Atomic collision calculations

Computer Physics Communications 26 (1982) 397—410 North-Holland Publishing Company

397

ATOMIC COLLISION CALCULATIONS K.A. BERRINGTON Department of Applied Mathematics and Theoretical Physics, Queen’s University, Belfast BT7 INN, Northern Ireland

and K.T. TAYLOR Science and Engineering Research Council, Daresbury Laboratory, Daresbury, Warringron WA 4 4AD, England

The basic theory of electron—atom/ion scattering and computer programs implementing this in the close-coupling and related approximations are described. Adaptations made to these codes enabling them to take advantage of the vector processing features on the CRAY-I are given, together with details of calculations thus made possible. An approach to solving the equations of electron—molecule scattering that specially makes use of vector machines is described. Current areas of interest in atomic physics such as atoms in strong magnetic fields, electron impact ionisation of atoms and ionic recombination in an ambient gas, where access to CRAY-I machines has stimulated progress are mentioned.

1. Infroduction Atomic collision theory has grown from the pre-second world war work of a few researchers such as Massey and Bates to be a subject involving several thousand workers throughout the world, Although it is still possible to make worthwhile contributions to the field that involve little more computing power than that provided by a standard pocket calculator there has been a need from the earliest days for access to the best computing facilities currently available. Thus Massey and Bates in the Belfast of the 1930’s had a differential analyser to work with. The years since then have seen the first electronic machines in the late forties progress to the vector processors of to-day. An account of the atomic collision problems tackled using machines such as the CRAY-i must necessarily be selective. Relatively few research groups have access to this latest generation of machines (though groups in the UK are very fortunate with a CRAY-I funded by the S.E.R.C. readily accessible by a large number of people 001 0-4655/82/0000—0000/$0275

©

involved in basic research). Thus, this paper will concentrate on those problems which people could see how to tackle immediately with advantage using a bigger and faster machine. Mention will also be made of how vector processing machines are influencing the development of new computer programs in the field and of the physical insight the results obtained with such machines have already provided into some of the most exciting problems. Finally, some account will be given of interesting areas in which progress may be expected over the next few years. The following topics will be considered in this paper. In section 2, there will be a detailed discussion of electron impact excitation using close coupling methods, where work is being done on CRAY-I machines at the S.E.R.C. Daresbury Laboratory in the UK and at NCAR and Los Alamos in the USA. Section 3 gives an outline of electron—molecule scattering, particularly using the linear algebraic formulation of the close-coupling equations, on CRAY’s at Los Alamos. Section 4 discusses electron impact ionisation of atoms using

1982 North-Holland

398

K.A. Berrington, K T. Taylor

a time-dependent wave packet method, on CRAY’s at the Magnetic Fusion Center in the USA. Section 5 outlines ionic recombination in an ambient gas using Monte Carlo techniques, on CRAY’s at Daresbury and Livermore, whilst calculations on the spectra of atoms in strong magnetic fields, using the CRAY-I at Daresbury, are discussed in section 6.

/ Atomic collision

calculations

Z is the nuclear charge. This Hamiltonian is diago-

nal in both the total orbital angular momentum L of the system and the total spin angular momenturn S. This allows us to write down a separate wavefunction ~ LS for each distinct choice of L and S. This is chosen to take the form NCHF ~4~LS(E)_~

~e(xi~.xN1~N÷10N+1)

)

Xr~ F(r

2. Electron collisions from atoms and ions

N +1

NCHB

An area of atomic collision physics that has received considerable attention over the years, and has taken advantage of the latest computers, is the electron impact excitation of atoms and ions s~tudied using variants of the close-coupling method. Related processes such as photoionisation have also been calculated by the same techniques. A number of computer programs have been written over the past ten years to deal with these collision processes. Some of these programs have been implemented and used on the CRAY- 1 cornputer, and only these programs will be discussed here. Two such are the University College London computer packages (IMPACT, COLALG, PHOTUC) [1] and the Queen’s University Belfast RMATRX package [2]. Two other programs, namely SUPERSTRUCTURE [3] and CIV3 [4] written at the respective institutions have also been implemented on CRAY-i computers to provide target wavefunctions necessary for input to the two packages.

+

The basic theory of the close-coupling approach was first set out by Seaton [5] and the computational aspects were reviewed by Burke and Seaton [6]. In general for the lighter atoms and ions considered a non-relativistic Hamiltonian is used. Written in atomic units this takes the form N±1

H~’~ =

2

~=

i

N±1

+

—

2

-~

t.,

+

~ 1>1=1

-~—

(1)

~

where there are N + 1 electrons in the system and

afXJ(xl...xN+l),

(2)

where ~ is the anti-symmetrization operator and E is the total energy of the system. The iii, are channel functions formed by coupling a wavefunction F, for the N atomic electrons, in the ground or some excited state, to the spin and angular parts of the wavefunction for the (N + I)th electron, which has a radial oscillatory part r~

1F,(rN+l). The x~are square integrable single configuration wavefunctions of all the N + 1 electronic coordinates: x1=r1, a~(i=l,2,...,N+ 1); with energy dependent coefficients a1. We see that wavefunctions describing various states of the target atom form an important part of the first summation in (2). Much effort has been devoted to developing computer programs that will provide these for any atom or ion required. In all cases the target wavefunctions are chosen to have the form

~~(xI...xN)= 2.1. Theory of electron atom/ion collisions in the close coupling approach

~

N +1

hCiJ~j(X1...XN),

(3)

where the c1 are normalised coefficients giving a linear combination of anti-symrnetrized single configuration wavefunctions 4~,.As in the wavefunctions describing the scattering system, each wavefunction ‘~, has associated with it a definite total orbital angular momentum L1and spin angular momentum S~.The are the same type of wavefunction as the in eq. (2) but describe a system of N bound electrons rather than N+ 1. It is, however, extremely convenient to form both sets of wavefunctions from the same set of bound orbitals. These are chosen to have standard spherical harmonics for angular parts, but the radial

KA. Berrington, K T. Taylor / Atomic collision calculations

parts P~1(r)satisfy the orthonormal relation

J

P~1(r)P~1(r) dr =

(4)

0

where in the standard notation n labels the principal quantum number and / the orbital angular momentum quantum number of the orbital, In the program SUPERSTRUCTURE of Eissner and Nussbaurner [3] these F,,, are calculated using a scaled statistical model potential and their numerical values are tabulated over a grid. On the other hand the program CIV3 of Hibbert [4] requires theorbitals. F,,, to have an analytic form as the sum of Slater Pn,(r)~bj(2~J)

_______

j

~/(2k)!

r~’e~”,

(5)

where k I + I. This latter program requires as input data some minimal set of F,,, often chosen to be those resulting from a Hartree—Fock calculation for the ground state of the target and allow a limited number of configurations 4~,to be formed. Additional F,,, can be supplied with incompletely specified coefficients b1 which the program varies to ensure normalisation and orthogonality to P~1 already included. With these many more configurations can be constructed. The coefficients C~1 in eq. (3) and an energy for a particular atomic state 1, are obtained by solving the equation ~‘

(6) where HN is a Hamiltonian with the same form as that specified in eq. (1) except that the coordinates of only N electrons are involved. In general the lowest energy eigenvalue and associated eigenvector of this equation are required. The program yields oscillator strengths for allowed transitions between states. Returning now to the total wavefunction for the (N + 1)-electron system, eq. (2), the problem is to determine the F, and a1. This is done in the 1M PACT formulation by applying the Kohn variational principle on the trial wavefunction ‘I’, defined by eq. (2), by requiring that

N’ I HN±tI ~)

(7) is stationary for variations in F, and a1, subject to

399

orthogonality conditions, HN~being defined by eq. (1). The subsequent transformation of the close coupling equations, eq. (2), to linear algebraic equations in the IMPACT method is outlined in subsection 2.2. If for the moment we restrict attention to some asymptotic region r ~ a, where all radial parts ~a of the bound orbitals included in the target description have become vanishingly small, the close-coupling equations reduce to d~ 1~(1~+ 2 ~ ~dr2 r NCHF ~‘

r

2 i~ ~~1~(r), il,...,NCHFandr~a, (8) where NCHF equals the total number of channels available to the colliding electron and V, 1 is the direct potential. The two computer programs RMATRX and IMPACT take advantage of this much simpler form of the close-coupling equations at large distances by using the program ASYMPT [7] developed at University College London which solves eq. (8). More will be given in the next section on the implementation of these codes on the CRAY- 1. 2.2. The IMPACT and RMA TRX methods and their implementation on the CRA Y

It is now appropriate to consider how the programs IMPACT and RMATRX treat the inner region r ~ a where the close-coupling equations cannot be simplified. 2.2.1. IMPACT IMPACT, published by Crees et al. [1], transforms the close-coupling equations in the inner region by replacing the integral operators by sums and the differential operators by difference operators [6] and solves the resulting linear algebraic equations by standard techniques. A mesh of P points spans the radial distance from the origin to r = a. To keep the set of equations to be solved as small as possible consistent with numerical accuracy a minimal number of well chosen points is used. They get further apart as r increases but each half wavelength of a radial

400

K.A. Berrington, K. T. Taylor

function should contain 4 or 5 points. Two further points P + 1 and P + 2 are used to fit to the solutions of eq. (8). In a starting region, points, 1, 2,...,c (where c 3, 4 or 5), the F,,..(r) are expanded as Zr = r~1{(1 + I, + 1 )A 11.. + r2gii.~(r)}, (9)

/ Atomic

collision calculations

solutions to the asymptotic forms yields the reactance matrix K and the scattering matrix S. For each SLir partial wave, the collision strength between state i and statef is given by ~SL~r(.f) =~(2s+i)(2L+ l)~Sjj_~jg~2.(12)

Linearly independent solutions started in this way must eventually be fitted to asymptotic forms, eq. (8), for r ~ 1’,,,, We can count up the number of unknowns in any solution using the IMPACT approach. One set of solutions is represented by NCHF radial functions F,(r); NCHB coefficients a1 and NLAM Lagrange multipliers, Thus, to specify the radial functions at the P2 = P + 2 tabular points in the inner and matching regions and the other quantities one requires an array of dimensions

In the implementation of IMPACT on the CRAY the linear algebraic equations (eq. (ii)) are solved by means of vectorized linear algebra machine language routines developed by Tom Jordan at Los Alamos, and this results in large increases in speed. For example, in a study (to be discussed more fully in section 2.3) of the electron excitation of S III carried out by C. Mendoza at University College London using the Daresbury IBM 370/165 and CRAY- 1 computers, the following timings were obtained for the 4pe partial wave (8 channels): 3 energies; CRAY = 56.5 s, IBM = 1210.5 s. This, however, includes the time to calculate the C-matrix in eq. (ii). If the energy independent part of the C-matrix is read from store rather than calculated in the run, the saving in time is even more dramatic: 11 energies; CRAY = 24.4 s, IBM = 3430.9 Applications of the IMPACT program to atomic physics will be discussed in section 2.3.

NTOT= NCHF*P2 + NCHB + NLAM. (10) The linear algebraic equations (described in detail in Burke and Seaton [6]) can be written in matrix form as

2.2.2. RMATRX We now move to the R-matrix method [8] embodied in the program package RMATRX, published by Berrington et al. [2]. Here the close-cou-

where the unknown g.,. (r) are represented by polynomials and may be shown to remain finite as r 0. The index i” labels the i” set of linearly independent solutions (i” = I~NCHF) which can be defined on specifying the constants A1. in eq. —~

C * G = X * A, (11) where C = coefficient matrix, of dimension NTOT by NTOT; X coefficient matrix, of dimension NTOT by NCHOP; A = matrix defining the linear independence of the solutions of dimension NCHOP by NCHOP; G = matrix of unknowns, of dimension NTOT by NLI, which on solution will yield values of the functions F,(r) at P + 2 points, values of the Lagrange multipliers and coefficients of the bound state functions in eq. (2). (NLI number of linearly independent solutions = NCHOP, the number of open channels, or = I when all channels are closed.) Solving eq. (11) at each energy and fitting the

~.

pling equations, as such, are not actually derived in the inner region but rather the wavefunction ‘I’ given in eq. (2) is considered to be represented 4k’ in terms of an R-matrix basis of wavefunctions ‘ viz.: ~ ~ (13) k

where the AEk are energy-dependent coefficients to be determined and the q,~take the general form Nd-IF NRANG2

~

=

C.J~

‘

~,

( ~,

X r7~~

j=i . . .

XN “N+ I GN ±i)

( rN±

NCHB

+ ~

dJkxJ(xl...xN÷l),

(14)

K.A. Berrington, K. T. Taylor

where the U~.J(rN+~)~f= I, NRANG2 form a set of radial basis orbitals. This set, complete if NRANG2 were infinite, is in practice truncated to between 10 and 20 terms, which is a further approximation of this approach beyond that introduced by the close coupling expansion (2). This truncation will be discussed later in connection with the evaluation of the R-matrix. The NRANG2 “continuum” orbitals, together with the “bound” orbitals (defined, e.g. by eq. (5)) form an orthonormal set for a given angular momentum in the region 0 ~ r ~ a. The coefficients cIk and dlk of eq. (14) are determined by diagonalising the Hamiltonian HN+~given by eq. (1) in this basis: 6kk., (15) (~k I HN+I I ‘Ilk.) = Ek where all the radial integrals are over the finite range 0 ~ r ~ a. The angular integrals are peformed using the same computer codes as used in the program CIV3 [4] in treating the N electron target problem. From the summations in eq. (14) we see the order of the Hamiltonian matrix is MNP1 = NRANG2 * NCHF + NCHB. It should be noted that the setting up of the Hamiltonian matrix and the diagonalization in eq. (15) is independent of the energy of the incident electron. This gives the method a great advantage when calculations are performed for a large number of energies. The solutions in the inner region must now be linked to those solutions of eqs. (8) appropriate in the outer region. This is done by invoking a matching condition, It can be shown (e.g. in Burke and Robb [8]) that on the boundary r = a between the regions the amplitude and derivative of F, ( r) are related by the equation: dF I F, (a) = ~ R a bF. I (16) dr 1,1

(

—

)

,

/

Atomic collision calculations

401

As mentioned in the description of eq. (14), the most important source of error in this method is the truncation of eqs. (17) and (18) to a finite, numerically manageable, number of terms. The distant neglected levels can play an important role in the diagonal elements of the R-matrix where they add coherently. The effect of these levels can, however, be included by applying a “Buttle” correction to the diagonal elements of the R-matrix at each energy (Burke and Robb [8]). We see in eq. (17) that R depends entirely on quantities derived from a treatment of the problem in the inner region. Obtaining the 15(a) and d .15/dr Ira that satisfy eq. (16) by matching to the solutions determined in the outer region at r = a, given by solving the coupled differential equations in eq. (8), yields the reactance matrix K the S-matrix and the collision strength. In the implementation of the RMATRX package on the CRAY, it is useful to examine each of the separate programs that make up the package STG1, STG2 and STG3 see fig. 1. As explained in section 2.1, the input data comes from the structure programs, CIV3 or SUPERSTRUCTURE, in the form of atomic orbitals and target state information. STG1 in fig. 1 evaluates all radial integrals required in the evaluation of the Hamiltonian matrix eq. (15) (and the dipole matrix for photoionization calculations) in the inner region r < a. The coding for these radial integrals is easily vectorizable and extremely advantageous. For example, in a study of the electron scattering of CIII (in a six-state calculation) carried out by the authors using the Daresbury IBM 370/165 and the CRAY-I computers, the following timings were obtained for the STG 1 program: —

—

STG1 unmodified; CRAY = 290 s, IBM = 2100 s; STG1 with vectorized radial integral routine;

‘-‘

‘~“

where the R matrix in this equation has elements

CRAY72s. STG2 in fig. 1 forms the Hamiltonian matrix

R, 1=

M~PI W,k (a) ~~5k( k= Ek — E

a)

(17)

and NRANG2 ~‘~~k(

r)

=

~ =1

cl/kU,1 ( r).

(18)

elements (and dipole matrix elements) by combining the radial integrals from STGI with the appropriate angular and spin integrals. The coding to evaluate these angular and spin integrals is not vectorizable at present, and can be very time-consuming. STG2 is thus a real bottleneck in CRAY

402

K.A. Berrington, K. T Taylor

Front-end machine

Vector processor eg. cs.s.Y~i I

/

Atomic collision calculations

been used for some years now on “conventional” computers, by many groups around the world. The collision codes have been used for calculations on electron collisions from light atoms and ions, and atomic photoionization. The structure codes have been used for oscillator strength and polarizability calculations. A wideHowever, variety of atoms and ions have been studied. until the advent of

[~~d~afrol I

structure programs eg. C1v3

I

vector processors, the calculations were generally restricted to low Z systems, with only a few low-lying states taken into account. More complex

~

~~TRx I

atomic systems can now be investigated and some CRAY applications are outlined for the two

r

I

STG1 _____--

packages, IMPACT and RMATRX. I

I

STC2

I

L~I1

amined by Mendoza [12]. The results are useful in

______________

LAI~_J~

2.3.1. IMPACT Work has centred on electron impact excitation calculations for forbidden transitions of ions of the second row of the periodic table. Specifically, 23p2) and phosphorus ions of isoelectronic the silicon (3s (3s23p3) sequences have been ex-

I

the interpretation of forbidden lines observed in the spectra of gaseous nebulae, novae, etc. One example is silicon-like S III for the forbidden transitions

Fig. 1. A block diagram of the RMATRX package.

3P—o’S, ‘D—~’S. A 7-state calculation including states (3s23p2)3P_~s1D,

calculations. A reformulation of the problem is required, and an alternative algorithm is discussed in a contributed paper to this conference by Burke and Scott [9]. STG3 in fig. I first diagonalizes the Hamiltonian matrices set up in STG2. A fully vectorized diagonalization routine is an obvious requirement since the size of the matrices can be around 500>< 500 in a typical CRAY calculation. The differential equations for the asymptotic region must be solved at each incident energy and this dominates a STG3 run. Work is in progress at University College London and at Queen’s University Belfast to vectorize this part of the calculation. 2.3. Calculations performed with the IMPACT and RMATRX packages on the CRAY The IMPACT and RMATRX packages, together with the associated structure programs, have

(3s23p2)3p, ‘D,

l~,

(3s3p~)5S°, 3D°,3P°,~D°

has been performed using the IMPACT package on the NCAR CRAY in Boulder. A pseudo 4s has been introduced to represent these target states by CI wavefunctions. Cross sections for photoionisation (a process closely related to electron impact excitation) are also of interest and can be calculated in the same run. For the example of photoionisation of S II we have hv+ S II(3s23p34S°)—sS III + e(4P”). The cross section for this process, calculated on a CRAY-l by Mendoza [12], is shown in fig. 2. Other members of the silicon sequence will be examined, specifically C1IV and Ar V, but since C matrices of the order of 800 are involved the

K.A. Berrington, K T. Taylor 25

I

I

/ Atomic collision calculations

I

403 I

I

sli 4s0_4p iso 20

-

—

I5S°)6p+ (3I~’)4p (S S°) ~15

-

-

C 0

(3QO)4p

0,

0510

-

-

(S S°)5p (5S0) 7p Bp

5—

0

I_I 0~1

___

02

0~3 04 Photoelectron Energy [RydI

Fig. 2. Photoionization cross sections for S II

(45O_4pe)

0~5

0~6

0~7

with IMPACT by Mendoza [12].

calculations will be done on the NCAR 1 MW machine. Both the RMATRX and IMPACT packages have been employed to examine transitions in members of the phosphorus sequence. As an example in fig. 3 Mendoza [12] used RMATR.X to calculate the electron excitation for the forbidden transitions in Si II: 4~0 -~~2D° 450 ...+2p° 2D°—+2P°.

cess:

Six states were used in the calculation:

2.3.2. RMATRX Electron collision, photoionisation and atomic structure data for a number of atoms and ions of laboratory and astrophysical interest are being calculated with the RMATRX package by the Belfast group. Extensive calculations were carried out on light atoms and ions before the availability of vector processing machines, e.g. refs. [13—16]. Current calculations using RMATRX on the Daresbury CRAY-I are outlined in table 1. As an

(3523p3)4s0

2D° 2p0

(

4pe 2j~e 2pe, 353p4)

This was run on the CRAY-1 at Daresbury. Pseudo orbitals 4s, 4p were introduced to represent these target states using CI. The results for forbidden transitions differed markedly from those of previous distorted wave calculations. Again photoionisation of S I is a related pro-

hv + S I(3s23p4 3pe)

—*

S II + e (3S0, 3P°,3D°).

So far the first two contributions (3S°and 3P°) have been calculated by Mendoza [12] on the Daresbury CRAY-l (see fig. 4), but the 3D°final symmetry requires a C matrix of order 600 and must be run at NCAR.

404

K.A. Berrington, K. T. Taylor I I

I

/

Atomic collision calculations I

I

I

12

s it

200

ao~o12o13o~4~5

20°

‘S°_

O!6 kI’Ffl2

!2po~

IkI2~I1~

IkI2~)i’

Fig. 3. Electron excitation collision strength for the forbidden transitions of S II using RMATRX by Mendoza [12], compared with earlier 3-state close coupling results (dots) and distorted-wave results (chain curve).

example of the most recent results, we show in fig. 5 electron collision strengths for the (Is2 )I 5— (ls2s)3S transition in helium-like 0 VII in an eleven state calculation [18]. The R-matrix codes have been extensively modified to enable the calculation of low-energy electron scattering [10] and photoionisation cross sections [23] for intermediate weight and highly ionized atomic systems. In such systems relativistic effects play an important role and are accounted for within the framework of the R-matrix theory through the use of model potentials and the Breit—Pauli Hamiltonian. A coupling scheme is used in which only the total angular momentum

and parity are good quantum numbers. Consistent with this the R-matrix theory of photoionization has been adapted to include relativistic effects in the initial and final state wavefunctions. This is achieved by application of unitary transformations to the Hamiltonian and dipole matrices, and thus is a very suitable procedure for a vector processor. This new suite of R-matrix codes has been exhaustively tested on the CRAY- 1 and is currently in preparation for submission to CPC. The new codes are being used to calculate: electron collision cross-sections for atomic potassium and the inert gases neon and argon; photoionisation of argon [23] and iron; and the electron

K.A. Berringlon, K T. Taylor / Atomic collision calculations I

I I

I

12P°I4d

200

I I

I

I

I

Sd

I

I

I

2p°

I

I

405

.~ C~

I

iT irt r

—

3r,O

r

—

40

~30 g

-

A

-

12P°I6s

7s

04 I’, -

05

~ 20

-

I

I~I

__~___

~

015

I

I

I

020

I

I

I

I

I

025

030

Photoelectron Energy IRydI

Fig. 4. Photoionization cross sections for S I (3P0_3PO) using IMPACT by Mendoza [12], compared with earlier 3-state close coupling results (dotted curve). The dashed line represents the Gailitis average.

excitation of the highly ionized systems of Ca XVII and Fe XXIII [10]. In beryllium-like Fe XXIII which have been observed in solar flares by Skylab, lines of all ten Table 1 Current RMATRX calculations on the CRAY-I for electron— atom and —ion collisions Atom/ion

References

He C III

OV OVII Si III Si IX Ca XVII Fe XXIII Ne Ar

[17] [18] [19—21]

[10] [22] [22]

Comments II state 12 state 12 state II state 12 state 10 state including relativistic effects (see section 2.4) 9 state 9 state

terms of the (1s22s2), (ls22s2p) and (1s22p2) configurations in intermediate coupling were included by Scott and Burke [10]. This gave up to 28 coupled channels for each total angular momentum and parity, and thus represents the biggest electron excitation calculation so far on the CRAY- 1. Fig. 6 shows the collision strength for the transition (1s22s2)’s~—(ls22s2p)3P~(which is spin-forbidden in LS coupling) with (curve E) and without (curve F) relativistic effects; and it is clear that relativistic effects increase the collision .

strength for this transition by a factor of two. Work is now underway to investigate relativistic effects in third-row series ions of interest in laboratory and astrophysical plasmas. For example, the lines of Fe II are seen in many astrophysical objects like Seyfert galaxies. In order to inter.

.

.

.

.

pret the data it is important to know the electron collision rates accurately. The Belfast group have embarked upon a very big claculation of electron

K.A. Berrington, K. T Taylor / Atomic collision calculations

406 0.020

nniiri

—

_________________________________ e — OVU 1S— 23S

—-

1111111 I

1

0.015 0

a, a

5)

~ 0.010 C

0

G5

Al

0.005

4’5

r,=2

n=3

5’O

Electron energy (Ryd.)

Fig. 5. Electron excitation collision strength for 0 VII (l’S—23S) using RMATRX by Berrington et al. [18], compared with earlier distorted-wave calculations.

0.03

~ 0.02

0.00~

4.9

5.5

6.1

6.7

1

~

Electron Energy (Rydbergs)

Fig. 6. Electron excitation collision strength for Fe XXIII without (curve F) relativistic effects.

(1 5~ —~ P~)using

RMATRX by Scott and Burke [10], with (curve E) and

K.A. Berrington, K T. Taylor

impact on Fe II on the Daresbury CRAY-i. The ten lowest states of Fe II are included in the R-matrix code. We are including configuration interaction wavefunctions for these target states. Previous calculations on e—Fe II used only single configuration wavefunctions, and these yielded very poor energiei for the target states where for example, the 3d7 and 4f states were about 30 times higher than the experimental value. But with the latest C.I. calculation, which includes all configurations arising from 3d7, 3d64s, 3d54d, 3d 4s4d and 3d54p4f, good energy splittings are obtained, Work is already underway for photoionisation of Fe I using these wavefunctions. This would be the biggest computational problem of its kind on electron impact excitation of any atom or ion, and could only be attempted on a vector processing machine such as a CRAY- 1. Finally, a new, fully relativistic code is under development at Oxford, in which the Dirac equation is solved in the R-matrix framework. This new code will allow electron excitation and photoionisation calculations to be extended to very heavy atoms and ions and to higher electron energies.

/ Atomic collision

calculations

407

equations can be written in a compact matrix form as in eq. (11): Mf=g,

forr~a,

(19)

and the solution of this is well suited to vector processing machines. The dimensions of the M, f and g matrices are (by analogy with eq. (11), NTOT>< NTOT, NTOT X NLI and NTOT X NLI. The matrix M thus dominates the storage requirement, since NTOT is defined in a similar way to eq. (10) and is typically up to 1000 for modern vector processing machines. The order of M can be kept as small as possible by reducing the number of mesh points (P2 in eq. (10)) in three ways: i)

by using high order quadrature methods such as Gauss—Legendre;

ii) by selecting the density of points to correspond with the strength of the interaction potential; iii) by selecting a different mesh for each channel or group of channels. So eq. (10) defining the order of the M matrix is modified: n

NTOT

~ Nc”~*P2°~,

(20)

1=1

3. Electron—molecuLe scattering

where Nc1’~is the number of channels in the ith group, with P2~mesh points. The linear algebraic method has been applied at the static exchange level by Collins and Schneider

The success of the close-coupling techniques used for electron—atom scattering have stimulated parallel work for electron—molecule scattering. A general R-matrix code for electron scattering from diatomic molecules is under development at the Daresbury Laboratory in the UK, and will shortly be implemented on the CRAY [24]. In the USA, Collins and Schneider [25] at Los Alamos have developed an electron—molecule code on a CRAY1 using the linear algebraic approach; motivated largely by the success of this method on the CRAY for electron—atom scattering. In the linear algebraic approach, Collins and Schneider [25] begin with the coupled differential equations within the single-centre expansion and close-coupling approximation. This set of equalions is converted to a set of coupled integral equations, which in turn is transformed to a set of linear algebraic equations by approximating these

~-

integrals by discrete numerical quadratures. These

[25].

25.0

200 ~

15,0

0

10,0

50

I —.—

I __.— - ~ —

.—

- —

—

-

0 0’

00

10

20

30 Electra,, 4.0 5.0 6.0 energy leVI

70

80

90

100

Fig. 7. Integrated cross sections as a function of energy for ~ and H 5 symmetries for e —CO2 by Collins and Schneider

408

K.A. Berrington, K. T. Taylor

to investigate electron scattering from H2, N2, LiH and CO2 molecules. For e —CO2 scattering, the biggest case so far attempted on the 1 Mword CRAY-l at Los Aiamos, the order of the M matrix for the ~ symmetry was reduced, by using the prescription (iii) above, from 1800 with equal numbers of mesh points per channel to 934 using eq. cross sections e—CO2 by 7.Collins (20). and Some Schneider [25] are for shown in fig. A

/ Atomic

collision calculations

The time-dependent Schrodinger equation, eq. (21), becomes (H— i.~—)tP(ri,r2t) where ‘I

0,

(23)

0 or r2 = 0. For t = 0: 2 (ri,r2o)2r2exp(r2)(/2d)~~~ x exp{—ikri + (r 1 a)2/2d2], (24) =

0 when r1

=

=

—

typical calculation for one energy took 35 s on the CRAY-I. Matrix partitioning procedures are currently being examined to extend the method to larger calculations.

4. Time dependent treatment of electron impact ionisation Recently, Bottcher [26] at Oak Ridge in the

i.e. the electron wave-packet is initially localized around r1 = a with a Gaussian spread of width d. If the Hamiltonian in eq. (22) is split into two parts representing each electron: H = H1 + H2, where 2 1 a H 1= + ~ V, —

USA, using a CRAY-i at the Magnetic Fusion Facility in Livermore, has developed a procedure for studying electron impact ionisation by numerically solving the time-dependent Schrodinger equation with Coulomb interactions. The time-dependent Schrodinger equation (in atomic units) of a system whose Hamiltonian is H, [H(q,p;

t)

i~~I(q;t)

—

0,

(21)

uniquely specifies the evolution of the wavefunction t1’ for all t>0 if its value at t = 0 is given. Consider the simplest system of electron impact of the hydrogen atom in its ground state: e +H(ls)

—~

e +e

+H~,

The approach used by Bottcher [26] is to follow numerically the time evolution of a wave-packet localised in space. To illustrate the method, a model Hamiltonian where the two elecrons are constrained to move on radial vectors at 180° to one another is used [27]: l(~+~\+v (22) -

H ~ where —

—

1 r 1

\ 8r,

2

ar2

1 r2

I r1

/

.~- —~

(25)

for i = 1, 2;

the Peaceford—Rachman algorithm is then employed. This sets 1L(r tp(t + T) = L(ri) 2)’L(r2)*L(r1)*s/i(t), (26) where L(r,) = 1 +~iTJ1~,for i = 1,2, so that each step has the form

f L(r)g,

(27) wheref is determined from g or vice versa. Requiring the integrated deviation from eq. (27) to be stationary with respect to small variations inf, say a~,yields a set of linear equations. In this way propagation of a wave-packet eq. (26) is reduced to multiplications by banded matrices or their inverses. Equal time steps ensures stability. The method is at present being used with more realistic Hamiltonians and results should soon be published on these calculations. 5. Ionic recombination in an ambient gas Ionic recombination in an ambient gas, for example X~+Y+Z~[XY]+Z,

K.A. Berrington, K T Taylor

where Z represents an ambient gas atom, and the alternative process of binary recombination: ±

-

/ Atomic collision calculations

409

defined to lie along the magnetic field so that B = BZ, then the projection of the orbital angular momentum M = L Z is a constant of the motion, with which is associated the linear Zeeman energy shift I~sE= ~t0BM. With the choice of A = ~r X B, the SchrOdinger equation can be written down from eq. (28) (in atomic units): 2 L2 1 + ~/3~r~sin2 8 I ‘P = Ia + 2 ar2 2r2 r / (29) •

X + Y —s x + Y are important processes in connection with the ,

—

development of lasers. These processes are being investigated on CRAY computers in the UK by Bates [28] Laboratory. and in the The USArecombination at the Lawrence Livermore rate coefficient is calculated by the Monte Carlo technique of computer experiments with different initial velocities and directions of X + and Y A semi-classical approach is used; where the X~,Y~ interaction is Coulombic and the X~,Z and Y, Z interactions are represented by hard spheres of fixed radii, and a Maxwellian distribution of velocities of the centre of mass of the ion pair is used. These Monte Carlo simulations are ideally suited to vector processing computers.

—

—

—

—

—

.

where $ = B/2c

To obtain approximate solutions to eq. (29), ‘P is expanded in a basis of Sturmian functions S,~P(r): tW...

—

Some processes studied by Bates [28] include the recombination of F and Kr + in the presence of helium or argon gases. This work has led to an investigation of mutual neutralization and ter-molecular recombination and their interaction, Bates has found [29] that the rate coefficient for mutual neutralization may be greatly enhanced by the presence of the ambient gas, and that mutual neutralization may suppress the rate coefficient for ter-molecular recombination to a very marked extent,

2.1 X l0’° au B(G).

=

“°“

‘~‘—

r

5(t)y ni

/M

p

30

p3,1

—

where the coefficients are to be determined. Now the overlap matrix ‘PP~~

B01

~

=f

drS,~(r)S,~(r)

(31)

0

vanishes except when n’ = n, n ± 1; so B can always be represented as a tridiagonal matrix. Thus the equation for the coefficients take the form of the generalised eigenvalue problem:

(32) 6. Atoms in strong magnetic fields Work is underway by Clark and Taylor [30] in the SERC Daresbury Laboratory in the UK to use the CRAY- 1 to investigate the quadratic Zeeman effect in high Rydberg states. There is much interest in this problem because of the wealth of uninterpreted experimental data and possible astrophysical applications. Clark and Taylor in ref. [30], examined the effect of a magnetic field (B) on the hydrogen atom, where the non-relativistic Hamiltonian is H=

2M

(~

+ ~A) C

2—

2 f—. r

(28)

Here, A is the vector potential. If the Z direction is

where the selection rules allow the Hamiltonian matrix H to be written as a banded matrix. By using an algorithm due to Crawford [31] implemented in CRAWAR by the UK National Physical Laboratory and modified slightly to exploit the vector processing capability on the CRAY-I, this equation was solved with the banded structure preserved throughout the calculation. This allows a basis of up to about 1500 Sturmian functions to be used within the fast addressable memory of the 0.5 Mword CRAY- 1 at the UK Daresbury Laboratory when eigenvectors are to be found; and up to about 3000 when only eigenvalues are calculated. This algorithm produces all the eigenvalues in about 90 s, and each eigenvector in about 1

S.

K.A. Berrington, K. T. Taylor / Atomic collision calculations

410 8

a

‘~

References

6

[1] MA. Crees, M.J. Seaton and P.M.H. Wilson, Comput.

4 2

Phys. Commun. 15 (1978) 23. [2] K.A. Berrington, PG. Burke, M. le Dourneuf, W.D. Robb, K.T. Taylor and Vo Ky Lan, Comput. Phys. Commun. 14 (1978) 367. [3] W. Eissner, M. Jones, H. Nussbaumer and P.J. Storey,

0 4

—

___________________

Comput. Phys. Cornmun. to be published. [4] A. Hibbert, Comput. Phys. Commun. 9 (1975)141.

LL

_______:J

I I ________________________________________ ~ 2j

I

Ii ~ 0 __________________________________ _______ ______________________.

~2l I I

0 8~

I I

I

I

___ ______

I I

I

dJ e

6] 4~

021 -10

[5] PG. Burke and M.J. Seaton, Meth. Comput. Phys. 10 [6] (1971) 1. [7] M. Crees, Comput. Phys. Commun. 23 (1981) 181. [8] PG. Burke and W.D. Robb, Advances in atomic and molecular physics, vol. New 11, eds. DR. and B.1975) Bederson 143. (Academic Press, York andBates London, p. [9] N.S. Scott and PG. Burke, Comput. Phys. Commun. 26 (1982) 419. [10] N.S. Scott and Phys. B Phys. 13 (1980) 4299. to [11] K.L. Baluja andPG. PG.Burke, Burke,J.Comput. Commun., be published.

~ ~ -8

~ILI~IIIIIIJIIJIJIIJI.II1IJIkL [~J -6 -4 Energy (au.) ~

-2

Fig. 8. The oscillator strengths for transitions from the ground state of hydrogen to high Rydberg levels with 1M1 I in a’ field of 47 kG by Clark and Taylor [30].

Clark and Taylor [30] thus determined oscillator strengths for transitions from the ground state of hydrogen to levels with M = 0 and M = 1 in the presence of a magnetic field of 47 kG. Fig. 8 shows the oscillator strengths for the M = 1 levels. The lowest lines here correspond to the perturbed ,~= 23 states. Note the structure consists of a principal line and its associated satellites in a cluster; with increasing energy the line clusters show widening and greater proximity to neighbours, eventually interpenetrating other clusters.

[13] Berrington, P.G. Burke, P.L. Dufton and A.E. King[12] K.A. C. Mendoza, in preparation. ston, J. Phys. B 10 (1977) 1465. [14] K.A. Berrington, P.G. Burke, P.L. Dufton and A.E. Kingston, At. NucI. Data 26 (1981) 1. [15] A. Phys.Berrington, B 7 (1974) PG. 1417.Burke and A.E. King[16] P.L.Hibbert, Dufton,J.K.A. ston, Astron. Astrophys. 62 (1978) 111. [17] K.A. Berrington, PG. Burke, P.L. Dufton, A.E. Kingston and AL. Sinfailam, J. Phys. B 12 (1979) L275. [18] K.A. Berrington, J.G. Doyle and A.E. Kingston, Abstracts of the XII Intern. Conf. on the Physics of electronic and atomic collisions, Gatlinburg, USA (1981) p. 455. [19] K.L. Baluja and A. Hibbert, J. Phys. B 13 (1980) L327. [20] K.L. Baluja, P.G. Burke and A.E. Kingston, J. Phys. B 13 (1980) L543. [21] K.L. (1981)Baluja, 1333. P.G. Burke and A.E. Kingston, J. Phys. B 14 [22] K.T. Taylor, C.W. Clark, K.A. Berrington, PG. Burke and P.C. Ojha, Abstracts of the XII Intern. Conf. on the physics of electronic and atomic collisions, Gatlinburg, USA (1981) p. 204. [23] K.T. Taylor and N.S. Scott, J. Phys. B 14 (1981) L237. [24] B.D. Buckley, PG. Burke and Vo Ky Lan, Comput. Phys. Commun. 17 (1979) 175. [25] L.A. Collins and B.I. Schneider, Phys. Rev. A (1981) to be

7. Conclusion This paper has outlined several areas in atomic physics where use is being made of vector processing computers and an attempt has been made to show how these areas will benefit from the new technology.

published. [26] C. Bottcher, Intern. J. Quant. Chem. (1981) to be published. [27] C. Bottcher, J. Phys. B 14 (1981) L349. [28] DR. Bates, J. Phys. B 14 (1981) LI 15. [29] DR. Bates, J. Phys. B 14 (1981) to be published. [30] C.W. Clark and K.T. Taylor, J. Phys. B 13 (1980) L737. [31] C.B. 41. Crawford, Commun. Assoc. Comput. Mach. 16 (1973)