T.HOLSTEIN
832
Vol. 56
MOBILITIES OF POSITIVE IONS IN THEIR PAREST GASES BY T. HOLSTEIN Westinghouse Research Laboratories, East Pitkburgh, Pennsylvania Recdud March 16, 1969
In this paper a method* is described for com uting the mobility of an ion in its parent gas. Formulas of kinetic theory give the mobility in terms of the cross-section k r momentum-transfer in an ion-atom collision. It is found that this crosssection is, in most cases, determined predominantly by charge exchange, which takes place readily between an ion and a parent atom. The comput.ations require knowledge of the “resonance” or chargeexcharge component of the total ionatom interaction; the latter is obtained b a new method whose sole prerequisite is a knowledge of the Hartree-Fock wave function of the outermost atomic ahell. Tge theory has been applied to the cases of neon and argon.
In calculating the mobilities of positive ions in gases, one .has essentially two problems to solve. The first of these is the computation of the crosssection for collision between ions and neutral atoms. Knowing this cross-section, one has then to carry out a kinetic theory analysis involving the energy distribution of the ions and gas atoms. In the case of sufficiently low external electric fields, to which the present paper applies, the kinetic theory part of the problem has been solved. In particular, for the case of ions in their parent gas, the mobility K , defined as the drift velocity per unit electric field, is given by the expression’ K = (3d/r/S)e/(MkT)’/¶N[ Q M ] ~ ~ .
(1)
In this relation N is the gas density, T,the absolute temperature, M , the common mass of an ion or , “average cross-section for atom and [ Q M ] ~ ~ . the momentum transfer’’ defined as
[&Isv.= ( 1 / 2 ) ( k T ) - s J m E 9 ~e - E l k T dE
(2)
where E is the kinetic energy of relative motion of an ion with respect to an atom, and QM the “monochromatic” momentum-transfer cross-section in an ion-atom collision. QM is itself defined by the relation Q~ =J‘q(e)(i - cos 0)dO (3) where q(e) de is the differential cross-section for scattering between angles e and e de in the center-of-gravity system of the colliding ion and atom.
+
QM differs from the conventional Ramsauer collision cross-section in that the latter does not contain the factor (1 cos e) which is a measure of the fraction of momentum lost by an ion in colliding with an atom. The difference between the two cross-sections will be particularly pronounced when the scattering, as given by q(@, is predominantly in the forward direction. It will be helpful in our further discussion to consider the mass motion of the atoms and ions as classical. This approximation not only considerably simplifies the treatment but is actually quite good except, perhaps, for the lightest atoms. The reason is that nuclear de Broglie wave lengths are generally small compared to the dimensions over which the ion-atom interaction suffers appreciable variation. The classical formulation .of the scattering process is illustrated in Fig. 1. Here are shown, in schematic fashion, two possible trajectories of the ion with respect to the atom. According to classical mechanics the scattering angle, e, is a unique function of the “impact-parameter,” b, i e . , the initial moment arm of the ionic orbit with respect to the atom. b is conveniently represented, as shown in Fig. 1, in a “target plane” perpendicular to the initial relative velocity. It is then clear that the chance that a particle be scattered through an angle between 0 and 0 de is equal to the chance that it impinge on the shaded strip indicated in Fig. 1. The area of this strip, i e . , 27r bdb may for our purposes be ’considered as the definition of the differential cross-section2 q(0) de. We are thus able to formulate the momentum-transfer crosssection in the alternate form
-
+
QM
Fig. 1.-Classical theory of atom-ion scattering: the trajectories depicted here are those of the ion relative to the atom (open circle).
* Preliminary notice T.Holstein, Phya. Rsv., 82, 567 (1951). (1) Equations (1) and (2) of our paper are transcriptions of equation3 (6). (7) and (9) of the article hy H. E. W. Massey, Reporla on Progress An equivalent formulation is given in Chapter XI1 of “The Theory of Atomic Collisions.” N . F. Mott and H. 8. W: Massey, Seoond Edition, Oxford, 1949.
i n Physics, 14, 248 (1949).
=lOD
(1
cos 8 ) 2 d db
(4 1
which will be more useful for our purposes. It would now appear that one has merely to determine 8 as a function of b in order to compute (4). However, in the case at hand, namely, the scattering of ions by atoms of the parent gas, the additional process of charge-exchange has to be taken into account; it turns out that this process affects the momentum transfer between a colliding ion-atom pair in a decisive way. The ease of occurrence of charge-exchange between an ion and a parent atom arises from the circumstance that, in the absence of interaction, the electronic energy of the system does not depend (2) As pointed out in the two references of footnote 1, q(0)de is defined forinally as the number of particlea scattered hetween B and B dB per second per unit inoident ourrent density by a single sratterer.
+
MOBILITIES OF POSITIVE IONSIN THEIR PARENT GASES
Oct., 1952
upon the location of the charge. In other words, the state in which atom A is neutral and atom B ionized is energetically equivalent to the state in which A is ionized and B is neutral. As a result, the weakest electron-transfer interaction between A and B, if given sufficient time, will cause chargeexchange. The effect of charge exchange upon the momentum-transfer cross-section is illustrated in Fig. 2. The scattering event therein depicted is one in which A, initially an ion, is scattered by B, which is initially neutral; the scattering angle in the centerof-gravity system is 0. It is then clear that, if, in the course of collision, charge exchange takes place, making A neutral and B the ion, the ej’ectite scattering angle will be ?r - 0. If, in particular, the probability of charge exchange is 1/2, the angular distribution of scattered ions is symmetrical about 0 = 90°, regardless of the angular scattering pattern of the atoms; the factor (1 - cos e) may then be equated to unity.
A (ION)
A
+F-\
833
change in momentum transfer between ions and parent gas atoms is thus established. We now come to the problem of computing the charge-exchange probability, Pex,as a function of impact parameter, b. This computation can readily be carried out once the atom-ion interaction is known. Figure 4 gives this interaction for the case of helium. For the sake of comparison the mean thermal energy of relative motion, SkTl2, for T a t room temperature (20”) is also given. vj
0
Lz
3, -
3
:2 - T k T > n
: -1
I I
I-
-
’Jr
1 0. z
!DISTANCE OF I C L O S E S T APPROACH :FOR CRITICAL ORBIT
lJ v
-
I N T E R N U C L E A R DISTANCE IN ATOMIC UNITS. B(AToM)
Fig. 4.-Atom-ion
interaction for helium.
A(AT0M)
The ion-atom interaction consists of two terms. The first, predominant at large internuclear disThe significance of these latest remarks becomes tances (> 9 Bohr radii, in the case of He) is the immediately apparent when we examine the nature ‘(polarization” interaction between the ion and the of the charge-exchange process. As will later be induced dipole moment of the neutral atom. It shown, there exists a critical impact-parameter, has the form -C/r4, where r = internuclear b,, which is the maximum b for which the prob- separation, C = e2a/2, and CY is the atomic polariaability; the latter quantity may be obtained either ability of charge-exchange, P e , , is equal to For b < b,, Pexis a rapidly oscillating function of b theoretically or from measurements of the dielecwith extremes at zero and unity and with an aver- tric constant of the gas.3 The second term, the soage of 1/2. For b > b,, on the other hand, Pex called ‘(charge-exchange” or ‘(resonance” interaction, actually consists of two branches, as shown in drops rapidly t o zero with increasing b. Fig. 4. This feature is connected with the degeneracy remarked above in which, to the first order, the *3P ,--/ = 1/2 energy of the system does not depend upon which of P ,, OSCILLATES RAPIDLY the atoms A , o r B is ionized; in the language / of quantum mechanics, the two states, $A and I B E T W E E N ZERO A N D UNITY I in which atoms A and B are ionized, respecI ,I WITH AVERAGE VALUE OF 1/2 +B, tively, are degenerate with regard to the (‘unper‘ \ u c ‘ - -/ - P e x DROPS OFF R A P I D L Y TO turbed” energy. This degeneracy is removed by , ZERO WITH INCREASING b the interaction between A and B, thus splitting the electronic energy of the system into two branches. Fig. 3.-Charge exchange in atom-ion colliaion. It is characteristic that in this, as well as other In view of this situation we may take (1 - cos e) cases of degeneracy removal, the states correspondequal to unity for b < b,, and may then write (4) as ing t o the two branches are not +A and $B, but linear combinations of these. In fact, the states +r and $a associated with the repulsive and attractive branches of Fig. 4 are Again anticipating results to be obtained below, d+ = 2-’/2 [+A - $B] exp - f i E , d t / h ] we may remark that the first of the two terms of the right-hand side of (5) gives by far the largest contribution t o QM. The second term contains the efh = 2-l/2 [$A f $B] exp - S ; & d L / h ] (5) fects of the residual small charge-exchange for b > bo plus the bona-fide scattering without charge- In (5) the time-dependence of and +a is written exchange; the latter is due essentially to the so- out explicitly, the functions #A and +B being taken called “polarization” component of the atom-ion (3) Higher order oorrections, such as terms in l/r’ (van der Waals interaction energy, and turns out to be quite small interactions) are negligible at internuclear dlstances of significance in (-1%). The crucial importance of charge-ex- our problem. Fig. 2.-Effect of charge exchange 011 scattering angle.
‘.___//
[ [
T. HOLSTEIN
834 time-independent.
Also
Vol. 56
paper.4 For our present purposes Vr and Va may be represented by the formulas
+ Vr + Va
Er EO Ea = EO
(6)
+
V , = -c/r4 VI e-=r V. = -c/r4 - VI e-ar (11) where VI and a are suitably chosen constants for
where Eo is the degenerate unperturbed level and Vr, V a the repulsive and attractive potential energy curves of Fig. 4 (inclusive of the polarization term). each gas. With regard to orders-of-magnitude, Vr and Va are, of course, functions of time via the these are indicated in Fig. 4; for the internuclear time dependence of the internuclear separation. separations with which we shall be concerned It should be pointed out that the combinations (> 8 Bohr radii in the case of He) both Vr and Va +A i +B are just those which occur in the theory are small compared to 3kT/2 when T is room of the molecular ion formed by the colliding ion temperature. This feature indicates the feasibility and atom. In the case of helium, for example, of using, as a first approximation, the simplest type and fir are the two lowest electronic states of of classical orbit, namely, straight-line trajectories. Hez+:22uand 22g;similar considerations apply to These are characterized analytically by the expression other gases. In the problem at hand, however, as contrasted r = (b2 + vZtZ)'/z (12) with the situation commonly encountered in molecwhere v is the relative velocity of the colliding ionular physics, the state of our diatomic system is not stationary, i.e., it is neither nor but atom pair. (12) incorporates the dependence on rather a linear superposition of them. It is this fea- impact-parameter which will ultimately appear in ture which, as we shall show immediately below, is P e x . We now insert (12) into (10); under the condiintimately involved in the theory of charge-extion that ab > > 1 (a condition which is always change between an ion and its parent atom. We choose a time ti sufficiently early so that A fulfilled in practice), the integral can be evaluated and B may be considered as non-interacting. approximately; one obtains ( t i will later be allowed to regress to -a). At P,, = sin2 [-hTV,e-ab (13) ti the state of the system is one in which either A or B is ionized. Choosing.for the sake of definite- We observe ness the former, we then have, using (5) (1) b, is defined implicitly by the equation
+,
(?)'I
+(ti) =
+A
Cr+r(t)
where
+
(7)
Ca$a(t)
H
(%)'la
4=-
( 2 ) With decreasing b the phase of Pe, increases exponentially; P,, thus oscillates rapidly between zero and unity for 2, < b,. A detailed argument, = 2-*/2 exp [ i S t i E a d t / h ] (8) not presented here, shows that, except in the immediate neighborhood of b,, the average value of Substituting (5) into (8), we then obtain, after a Pe, is little algebra, the result, valid for arbitrary t , (3) For b > b,, Pe, G (2r bV;/ahv2)e-2ab which diminishes rapidly with increasing b. + ( t ) = exp L-i ,'(E, + Ea) dt/2h It will be remembered that these properties were utilized earlier in this paper to demonstrate the importance of charge-exchange for momentum transfer between ions and parent atoms. i $ sin ~ [J,t(.Er - E.)dt/2h We may here remark that, in the case of helium, (9) b, turns out to be of the order of 8 Bohr radii. For We now introduce a time ti sufficiently later than other gases, bc is even greater. The effective "colthe encounter between A and B so that the atoms lision radius" is thus seen to be large compared to may be considered as no longer interacting. (ti atomic dimensions. In particular, it is definitely may, in fact, be taken positively infinite.) The larger than the radius for momentum transfer beprobability of charge-exchange during the collision tween an ion and a foreign-gas atom, in which is then equal to the absolute square of the coefficient charge-exchange plays no significant role. Referring back to (5), we may estimate the magof +B in (9) at time tf. Passing to the limit ti -+ - and tf -+ 03, we obtain nitude of the second term of that equation under the assumption that actual deflections for b > 6, are negligible and that the whole effect ariscs from l j C x= sill' [J-Lm(flr - , ~ , ~ ~ i t / 2 / & ] charge-exchange. In this case, it is easily showri or that Pex= sin* V, - Va)dt/2h] (10) Jcm(l - cos 0 ) 2H bdb = 4nPexbdb & nb,/a (15)
-
P
I
Jtl
-
1
11
+
[J-y(
In order t o make further progress, we require knowledge of Vr - Ira as well as of the classical orbit of relative motion. With regard to the former, a brief discussion will be given below, a complete treatment being reserved for a later
J:
roughly a 10% correction in specific cases. (4) To be published in The Phgsical Review. In this paper we shall treat the nuclear motion quantum mechanically and thereby provide the formal justification for the semi-classical approach of the preaent paper.
MOBILITIES OF POSITIVE IONSIN
Oct., 1952
The contribution to QM of true scattering-deflection of the ions by the atoms without charge-exchange-has been estimated for specific cases and turns out to be small (-1%). Up t o the present we have not taken into account the polarization component of the ion-atom interaction. The primary effect of this component is a modification of the orbit of relative motion so that (12) is no longer valid. In particular, the distance of closest approach, r,,,in, is less than b; it is given implicitly by the relation b2/rLi. = 1
+ c/ErAi,
IL W
z
4.
A
---------- aa--------------J
z
5
n W
E
+
529.
835
3
+ +
(.5) H. S. W. Mrtssey and C. B. 0. Mohr, Proc. Roy. Sac. (London), 144, 188 (1934). (6) E.P., H. A . Bethe, Handbuch Phueile, 24, 1, Chapter 3, pp. 528-
PARENT GASES
I
(16)
E being the kinetic energy of relative motion as defined at the beginning of this paper. It then turns out that one obtains substantially the right result by replacing b by r,in in (14). In practical cases the effect of this replacement on QM is of the order of 10%. Equations (5), (14), (15), (16), and the above paragraph provide the recipe for obtaining QM as a function of E once the constants VI and a,which chiracterize the resonance interaction, are known. The determination of this interaction is nothing more than the problem of calculating electronenergy curves of diatomic molecular ions at large internuclear separations. For the case of helium, Vr and Va ha,ve been computed by Massey and Mohrs utilizing a Perturbation treatment based on the approximation of electronic wave functions by combinations of atomic orbitals. (The simplest calculation of this type, that for H2+, is given in standard texts.6) In our treatment, we have employed an alternate method which we believe provides more accurate results a t the large internuclear separations of interest in the mobility problem. A complete description of this method will appear in a later paper.4 We give below a very brief summary for the simplest possible case: the interaction of a proton with a hydrogen atom, i.e., the Hz+ problem. Referring to Fig. 5, we note that the Hamiltonian of the electron is unchanged by reflection of the electron coordinates through the median plane, M, which bisects the internuclear axis. From this symmetry feature it follows that either the wave function, $, or its normal derivative, d+/dn, vanish on M. This condition permits us to formulate the problem as follows: Solve Schrodinger’s equation in a half-space, e.g., region I of Fig. 5, with the boundary conditions, = 0 or d+/dn = 0 on M and the regularity of a t infinity. The solution in region I1 is then obtained by reflection through M ; in this operation fj either changes sign or does not, depending on whether or d$/dn is equal to zero on M. The (ground-state) hydrogen wave-function diffew from the solution of this problem in two respects. Firstly, the potential energy is that due solely t o proton A, whereas the potential in our problem is that arising from both protons. Secondarily, the hydrogen wave-function does not satisfy either of the boimdnry Conditions $ = 0 or d$/dn = OonM.
THEIR
’
4.
‘ B
T.HOLSTEIN
836
where N and Ze are suitably chosen constants. One then has, for the electronic energy, the expres~on V = -C/r4 i ( r r N 2 / 2 ) (4/e)’/Zee-Zar
This expression is not altogether of the form of equation (11) because of the presence of the factor r in the resonance term. However, over a limited region of interatomic separation-and this is all that is reqyired for the computation of Pe,-this factor can be approximated by a suitable exponential. As pointed out above, once the resonance interaction is known, one can proceed straightforwardly to a determination of QM as a function of E. Inserting the results into equation (1) and (2) we may then compute the ionic mobility. Specific calculations have been carried out for neon and argon. The results are K = 4.2 and 1.64 cm./sec. per volt/cm., respectively, under conditions of standard gas density (2.69 X 101g/cc.) and T = 293°K. We remark here t.hat neon and argon were chosen specifically for the purpose of comparison with the recent measurements of Hornbecks; from the theoretical standpoint, the choice is not too advantageous since the outer shells of both these elements are p-shells, rather than sshells, as required by our theory. The calculations therefore entailed further approximations, in addition to those already introduced in the theory. However, it is estimated that the errors therewith incurred do not exceed 10%.
DISCUSSION R. M. NOYES(Columbia University).-Is not mobility reduced by charge transfer because the cross section is so much greater than the gas viscosity cross section? It would seem that if char e transfer occurred only when the particles arc as close as t f e gas viscosity collision diameter, charge transfer should not affect the mobility. ( 8 ) J. A. Hornbeck, P h y s . Rev., 84. 615 (1951).
Vol. 56
T.HOLSTEIN.-It is true that, if the charge transfer took place only within the gas viscosity collision diameter, its effect on mobility would be negligible. In the case of ionatom collisions, however, there exists the additional polariaation interaction whose effective collision diameter is larger than that for ordinary gas viscosity. It is this diameter which must be exceeded by the diameter for charge-exchange in order for the latter to be significant. H. RIES (Bell Lab.).-In connection with Professor Franck’s question concerning the fact that H + in Hs0 achieves a higher mobility through charge transfer while in gaseous conductivity the mobility is reduced by transfer, I would like to oint out that in the gas the transferred particle only carries t i e charge at the moment of its transfer. This means in the case of long free paths that the carrier is not mainly the exchanged particle. The effect of transfer is to produce larger deflections and so reduce mobility. In the case of H + in HzO the H i 0 molecules do very little moving and the exchan ed particle, Le., H+, is the principal carrier. Hence, here, t8e transfer phenomenon leads to greater mobility.
K. RUEDENBERG (University of Chicago).-The author reports a new method for computing the energy of the Hz+ ion at large internuclear distances. I would like to raise the following points: 1. What would be the closest internuclear distance when this approximation is still valid? Would it have any bearing on molecular structure calculations? 2. It is well known that wave functions of a rather simple analytical form [Such solutions have been given by James and by Svartholm.] reproduce Hylleraas’ result for the IIz+-energy up to about 5 significant figures. It seems possible that these energy curves might give an even simpler solution than the one given by Dr. Holstein. T. HOLSTEIN.-^. The internuclear distance in question is of the order of 5 Bohr radii. Since this distance is large compared to the usual interatomic distances in molecules, the applicability of the approximation to molecular structure calculations is rather dubious. 2. The quoted papers [H. M. James, J. Chem. Phys., 3 , 9 (1935); N. Svartholni, Z . Physik, 111, 186 (1938-39)] claim only to have reproduced Hyllerrtrts’ result for H+z in the neighborhood of the energy minimum (-2 Bohr radii). No results have been obtained for large internuclear separations; in fact James states [loc. cit., p. 141 that his wave function is “of course not so well adapted to the treatment of molecules for very iarge separations of the nuclei.”
