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258

The Collision of Electrons with Molecules.

By H. S. W. Ma ssey , B.A., M.Sc., Trinity College, Cambridge, Exhibition of 1851 Senior Student, and C. B. 0. Mohr, B.A., M.Sc., Trinity College, Cambridge, Exhibition of 1851 Student (Melbourne).

(Communicated by R. H. Fowler, F.R.S.—Received November 13, 1931.)

Introduction.

Since the development of the Born theory of collisions* in 1926, its application to collisions of electrons with atoms has been treated in considerable detail. Recently the authorsf have included also the modification due to Oppenheimer, J in which account is taken of electron exchange. However, for the collision of electrons with molecules very little has been doue beyond the calculation by Massey§ of the elastic scattering of electrons in molecular hydrogen.

In the present paper use is made of the collision theory of Born and of Oppenheimer ( loc.cit.) in order to consider various phenomena occurring on electron impact with molecules.

Firstly the elastic scattering is considered. General formulae for the case of diatomic molecules are obtained, including the relation between X-ray and electron scattering. One result is that the average scattered intensity due to a number of axially symmetrical fields with random orientation is a function of v sin where v is the velocity of the incident electrons and § the angle of scattering. The case of molecular hydrogen is then treated in detail, and the intensity of elastic scattering calculated for all angles and velocities for which the Born formula is valid. Curves are also given for nitrogen, a simplified model being used.

In the second section, energy interchange between electrons and molecules is discussed. The importance of the Franck-Condon|| principle is considered, as inferring the diffraction of inelastically scattered electrons and the excitation of the B-state of hydrogen calculated approximately. Finally, the probability of dissociation of the molecule into two atoms in the ground state (corre- * * * §

* ‘ Z. Physik,’ vol. 39, p. 803 (1926).t ‘ Proc. Roy. Soc.,’ A, vol. 132, p. 605 (1931).X 1 Phys. Rev.,’ vol. 32, p. 361 (1928).§ ‘ Proc. Roy. Soc.,’ A, vol. 129, p. 616 (1930).|| ‘ Trans. Far. Soc.,’ vol. 21, Part 3 (1925).

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Collision o f Electrons with Molecules. 259

sponding to a 1 42 —1 32 transition) is calculated using Oppenheimer’s theory. We now proceed to the detailed discussion of these questions.

A. Elastic Scattering.

The Scattering of Electrons by Axially Symmetrical Fields with Random Orientation.—Assuming the validity of the Born formula, the intensity of elastic scattering by a potential V (r, 0) is given by

8 n?m A4 "

- *»i) 2 (1 )

where khj^Tim is the velocity of the incident electron and n 0 and n x are unit vectors in the initial and final directions of motion of the electron.

«0

F ig. 1.

Let us now take the axis of z along the vector n 0, and suppose the vector nx in the yOz plane. Then if / are the angular co-ordinates of the axis of symmetry of Y referred to this co-ordinate system (fig. 1) and 8 the angle between n 0 and nx,

e ik (n0 - n,) . r _ eX p j - ^ . { — gjn 0 gjn g in S -{- COS 0 (1 — COS S )} ] .

Also we may expand V (r, 0) in a series of harmonics referred to the axis of symmetry, as follows

V (r, 0) = £V W (r)PM (cos

where u is the angle between the radius vector r and the axis of symmetry. Since

cos u — cos 0 cos ^ -j- sin 0 sin ^ cos (y — <f>),



260 H. S. W. Massey and C. B. O. Mohr.

the use of the addition theorem for spherical harmonics givesm = n

V (r, 0) = 2 2 Vn(r)Pnm (cos <Jd (cos 6) eim *>.n m = — n

Substituting in (1) gives

I = |S S P„» (cos +) e-<~* I„ „ \ \h r n m

whereIn m = I Y n(r) Pnm(cos 6) eimii> exp [ikr {— sin 0 sin ^ sin §

-f- cos 0 (1 — cos 8)}] r2 sin 0 dr d0 d<f>

where*

Hence

= F w (ksin P ) i m Pnm (sin P ) ,

F w (k sin |8 ) = 2 7iin -J j Yn (r) J n+i sin P ) ^ 2 dr.

1 = &n3m?2 2 P„m (cos i\>) e~imxF w (A sin P ) Pwm (sin P ) | 2.

If we consider the axis of the system as oriented at random we must now average over all the possible values of <J; and y. This givesf

i = ^ i f „2 ? ( - 1)’h n — 0 m —--n

1 (n -f- m )!2 % - f - 1 ( w — m ) !

[P ,“ (sin |S)]2,

and using the addition theorem for Legendre polynomials J we obtain at once

Q—3^,2 ool = ~ S [Fw (& sin P ) ]2. (2)

h' 7i=o

Thus we have at once the result that the intensity of scattering is a function only of the product k sin p , i.e.,of v sin p .

The Relation between X-ray Scattering and Electron Scattering.—Consider a molecule with n electrons in a state represented by the wave-function

(1, 2, n). The interaction energy between the molecule and an incidentelectron (which we denote by n -f- 1), is given by

V = [ { — + — - S - £ — ) I + (1, 2, n) pdx , dr„, (3)* L i -KL P n + 1 *•=! r 8,n+ 1

* Vide Watson, “ Theory of Bessel Functions,” Cambridge, 1922. As all further formulae involving Bessel functions are readily accessible in Watson no further references will be given.

f Whittaker and Watson, “ A Course of Modern Analysis,” Cambridge, 1927, p. 324.J Whittaker and Watson, p. 326.




where rn+1, Vn+i are the distances of electron n -j- 1 from the two nuclei, and ra n+1 the distance between the electrons s and -f- 1. Hence substituting the value of V given by relation (3) in equation (1) we have, for the elastic scattering,

1 = 8Tc3m2 — + — — S )\ (1 ,2 , ...,w )|2ei*<n»-n»>-R«+i^n + 1 Pn+1 « = n+1

dxr dx2, dxn dx»+l 2, (4)

where R denotes the vector distance of the electron from the centre of the molecule.

Now Bethe* has shown tha t

n r * gik n . r,k2 (5)

where r 2 is the vector distance from a fixed origin and n a unit vector. Using this formula with the origin a t the centre of the molecule reduces (4) to the form

1 = 8 7 i3m2 I f / Ze2 , Ze2f ( -J \r,w+l

| (1, 2, ..., n)j2 elk(-n° Dl)-Ril+1 ...

- fj 1 ^ (1’ 2> •••.'»)i2eii<n,' n‘) • R - %

where capital letters denote distances from the centre of the molecule. The series term is equal to

4tl4mv2 sin2 P

F,

where F is the X-ray structure factor for the molecule. Consider now the first term. Integrating over the elements of volume, the first term is simply

l

= 2Ze2 cos {\k (n0 — nx) . d} j i (n°

= 2Ze2n: cos {§& (n0 — n2) . sin2p ,

' Z£ l 4 - \ gik (n0—nl} . R «+i

i+l Pn+1

where d is the nuclear separation. Hence

8r:4m2e41 =W& sin4 P

| 2Z cos {\k(n0 — iq) . d} — F j2.

* ‘ Ann. Physik,’ vol. 5, p. 325 (1930).



262 H. S. W. Massey and C. B. O. Mohr.

I t is obvious from the above calculations that there will exist a similar relation for polyatomic molecules.

The Case of Molecular Hydrogen.—This case has been considered by Massey (loc. tit.)* but the calculations were not then carried out in detail for low velocities and small angles of scattering.

Making use of the Born formula without exchange, we have

j 87c 3m 2~

^O2 (r2r3) e%k (n° ‘ R* ^T1 ^ t 2 ^ t 32

(6)

where suffixes 2 and 3 denote the two bound electrons, and 1 the scattered electron, while r, p, R denote the distance of an electron from either of the two nuclei or the centre of the molecule respectively. For (r2r3), the wave- function of an electron in the ground state of the molecule, we take

(W s) = N {e"z<'•+*•> + e~Z(r‘+^}, (7)

Z being given by Wang’s variation methodf as 1 • 166/a0, where a0 is the radius of the ground orbit of the hydrogen atom, while N is the normalising factor given by

W = ~ Z6 {1 + e- 2Z* (1 + Z \Z 2d2)2}-1. (8)2tt

As a result of the symmetry of the above expressions in the suffixes 2 and 3 and in the nuclear co-ordinates, and with the use of the simple vector relations

Ri = r, + id = Pl - |d , (9)

the integral I is a t once reduced to the form

I = [cog {ik (no _ nj) _ d}Ii + v /gl2?;

whereS = e~2M (1 + U + \Z 2d2)2,

11 = [[(-- — ) («.-«!)• «•. dT-.dx.,J ' h

1 2 = f f ( - - f — — — ) e~Z(r*+Pl) R s faJJ Xrj. px r12

* Owing to a numerical error the ordinates of fig. 2 of this paper should be multiplied by a factor of 1-51.

t ‘ Phys. Rev.,’ vol. 31, p. 579 (1928).




Now Ij is merely the integral occurring in the elastic scattering of electrons in molecular hydrogen,* and gives simply

j _ 7T2 (2Z2 -j- 2 sin2 P )1 — Z3 (Z2 -f- ¥• sin2 P ) 2 * ( 10)

The calculation of I 2 may be most conveniently carried out in polar co-ordinates as follows. Changing the origin of co-ordinates to the nuclei 1 and 2 respectively gives

__ gi»*(n0- n , ) -d | _ 1 j g - Z {r2 + y(r,«+2dr,cosw + rf8)} (n0 —n ,) . rx dx<>

J r12

_L g - i tk C n o -n ^ .d | if_1 _ \ g - Z |r 2 +v'(r22-2rfrscosM+rf1!)} e ifc(n0- n , ) . r , ^ ^

J Vr, rJ 1 ' 2’where

^1 *W

cos u = cos 0 cos 4 + sin 0 sin 4 cos

the axis of z being along n0 and 4s X being defined as in § 1. Using now theexpansions

where

e z«/(r1*+2<Jr, costt+d*) _ ^ f n( (2w -J- 1) P n (cos w),n = 0

/« (»i» <*) = ~ ^ < dZV drx

__ d f ^ n + l / 2 (Z ^ l) I n + l/2 ( 2 ^ ) \ r \

- ~ d z \ zVSJ r h > dand using the integrals

r2n ----------ea sia 0 sin * + b sin 0 cos * d _ 2 tt J q ( V a 2 - f 6 2) ,

Jo

f etircos(,cos P M (cos 0) J 0(&rsin 0sin 4) sin 0d0 — J n+1/2 (^r) Prt (cos 4),Jo v'o gives

~&2sin2P = cos{P (n0 —nx) . d}

•sm

^ - A / f r ^ V l S S PS, (cos «),) -Z3 v \A *sm |8/n = 0m =-2»

P ^ (sin P ) eim* j e~Zr f 2n (r, d) J 2n+l/2 (2&r sin P ) r^2 drj

(P K -n j) • d} v (TT—T?) s 2 p?»+i<cos +)P?»+1 (™iP)e*”ACSIII2O =x0 mi — — 2n-{- 1

f e ~ Zr fzn(r > J 2n+3/2 (2 k r s i n P ) ^3/2Jo

* Elsasser, ‘ Z. Physik,’ vol. 45, p. 522 (1927).



264 H. S. W. Massey and C. B. 0 . Mohr.

The series occurring cannot be summed, but the convergence of the successive terms is satisfactory. The first term of the even series can be evaluated analytically, but the other terms must be numerically calculated as they involve the integral

f00 e~kr sin ^Ja V

Actually only a small fraction of the scattered intensity is due to these higher terms. Considering for convenience only the first term, we see that averaging the scattered intensity over all possible orientations of the molecular axis introduces a term

\(l + S—-- ], where x — 2led sin = g-n

This term is just the factor required by the diffraction of waves at two similar obstacles distant d apart, which scatter coherently.

We have finally for the averaged intensity of scattering

z w i l - i* - l e~zr /o (r> d) (2kr sia i8) r+ higher terms.

1 +sm

In fig. 2 the scattering by a hydrogen molecule and by two separate hydrogen atoms is compared and the ratio illustrated graphically as a function of

F ig. 2.—Illustrating the ratio of scattering of H 2 to the scattering by 2H. I. Without the diffraction factor 1 + sin x/x.II. With the diffraction factor 1 + sin xjx.

a0k sin and it is seen that the ratio approaches unity for increasing values of the argument, as is expected. Also it is seen that the ratio, not considering the diffraction factor, decreases slowly below unity with decreasing velocity of the incident electrons ; this is due to the increased concentration of charge in the hydrogen molecule consequent on the effectiveness of the binding resulting




in a decrease of effective cross-section for electrons of such velocities tha t the major fraction of the scattering comes from the outer levels of the system. The oscillations in the curve are, of course, due to the diffraction effect referred to above.

I t is of interest to compare the results of this theory with the various experimental investigations of the angular distributions of electrons scattered elastically in molecular hydrogen at various velocities. Angular distributions have been measured by Bullard and Massey* for 4, 10, 20 and 30 volt electrons ; by A m otf for 30, 80, 200, 400 and 800 volt electrons ; by Macmillan^ for 50, 75, 100 and 150 volt electrons, and by Harnwell§ for 120 and 180 volt electrons.

Comparison of theory and experiment in the case of the early work of Harn- well and Macmillan shows tha t the observed angular distributions fall off much more rapidly with angle of scattering than the calculated curves. Comparison with the more recent work of Bullard and Massey, and Arnot shows th a t a t 80 and 200 volts there is very satisfactory agreement, but the close similarity of the calculated curves for H and H 2 prevents a definite decision as to which of the latter two curves is favoured by experiment. At and below 30 volts, however, there is no longer any trace of fit between the experimental and calculated curves, indicating the failure of the Bom formula a t these velocities. Curves for three different voltages in hydrogen are given in fig. 3.

Theoretical angular distribution curves for nitrogen for comparison with experiment have been calculated as follows. The elastic scattering by any atom has been treated by Bullard and Massey|| by applying the Thomas-Fermi electron distribution to the Born formula, and from a table which is given in their paper angular distributions are at once obtainable. I t is then assumed that the scattering by the nitrogen molecule is closely similar to tha t by two nitrogen atoms 1 • 1 A.U. apart, the former being obtained from the latter by the use of the diffraction factor discussed above. This assumption is shown above to be fairly accurate for the case of N 2, owing to the fact tha t only 6 of the 28 electrons in the nitrogen molecule are shared and take part in the binding.

Angular distributions have been obtained experimentally in nitrogen by Bullard and Massey ( loc. cit.) at 10, 30 and 60 volts, and by Arnot ( . cit.) * * * §

* ‘ Proc. Roy. Soc.,’ A, vol. 133, p. 637 (1931).t ‘ Proc. Roy. Soc.,’ A, vol. 133, p. 615 (1931).

t‘ Phys. Rev.,’ vol. 36, p. 1034 (1930).§ * Phys. Rev.,’ vol. 34, p. 661 (1929).|| * Proc. Camb. Phil. Soc.,’ vol. 26, p. 556 (1930).




at 200, 400 and 800 volts. I t is found that there is satisfactory agreement between theory and experiment at 30, 60, 200, 400 and 800 volts, but no trace of agreement a t 10 volts or lower, indicating the failure of the Born formula at these low voltages in nitrogen. These results are illustrated in fig. 3.

F ig. 3.—-Comparison of observed and calculated angular distributions of scattering in nitrogen and hydrogen, showing the failure of the Born formula at low velocities. — Calculated curve. ... Experimental points. The upper of the two calculated curves for hydrogen is that calculated for the atom ; the lower, for the molecule.

B. Inelastic Collisions.

In considering the excitation of molecules by electronic impact, the question arises as to how the nuclear distance of the final state to which the molecule is raised by the impact will be related to that of the initial state. According to the Franck-Condon principle ( loc. cit.), the nuclear separation is instantaneously unaltered in a collision and strong evidence in favour of this is obtained from the measurement of the excitation potentials of molecules. Referring to fig. 4, which gives the potential energy curves of the ground state and an excited state of a molecule (actually the curves for the ground state and the B state of the hydrogen molecule are shown in fig. 4), it is found that excitation of this state requires energy corresponding to the switch AB instead of AD. The comparative sharpness of the effect found also seems to indicate that there is no appreciable change of nuclear distance on impact. Immediately after impact the molecule can be taken as in the unstable state represented



Collision o f Electrons with . 267

by B and falls by quantum jumps to the state D ; these jumps are, however, of no account in considering the probability of the initial process.

Viewed from the standpoint of quantum mechanics, the Franck-Condon principle simply states tha t the excited state which is most probable instantaneously after the impact is th a t in which the overlapping of the initial and final wave-functions is a maximum. At low temperatures we then expect little deviation from the Franck-Condon principle, the spread of the vibrational wave-functions being then very small. Also in the classical picture the

Volts

Nuclear separation, in units of a 0 —*-

F ig. 4.—Illustrating transition from ground state of H 2 to 1 3E and B states.

principle is justified by the consideration tha t the time taken by an electron of moderate velocity to traverse the distance of separation of a molecule is quite small compared with the period of vibration of the molecule.

Assuming, then, tha t the principle is strictly valid, application of Born’s collision theory shows tha t the chance tha t an electron will be scattered between angles 8 and S -j- d$, after exciting a molecular level, is given approximately

byII (8) = A ~j~[~ ~ | | V ^0 4*1 ^ ^ ^

where tj;0, tjq are the wave-functions of the initial and final states of the molecule with the nuclear separation tha t of the initial state, V the interaction energy,

Jc'hfZixm the velocities of the scattered electron before and after impact, and A denotes the averaging over all orientations of the axes.




For simplicity, let us now consider the excitation of an electron from a single homopolar bond of a homonuclear diatomic molecule. The wave- functions ^2 may have either the same or opposite symmetry with respect to the nuclear co-ordinates, so we may write in. the form

M i = f ( r> p

where r and p denote co-ordinates referred to the nuclei 1 and 2 respectively. Denoting the molecular electrons by suffixes 2 and 3, and the incident electron by suffix 1, we have

V = e2( —— f- 1r 12 r 13

Substitution in (11) gives then for the scattering probability

8 k'Ii(6) = A

whereh*

1

j J 2 —j- J 3 j2 sin S dS,

J f12[ f { r , p ) ± f ( P, rj] e* (kn° *Rl dx2 dx3,

— \ f (r, p) ± f ( p , r)] ei( fc'n,) Rl dv! dx2 dx3.r is

Consider now the first of these two integrals. One of the functions, say f(r, p), will contain the terms corresponding to electron 2 around nucleus 1, the other terms corresponding to electron 1 around nucleus 2. I t will then be convenient to calculate

| | j — f (r, p) el {kn<> • R‘ dxj d r2 dx3,

by changing the origin to nucleus 1, the other integral by changing to nucleus 2. We then find

J 2 = | | | _L f (r, p) , dx3

± ni).d j j j _L f(p, ei(kn0-k'nL).Pl dT ^X3.

Now

| | | _L j (/5 p) ). r,

j | | (p,

\ — / {r, (r2+ 2 dr cos w-j-d2)} (kn° * n,) *r' dx, (A)J»12

[— f{ r , \/{r2—2 dr cos u-\-d2)} el (kn° ~ fc ni)' r* d x, (B) J 12




where u is the angle between the radius vector r and the axis of symmetry of the molecule.

The integral on the right may be considered expanded in series by expanding f(r , p) in spherical harmonics of u. The zero order term in this expression will be the same for both integrals A and B, so to this order we have

J 2 = « ^ — *'n i • d )}.

where a is a function of k, k', 8, but independent of the angles of orientation the cosine or sine occurring according as the wave-functions ^ 0, <jq have the same or opposite symmetry in the nuclear co-ordinates. Similarly

We see, then, th a t we have for the scattering probability

8 v?m2k!Ix (8) = A JI a /s in ^ ^ n ° — ^ ni) * d} j 2 + higher oscillating terms.

The averaging over all angles gives then

I(S) = 8n?m2 k'~ 2 1a r ( i i

sm x

wherehx k ’ ‘ \

x = d (k2 + — 2 cos

) -f- higher oscillating terms, (C)

As a consequence, diffraction effects must occur in the excitation of molecules by electron collisions. If we have the -(- sign, this will result in greater scattering at small angles, and if we have the — sign, there will be less scattering at small angles. In order to detect such an effect, it would be necessary to use a heavy molecule so tha t the term a of (C) does not fall off too rapidly with increasing angles of scattering, and to employ an accurate method of velocity resolution of the scattered electrons. The lower the velocity of the electrons used, the more noticeable the effect should be.

It is usually stated th a t inelastically scattered waves are incoherent, but this applies only to the totality of such waves, not to the particular waves which are scattered with a given wave-length change. In the case of a crystal, the states which may be excited lie very close together owing to the very large number of similar components of the c ry sta l; as a consequence, no diffraction effects will be expected in such waves. In a diatomic molecule, however, the




energy states form a discrete set, and it is possible to separate out the different inelastically scattered waves, each of which will show diffraction effects.

The Excitation of the B State.

As an illustration of the above ideas, we will consider the excitation of the B state of molecular hydrogen by electronic impact. This state is symmetric in the electrons, antisymmetric in the nuclei; the wave-function has been determined fairly accurately by Guillemin and Zener* by a variation method similar to that used by Wang ( loc. cit.) in determining the wave-function of the ground state. They express the wave-function 4*i in elliptic co-ordinates p, p, where p, p, are defined by

p = (r + p)/d , (x =

r, p, being the distances from nuclei 1 and 2 respectively, and they find for the value

M e ' -ap2 — frp3 (g/M2+CM3

where Nx is the normalising factor given by

N, - 2 7r2# 64 a363c3

e 2a| 2a-f- 1 — - a2 i (sinh 2c (26c2 -f- c2 — 62) -f- 262c cosh c} 3 /

Ar e 2& ( 26 + 1 — t 6a ) {sinh 2c (2ac2 -f- c2 — 2) + 2a2c cosh c}\ O f

— 4c3 e~a~b [ 2 aH~ 1 — ^ a2) (26 + 1 — - 62) ,\ 3 / \ 3 /J

and a = l-5 , 6 = 0*8, c — 1-2, / = 0 in terms of a nuclear separation of 1*25A.U. As we assume that during the collision the nuclear separation remains that for the ground state, namely, 0*75 A.U., the parameters were slightly adjusted to what was considered their most probable values for this separation, the values a — 1 • 4, 6 = 0- 65, c = 1 • 2, = 0, being finallyadopted.

For the interaction energy V we may take the potential

*12 *-13

as the interaction between the colliding electron and the nuclei gives no contribution, owing to the orthogonality of the functions <!»0 (r2r3) and (**2**3)-

* ‘ Phys. Rev.,’ vol. 34, p. 999 (1929).




The probability of excitation of the B state is then obtained by substituting the above values of V, ^ the expression (11). Substituting thenumerical values of the parameters a, b, c, f , and as a result of the symmetry in the expressions between electrons 2 and 3, we find th a t we have to evaluate the integral

I J _ e - A „ —Bm, j ( in 0- * ' n i) . r, ^ ( 1 2 )

J *12

for the following three different cases :—

(1) B — 0(2) B - A(3) 2B - A.

The second of these is just the integral corresponding to the case of atomic hydrogen, while the first corresponds to the integral occurring in the elastic collisions in molecular hydrogen ; the third is characteristic of the B state.

Now, using Bethe’s formula (loc. cit.) and changing to polar co-ordinates throughout, we obtain for expression (12) the form

47Ck2 + k'2 — 2 cos 3 i

e -C r 2- D p 2 &i (knB-k'n, ) . r2 ^1 2 *

This integral can be evaluated in precisely similar maimer to the integral I2 occurring above in the case of the elastic scattering. In this maimer an expression is obtained for the probability which has to be integrated over all angles of scattering ; this is most rapidly and conveniently accomplished by graphical means. The probability of excitation of the B state for various velocities of the incident electron is thus finally obtained, and the result is illustrated in fig. 5. I t is seen tha t the probability rises to a maximum at about 5 volts above the excitation potential of the B state, and then falls off quite slowly, being still appreciable a t comparatively high velocities.

0-0001real

Volts —Fig . 5.—-Probability of Excitation of B state of H 2 by Electron Impact.



272 H. S. W. Massey and C. B. G. Mohr.

In the above calculations for the elastic scattering and for the excitation of the B state, the effect of electron exchange has been neglected ; the process of exchange, however, will only be appreciable in its effects over a small range of voltages just above zero volts in the case of the elastic scattering, and above the excitation potential in the case of the excitation of the B state.

As an illustration of the general conclusions of § 4, fig. 6 illustrates approximate angular distributions of electrons scattered after exciting the B state.

85 volts30 volts

Angle of scattering-

Fig. 6.—Approximate Angular Distributions of Electrons scattered after exciting theB state of H 2.

As the B state is antisymmetric in the nuclei, these angular distributions show a maximum due to the diffraction factor 1 — sin

The Dissociation of the Hydrogen Molecule by Electron Impact.

The quantum theory of the interaction of atoms due to Heitler and London* introduced the conception of unstable states of molecules due to possibilities of atomic interaction giving a potential curve with no minimum. Thus for two hydrogen atoms in the ground state, there are two possible modes of interaction according as the atomic electrons have the same or opposite sp in ; the potential curves of the two resulting states of the hydrogen molecule are shown in fig. 4. The stable one of these, the 1 1S state, is the ground state of the molecule ; the unstable one, the 1 32 state ; thus a transition from the ground state to the 1 3S state will result in dissociation of the molecule. The two states are analogous to the singlet and triplet states of helium, the stable state being the one with opposite electron spins, the unstable state having electrons with the same spins. Transitions between the two can only take place by electron impact with appreciable probability if electron exchange takes place, and Oppenheimer’s theory must be used to calculate the probability.

* ‘ Z. Physik,’ vol. 44, p. 455 (1927).




Assuming the validity of the Franck-Condon principle, we must take a transition from A to D in fig. 4, corresponding to incident electrons of 11-5 volts energy,* and producing two neutral hydrogen atoms with mutual kinetic energy 7 • 2 volts.

The probability of the process is given b y |

8-nPm? I V4>o M (V i) ' «>-*'»■ ■ «• (Zx, <JTj d r3 j *,

where the interaction energy V is taken to be

e2 e2 _ e2 __ e2*3 Ps *28 ’

v|;0 (r2r3) = e~Z(r*+p*) + e~z<r»+p>\ and <pf ( r ^ ) = e~Z(r^ - e- z <*+*>,

the same Z being used in both the initial and the final wave-function for simplicity in calculation, the mean value of 1 •G8/a0 being taken. I t is then easily seen that

32tt%i2Ci*

a sin (kn0 — k'lij) . + 6 sin j 0 + ife'nj). “ jd)

where a = (Ia — 1 ) I r(2Z) — I 8F (Z, k),& = (!« — Iy) I r’ p(Z) — I«Ir (Z, k),

F(Z) = j e - ZrdT = 8uZ"3,

F ’ v(Z) = f e-zr-zpdx = 7 ie ~ zdZ~3 (1 + Z 2d2),

F (Z, A;) = J e - Zreikn' r dT = 8rrZ (Z2 + B )~‘\

I a = | j / i —

I 3 = I f — e -Z n -Z r ii e« m ,.r1- i* 'n 1.r 2 ^J J *12

IY = [ f — e-Zn-Zj>,eiA:n0.r1-i&'n1.r8 dTi ^- J rl2

I 8 = f f — e - Z f l “ 2Z** e i&n» • r ‘ d T l d v 2 ,JJ ri2

I< = | j ~ - e~Zr*-zP> e-r - dv2.

* t his value is uncertain by at least two volts owing to the approximate nature of the potential curves.

t ‘ Proc. Roy. Soc.,’ A, vol. 132, p. 605 (1931).

VOL. CXXXV.— A. T




The latter five integrals are readily evaluable by expanding the terms 1 /r12,e~Zp, eikn°-r, in spherical harmonics as before, the series being sufficiently convergent at low velocities. Averaging over all orientations of the axis of the molecule, and then integrating over all angles of scattering gives

¥132tc

= (a2 + [ 1 sin kd sin k'd \~W ~YT -f- 2 /sin kd sin k'd ,

\ kd ~ W r

The probability of dissociation of the hydrogen molecule so calculated as a function of the velocity of the incident electron in volts is shown in fig. 7. I t is seen that the probability rises very steeply just at the excitation potential of the process, soon attains a maximum, and falls quite quickly to

Electron energy, in volts

Fig. 7.—Probability of Dissociation of H 2 into 2H by Electron Impact.

smaller values at a few volts above the excitation potential; this is in sharp comparison with the slower fall off in the probability of excitation of the B state. Also the maximum probability of the process is quite appreciable, being 0 • 7 of the total cross section at tha t voltage.

Experimental investigations on this subject have been carried out by Glocker, Baxter and Dalton,* and by Hughes and Skellet.f Their observations showed that the process of dissociation of the hydrogen molecule by electron impact sets in at 11 *5 volts ; this is the calculated voltage at which dissociation into neutral atoms in the ground state occurs, and is also well below the voltage at which dissociation might occur in any other manner. The fact that the process was found by these observers to have a measurable intensity just above the excitation potential is in agreement with the sharp rise of the calculated curve shown in fig. 7 ; also the fact that Glocker, Baxter and Dalton were

* ‘ Journ. Amer. Chem. Soc.,’ vol. 49, p. 58 (1927). f ‘ Phys. Rev.,’ vol. 30, p. 11 (1927).




unable to detect any trace of dissociation at 100 volts is in agreement with the sharp fall off in the calculated curve.*

Summary.

The theory of collisions due to Born and Oppenheimer is applied to several phenomena involving the impact of electrons with molecules.

Elastic collisions are first considered within the range of the validity of Born’s formula. The scattered intensity due to a number of axially symmetric fields with random orientation is shown to be a function of the product of the velocity and the sine of half the angle of scattering. A close relation is seen to exist between electron and X-ray scattering, and a simple formula found to relate them. The intensity of elastic scattering in molecular hydrogen is treated in detail for all angles and velocities for which the Born formula is valid.

Inelastic collisions are then considered, and the importance of the Franck- Condon principle discussed. I t is shown th a t the latter implies the diffraction of electrons which have excited a particular electronic state of the molecule. This is illustrated by considering the probability of excitation of the B state. Finally, the probability of dissociation of the hydrogen molecule by electronic impact into two neutral atoms in the ground state is calculated.

* [Note added in proof.—Further experimental evidence on this subject is provided by the work of Jones and Whiddington (‘ Phil. Mag.,’ vol. 6, p. 889 (1928), who investigated the energy losses of electrons in hydrogen for zero angle of scattering. They find an energy loss of 9 • 5 volts which probably corresponds to dissociation. The probability of this loss varies in much the same way as the calculated, but as only non-deviated electrons were investigated, no exact comparison is possible.]



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