Ionization and similitude

Physica XT, 110.4

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December 1945

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by P. H. CLAY 7 $ 1. An ionization chamber is an instrument in general use for the determination of the intensity of cosmic rays, X-rays, rays in nuclear physics etc. These rays ionize the gas in the ionization chamber, either directly or indirectly, by means of fast charged particles, and with the aid of an electrical field with a power F the two kinds of ions may be separated and driven towards two electrodes, so that their charge, and, therefore, their number may be measured according to the customary electrostatic methods. The number of pairs of ions formed by radiation in the ionization chamber is a reliable measure of the intensity of the radiation. Experience teaches us that, in the ionization chambers in general use, not all of the ions reach the electrodes. The measured current of ions i is smaller than the ions charge I liberated per sec. ; we accordingly find the formula

in which K stands for the lack of saturation. This lack is caused by the recombining of ions before they reach the electrodes. We may dinstinguish the following ; initial recombination, caused by the dense accumulation of the ions immediately after their formation by a fast charged particle along the-track of this particle, and volume recombination between ions of different tracks, which has no relation to this original accumulation. Lack of saturation through volume recombination is bound to increase with the intensity of radiation, whilst, in the case of low intensities, the lack through initional recombination, being independent of intensity, will coml ) The manuscript of this paper was given to me a few days before the author’s death, July 16, 1943 and we had no occasion to discuss it. I am much indepted to Mr. S. l7. d e G r o o t for carefully reading it and correcting J. CLAY. the translation.

Physica

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pletely prevail. This is how matters stand in the case of the customary ionization measurements. In this case, it appears that K still depends, in a very complicated way, upon F, upon the density d of the gas, upon the temperature T, as well as upon the nature of both the radiation and the gas. A knowledge of K is imperative in order to obtain satisfactory results with ionization measurements. Conversely, the influence of F, d and T upon K should enlighten us concerning the nature of the radiation and that of the gas. In the following we shall attempt, by applying considerations of similitude, to obtain an insight into the influence of F, d and T upon the lack of saturation through initial recombination only, which we shall indicate by K. In this way we shall also come to an understanding of the possible influence of the nature of the gas and the nature of the radiation upon K. The result of our considerations may be directly checked by experiment ; it does not contain any complicating approximations. 5 2. An ionizing particle forms in the course of its path loose pairs of ions and groups, or clusters, containing 2 or more pairs of ions through the intermediary of 6 electrons (and possibly also of photons) Primarily, S processes are set up per cm length of track, ultimately giving rise to N pairs of (singly-charges) ions. The elementary ionization process consists either in the drawing of one electron (ev. more than one electron) from a molecule (ev. atom), which remains behind as positive ion, or in splitting up a molecule into a positive and a negative ion (and possibly also other parts). After haivng covered a certain distance the electron attaches itself to a molecule, thus giving rise to a negative ion. In an electropositive gas, in which negative ions are also directly formed, the range of these electrons is smaller than that of the 6 electrons (or photons), so that, for the initial recombination process, the path of the ions consists of S groups of one or more pairs of ions per cm. The positive groups have dimensions Y,, and the negative groups .have dimensions r2, whilst, within each group, there is only a slight difference between rl and r2. If l/S < 2r, then 1IS < 2r2 may also be said to apply, so that we may speak of homogeneous positive and negative columns of ions. If 1IS > 2r,, and 1IS > 2r2, then we have to do with columns of separate clusters. In a perfectly pure electronegative gas the electrons would remain free and no negative ions could possibly arise at all. In

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199

reality, however, there always exist electropositive impurities, so that the electrons are able to attach themselves to such a molecule, after having covered some distance;.partly through diffusion, which may become much greater than the range of the electrons. The situation prior to the recombination process resembles that in electropositive gas; now, however r, > rl, and may become r2 > ri, so that it is possible for the case 2ri < l/S < 2r2 to arise. In this case we have a negative homogeneous column, and a positive series of clusters: in electronegative gases lack of saturation is considerably less than in electropositive ones. $3. Ions tend to obtain chemical and physical equilibrium with the surrounding gas. We will assume that, they are singly charged during the recombination process with which we shall now deal, that the positive ions have a mobility ul, and the negative ions a mobility u2. During the recombination process they are subject to a diffusion with a coeffiicent D. According to T o w n s e n d ‘) D -=ZL

kT e

(2)

in which k is the constant of B o 1 t z m a n n; e the charge of an electron quite apart from the sign (whether it is a negative or a positive one). Assuming positive and negative ions to be equally divided in the space as between concentrations n2 and nl, we shall call the recombination process normal. In this case it may be described with the aid of the recombination coefficient tc, according to the formula dn, -z-Edt

dn, dt

0: w2

by J a f f C “). Characteristic of the nature of this process is the relation between the free length of the path A of the ions, the most probably relative distance between the ions 1 = 5/9n*, if n, = n2 = n, and that distance a between a positive and a negative ion, at which the decrease in potential energy through C o u 1 o m b-attraction e2/ca equals their thermal energy 3/2 kT 2e2 a=3EkT At room-temperature

(3)

(and E = 1) we get a a 4 , 1o-6 cm. In the

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P. H.

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case of h/a < 1 we get, both with I> tc =

dxe

@l

a and with 1 < a: + E

s12) (4

In case of greater values of ?,/a, tc falls below the value (4), and we can put tl = 4xef (Ul + 4 m * which f 5 1. (5) E In air, at room temperature f = 1 above 7 atm. For normal recombination to be effected it is sufficient if every positive ion is surrounded by negative ions in the same way as with equal distribution of concentration, and vice vema. Should any deviation occur in these relative positions, this will influence the recombination process, which, in this case, we shall call anormal. Such deviations may especially occur in the dense accumulations of positive and negative ions in clusters, directly after their formation. Owing to the action of the diffusion, however the mutual relative position of the ions will always strive after the normal. In the beginning the anormal recombination will be greater than might normally be expected. The anormal recombination will become particularly significant when local distances smaller than a occur between the ions. In the case of greater densities and larger numbers of pairs of ions per cluster, this significance for the initial recombination is increased. This form of recombination between the positive and the negative ion of one original pair is called preferential recombination 3). For an adequate discussion of the anormal recombination process we should have to be in possession of data concerning the particular relative position of the ions, and this we are not. One thing is certain, however : at lower pressures normal recombination is bound to prevail, and may well be dealt with in the manner described below. The difference between our own results and experimental data will have to provide the answer to the question, in what circumstances and in what way anormal recombination may occur in addition; a certain amount of difference, however, may also be caused by the fact that our supposition of singly charged ions (with unchangeable mobilities) has not been altogether complied with. 5 4. In accordance with No. 3 we will assume that the initial recombination process, as we take the average either over a large number of clusters or over a great column length, has a normal cour-

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se. For a concentration nl of the positive ions the following general equation applies ; we shall neglect the influence of ‘d and T on e and confine ourselves to the case E = 1 : -dnl = D, A nl + z~i F grad nl + ulnl div F dt

u n1n2

(6)

In this, the vectcr of the electric field F contains the (homogeneous), induced field F,, plus the contribution to the field by the space charge of the ions, so that div F = 47~ (n, -

n,), for div F, = 0

(7) An equation similar to (6) applies to the concentration n2 of the negative ions, with the contrary sign for the 2nd and 3rd term of the right hand member. We shall first consider the initial recombination which arises in a straight (homogeneous) column in an electropositive gas, formed under an angle ‘p with F,,, at the moment t = 0. The distribution of the ion concentrations around the column-axis, when t = 0, is rotational symmetrical; the concentration of the positive ions in de column-axis we shall call noI, and we introduce a characteristic thickness of the column 2ri. We assume that, when t = 0, the following formula applies: nr for the positive and n2 for the negative column, n1

=

no1

fl

(

;

1)

;

n2

=

no2

f2

(

$

)

in which Y indicates the distance from the column-axis. We shall think of ri and r, as defined in such a way that the specific ionization N complies with the formula N = 7~3 no1 = x+j no2.

(84

There is always a correlation, therefore, between n,ol, no2, yl, r2, and only 3 out of the 4 are independent quantities. Now the recombination process, as a function of place and time, always takes place in a corresponding way, assuming different values for noI, rl, F,,, u1 etc. if cp,fi, and f2 are the same and if the following dimensionless parameters retain their values:

The parameters

p, q and s owe their existence to the equation

of

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the first right-hand term in (6) with the other terms in the second right-hand terms. Thereby s, for instacne, represents the relative influence of the space charge of the ions on the growing through diffusion, of the column. In several cases growing has advanced to the same extent if the time is measured with r:/D, as unit. We can here introduce the relationships (2) and (5) (with E = 1) in (9): p=

F erl ‘---; kT

e2 q = 4x ?zopf f(g + 1) p =-zzT

Here we observe that,

according

nolrf

s = e2

kT

4x/(;+

1)

(lo)

to (3), c2/kT represents a length

312 a.

The significance of the parameters becomes more easily surveyable if we intrdouce (8a), and, in the place of q and s, introduce the following : * e2 e2 q*=4dVf(g+1)m; s*=NhT=4rrf(~+l)forcolumns (11) The initial recombination process, therefore, leacls to the incidence of a lack of saturation K, which would be a function of ‘p, f,, f2, r2fr1, p, q*, s* and g. $5. The above result is stillextremely complicated as well as difficult to verify experimentally. It may, however, be considerably simplified by applying it to the following, practically possible investigation. Imagine an ionization vessel and a radiation source in a fixed position. The vessel is always filled with the same gas, but we are able to change the pressure. The positions are arranged in such a way, that the distribution of energy and direction over the directly ionizing components in the radiation complex is independent of d and T. The reasoning followed in 9 4 applies to each of the ionizing components of the radiation complex. We may assume that, for each component, the fl and f2 in (8) depend so little upon d and T as to be of no influence upon K. The relationships between the specific ionizations N of the components will be independent of d and T. We assume the same to be true of the relationships between the thicknesses of the columns 2r, (2ri or 2~~). In this case the lack of saturation c (the mean of K over all values of ‘p) in the total ionization I merely become a function of the parameters 9, q, s and g. For this we

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use the form (1 l), fill in for N and r. the average values ?? and To, thus getting : s=N’d; Thus, j? acquires the following

;o=$

(12)

form: N’d , -, T

g for columns

(13)

We now assume the following: (u) We take N’ to be independent of d and T, which, in all probability, corresponds fairly well to reality. (b) We also take ?A,to be independent of d and T. We may expect that if 7; is dependent upon d, anormal recombination will also take place. The manner, in which r. occurs in electropositive gases leads us to assume that T is of no influence upon ?A. (c) We take g to be independent of d and T, in so far as this should have any influence upon K. (d) We confine ourselves to a case in which F = z/d is so low that the influence of s, in (13), or ??‘d/T is negligible within the field of f c 1. The meaning of this will be apparent from No. 8. Thus, in the proposed investigation,

K=qs,

!A)

(14)

applies to Z insteacl of (13). We determine now i as a function of d, T and F,‘ in order to control the result, given in (14). On account of the conditions, which are certainly fullfilled, the saturation current I will be independent of T and be known as a function of d. Is the source of the radiation outside the ionizationvessel I N d, as the absorption of the rays in the gas can be neglected. As d is small, I can be measured and the saturation may be calculated. As there is no interaction of the walls of the vessel, I is independent of d. From i we find Kand as we dispose of the 3 variables d, T and F,‘ (from (14), we can control the existence of the functional relations. This becomes still more simple in the region, where F,/dT is so large and fdfT so small that for every ionisationfactor K << 1. From (6) then follows that K N 4 and (15)

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It is only necessary to check i as a function of d and F,,; a possible dependence of Fi and N’,from d and T can be deduced and control is not disturbed by the influence of T on r;, ?’ and g (vide (a), (b) and (c). In the confirmed circumstances of the proposed measurements the validity of (14) and (15) can be tested. Deviations between (14) and (15) and the experimental results, which set in by high densities and low temperatures will be produced by anormal recombination, because the herefore mentioned suppositions are probably fulfilled in the field of the normal recombinations. So far as the limitation (d) is concerned, we may observe that the elementary ionization process (through .electrons or protons) has hardly any connexion with the value of either S or N (vide @ 3 and 8), so that this will also apply to the incidence of a normal recombination. The a normal recombination will give a part of the lack of saturation, which is scarcely dependent of N and F,,. By the proposed measurement we learn the anormal recombination of a certain gas as a function of d and T; also for other conditions of the ionization. 3 6. The corresponding treatment of the recombination, in clusters in an electropositive gas is also a reason for the application of parameters (9). Taking v to be the number of pairs of ions in the cluster, and 2r, as the dimension of the cluster, we get, in the place of (1 1). q* = 4x (4)

/(g + 1) (&)

for clusters

(16)

s*+)(2&L---4xf:g

+ 1)

The experiment proposed in No. 5 and the simplifcations duced in No. 5 produce, in the place of (I 3), for clusters.

intro-

(13~)

The reasoning corresponds completely with the foregoing. and the suppositions put forward in 3 5, sub a, b, c and d (we cannot here have any .influence upon d, but according to 5 8 this is always adequately complied with) lead, in the present case, to the same result (14) and (15). The distinction between cluster recombination and column recombination consists only in the different influence

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of i$., and can be ascertained (determined) only if 76 is dependent upon d and T. In reality, pure cluster recombination occurs only rarely, since, during this process, the clusters expand, untill, in the end, the clusters of one and the same track become diffused among themselves with the result that cluster recombination changes into column recombination. $7. Ion constants do not figure in the results obtained above (vide 10, 11, 13, 14 and 15). If the ions possess mobility spectra, which, taken relatively, are not dependent upon d and T, this would not in any way influence the validity of the result given in (14) and (15). In column recombination, the influence,on K of the nature of the radiation is expressed by the distribution of energy (which determines the specific ionization N) and direction amongst the ionizing components. For each component, K is determined by the value of ‘p and N. Since neither fi nor fi, from (8), will depend to any large extent upon N, it will be easy to read, from (13), the influence of N upon K, if, for N’d, we substitute N. Within the field K < 1, and according to (1 S), K - N. When the columns consist of sufficiently long, straight lines, only the component of F,,, at right angles upon the columns, F,, sin ‘p, will have any influence upon K. In cluster recombination, K will be independent of the distribution of both energy and direction, when the clusters are, on an average, not directed and when the division of their size is not dependent upon N; this condition is sufficiently well complied, except in the case of extremely soft radiation. When column recombination and cluster recombination occur in a mixed form it is not possible to ascertain the extent of the influence of the distribution of energy and direction upon K. According to No. 5 (cf. (13)), the influence of the nature of the gas upon K is expressed by the values of N’, i;;, F;/Ti, f and g, and of the functions f, and f2 from (8), in column recombination, if we assume that the subdivisions (distributions) of specific ionization and direction amongst the ionizing components correspond to each other. In this, both fl and fz will depend only little upon the nature of the gas, whilst in all cases g = 1, and F$/?;, remain valid. The influence of the gas upon K may approximately (according to (14) and (15)) be accounted for by fd/T and F,JdT, to be multiplied by the proportions

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of N and of r1 in both gases. A corresponding cluster recombination.

reasoning

applies to

5 8. We shall now try to ascertain under what conditions we may expect the occurrence of anormal recombination in electropositive gases, on the basis of existing experimental data. According to K 1 e m p e r e r 4) and J a f f C “) the thickness of ion columns producing u-rays in air under a pressure of 1 atmosphere and at room temperature, is 3,6. lOA cm. In Cz, and even in very much polluted H,, this thickness is about the same. This result, therefore, applies both to positieve and to negative ions in electropositive gases (cf. No. 9). It may also be applied to clusters; these contain, according to W i 1 s o n E), B r o d e ‘) and B e e k m a n *) on an average from 2 to 34 pairs of ions, which are but little dependent upon the specific ionization and the nature of the gas. Under room temperature, the most probable distance, in air, etc. between the ions in a cluster I, will therefore, be equal to a = 4. 10Pb cm (vide (3) in No. 3), under a pressure of nearly 100 atm. We may already expect some degree of anormal recombination considerably below 100 atm. In lower temperatures anormal recombination occurs also when the density of the gas is proportionately lower. In the case of a-rays in air of 1 atm. and room temperature, we may expect on an average, a primary specific ionization S = 1O4clusters/cm, whilst for hard p-rays this would be about 20 to 30 clusters/ cm. It follows from the cluster dimension given that, we get column recombination in the case of a-rays and in the case of hard p-rays cluster recombination which, however, may change into column recombination. We shall now try to ascertain the significance of the limitation (d) in 3 5, which applies to the results (14) and (I 5). In the case of rays with N = 3.1 O4 pairs of ions/cm (vide (12)) the parameter becomes s = 1 at room temperature and 8 atm. pressure (vide (11)). In that case the field of pressure in which the space charge of the ions influences the diffusion process, still penetrates into the field of pressure in which f < 1. In the case of p-rays producing cluster recombination v = 37 and ?,, = 2. 10J cm. s* = 1, under about 200 atm. and at room temperature. For column recombination with dN’ = 100 pairs of ions/cm of hard P-rays, we shall, of course, find a much higher pressure still.

IONIZATION

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207

5 9. We have, so far, considered those electropositive gases in which the axes, or centres, of the positive and negative columns of clusters coincide at the beginning of the recombination. In polluted electronegative gases, however, the negative ions arise from electrons which, prior to their being attached to a molecule are driven along for some distance by the electric field, so that the positive and negative ioncolumns or clusters already lie apart from each other at the beginning of the ionrecombination. During the attachment phase, however, (6) is also valid for the positive ions, as well as a corresponding equation for the electrons, so that this phase, too, is ruled by the parameters $, 4, s, and g from (9) and by the other quantities and functions enumerated there, as applied to positive ions and electrons. If,. during this attachment phase, recombination between electrons and positive ions should takeplace, the problem would become very complicated indeed. In the conditions with which we are concerned here, this recombination may probably be ignored. Electrons, however, are so much more mobile than ions that the latter hardly leave their place during the attachment phase. We need not, therefore, take the parameters q and g into account, during the attachment phase, and the other parameters are not different in respect either to the positive ions and electrons, or to the negative ions. The entire process, therefore, may be treated in the same way as we did in $5 5 and 6 in regard to electropositive gases. The distribution of concentration in the negative columns or clusters

(vide (8)), however, now depends not only upon the spatial distribution of the elementary ionization processes, but also upon the energy distribution of the electrons shooting off upon the diffusion process of the electrons, and upon their attachment process. The distributive function will always show the same characteristics ; but the negative column- or clusterradius will depend to a much greater extent upon T than in the electropositive gases, whilst it is not possible to foresee the peculiar nature of this process. This radius will also depend upon d if, during the diffusion process, the space charge of the ions begins to play its part. In electronegative gases, therefore both (13) and (13~) are also valid ; but the simplifying supposition b in 3 5 is inadmissable, whilst $/Fi will also depend upon d and T. Since, however, FG > F;, Y, and therefore l;;/?l will play only a

208

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subsidiary part. In the case K < 1 it is possible to study the influence of F, and d upon K, with the aid of (15), in the same way as in electropositive gases, when it will be posible to ascertain an influence of d upon r,. Since in electronegative gases the lack of saturation is so much slighter than in electropositive gases we are much more in the K < 1. Further the influence of T upon K provides us witl?a means of distinguishing, experimentally, cluster recombination in an electronegative gas, column recombination in an electronegative gas, and cluster- or column recombination in an electropositive gas. In an electronegative gas there is a marked difference between the rays of the negative and the positive columns or clusters; within the field of validity of (15), the incident function K(F,/dT) will be different there from what it is in an electropositive gas, other things being equal. On comparing electronegative gases the same reasoning applies that we gave in 5 7, accepted that f$$ << 1 and 7, is, therefore, insignificant. 9 10. To test the result obtained above experimentally according to the method discussed in § 5, extremely accurate measurements of i, as a function of d, F,‘ and T are necessary; especially in the field K < 1. The accuracy required, will necessitate a special measuring technique for i. We hope before long to start investigations in this direction. Also for arbitrary cases of ionization in a certain gas, the result of the proposed measurements, together with the formulae in (13), (14) and (15) that follow from our considerations will make it possible to estimate in good approximation the lack of saturation. Received

August

27th,

July

1945.

1943.

REFERENCES I) 2) 3) 4) 5) 6) 7) 8)

J. G. G. 0. G. C. R. W.

S. J Lo K J T. B. J.

Towns e n d, Electricity in Gases, Oxford 1915. a f f 8, Phys. Rev. 88, 968, 1940; 58, 652, 1941. e b, Fundamental Processes of Electrical Discharge in Gases, 1 e m p e r e r, Z. f. Phys. 48,225, 1927. a f f 8, Ann. d. Phys. 42,303, 1913; Phys. 2. 30,977, 1929. R. W i 1 s o n, Proc. Roy. Sot. London, A 104, 192, 1923. B r o d e, Rev. Mod. Phys. 1’1, 222, 1939: B e e k m a n, Hand. XXVIIIe Nat. en Gen. Congres Utrecht

N. Y.,

1941,

1939.

p. 103.