier expression for the differential cross section given, for example, by Blatt and Biedenharn [116], by a single incoherent summation over ℓ t . The corresponding evaluation of the summation is very much more efficient and has been incorporated into several computer programs (e.g. by Salvini [808]).
A second advantage of the angular momentum transfer formalism is that it enables simple qualitative features of the angular distribution to be readily described and understood. For example, it is useful to introduce the concept of parity-favoured and parity-unfavoured transitions by the equations
An example of a parity-favoured transition is
In this case \(L_i=0\) and \(L_j=1\) so from Fig. 2.2 we see that \(\ell_t=1\). On the other hand, from the conservation of total parity, \(\ell_i-\ell_j\) must be odd. An example of a parity-unfavoured transition is
In this case \(L_i=0\) and \(L_j=1\) so again \(\ell_t=1\). However, from conservation of total parity, \(\ell_i-\ell_j\) is even.
One of the most interesting features of parity-unfavoured transitions is that the differential cross sections in the forward and backward directions vanish. This follows by considering the factor
in expression (2.135) for the scattering amplitude. The spherical harmonic \(Y_{\ell_jm_{\ell_j}}(\theta,\phi)\) contains a factor \((\sin \theta)^{m_{\ell_j}}\) which causes the angular distribution to vanish at \(\theta =0\) and π when \(m_{\ell_j}\neq 0\). However, when \(m_{\ell_j}= 0\), the Clebsch– Gordan coefficient in (2.141) reduces to \((\ell_i 0 \ell_j 0| \ell_t 0)\) which vanishes when \(\ell_i + \ell_j + \ell_t \) is odd, which proves this result.
The preceding theory must be extended to describe electron collisions with ions. As before, we rewrite the wave function Ψ i in (2.6) in terms of incident and scattered waves as in (2.122), which now has the following asymptotic form:
The incident Coulomb-distorted plane wave term in \(\Psi_i^{\textrm{inc}}\) can be decomposed into partial waves using (1.49) and (1.64). This gives
where \(f_{c}(\theta)\) is the Coulomb scattering amplitude defined by (1.52), and where we note that (2.143) applies except when \(\theta=0\), since in this case \(r\rightarrow \infty\) does not imply \(|r-z|\rightarrow \infty\). After substituting (2.143) into (2.142) we then find that
where
We then carry out a partial wave decomposition of Ψ i using (2.126), where the coefficients \(B_i^\Gamma(E)\) are now chosen so that the ingoing wave terms in (2.126) and (2.144) are the same. The scattered wave function \(\Psi_i^{\textrm{scatt}}\) is then
The scattering amplitude for electron–ion collisions is obtained by expanding the channel functions \(\overline{\Phi}_i^\Gamma(\textbf{ X}_N;\hat{\textbf{r}}_{N+1}\sigma_{N+1})\) in (2.146) and comparing with (2.142). We obtain
where, as in (1.75), \(f_{c}(\theta)\) is the Coulomb scattering amplitude and \(f_{ji}^{S}(\theta,\phi)\) is the scattering amplitude arising from the additional short-range potential. We find that
where, as in (2.119), the T-matrix is defined by \(T_{ji}^\Gamma=S_{ji}^\Gamma-\delta_{ji}\). Equation (2.148) describes a transition from a state defined by the quantum numbers \(i\equiv\alpha_i L_i S_iM_{L_i}M_{S_i}\pi_im_i\) to a state defined by the quantum numbers \(j\equiv\alpha_j L_j S_jM_{L_j}M_{S_j}\pi_jm_j\).
The differential cross section for a transition from a state denoted by the quantum numbers \(\alpha_i L_i S_i\pi_i\) to a state represented by the quantum numbers \(\alpha_j L_j S_j\pi_j\) is obtained by substituting (2.147) into the expression for the differential cross section, given by (2.8), averaging over the initial quantum numbers and summing over the final quantum numbers of the target state and scattered electron. The total cross section for inelastic collisions is then obtained by integrating over all scattering angles of the outgoing electron and summing over all LSπ values giving (2.131) and (2.132).
Finally, in applications involving electron–ion collisions it is often necessary to determine a quantity \(\Omega(i,j)\), first introduced by Hebb and Menzel [447] and subsequently called the collision strength by Seaton [849, 850, 857]. It is defined in terms of the total cross section \(\sigma^{\textrm{Tot}}(i\rightarrow j)\) measured in units of \(\pi a_0^2\) by
where \(\omega_i=(2L_i+1)(2S_i+1)\) is the statistical weight of the initial state, denoted by the quantum numbers \(\alpha_i L_i S_i\). Since k i has the dimensions of a reciprocal length, \(\Omega(i,j)\) is dimensionless. It is also symmetric so that \(\Omega(i,j)=\Omega(j,i)\). In an ionized plasma, we also need to consider the electron–ion collision cross section averaged over a Maxwell distribution of electron velocities. We introduce the collisional transition probability \(q(i\rightarrow j)N_{e}\) where
Here \(f(v_i,T_e)\) is the Maxwell velocity distribution function, normalized according to
v i is the velocity of the incident electron, N e is the electron density and T e is the electron temperature of the plasma. Expressing \(\sigma^{\textrm{Tot}}(i\rightarrow j)\) in terms of the collision strength we find that the probability of de-excitation is
in cubic centimetres per second, where T e is in degrees Kelvin, ω j is the statistical weight of the jth target state and e i and e j are the target energies defined by (2.5). The effective collision strength \(\Upsilon(j,i)\) introduced in (2.152) is defined by
where ε j is the energy of the scattered electron in the jth state in Rydbergs and \(k=6.339\times 10^{-6}\) Rydbergs/°K is Boltzmann’s constant. Clearly if \(\Omega(j,i)\) is independent of energy, then \(\Upsilon(j,i)=\Omega(j,i)\). Also we find that the probability of excitation is
We also note that in many applications in plasma physics and astrophysics it is sufficient to know the effective collision strength \(\Upsilon(j,i)\) rather than the collision strength \(\Omega(j,i)\) for the transitions of interest. This can be important since we will see when we discuss recent low-energy electron–ion collision calculations in Sect. 5.6 that in many cases of interest the collision strength is dominated by resonance structure requiring a very large number of energy values to fully resolve. However, the corresponding effective collision strength is usually a smoothly varying function of temperature that can be accurately represented by a few well-chosen parameters.
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Burke, P.G. (2011). Multichannel Collision Theory. In: R-Matrix Theory of Atomic Collisions. Springer Series on Atomic, Optical, and Plasma Physics, vol 61. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15931-2_2
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