NASA Technical Memorandum 101559 

? 

Charge Exchange Trmsition Probability for Collisions 

Between Unlike Ions and Atm Within the Adiabatic 

Approximat ion 

Y, J, Xu 

GI SI Khandelwal 

J, W, Wilson 

PROBABILITY (NASA-TU-101559) FOR COLLISIONS C HARCE EXCHANGE BETUEEI TRANSITIOB U19 LIKE IV 89- 257 04 

IONS AND ATOHS UITEIY THE ADIABATIC 

BPPROXIMATIOB @AS A. Langley fesearch Unclas 

Center) 43 p CSCL 03B 63/93 0211549 

April 1989 

National Aeronautics and 

Space Administration 

~ley~center 

Hampton, Virginia 23665-5225 
CHARGE EXCHANGE TRANSITION PROBABILITY FOR COLLISIONS BETWEEN UNLIKE IONS 

AND ATOMS WITHIN THE ADIABATIC APPROXIMATION 

Y. 3. Xu and G. S. Khandelwal 

Department of Physi cs , 01 d Domi ni on Uni versi ty 

Norfol k , Vi rgi ni a 23508 

and 

3. . . Wi son 

Langley Research Center 

Nat i onal Aeronaut i cs and Space Admi n i s t r a t i on 

Hampton, Virginia 23665-5225 

4BSTRACT 

A simple formula for the transition probability for electron exchange 

between unlike ions and atoms is established within the adiabatic 

approximation by employing the Linear Combination of Atomic Orbitals (LCAO) 

method. The formula also involves an adiabatic parameter, introduced by 

Massey, and thus the difficulties arising from the internal energy defect and 

the adiabatic approximation are avoided. Specific reactions 

Li3' t H +Lit+ + H+ and Be4' + H +Be3+ t Ht are considered as examples. 

calculated capture cross section results of the present work are compared with 

the experimental data and with the calculations of other authors over the 

velocity range of 10 cm/sec to 108cm/sec. 

The 

7 
2 

I. INTKODUCTION 

Charge transfer collisions provide an interesting three-body system which 

is amenable to theoretical analysis in addition to their importance in the 

studies of fusion plasmas, astroph.ysica1 processes, and more recent studies o 

short wavelength lasers. Traditional interest still looms as important 

applications in heavy ion accelerators and radiation-induced chemistry as we1 

as charged particle transports in matter. 

In the past, the perturbed stationary state (PSS) method has been applied 

to study a symmetric resonance process such as A+ + A +A + A+. 

theoretical approaches rest on obtaining stationary molecular states 

(refs. 1-3). These results are usually in good agreement with their 

experimental counterparts. Encouraged with this success, in the above 

resonant process, extending these calculations to the collisions between 

unlike ions and atoms is presently studied. In this context, the reaction of 

the type 

Various 

A'+ B + Bq+ AE 

in which AE is the internal energy defect would be of great interest. 

Unfortunately, the formulation of the problem of the non-symmetrical 

charge transfer collisions of equation (1.1) is rather difficult using the 

procedures similar to the ones appl ed to the symmetric case. The correct 

procedure should involve the expans on of the wave function in terns of 

molecular wave functions. Rapp and Francis (ref. 4) have pointed out that the 

connection between the molecular and atomic approach in non-symmetrical charge 

transfer is not easily obtained. These authors in their work abandoned the 

molecular expansion, and instead, within the lowest order approximation, they 
3 

replaced the molecular wave funct 

atomic energi es. 

ons and energ es by atomic orbitals and 

In this paper, we adopt a correct molecular expansion procedure by 

employing the Linear Combination of Atomic Orbitals (LCAO) method. 

symmetrical charge transfer process of equation (1.1) is calculated for low 

and intermediate energies using an adiabatic approximation together with 

Massey's (refs. 5 and 6) adiabatic criterion resulting in a limited collision 

range. A simple formula for the transition probability is obtained under the 

The nontwo-state approximation. The formula is expressed in terms of overlap 

integrals, exchange integrals, and Coulomb integrals. The internal energy 

defect AE and the collision range Rc also appear in these formulas. 

Specific reactions Li3+ t H +Lit' t H+ and Be4' t H +Be3' + Ht are 

considered as examples. The results are comDared with the experimental data 

and with the calculations of other authors, and a brief discussion is 

presented. All these overlap integrals, exchange intergrals, and the Coulomb 

integrals can be calculated analytically by employing the hydrogenic wave 

functions. The simplified picture presented in this paper gives insight into 

the essential nature of the charge transfer process. 

11. QUASI-CLASSICAL APPROXIMATION 

As is known, the passage from quantum mechanics to the limit of classical 

mechanics is similar to the passage from wave optics to geometric optics 

(ref. 7). 

as U = sei', where a is the amplitude, 

of minus one. 

wavelength i.e., to a large value of the phase I$. Similarly, in the limiting 

case of classical mechanics, one writes the electronic wave function as 

The field components U in the electromagnetic wave can be written 

+is the phase and i is the square root 

The limiting case of geometric optics corresponds to a short 
4 

P=ae where a is a slowly varying function, S is the action which 

takes on large values, and h is the Planck constant divided by h. 

Henceforth, we will use atomic units in which h = 1. This use of the action 

variable is determined after considering the analogy of the principle of least 

action in classical mechanics and Fermat's principle in optics. As is known, 

the action function is given by 

S = - Et + C I Pidqi 

ER 

U 

- - - + ' Pidqi 

and becomes large for small velocity u. Thus, Landau (ref. 7) stated that 

JI = aeiS is the wave function of an "almost classical" or "quasi-classical" 

physical system. Furthermore, by absorbing the second term of the above 

equations into the quantity a' and retaining the adiabatic approximation 

(i. e., u extremely small), the wave function JI becomes 

il Edt = a' e to 

where a' is a slowly varying function of time t. 

to the capture process of present interest. 

Let us now apply the above 

Consider the situation in which the projectile captures an electron from 

the target. Then I# is the wave function of the electron long before the 

collision and is essentially the electronic wave function of the target. 

the other hand, after the collision the wave function of the electron would be 

the electronic wave function J12 of the projectile. 

1 

On 

These states are not 
5 

stationary states in the sense that S - Et is an extremum along the 

trajectory. The stationary states of this system are the two molecular states 

denoted by J, and JI,. (In a symmetric case, g refers to a gerade state and u to 

ungerade state.) Thus, according to equation (2.2), within the adiabatic 

approximation, the wave function at anytime t can be expanded as 

9 

JI = A(t) JI exp(-i It -to E g dt) + B(t) JIuexp(-i ITto Eudt) 9 

where E and E, are eigen energies; J1 and JI, are the corresponding two eigen 

functions. (Two-state approximation is employed.) Equation (2.3) is similar 

to the one usually derived under the PSS method (ref. 8). 

and J, are slowly varying functions of 

9 9 

The quantities A(t), B(t), J, 9’ U 

time. 

independent are discussed in the appendix. 

one can ignore the terms 4 

pertinent equations. This leads to the conclusion that the quantities A(t) 

and R(t) are independent of time. 

The conditions under which A(t) and B(t) can be treated as time 

. Under the adiabatic approximation, 

and 4, relative to the other terms in the 9 

For simplicity, consider a fully stripped ion of atomic number 2, moving 

slowly through the target hydroqen atoms. 

electron to the target nucleus and r2 the distance from the projectile nucleus 

(fig. 1). The internuclear distance of the projectile and the target is 

Let rl be the distance of the 

denoted by R. The Hamiltonian of the system with the electron and two nuclei 

in atomic units is 

z1 

2 1 

- -- -- z 1 v2 

H=R-9 r r. 
6 

Wave functions of interest are 

1 

61= 7; exp (-r1) 

with (2.7) 

The JI is the wave function of the electron when it is localized near the 

first nucleus (target), ignoring the presence of the fully stripped projectile 

ion. 

localized near the projectile nucleus, ignoring the presence of the target 

nucleus. 

1 

On the other hand, I), is the wave function of the electron when it is 

As noted earlier, in arriving at equation (2.3), the wave function at any 

,time is given by the linear combination of two quasi-molecular stationary 

states J, and Jtu . These states are determined by the variational method by 

employing the Hamiltonian H with their form given as 

9 

(2.8) 

(2.9) 

where S = I + $ dT is the well known overlap integral. The eigen value of. 

the Hamiltonian is the potential function for the nuclear motion in the BornOppenheimer approximation. 

similar to the usual potential crossing problem. 

12 

We note that the charge transfer problem is 
7 

The quantities A(t) and R(t) in equation (2.3) are determined from the 

initial condition J, -+ $,as t + -to to be satisfied by the value 1 

Lett 

where 

ng t -+ 00, using equat on (2.3)y we obtain 

JI = C" J, t C" J, 11 22 

..dnce, the transition probability 

w12= 

+= E dt e -i I ~ E dt 9 ul 

is g,ven by 

(2.10) 

(2.11) 

(2.12) 

(2.13) 

(2.14) 

where 1* = (E - Eu) dt 

9 (2.15) 

For any symmetric resonance process, the quantity AE = E - Eu plays the 

ke,y role in determining the transition prohability. We will determine it by 

the variational method. 

obtained by the LCAO method as 

9 

We note that the molecular wave function Q can be 
8 

The expectation value of the Hamiltonian H is 

2 2 

& = LA-&. pz--. = c1 H1l+ __ C1C2(H12+ ___ H21) + c2 H22 (2.16) cl+ 22 c2+ 2 c1c2s I 4)2dr 

where 

= $IT H $.dT (2.17) "i jJ 

The stationary states are obtained by the stability condition 

which leads to 

C1(HIl- E) + C2 (H12- ES) = 0 

Cl(Hzl- ES) + C2(H22- E) = 0 

(2.18) 

(2.19) 

(2.20) 

For the symmetric case (Z=n=l) we have Since H is a Hermitian 

operator, 6 and $* are real functions, thus H12 = H21. The secular equation 1 

of equations (2.19) and (2.20) is 

Hll = H22 = H. 

2 (H - E) (H - E) = (HI2- ES) (2.21) 

which leads to two eigen energies 

- H12 

El= 1 - s (2.22) 
from which 

H12 

&2= 1 t s 

9 

(2.23) 

(2.24) 

Using the Hamiltonian and the wave functions (Z=n=l) results in 

H = El,"+ 3 (2.25) 

H12= E2,0 StK (2.26) 

where and Epln are the atomic ground state energies of the electron in 

the target and projectile orbitals, respectively. The S is the overlap 

integral given (ref. 9) as 

s =I +pl dT 

R2 -R = (1 + R +-) e 3 

(2.27) 

and K is the exchange integral given (ref. 9) as 

- 1. 2 -R - (~-3R)e 

(2.28) 
10 

and J is the coulomb integral given (ref. 9) as 

(2.29) 

1 -2R = (1 t a) e 

Thus, 

so that, 

HI2- HS = K - JS (2.30) 

(F 2 - 3) 4R exp(-R) - (- 2 t 4 t - '! t $) exp(-3R) 

1 - (1 t R t exp (-2~) 

(2.31) R E2- E = __-___ _-___ - 1 2 

For, R >> 1, we have from the above: 

(2.32) -R €2- E 1 LLT Re 

- 3 

which is on the order of the result given by Firsov, Landau, and Herring (see 

discussion in ref. 10) of -4Rexp (-R-1). We now consider the extension of 

these results for symmetric resonance transfer to non-symnetri c resonance 

transfer. 

111. NON-SYMMETRICAL RESONANCE TRANSFER 

In recent years, because of its usefulness in many areas of physics, the . 

2 reaction A t B + Bqt AE and, in particular, the fully stripped ion 

charge transfer process have received great attention. 

has been made in hoth the experimental and theoretical studies of this 

Considerable progress 

I. . 

problem. The unitarized Distorted Wave Approximation (UDWA) of ref. 11 has 
11 

been a successful numerical method, since it covers a wide energy region. For 

extremely lov~ energies, the classical Rarrier model of ref. 12 and the 

absorbing sphere model of ref. 13 have been of interest to experimental 

physicists because of their considerable simplicity. The other numerical 

models, on the other hand, involve complicated calculations. Still, the early 

Landau-Zener formula .is widely used by many authors, since it has been studied 

fully for a long time (refs. 14-17). However, the Landau-Zener formula has 

its limitations. In this model the transition is assumed to be limited to a 

narrow region around the crossing point Rcp. 

varies linearly with the internuclear distance in the narrow region, namely, 

The potential energy separation 

The coupling matrix element Hif has a constant value 6. 

In 1962, Rapp and Francis (ref. 4) studied the charge exchange between 

unlike atoms. 

states and the atomic orbitals is not easily obtained. Therefore, as a lowest 

They pointed out that the connection between the molecular 

order approximation, they replaced the wave functions of the stationary 

molecular states by the atomic orbitals. This connection was found by 

employing the LCAO method. The physical difficulties arise when one 

simultaneously retains the internal energy defect and the infinite interact 

range within an adiabatic approximation or a quasi-classical approximation, 

because such a situation renders transition probabilities small. Actually, 

the infinite interaction tine does not exist at all, since there are so many 

target atoms in the material that the average interatomic distance probably 

on 

will be a suitable estimate of the upper limit of the effective interaction 

range. After such an interaction time, the projectile undergoes another 
12 

collision process. Massey (refs. 5 and 6) introduced an adiabatic criterion 

based on the following argument. Classically, the transition probability is 

proportional to the ratio of transition time and the collision time. Since 

now there exists an energy defect, AE, the transition time can be estimated 

as l/AE. Meanwhile, the collision time is proportional to an adiabatic 

parameter divided by the velocity of the projectile. 

time can be comparable to the collision time does there exist a significant 

transition probability. The adiabatic parameter thus cuts off the interaction 

time. This parameter is of the order of the atomic dimension. Hasted pointed 

out in ref. 6, that "the analysis of a large volume of experimental data leads 

to surprisingly small probable errors (10-20 percent) with a value R,= 7A 

common to different types of reactions." This adiabatic parameter of 7A is 

the range of internuclear separation over which the charge transfer is deemed 

possible (ref. 11). Actually, the cross sections were measured as functions 

of projectile velocities. 

maximum cross sections v, were almost a linear function of AE . leads to an adiabatic paraneter Rc = 7A . as a "cut-off" range in our calculations. 

Only when the transition 

Surprisingly, the velocities corresponding to 

The slope 

We apply this adiabatic parameter 

The capture cross section is thus determined in terms of the adiabatic 

parameter Rc, the energy defect AE, and the velocity of the projectile. It 

also depends on the atonic wave functions of both the projectile and the 

target and the interaction between them. 

stationary molecular states are still denoted by J) and J) where the 

subscripts g and u are used here only for convenience. These states assume a 

In the non-symmetrical case, the two 

9 U, 
13 

more general form than the corresponding symmetric resonance case. These more 

general states are defined as 

and 

$,= Ci ($I+ f24~2) 

where C1 and ClI are determined by the normalization condition 

so that 

1 

1/2 c= 

1 (1 + 2Sfp f12) 

Similarly, 

1 

1/2 Ci= 

(1 + 2Sf2+ f22) 

(3.3) 

(3.4) 

(3.5) 

The quantities S, fl, and f2 are functions of internuclear distance R. 

Under the adiabatic or quasi-classical approximation, the quantites A(t) 

and B(t) in equation (2.3) can be treated as time independent. 

equation (2.3) with equations (3.1) and (3.2) becomes 

Thus, 

t 

$ = A C1($1+ fp2) exp (-i I-t Eg dt) 

0 
14 

A' = .f2 

f2- fl 

and 

-to 

Since J, must tend to J,l when t +- to , we obtain from (3.5): 

fl 

f2- fl 

B' = - 

0 

-t 

with A' = A C1 and B' = C'] 

(3.7) 

(3.8) . 

The quantities A', B', fl, and f2 are all functions of internuclear 

distance R. Also, these quantities should be symmetric. 

A'(to), B'(-to) = B'(to), fl(-to) = fl(to), and f2(-t,) = f2(to), since the 

variational method is invoked, by which they depend only on the distances 

between the projectile and target nuclei. 

can be deemed to be time independent 

Thus, A'(-to) = 

Also, since the quantities A and B 

0 Abfllto = A'fl I -t 

Equation (3.6) can now be written in a general form as 

J, = C" J, + C" $ 11 22 

(3.9) 
15 

where, 

C; = A' fl exp(-i I t e E dt) + B'f 2exp(-i t -e E udt) 

dt) - exp(-i j!p dt)) 

0 -9 0 

(3.11) 

- f2 fl iexp(-i I-? t 

f2- fl 0 0 

where the coefficients are to be determined by the stationary phase method 

applied to the expectation value of the Hamiltonian. Hence, the transition 

probability is given by 

ff 

w12= ( f2- * fl) sin'n (3.12) 

where 

Notice that the transition probability of (3.12) differs from the 

symmetric case by an additional factor 4 ( i:-fil )* . This factor is 

determined by the LCAO method, i.e:, by the solutions of the basic equations 

(2.19). 

Cy (H1- E) + C~(B-ES) = 0 (3.14) 

and 

Cy (6-ES) + C" hH -2") = 0 . (3.15) 

where E. = H = J because H is Hermitian and $1 and tc, are.rea.1; H1 = Hl and 12 21 2 

H2 = H22. 
16 

In order to obtain a simple and explicit solution for the transition 

probability, the values of 13, S, and E will be discussed. When the 

internuclear distance is not too small, the quantities 4 and S are of the 

same order, whereas E is about the order of -0.5. Thus, by replacing the 

quantity B - ES in equations (3.14) and (3.15) by B, the problem is amply 

simplified. Later in this section, we will explain in detail the 

1 

on for this approximation. Thus, equations (3.14) and (3.15) can 

as 

(3.16) C; (H1- E) + Ci 7 1 B = 0 

justi fi cat 

be written 

and 

C" - 1B + Ci (H2- E) = 0 

12 

which give 

12 (H1- E) (H2- E) = (2 B) 

For the case H2 - H1 > 0, the above equation when solved for E yields 

where 

H2- H1 

2 A= 

B1= { 1 - (B/ZA) 2 - 1) A 

(3.17) 

(3.18) 

(3.19) 

(3.20) 

(3.21) 
17 

When the internuclear distance is not too small, B is rather small. Also, in 

the non-symmetrical process, except for a resonant channel, HI and H2 are 

usually not very close to each other. Thus, 

I 

2 

H1- H2 1 > 6 

Hence, under the above condition, B1 can be simplified as 

The equation (3.19) leads to 

E~= H1- B1 

1 E?= H2t R 

The equation (3.16) for E = E? becones 

C; (H1- H2- B1) + - 1C" B = 0 

22 

thus, 

C; (HI- 5- B1) H1- 

fl= 7 = - B /2 -= -B/i 1 

For E = E , the equation (3.17) becomes 1 

r. I1 

-- "i + C;(H2- H1t B1) = 0 2 

(3.22) 

(3.23) 

* (3.24) 

(3.25) 

(3.26) 

(3.27) 

(3.28) 
18 

Thus, 

f=-- 2 - - B/2 -B /2 

H2- H1 2 Ci H2- H1t B1 

Thus, the transition probability is 

12 2 W12= 4 sin TI 

where 

and 

(3.29) 

(3.30) 

(3.31) 

(3.32) 

In the case Hlf H2, or where HI and H2 are not close to tqle same value, 

applying (3.22), and remembering that B1 is of the order of fi2, we have 

H2- H1+ B H - H1 

1% 2 (3.33) h 

-.- 

B /2 B /2 A= 

and the transition probability is 

(3.34) 

This is the required simple transition probability formula which is 

determined by H12, (H2- H1), and the value of TI. 

related to the overlap integral. Physically, it is reasonable to expect 

The quantity H12 is closely 
19 

higher transition probability with more overlap. 

strongly related to the energy defect AE (at a large distance). 

there will be less transition probability with more energy defect. 

The quantity H2- H1 is 

Obviously 

For the case ti1= HE, we have from equation (3.18) 

2 2 (H - E) = (B/2) 

which yiel ds 

E = H f 8/2 

f=l,f--landA=2 1 2- 

(3.35) 

(3.36) 

(3.37) 

2 The transition probability for this case is given by W = sin n, which is the 

familiar formula (2.14) of the symmetric case obtained in Section 11. 

Let us now discuss the assumption of the replacement of the quantity 

8 - SE by B/2 in equations (3.14) and (3.15). Initially, this approximation 

seems somewhat arbitrary. However, B/2 is not strictly equal to B - SE in 

equations (3.14) and (3.15). Recall that the equation (2.19) is derived from 

the variational principle, which implies that the LCAO molecular state 

function is made the most stable state at that particular interatomic 

separation. On the other hand, equations (3.14) and (3.15) were altered by 

the replacement B - !k -+ B/2, which implies that a set of molecular states 

are chosen that, although not the most stable, nevertheless approach the most 

stable states for large separation. 

molecular states are chosen. 

In other words, a set of quasi-stationary 

I n t h i s way we may refer to a quasi-stationary 

molecular state expansion procedure. Actually, in chemistry, at a large 

distance, the molecular states are not stable at all. However, we are more 

interested in looking for a set of nathematically permitted basis states. 
20 

Prohab 

(refs. 

states 

approx 

y, this is one of the reasons why the atomic expansions are widely used 

1, 2, 4, and 18). Thus, it is nore important that these molecular 

be orthogonal. Fortunately, our quasi-stationary molecular states 

mately satisfy the requi rement of orthogonality. In the foll owing, we 

wi 11 expl i ci tly show how the orthogonal i ty requi renent is approximately 

satisfied. Consider 

where 

and 

Now , 

1 

1/2 C1= 1 1,2 and Ci = 

(1 + Sfl+ fl) 2 (1 + Sf2+ 2 f2) 

(3.38) 

5' - 0.9 

H2- H1 2 CY' - - -- and f = -- - --- 

(3.39) 

h 

Remembering flf2 = - 1, we have 

(3.40) 

For the case H1 #HE, when the internuclear distance R is not too small, . 

usually IH2 - Hll >> 10.5 61; thus (fl( >> 1 and )f21 << 1. Also, fl + 

1 1 

2 1/2 -b + -and 1 2 1/2 f 

(1 + 2Sf + f ) 1 (1 + 2Sf + f ) 11 22 
21 

One sees from these that 

f. 

Now, the overlap integral of the two atomic orbitals between the 

projectile and the target is small except when the internuclear distance 

approaches zero. Thus, J, J, d~ = 0. 

9u 

For the case H1 = H2, we have fl = 1 and f2 = -1. Thus, 

1 

1 I JlgJladT = ----- I (fl+ f2) J11J12dr = 0 

( 4- S2) 1'2 i 

For the case Hl- H2, we have fl = 1 and f2 * -1. Thus, 

I JlgJl,dr - (fr + f2) s = 0 

We observe that for the cases H1 = H2 and HI- Hp, because of the 

cancellation between fl and f2, the orthogonality is preserved. 

orthogonality point of view, this expansion is at least not worse than the 

atomic expansion. 

From the 

The capture cross section Q, according to the impact parameter method, is 

given by 

u = 291 PIt bdb 

0 

(3.41) 

(see fig. 2) where b is the impact parameter, and P refers to the transition 

probability. Usually, P takes the value P + -; but as discussed earlier, an 

adiabatic parameter will be employed to evaluate the cross section. 

transition probability P is given by equation (3.34) as 

t 

The 
22 

(3.42) 

where all the quantities appearing in this equation are functions of the 

internuclear distance R = (b + 

parameter defined by 2 u to= Rc. 

be 7A. 

Rc) '" in which R, is the adiabatic 

The value of the parameter Rc is chosen to 

IV. CHARGE TRANSFER IN ~i3+ + H 

Let us evaluate the magnitude of the transition probability for the 

following reacton: 

+ Li3+ + H + Li 2+(n=3) + H 

The magnitude of the transition probability is 

P= 

0 

4 (kl2 

(H=H) 12 1 

where 

sz =E S+--K 2,n R 
23 

lV2 1 z z H1= I 7 - - - - 

r r2 +E) $1 * 

Z 

2R 

- Ep,, - 3 +- 

(4.4) 

The quantities B, H1, H2, and n are calculated by evaluating the overlap 

integral S, the exchange integral K, and the Coulomb integral J, defined as 

foll ows: 

, and v2 * s = I J11J12 *, K =I 

rl 

The integrals S and K can be evaluated by introducing well known 

transformations of ref. 9 (fig. 3): 

and 

1 X =E (r2+ rl); 1 c x <- 

1 

IJ = E (r2- rl); -1 c IJ c 1 

(4.5) 
24 

For our special case, Z2= 3, Z1 = 1, and n = 3, we have 

s = -I JIp2 cfr 

Now, from table 1, the value of A3 = 0,013; thus, we obtain 

43 2 S = 0.013 (1.6 R + 8R + 24 R + 48R + 48) e-R 

Similarly, the exchange integral is 

32 K = 0.026 (4R + 8R + 12R + 12) e-R 

and 

s2 Y2 = R - I-- dT 

rl 

3s 

R = -0.5s + - - K 

The calculations of the Coulomb integrals J1 and J2 are done in the 

following way. (See fig. 4.) 

n 

(4.9) 

2 L 

$1 dT - 2n -2r -- sin _--- e de r dr J1= -I ----- - --I TI e 

r2 (R 22 + r - 2rR cos e) 1/2 
(4.10) 

where P = r/R. Finally, one obtains 

= 4R2 { e-2PR p2 Q t Jy e-'R p Q) J1 

(4.11) 

1 -2R 

- (1 + R) e 

1 

R =- 

1 One notes that J1+ E (R > 1)and J1+ 1 (R + 0). 

as follows (fig. 5): 

The quantity J2 is calculated 

2 2 -2r sin - - 0 - do . r ._ dr - 

(R + r - 2rR cos 0) 22 1/2 

dT = 2n (A3x 9) ] e -- 

(4.12) 

- 

-E- (0.043 R 4 + 0.215R 3 t 0.645R 2 t 1.29R + 1.61 + ~1 1 e -2R 

For the calculation of n, one notes for the Z = 3 and n = 3 case 

3 HI= -0.5 - 35 t -- 1R 

(resonance) that 

3 H2= -0.5 - 3 + .- 2R 
26 

Thus , 

1t 0 =- AE (R') dt = ?I o (H2- H1+ 2B1) dt 

-t 

0 

--- where, R' = 2 2 2 b +ut 

The capture cross section, u, is given by 

u = J; P(R) 2trb db 

where P = Po (R) sin% in which Po is taken from equation (4.1) and 

H12, H2- H1, A, and B1 are all functions of R2= b2 + - R 12 

4 c* 

(4.13) 

(4.14) 

V. RESULTS AND DISCUSSIONS 

The results of the calculations for the reactions Li+" + H +Lit+ t H+ 

are exhibited in figure 6. They are compared with the results based on the 

Uni tarized Di storted-Wave Approximations of Ryufuku and Watanabe (ref. 18), 

the ten molecular-state calculations of Kimura and Torson (ref. 19), and the 

calculatiorts of Stollherg and Hai-Woong Lee (ref. 17) based on the LandauZener formula. The results are also compared with the experimental data of 

Sein et al. (ref. 20). Also, the results of the calculations for the 

reaction Be 

the UDWA approximation and Exponential Di storted-Wave Approximation 

calculations of Suzuki et al. (ref. 21) and with the molecular orbital close 

coupling calculation of Hare1 and Salin (ref. 22). No experimental data were 

available for this process. 

4t 

t H + Re3++ H+ are exhibited in figure 7 and are compared with 
27 

In figure 6, both the results of our calculations as well as those based 

on the Landau-Zener formula do not appear to show the tendency of the cross 

section to increase toward the high-velocity region. This is partly due to 

the implied employment by us of the Born-Oppenheimer approximation in which 

the effect of the nuclear kinetic energy terms are ignored in the 

Hamiltonian. 

Oppenheimer approximation should break down. When nuclear motion is 

8 For a large projectile velocity (u2> 10 cm/sec), the Born 

incorporated, there will appear an additional electron kinetic energy term 

because of the relative motion between the projectile and the target. 

becomes more obvious when our coordinate systm is set on the projectile. 

Then the kinetic energy of the electron becomes -!j me(G2+ Go)’, where Go is the 

velocity of the electron in the laboratory frame. 

This 

Hence, the average kinetic 

I 22 

2eo energy becomes - m (u t u2). A significant additional kinetic energy term is 

obtained when u2 is comparable to the electron velocity that is the first Bohr 

velocity. Also, if we choose the laboratory frame, the electron will receive 

an additional velocity fron the momentum transfer. Thus, because of the above 

reasons, some higher states will be excited, causing a significant 

contribution to the total cross section. 

Let us now discuss the general energy dependence of the cross section. 

One expects that toward the high velocity region, the cross section should 

decrease as a result of the increasing relative velocity between the 

projectile and the target. In the extremely low energy region, the cross 

section for these processes should not rapidly approach zero with a decrease 

in u, but rather approach a not too small constant. The calculated cross 

section in the energy region of interest (below a few keV) approaches a 

constant value due to the term H2- HI in the argument of the sine function. 

This makes the value of the sine function very large, and the transition 
28 

probability oscillates rapidly. The summation over the impact parameter 

2 approaches an average number, sin rl - 0.5. When u becomes larger, there will 

be a slow oscillating behavior, but this velocity region is beyond our 

adiabatic approximati on. 

In the symmetric case, since HI= H2, only when the impact parameter b is 

small does the transition probability exhibit the rapidly oscillating behavior 

different from the non-symmetric situation. 

Finally, experimental physicists (refs. 23 and 24) have recently noticed 

ndependent 

results wi 11 

on 

that below a few keV, the capture cross section shows velocity 

behavior in helium and lithium materials. It is hoped that our 

help in classifying these and other relevant capture cross sect 

observations. 

CONCLUDING REMARKS 

A semiclassical approach is used to derive a quasi-stationary state 

method to approximate the charge transfer cross section. 

slightly more complicated than the usual Landau-Zeuer formalism and greatly 

simpler than the coupled channel and various distorted wave formalisms. 

Unl i ke the Landau-Zener result , we predict nearly constant cross sections 

below a few KeV in agreement with recent experiments and the more complex 

formalisms. 

The result is only 
REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

20. 

0. R. Rates and N. Lvnn, Proc. Roy. SOC. (London) A253, 141 (1959). 

E. F. Gurnee and J. L. Magee, J. Chem. Phys. 26, 1237 (1957). 

D. Rapp and I. B. Ortenburger, J. Chem. Phys. - 33, 1230 (1960). 

D. Rapp and bl. E. Francis. 3. Chen. Phys. 37, 2631 (1962). 

H. S. W. Massey, Rept. Progr. Phys. - 12, 248 (1949). 

3. B. Hasted, Physics of Atonic Collisions (Butterworths, London, 1964), 

-- 

- 

- 

pp. 420-424. 

L. D. Landau and E. M. Lifshitz, Quantum Mechanics Non-Relativisitic 

Theory (Pergamon Press, 1977), pp. 19-20. 

D. R. Bates and R. McCarroll, Proc. Roy. SOC. A, 245, 175 (1958). 

N. V. Riggs, Quantum Chemistry, (The Macmillan Company Collier-Macmillan 

- 

Limited, London, 1969) , pp. 72-75. 

Reference 7, p. 313. 

H. Ryufuku and T. Watanabe, Phys. Rev. AlS, 2005 (1978). 

H. Ryufuku, K. Sasaki and T. Watanabe, Phys. Rev. A21, 745 (1980). 

R. E. Olson and A. Salop, Phys. Rev. - A14, 579 (1976). 

L. Landau, Z. Phys. Soviet, - 2, 46 (1932). 

C. Zener, Proc. Roy. SOC. A, - 137, 696 (1932). 

R. K. Janev, 0. S. Belic and B. H. Bransden, Phys. Rev., A28, 1293 

- 

- 

- 

(1983). 

M. T. Stollberg and Hai-Woong Lee, Phys. Rev. - A29, 2448 (1984). 

H. Ryufuku and T. Watanabe, Phys. Rev. A19, 1538 (1979). 

M. Kimura and W. R. Torson, 3. Phys. - 816, 1471 (1983). 

W. Seim, A. Muller, I. Wirkner-Bott and E. Salzborn, J. Phys. By 14, 

- 

- 

3475 (1981). 
30 

21. H. Suzuki, N. Toshima, T. Watanabe and H. Ryufuku, Phys. Rev. - A29, 

529 (1984). 

22. 

23. E. Justiniano, C. L. Cocke, T. J. Gray, R. Dubois, C. Can, W. Waggoner, 

C. Hare1 and A. Salin, Phys. Rev. - B10, 3511 (1977). 

R. Schuch, H. Schmidt-Bocking and H. Ingwersen, Phys. Rev. - A29, 1088 

(1984). 

24. W. Waggoner, C. L. Cocke, S. L. Varghes and M. Stockli, Phys. Rev. 

-- A29, 2457 (1984). 

e* . 
31 

APPEND I X 

In this appendix, the conditions under which the quantities A(t) and B(t) 

can be treated as time independent in the equation (2.3) are obtained. The 

central point in this treatment is the observation that when u -+ 0, compared 

with the phase change, a is a slowly varying function in $ a e . As 

discussed in Section 11, this leads to 

is 

We have shown previously that Y and Yu are slowly varying functions and 

9 

had assumed that A and B vary so slowly that they can be treated as time 

independent. Now, let us estimate how slowly \y and lU are varying and under 

what conditions A and B can be treated as time independent. Having done this, 

we will know the conditions under which our analytical model of this paper 

works. 

9 

The wave functions of the stationary states can be formed by linear 

combination of atomic wave functions, i.e., 

and 

where JI and JI 

the target, respectively. They are independent of the internuclear distance 

R. 

condition and the LCAO method to he 

are the atomic electronic wave functions of the projectile and 1 2 

The coefficients C1, C2, CI1, and CI2 are determined by the normalization 
32 

1 1 

2 1/2 (1 + 2Sfl+ fl) 2 1/2 ; c2= (1 + 2Sfl+ fl) 

C1= 

where fl and f2 are solutions of equations (2.18) and (2.20) and are known 

functions of the internuclear distance R. 

Now, first let us estimate how slowly the molecular eigen functions 

vary. For this consider 

d C. dC 

where b is the impact parameter and u 

notes that if u is small, then $ and similarly tu should be small. Now, 

consider the following 

is the velocity of the projectile. One 2 

2 9 

In the above, the second tern contains a factor u because of $ and the 

2 9’ 

third tern contains a factor of E which is about the order of unity. 

Therefore, if u g 1, compared to the terms containing E and E, we can 

neglect the terms 5, 

9 

2 9 

and GU . 9 
33 

Next, let us discuss the behavior of the quantities A(t) and B(t). For 

this we assume 4,- $,= 0, then using the Schroedinger equation i$ = H$ and 

equations (B.l) and (B.2), we obtain 

Recall the basic equation of LCAO (eqs. (2.19) and (2.20)) 

C1(Hll- E) + C2(H12- ES) = 0 

C1(H?l- ES) + C2(H22- E) = 0 

and the two sets of solutions C1, C2 and CI1, CI2 corresponding to eigen 

energies E and E,, respectively. Thus, the equations 9 

Ci (Eu- Hl) + C'(E S - H ) = 0 2u 12 

lead to 

t t 

i k (cl+ C~S) e -i Egdt + i B (ci+ C; SI e - i I 0 (B.4) 
34 

Similarly, starting again from the Schroedinger equation i$ = ti$,, we will 

obtain L 

The above two equations can be written as 

and 

t 

'A (c2+ C~S) + B (c;+ cis) e- I (Eu- Eg) dt = o 

Now, h and 'B have nonzero solutions if, and only if, the secular equation is 

equal to zero, as follows: 

which leads to 

f*- fl= (f2- fl)S 2 

where S is the overlap integral and is always less than unity except when 

R = 0 and $1= $2, and where fl and f2, defined by fl = C2/C1 and f2 = C12/CIl, 

are obviously not always equal. 

zero. 

Thus, the secular equation cannot always be 

Therefore, the only solution of equations (B.5) and (B.6) is 'A = g = 0. 
35 

Thus, if we assume $ = $,= 0, then A and B are independent of time. 

9 

A(t), and B(t) are very slowly g’ $us However, if u < 1, we can say that J, 

varying functions. In practice, if u < 5.x 10 cm/sec , i. e., in atomic 7 

units u < 0.2 , the condition u < 1 is satisfied. It is noticed in figure 6 

for values u > 6 x 10 cm/sec, our results do not exhibit the increasing cross 

section as observed in experiment. 

7 
36 

TABLE 1. A list of functions for the reaction of 

3+ 

Li + H + L!+ (n=3) + H+ 

FUNCTION EXPRESSION 

S 

K 

0.013 e-R(1.6 R4 + 8R3 + 24R2 + 48R + 48) 

0.026 e-R(4R3 + 8R2 + 12R + 12) 

1 - 2R 1 

- - e (1+ -) 

R R Jl 

1 

- - (0.043 R4 + 0.215 R3 

.R 

J2 

3s 8 =H - 0.5s + - -K 

12 R 

Hl 

3 - .OS - 3.J + - 

R 

3 

R 

- 0.5 - J2 + - "2 

AE(R) (H2 - H1) + 2 Rl 

Po 

An 

1 -2R 

R 

+ 0.645R2 + 1.29R + 1.61 + -) e 

H2 -HI +B1 4 A-2 where, A = - ---- Oo5 + 

HI -M -B1 0.5 8 

Hi = H2 I1 
37 

2 

1 

Figure 1. Coordinate vectors of the three-body system. 

2 

T 

b 

Figure 2. Oornain of collision at impact parameter b over time domain 

-to t to. 
38 

Figure 3. Coordinates used for transformation Equations (4.5) - (4.7). 

I 

Figure 4. Coordinates used for evaluation of coulomb integral J1. 
39 

R 

1 

Figure 5. Coordinates used for evaluation of coulomb integral J2. 
40 

a 

0 

a 

a 

- 

0 A 

h LN 

Y 

a 

X 

X 

w 

X 

I 

I 

I 

I 

I 

'I 

I 

I 

I 

I 

I 

@I 

@I 

1 

I 

I 

I 

I 

\ 

\ 

\ 

0 

I I 

0 ln 

0 

4 

+ cn+ 

cn .I0-l 

L 
41 

I 

1 

i 

I 

i 

Y 

L 

3” 

U 

C 

W 

v) 

W 

L 

n 

‘ ‘\i 

WG 0 aJ 

JI \, ,E\ I,, I I 1 

In W 4 4 I 

d 

4 

0 

4 

I I 0 0 4 4 

a 

L 

W 

C 

W 
Report Documentation Page 

1. Report No. 

NASA TM-101559 

4. Title and Subtitle . 

Charge Exchange Transition Probability for Collisions 

Between Unlike Ions and Atoms Within the Adiabatic 

Ap p r oxima t ion 

2. Government Accession No. 3. Recipient's Catalog No. 

5. Report Date 

April 1989 

6. Performing Organization Code 

7. Authorkl 

Y. J. Xu, G. S. Khandelwal, J. W. Wilson 

9. Performing Organization Name and Address 

NASA Langley Research Center 

Hampton, VA 23665-5225 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, D.C. 20546-0001 

~~ 

15. Supplementary Notes 

8. Performing Organization Report No. 

10. Work Unit No. 

199-22-76-02 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14. Sponsoring bgency Code 

Y. J. Xu and G. S. Khandelwal: Old Dominion University, Norfolk, Virginia. 

J. W. Wilson: Langley Research Center, Hampton, Virginia. 

J Pagal 21. No. of pages 

42 

16. Abstract 

22. Price 

A03 

A simple formula for the transition probability for electron exchange between 

unlike ions and atoms is established within the adiabatic appraximation by 

employing the Linear Combination of Atomic Orbitals (LCAO) method. 

also involves an adiabatic parameter, introduced by Massey, and thus the 

mat'on are avoided. Specific reaction Li + H+Li + H and Be + H+Be 

+ H are considered as examples. The calculated capture cross section results 

of the present work are compared with the experimen a1 data and w'th the calculation of other authors over the velocity range of 10 

The formula 

difficulties arising from the internal enwy defectdnd tp adiabztic appro 3+- + 

f 8 cm/sec to 10 cm/sec. 

19. Security Classif. (of this report) 

Unclassified 

17. Key Words (Suggested by Authorls)) 

Charge exchange 

Ion react ions 

Space Radiation 

20. Security Classif. (of 1 

Unclassified 

18. Distribution Statement 

Unclassified-Unlimited 

Subject Category - 93 

NASA FORM lm OCT 86