T MOLECULES AT TEMPERATURES COLLISIONS BETWEEN ATOMS

COLLISIONS BETWEEN ATOMS AND MOLECULES AT ORDINARY TEMPERATURES

BY H. S. W. !VLASSEY

University College, London C O N T E N T S

PAGE

5 1. Introduction . . .

2.2. T h e interaction between helium atoms, . . . 231 2.3. Measurement of total collision cross-sec lar ray method . ²⁵⁴

2.4. T h e mobility of positive ions in gases .

3.1. Inelastic collisions b

rotational and translational energy . . . ²⁶⁷

$ 1 . I S T R O D U C T I O N

H E experimental study of the phenomena associated with the slow collisions of atoms and molecules must usually be carried out under conditions in

T

These determine the coefficient of viscosity, and the thermal conductivity of a gas, as well as the coefficient of diffusion of one gas in another or of ions through a gas (ionic mobility). It is from an analysis of the magnitude and temperature dependence of these coefficients that much information has been derived about the interactions between atoms.

T h e quantum theory has shown that a further quantity, which is capable of direct measurement, and may also be used for this purpose, is the total collision cross-section for impacts between the atoms. $ 2 will be devoted to a review of these various aspects of elastic collisions.

I n collisions between molecular systems, one of which possesses excess energy of internal motion, it is of interest to discover the probability that, on impact with a normal molecule, the energy be redistributed in other degrees of freedom.

T h e problem is essentially that of determining the extent to which energy transfer between relative translation, electronic motion, vibration and rotation takes place. This is of considerable importance in such phenomena as the quenching of fluorescence, sensitized fluorescence, the dispersion of sound and the rate of unimolecular reactions. 5 3 will include a brief review of these transfer collisions and their importance in certain branches of physics.

We shall first discuss elastic collisions,

Collisions between Atoms and Molecules 249

§ 2. E L A S T I C C O L L I S I O K S 2.1. The Effective Cross-Sections

It is convenient to discuss first some of the features of the quantum theory of collisions. We shall throughout refer the motion of the two colliding atoms or molecules to the centre of mass as origin, so that 8, the angle of scattering in this system, is the angle through which the direction of relative motion is turned in the impact. T h e differential cross-section is such that the number of collisions which occur per cm3 per sec., between two kinds of atoms which are moving with relative velocity v, in which the direction of relative motion is turned through an angle between 0 and 8 +do, is

2nN,N2v1(8) sin 6' do, . . . . ^* . ⁽¹⁾

N I , iV, being the number of each kind of atoms present per cm3.

elastic collision cross-section Qo is given by

T h e total

Qo ⁼ 2n f I(8) sin 8 do. . . . ^, (2)

0

I t is such that the number of collisions taking place can be calculated by treating the atoms as rigid spheres of radius (Q0/4n)h.

ilccording to the classical theory Qo, defined in this way, would not be finite if the interaction between the atoms falls gradually to zero and has no sharp boun- dary. T h e deviation produced by even a weak interaction at very great distances would be finiteand theeffective sizes oftheatomsunbounded. I n quantumtheory, owing to the uncertainty principle, no meaning can be attached to deviations through less than a certain angle-they are unobservable. As a result of this the total cross-section Qo defined by (2) is finite in the quantum theory, provided the interaction between the atoms falls off more rapidly, at large distances, than the square of the inverse distance between them (Mott 1930, Mott and Massey 1949a).

If means can be found for measuring so another useful quantity for providing information about interatomic forces is available. It is important to notice that Qo can never be calculated by classical theory as the contribution from very small angles of scattering is always given incorrectly by that theory.*

A remarkable example of the fact that the classical theory can never give I(8) correctly for all 8, except for the very special case of the Coulomb field, is provided by the problem of the rigid sphere. If the atoms are regarded as rigid spheres of diameter a, with no other interaction, the classical collision cross- section is, of course, finite and equal to na2. When the wavelength h of the relative motion is large compared with a, the quantum theory gives a different value, 4na2, which is not surprising as the wavelength is large compared with the size of the obstacle. But what is rather more unexpected is that the quantum theory, in the limit of very short wavelengths, gives, not a cross-section nu2, but

one of twice that value. T h e

quantum theory gives the same value of I(8) for 8Bh/2xa, but for smaller values of 8 it predicts that I ( 0 ) should increase rapidly to reach n2a4,X2 at 8 = O (see the schematic representation, Figure 1). It is the additional contribution from this

T h e classical form for I(8) is a constant, a2,4.

* It is possible to eTaluate the cross-section Q(Q,) for scattering through angles greater than 8, bl clacsical theory probided that theory is valid for calculating the scattering through angles as small as 8,. T h i s may be made use of in practice by ensuring that deviations through angles less than 8, vi11 not be observed.

250 H. S. W. Massey

region which doubles the classical cross-section. T h e effect is an example of edge diffraction and it has been discussed by Massey and Mohr (1933) and by Wergeland (1945).

T h e viscosity and heat conductivity of a gas depends on the ease of trapsport of momentum from one part of a gas, in which the mean kinetic energy is higher, to another. T h e more frequent the collisions the less easy the transport, but collisions in which the energy of the colliding partners is evenly shared after impact are more effective in this regard than others. T h e collision cross-section Q, effective in viscosity is given by

Q, ^{= .zT} j ^{'. I(@> sin3 6' do,} . . . . . . (3)

0

greater weight being given to deviations through 90". T h e complete formula for the viscosity coefficient y~ of a gas has been given by Chapman (1916, 1917)' and by Enskog (1917). I t is

. . . ^{* (4)}

10 4r;kT 3 2 l + ~

v = m ( T - ) K '

where M is the mass of a gas atom, T the absolute temperature, k Boltzmann's constant and

m

R,, = 3 _. 1 - - m ^{u7Q,(v) exp { - Mu2/4kT) dv. , . . . . ( 5 )} E is a small correction, never greater than 0.01'7.

This is defined by

A further effective cross-section QD is required for the theory of diffusion.

QD = 2 x i" ^{(1 -} cos 8)1(8) sin 6' d6'. . . . . . . ⁽⁶⁾

- 0

Diffusion is retarded more effectively by those collisions in which 8-180"

so that the paths of the colliding atoms are nearly reversed in sense-hence the additional weighting at these angles T h e coefficient of diffusion D for two gases is given by (Chapman 1916, 1917, Enskog 1917)

.... ^..(7)

where

1

P12= J u5&)(v) exp { - ~ ~ l M 2 ~ 2 / & T ( ~ l + i % ? 2 ) ) dv. . . . . . . (8)

-cc

M I , M 2 are the masses of the atoms of the respective gases and n,, n2 are the concentrations in number of atoms/cm3. c0 is again an unimportant correction, never greater than 0.136.

T h e mobility of an ion in a pure gas is given by the relation

K ⁼ e D / k T , . . . (9) where D is given by the formula (7).

usually be calculated with accuracy by classical methods.

Unlike the total cross-section the viscosity and diffusion cross-sections may Of all monatomic

Collisions between Atoms and Molecules 251.

gases, helium is the only one for which quantum theory must be used to analyse,, in terms of atomic interactions, the viscosity at temperatures above the liquefying point. At temperatures below about 8 0 " ~ . the classical theory becomes very inaccurate for this gas. A discussion of the quantum modifications for rigid spherical atoms has been given by Massey and Mohr (1933).

We now discuss certain applications of these formulae, 2.2. The Interaction between Helium A t o m

An accurate determination of the interaction energy between helium atoms at various separations is important for several reasons. I n particular, a satis- factory theory of the peculiar behaviour of liquid helium may well depend on the detailed nature of this interaction. Evidence is available from three sources, measurements of the viscosity and second coefficient over a wide range of tempera- tures and study of the scattering of helium atom beams by helium. T h e informa- tion obtained is still incomplete and not very consistent.

Figure 1. Schematic form of the relative Figure 2. Illustrating various assumed potentials V ( r ) for interaction between helium atoms plotted as functions of r/vo where yo = 2.945 A .

angular distribution function I(@) for collisions of rigid spheres of diameter a, h being the de Broglie wavelength of the relative motion.

At la'rge separation there is a weak van der Waals attraction which changes rapidly to a strong repulsion. T h e depth of the attractive potential is so small that it is only from low temperature data that information can b e derived about this part of the interaction.

There exist seven deter- minations of the viscosity (van Itterbeek and Keesom 1938), and thirteen of the second virial coefficient (Holborn and Otto 1926, Nijhoff, Iliin and Keesom 1928, Keesom and Kraak 1935, Keesom and Walstra 1939), of helium at temperatures below 30" K.

T h e first analysis of the viscosity data using quantum theory to calculate the viscosity cross-sections was made by Massey and Mohr (1933, 1934). They assumed the interaction to be of the form given by Slater (1928), viz.

2.21. Evidence f r o m low temperature experiments.

V ( r ) ⁼ be-ar ^- CY-^, ^{. a . .} . . ( l o ) where h = 7.7 x 10-10 erg, a = 4.6 x lo8 cm-1 and c = 1.37 x 10-60 erg cm6, and

2 5 2 H . S. W. Massey

found fair agreement with the observations. Since then a number of attempts have been made to improve on this potential, taking into account the second virial coefficient data also, the quantum formula for this coefficient having been derived by Gropper (1937) and by Beth and Uhlenbeck (1937). T h e most recent analysis is that of Buckingham, Hamilton and Massey (1941). They considered the family of interactions illustrated in Figure 2 in all of which the zero occurs at 2.945 A. T h e asymptotic form of these interactions was given by - c, y6 but with different constants c for each. Curve I V corresponds to the Slater potential (lo), curve I to one of the same form but 30% stronger, and curve 111 to a potential assumed by de Boer and Michels (1939) in their calcula- tions. Of these interactions I and I1 give rise to no discrete stationary state whereas I V and V give one such state with binding energy 6 x low6 and 1.5 x 10-*ev. respectively. It is not certain whether interaction I11 does or does not give a bound state but if so it must have extremely small binding energy.

T h e T h e evidence from the virial coefficient data is illustrated in Figure 3.

-2@o t

3 1

k

1 ^//'

Y E

/ / -400.

, ^{I I L}

4 8 12 16

q,2 = 4n2Mi-,-,2kT/h2

.Figure 3. Comparison of observed and calculated virial coefficients ( B ) plotted as functions of q i = 4 n 2 i W i k T , h 2 where ill is the mass of a helium atom and ~ , = 2 . 9 4 5 A.

Full curves, calculated results from corresponding assumed potentials I-V illustrated in Figure 2.

- - - - - - - Keyes' smoothing function.

Experimental points : x I

0 Keesom and Kraak (1935).

+ Keesom and Walstra (1939).

Holborn and Otto (1926).

Nijhoff, Iliin and Keesom (1928).

data do not lie on a smooth curve and a considerable gap exists between 5" and 15".

Keyes * has constructed an empirical function which provides a representation of the data between 2" K. and 650" K. which gives a good fit to the experimental points over the full temperature range except near 1 5 " ~ . T h e virial coefficients calculated by use of the interactions I-IV are illustrated in the curves I-IV of Figure 3. It will be seen that the best fit is likely to be obtained by an interaction

;falling between I1 and 111 of Figure 2.

* Private communication to Dr. R. A. Buckingham.

Collisions between Atoms and Molecules _2.5 3

This conclusion is not consistent with the viscosity data. T h e observed viscosity between 1 . 6 4 " ~ . and 5 " ~ . is fitted best by interaction V of Figure 2 and, if anything, suggests an interaction slightly weaker still. On the other hand, at temperatures between 1.5" and 30" I<., the best fit is with the interaction I of Figure 2.

It is not clear whether the lack of consistent agreement is due to imperfection in the forms of the assumed interaction or to inaccuracy in the experimental data, Although the calculations were carried out with a fixed value of the distance ro at which the zero occurs it is unlikely that any adjustment of this distance will eliminate the inconsistencies. There is clearly a need here for further experimental work *.

I n carrying out the calculations by the quantum theory allowance is made for the fact that helium atoms obey the Bose-Einstein statistics. T h e importance of this may be judged from the calculations carried out by 3Iassey and Buckingham ( 1 9 3 8 ) with the field I of Figure 2. They compared the viscosity cross-sections Qq obtained on the respective assumptions of Bose-Einstein, classical and Fermi- Dirac statistics. Their results are illustrated in Figure 4 and it will be seen that

I ^I

0.1 0.5 1 2 5 IO 25

T e m p e r a t u r e (DK)

Figure 4. Calculated viscosity cross-sections Q O for helium using the interaction I of Figure 2.

T h e relative velocity z> of the atom is given in terms of a temperature Twhere ~ = 3 ( k T , ' ; Z l ) l , M being the mass of a helium atom. z; is thus the most important relative velocity in determining the viscosity at temperature 2'.

Curve I-assuming Bose-Einstein statistics.

Curve 11-assuming Fermi-Dirac statistics.

Curve 111-assuming classical (Boltzmann) statistics.

Energy of Relative Motion (e.volts)

Figure 5. Illustrating the variation of the cross- section for transfer of excitation betu.een a t o m s as a function of the energy of relative motion, for three energy discrepancies A E .

it is only at temperatures below 3 " ~ . that the nature of the statistics is at all important.

The evidence from the measured viscosity of helium in the temperature range from 20" to 2 0 0 " ~ . has been discussed most recently by Amdur (1947). H e calculated the viscosity from the formulae ( 3 ) , ( 4 ) and ( 5 ) using classical theory and the interaction energy- proposed by Margenau (1939). This is given by

- 0 . 0 1 3 9 ~ - ~ - 0 . 0 3 0 ~ - ~ - 0.035r-1°] x 10-10 ergs, 2.22. The evidence f r o m medium and high temperatures.

V ( r ) = [7.70 - 5.60

* Some clarification is provided from measurements by Cbbink and de Haas (1943) of the Their results are consistent with an interaction thermal conductivity of helium gas at 1.5 and 3" K.

between I1 and I11 (Figure 2). agreeing with the evidence from the second virial coefficient.

254 H . S. W. Massey

where Y is in A . , and is illustrated by curve M of Figure 2. Amdur obtained agreement within 5 yo for the viscosity over the whole temperature range considered.

This leaves the situation rather more confused than ever because the low tempera- ture evidence does not appear to be consistent with Margenau's interaction.

This may easily be seen by reference to Figure 2, which shows that this inter- action includes a considerably stronger long range attraction than any used in the calculations of Buckingham, Hamilton and Massey. T h u s the minimum is rather deeper and the zero occurs at a smaller distance. Furthermore, the analysis of the second virial coefficient of helium in the temperature range 70-673"~., carried out by Buckingham (1938), does not support the Margenau potential. T h u s he concludes that the zero occurs for r = 2.65 A. and finds the data consistent with the interaction

V(Y) =[8.71e-468'-0.0147r-6] x 10-lOergs. . . . ^* . . ⁽¹¹⁾

This interaction gives a slightly harder repulsion than that, V(Y) = [4.39r-12 - 0.0152~-~] x 10-IOergs,

derived by Hirschfelder, Ewe11 and Roebuck (1938) from an analysis of the Joule-Thomson coefficients of helium.

It must be remembered that all the interactions derived by assuming special functional forms including adjustable constants are only likely to be reasonably accurate over the limited range of Y which is effective in determining the physical properties used for the analysis, over the temperature range available. Extra- polation to other distances can only be carried out with caution.

A severe test of interactions derived from properties measured up to tempera- tures not greater than 7 0 0 " ~ . is provided by the experiments of Amdur and Pearlman (1941) on the scattering of energetic helium atoms in helium. I n these experiments, beams of helium atoms, with homogeneous energies between 300 and 1,000 ev., were prepared from helium ion beams of the same energy, by charge transfer. T h e cross-section for scattering of these atoms through angles greater than 4.5" in collision with helium atoms at room temperature was measured by observing the absorption of the beam in helium gas. By comparing these observations with calculations based on Buckingham's potential (equation

r (1 1)) it was found that this potential gives rather too large a repulsion for Y <0.8 A.

but, even at 0.55 A., it is only about twice too large. This indicates a wider range of validity of the potential than would be expected.

2.3. Measurement of Total Collision Cross-Sections by the Moleculzr Ray Method T h e measurement of the total collision cross-section Qo by molecular ray technique is simple in principle. T h u s to measure Qo for two kinds of atoms A and B, averaged over the distribution of relative velocities fixed by the temperature of the gas, a narrow molecular beam composed of the atoms A is fired through the gas of atoms B contained in a small scattering chamber. T h e intensity of the beam is measured after passage through this chamber, both when the gas B is present at a measured pressure and when it is absent. From the ratio of these two intensities the mean cross-section may be determined, using the standard formulae of the kinetic theory of gases.

I n order that a definite cross-section be obtained which is effectively inde- pendent of the angular resolving power of the apparatus it is necessary that this resolving power be very high. As a rough guide for testing whether any parti-

Collisions between Atoms and Molecules ₂₅ 5

Alkali metal atom Li hTa K R b

Cs

cular design is adequate a schematic angular distribution function I(6), illustrated in Figure 1, may be used. This is based on the analogy with the rigid sphere problem. If a is the sum of the gas kinetic radii of the atoms, estimated from viscosity measurements or in some other way, then I(6) may be taken as equal to a*, 4 for 0 >42na and to rise linearly from this value to n2a4/X2 at 6 = 0, X being the wavelength of mean relative motion of the atoms. T h e apparatus should be adequate for the purpose if, with this distribution, the ratio

He

Q CUI C(2)

106 14.4 17

130 17.8 (26)

165 31.7 35

152 25.4 (40)

162 29.4 44

fI(0) sinHdO/rI(B) 0 sin8d6,

6 0

. . . . . (12) is sufficiently close to unity, Bo being the smallest angular deviation which can be detected by the apparatus.

Almost all experimental work in this direction has been carried out using molecular beams of alkali metal atoms. This is because the intensity of such a beam may be measured very conveniently and accurately by the hot filament method. T h e atoms undergo surface ionization and the current of ions produced is easily measured.

T h e earliest work on the collisions of atoms was that of Broadway (1933).

It was directed to establishing the validity of the quantum result that Lim,,,I(B) is finite. H e observed the scattering, through angles between 0.2” and l o , of a beam of sodium atoms by a beam of mercury atoms. Evidence in favour of the finite limit was found.

Two detailed experimental measurements of the cross-sections of the alkali metal atoms for collision with various gas atoms have been carried out, the first and most extensive by Rosin and Rabi (1935) and the most recent by Estermann, Foner and Stern (1947). T h e resolving power in the latter experiments was very high, the angle 6, of (12) being about 5”. I n Rosin and Rabi’s apparatus it was about 1’.

It was an interesting feature of Rosin and Rabi’s experiments that the distri- bution of intensity across the beam remained the same in form with and without gas present in the scattering chamber. This absence of broadening by the scat- tering gas indicated that the probability of deviations taking place through angles less than the resolving power of the apparatus was small.

Rosin and Rabi measured cross-sections for collisions of each of the alkali metal atoms Li, Na, K, Rb and Cs with each of the rare gas atoms He, Ne and A.

Their results are given in Table 1. T h e only measurements of Estermann, Foner and Stern which can be compared are those for Cs and H e collisions.

They find a cross-section 2; times greater than that of Rosin and Rabi (see Table 1) Table 1. Observed Collision Cross- Sections for Alkali metal-Rare gas Atom

Impacts and t h e derived van der Waals Constants Ne

Q C(1) C ( 2 )

120 18.7 32

213 40.4 (51)

259 50.5 68

268 49.0 (77)

287 56.7 87

.4

--

Q C(1) C!2)

303 188 125

40 1 192 (200)

580 356 260

572 249 (290)

572 235 325

Q , cross section in cmz.

C‘, derived van der Waals’ constant in cmz : (1) from observed Q, (2) from polarizabilities.

256 H. S. JV. Massey

and it is not clear how the discrepancy arises. T h e resolving power used by the latter authors should have been adequate.

A theoretical analysis of these results (Massey and Buckingham 1936), may be made on the plausible assumption that the van der Waals attraction, of the form

- c / y 6 , where Y is the distance between the atoms and c a certain constant, is re-

sponsible for the scattering and is so large that the effect of the repulsion at short distances is quite negligible. For these conditions Massey and Mohr (1934) derived the formula

Q ---(-CA) 9 3 2 / 5 '

' - 4 32

where h is the wavelength of the relative motion. Using this formula and a suitable mean for A, values of the constant C for different pairs of interacting atoms may be derived. These are given in Table 1. It is further possible to estimate Cif the polarizabilities of the atoms are known (Hellmann 1935). Those of the rare gas atoms are available from their refractivities (C. and M. Cuthbertson 1911) and those of lithium, potassium and caesium have been measured directly by Scheffers and Stark (1934) using molecular ray methods. T h e estimated values of C obtained in this way are also given in Table 1, interpolated values being included for sodium and rubidium. I t will be seen that, on the whole, the agreement between estimated and derived values of C is not unsatisfactory.

There is a tendency throughout for the derived values to be too small for rubidium and caesium. On the other hand, the value derived for the Cs-He case from the measurements of Estermann, Foner and Stern is many times greater than the estimated.

It seems clear that the measurement of total collision by molecular ray methods is likely to provide a powerful tool for investigating the interactions bet\+een atoms but much more experimental work is still required to establish it on a reliable foundation.

2.4. The Mobility of Positive Ions in Gases

A considerable amount of experimental work has been carried out on the mobilities of positive ions in gases. It has been described by Tyndall (1938) in his book The Mobility of Positive Ions in Gases.

T h e interaction energy between the ion and atom may be represented as of the rigid sphere form with a superposed attraction due to the polarization of the atom by the ion. T h u s the interaction when the ion and atom are at a distance Y apart is taken to be

where cc is the polarizability of the atom. T h e radius a may be obtained from gas kinetic data and ^c( from refractivity data or by molecular ray experiment (see $2.3). Alternatively the interaction may be taken to be of the form V ( r ) =ay+ - br-6 - $xe2/r4, where n is a positive integer greater than 6, the value 10 being mathematically most convenient. T h e first two terms are taken to be the same as for interaction between the neutral atoms, the form being that assumed by Lennard-Jones (1931) in his study of the equation of state of mon- atomic gases. Given V(Y), I( 0 ) may be calculated by classical methods, and hence the mobility by means of formulae (6), ( 7 ) , (8) and (9). Calculations carried out in these ways by HassC and Cook (1931) have been found to give satisfactory agreement with experiment for a number of cases.

For most cases a classical theory is adequate.

V ( Y ) = - + ~ e 2 / ~ ~ , r > a ; V(Y)+CO, r<a,

When considering the niohilitp of an ion such a s We 1 in a gas of the same kind it is necessary to allow for the fact that charge transfer may occur on collision : H e ” + H e - + H e + H e - . Because of this, it is not a good approximation to calculate th e mobility assuming that the interaction between, say, He-- and H e is given by adding a polarization potential to that between two helium atoms.

‘The correct procedure was given by Massey and Smith (1933) and applied by Massey and Mohr (1934) to calculate the mobility of He-? in He.

,kcording t o quantum mechanics there are two possible interactions between a helium ion and atom in their ground states, depending on whether the electron distribution is symmetrical or antisymmetrical with respect to the two nuclei.

Corresponding to each interaction there is associated an amplitude for scattering between angles 0 and 0 +do. T h e correct amplitude to take is a suitable linear combination of these two. T h e symmetrical interaction includes a powerful long range attraction which makes such a large contribution to the scattered amplitude that the effective diffusion cross-section is considerably larger than would he the case if charge exchange were ignored.

T h e mobility c.a,lculated i n this way for H e in He was found to be 12 ciyi2, sec.

volt. Th is is in marked disagreement with the value 21.4 cm”. sec. volt, for the ions from a helium glow discharge, observed h y ‘l‘yndall and Powell (1931).

‘ r h e reason for this discrepancy is still not clear. Yleyerctt (1944) has suggested that the ions concerned in the experiments were not He+- but He,+. H e points out further that, in earlier measurements by the same authors (Tyndall and Powell 1930) in which an x-particle source of ions was used, the value 1 3cm2 sec.

volt was found, in much clcser agreement with the calculated value. On the other hand, 3Crott (private communication) has drawn attention to the possibility that the neglect of certain terms in the calculation might be serious. These terms were ignored on the assumption that the relative velocity of the ion and atom is so slow that electron exchange can frequently occur during a collision.

T h e frequency of exchange is equal to ?,E h where I E is the energy difference between the symmetrical and antisymmetrical interactions. At large distances 1 E is small and exchange occurs very infrequently. ‘The effect of the relative motion may then be to reduce the magnitude of th e exchange interaction at these

distances. It is doubtful

however whether the correction arising in this via!; is large enough. 3Ieyerott (1944) has attempted to obtain some information on this point by calculating th e interaction between a lithium ion Li- and a helium atom, by quantal methods, and hence obtaining a theoretical Yalue for the mobility of Li-- in helium to compare with the observations of Hoselitz (1940). ‘The lithium ion has a similar electronic structure to that of helium but the exchange effect is small and better agreement with experiment should be obtained if it is &is effect which has been mi:calculated with He-i-. Only moderate agreement was found, as is clear from reference to Table 2, so that n o definite conclusions can be drawn.

Table 2.

T hi s would reduce e,, and hence increase A’ in (9).

Comparison of Observed and Theoretical T-alues for the Mobility of Lij- in H e

Temp. (’ IC.) . . ^{. . . . 20 58 90 195 29 1 389 483}

Mobility Observed , , 20 21,s 22.2 21.9 25.8 27.8 29.2 (cm”l-olt. sec.) { c a l c u l a t e d . . 1 6 4 17.4 15.5 18.6 19.4 20.1 20.9

T o clarify the position the mobility of He-+ ions n-ith established identity

P S P R 17

2 j 8 H . S . W . Massey

should be measured. A further possibility would be to measure the coefficient of diffusion of metastable atoms of helium in helium (see Dorrestein and Smit 1938), as in this case also electron and excitation transfer must play an important r61e. T h e theoretical importance of a satisfactory solution is considerable because these cases of charge and excitation transfer, in which no change in relative kinetic energy occurs, can be regarded as limiting cases of collisions in which charge and excitation transfer can only be effected if the energy of relative motion is changed. A satisfactory theory of these elastic processes must be expected to precede development of such a theory for the more com- plicated inelastic cases.

5 3. C O L 1 , I S I O S S I K V O L V I N G C H A N G E O F I N T E R N A L E N E R G Y

A considerable volume of evidence is available concerning the ease with which energy exchange may occur between that of relative translation and various degrees of internal freedom of the colliding systems. Nevertheless, although it is possible to give general rules concerning these energy transformations, we are still far from possessing an interpretation so complete as to make possible the prediction of the rates of a particular reaction except in very few cases.

3.1. Inelastic Collisions between Atonzs-Exchange between electronic and translational Energy

I n collisions between atoms, the only internal motion which can exchange energy with that of relative translation is that of the electrons. ,4s we are con- cerned with impacts in which the relative kinetic energy is of the order kT,

inelastic collisions between atoms cannot occur unless one of the atoms is in an excited state.

T h e occurrence of inelastic (and superelastic) collisions between excited atoms leads to the phenomena of the quenching of resonance fluorescence by the presence of foreign atoms, and of sensitized fluorescence.

Resonance fluorescence is produced when a gas or vapour of atoms 4 is excited by radiation, from a suitable lamp, which arises from transitions from an excited (nth) state to the ground state of the atoms X in the lamp. Resonance absorption of this radiation raises atoms A in the resonance chamber to the nth state, from which they return to lower states by emission of the resonance radiation. T h e intensity of this resonance radiation for a given intensity of irradiation from the lamp is often reduced very considerably, or quenched, by the addition of foreign gas to the resonance chamber. This is because the atoms of this gas, in collision with excited atoms A , remove the excitation energy. With suitable experimental precautions, necessary to avoid complications due to imprisonment of resonance radiation and to line broadening by the foreign gas, it is possible to measure the effective cross-section for the quenching due to different atoms and molecules (see llitchell and Zemansky 1934a, b).

Cross-sections for the quenching of the resonance radiation of mercury (Zemansky 1930, Bates 1930, 1932, Duffendack and Owens 1934, Evans 1934, Olsen 1941, Olsen and Kerr 1947), cadmium (Lifson and Mitchell 1935), thallium (I’rileschajeua 1932, 1935), and sodium (Winans 1930, Kisilbasch, Kondratjew ancl Leipunsky 1930, Terenin and Prileschajewa 1931, Kondrattjew and Susken

Collisions between At" and Molecules 259

1935, Norrisli and Smith I WO), l y tiitferent gascs have bccu measured. 'I'hc data for quenching by molecules are disccissed in $3.2. For all the atomic gases investigated the quenching cross-sections have been found to be very small ( <10-l'cm2). Some authors have given actual values for some of the cross- sections but usually it is merely said that they are too small to be observed with the existing experimental limitations. T h u s Olsen (1941) finds negligible quenching of mercury radiation by helium and. krypton but gives cross-sections 3 and 2~ 10-17cmp, for quenching by neon and argon respectively. I t seems quite certain, by comparison with the data for quenching by molecules, that atoms are very ineffective in quenching, the chance per gas kinetic collision being much less than 1,'lO.

Sensitized fluorescence occurs when the irradiation of a mixture of gases and B, by radiation which excites the atoms -4 by resonance absorption, leads to emission of radiation by the atoms B. This is due to transfer of excitation from the atoms *4 to the atoms B in collisions (see Mitchell and Zemansky 1934b).

T h e most important general rule, n-hicli has emerged from the experimental study of these phenomena, is that transfer of excitation is most effective when the minimum amount of energy is transferred to or from relative translation, i.e.

when the energy difference betvieen the initial and final states of the first atom is equal to that between the final and initial states of the second. I n these circum- stances the colliding systems are said to be in energy resonance. A most definite illustration of this resonance effect was provided by some experiments of Beutler and Josephy ( 1 929) who investigated the mercury sensitized fluorescence of sodium.

They determined the relative population of different excited levels of sodium and found a sharp maximum for the 9s level. T h e excitation energy of this level is only 0 . 0 2 0 e ~ . higher than that of the 6"P, level of mercury from which the excitation \yap transferred. -4 small subsidiary maximum was also observed for the 7s level whose excitation energy !ies closest to that of the metastable 63P, level of mercury.

T h u s Gran and Duffendack (1937) observed the enhancement of the s, p and d levels of the Pb+ spark lines when neon was added to an electric discharge in lead vapour.

T h e enhancement is due to the transfer of charge and excitation between neon and lead by the reaction

Similar resonance effects have been observed in other reactions.

Xe- + Pb -tPbT' + X e .

It was found, again, that the maximum enhancement in any series of lines, starting from levels with the same azimuthal quantum number, occurred when the energy resonance was closest.

h second general rule, due to Wigner (1929), is that the total. electronic spin is conserved in a transfer reaction. Although this rule is useful as a general guide to the most important processes it is not accurately valid. T h e most remarkable cases in which it fails occur with helium atoms. I n the course of an investigation of the excitation of the helium spectrum by electron impact Lees and Skinner (1932) produced strong evidence to show that the cross-section for the reaction He(4IP) + H e ( l ' S ) +He('S) +He(43D) + 0 4 0 6 ev., is not small compared with the gas kinetic. Maurer and Wolf (1934, ?940) studied these

. . . . . . (13)

17-2

2 60 H. S. M7. Massey

reactions in great detail and obtained the eft'ective cross-sections (in terms of the gas kinetic) given in Table 3.

Table 3

Resonance ReaLtion discrepancy

ev.

0.013

~ ~ ^0.006 ~ - ~ ~ ~ ~ {

4'1' -4'11J 0.006 4'P-- 4 'SI 0.068 3'P-> 3 3P 'i ^{0 4 7 9}

4'P-2 4 'SJ 0,148

Cross- section

1.4 8 45 10

< 2

< 1

Resonance Cross- Reaction discrepancy scction

j1p+ j ID 0.003 34

ev .

SIP-, 5 :q) ) 0 ~ 0 0 3 18 j'P-> 5 'SI 0 4 3 4 ^{, '} 1

sir>+ ^:ISJ 0 .( )7 1 ' 0 . 6

These results exhibit the resonance behaviour quite clearly but Wigner's rule only holds in a weak form. T h e cross-section for a process in which the total spin is conserved is larger than for the corresponding process in which it is not, biit only by a factor of 2 or 3.

A further source of information about atomic collisions in which electron transitions occur is from the study of the formation and destruction of metastahlc atoms in an electrically excited gas. These experiments are rather difficult to interpret accurately because a number of factors are involved. An account of experiments in the rare gases have been given by llitchell and Zemansky (1934a) in Resonance Radiation and Excited Atom. -1s an example it is found (31eissner and Graffunder 1927, Zemansky 1929) that the cross-section for excitation of metastable neon atoms in either one of the 3P levels to the IF', level which lies 0.15 ev. above the uppermost, 3P0, level, by impact of normal neon atoms at room temperature, is 4.2 x l0-l9y cm2. y , the fraction of neon atoms possessing sufficient energy to produce the excitation, is likely to be of order lO-'so that the cross-section is between 0.1 and 0.01 of the gas kinetic. -1 somewhat smaller but comparable result has been found by de Groot and Penning (1933) from an analysis of some earlier experiments by Dorgelo and Washington (1927). T h e value they derive is 3.9 x 10-*o:y. For argon the cross-section for the excitation 3P0-t1P, appears to be considerably smaller (Anderson 1931). O n the other hand, doubt is thrown on the validity of the method by the impossibly large cross-section found (Ebbinghaus 1930) for the excitation 2 3 S +2'S, in helium.

Recently Kvifte and Vegard (1947) have reported experimental estimates of the cross-sections for de-activation of oxygen atoms in the metastable IS state by impact of neon atoms. I n this case the process involves a superelastic collision and the factor y does not arise. T h e very low probability of lo-' per gas kinetic collision was found for this process which involves transfer of 2.0 ev. internal energy into energy of relative translation.

Finally, Kruithof and Druyvesteyn (1937) have obtained an estimate of the cross-section for the reaction

Ne' -k A +Ne + A + +e, . . . . . . (14) S e ' being a metastable atom, from a study of the electron ionization coefficient

cif neon containing a small admixture of argon. This gave the ratio of the cross- section for the above reaction to that for destruction of metastable neon atoms in collisjon with normal neon atoms, Using de Groot and Penning's value (1933)