On the Nature of X-Rays

On the Nature of X-RaysAuthor(s): Jarl A. WasastjernaSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 136, No. 829 (May 2, 1932), pp. 233-242Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/95764 .

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233

On the Nature of X-Rays.

By JARL A. WASASTJERNA, Helsingfors University.

(Communicated by W. L. Bragg, F.R.S.-Received December 8, 1931.)

? 1. Einstein's* statistical-thermodynamic calculations of the random variations of the radiation density constitute the ground upon which the light quantum hypothesis was originally based. According to these calculations such a variation of the radiation energy prevails-superposed above the- fluctuations caused by the interferences calculated according to the classical theory-as if the radiation consisted of mutually independently mobile quanta hv of energy. According to Einstein, Maxwell's theory correctly renders mean time values, which alone have been directly observable, as proved by the complete agreement between theory and experiment in optics; but Maxwell's theory leads to laws respecting the thermic properties of radiation which are incompatible with the entropy-probability relation. After the wave- mechanical theory of de Broglie and Schrodinger had been generally accepted, the idea concerning the statistical character of the wave-field-already presented by Einstein in 1905-has, of course, been taken up again by Born in a more general way.

As is well known, Bohr and Heisenberg have tried to conquer the difficulties to which the radiation theory has led by a radical change in our conception of energy.

According to Bohr every idea which we can form of the radiation phenomenon consists only of a construction which in a more or less convenient way sum- marises our experiences regarding the statistical connection between certlain real elementary processes, separated from each other in regard to space as well as time. We can only state that mutually linked elementary processes take place, whereas the real linking mechanism cannot as a matter of fact be brought into concordance with our conception of space and time. Bohr's conception agrees closely with a way of putting the matter, suggested by Ritz amongst others, with retarded functions, and the same objections may be raised in both cases. Thus, considered philosophically, in Bohr's case all problems connected with the nature of the radiation are eliminated as meaningless but yet, on account of the statistical, irrational linking between the elementary

* A. Einstein, 'Ann. Physik,' vol. 17, p. 132 (1905); vol. 20, p. 199 (1906); ' Phys., Z.,' vol. 10, p. 185 (1909); vol. 10, p. 817 (1909).



234 J. A. Wasastjerna.

processes, considered in Bohr's theory, that theory leads to the same variations -possible to state in principle-as Einstein's light quantum hypothesis.

Considering how important for our entire present mode of physical reasoning is the conception of a propagation of energy with definite velocity, it appears as if it were most practical, in any case provisionally, to sum up the experimental results by assuming localisation of energy as to space and time.* This will give us a clear idea of the position held by the observed phenomena in the entire complex, affected by the Bohr and Heisenberg theory. At the same time this will also, as it were, automatically allow us later on to adopt without difficulty an interpretation compatible with Bohr's standpoint.

Thus we will start by assuming that the energy is actually propagated with definite velocity from point to point in the field of radiation. In other respects the radiation mechanism may be considered as unknown.

? 2. When the intensity of a radioactive cx- or ,8-radiation is measured by the ionisation produced in an absorbing gas, we know that certain probability variations occur, that is to say the Schweidler variations, which are characteristic of the radioactive disintegration processes, and simultaneously exhibit the corpuscular nature of the oc- and 5-radiation. These variations, which are closely related to the density variations, according to Einstein characteristic of black body radiation, have been established by Meyer and Regener,t Kohl-

rausch,j and Geiger,? and they have been carefully investigated by Ernst,II Muszkat and Wertenstein? and by Borman.** The Schweidler variations have been studied by means of various differential methods, the principle of which may be applied likewise when investigating the variations which, according to the light quantum hypothesis, are to be expected in the case of

the ionisation effects caused by X-rays. Similar experiments (for y-rays) were proposed by v. Schweidlertt as early

as 1910 and they have actually been carried out by E. Meyer,fl Laby and

* J. A. Wasastjerna, 'Soc. Sci. fenn. Comm. Phys.-Math.,' vol. 5, No. 19 (1930).

t E. Meyer and E. Regener, 'Verh. denLts. phys. Ges.,' vol. 10, p. 1 (1908); 'Ann. Physik,' vol. 25, p. 757 (1908).

: K. W. F. Kohlrausch, ' SitzBer. Akad. Wiss. Wien,' vol. 115, p. 673 (1906). ? H. Geiger, ' Phil. Mag.,' vol. 15, p. 539 (1908). A. Ernst, 'Ann. Physik,' vol. 48, p. 877 (1915). A. Muszkat and L. Wertenstein, ' J. Phys. Radium,' vol. 2, p. 119 (1921).

** E. Borman, ' SitzBer. Akad. Wiss. Wien,' vol. 127, p. 2347 (1918). tt 'Phys. Z.,' vol. 11, p. 225 and p. 614 (1910). $$ 'Ber. deuts. phys. Ges.,' vol. 32, p. 647 (1910); 'Jahrb. Rad. Elektr.,' vol. 7, p.

279 (1910); 'Phys. Z.,' vol. 11, p. 1022 (1910); 'Phys. Z.,' vol. 13, p. 73 and p. 253 (1912); 'Ann. Physik,' vol. 37, p. 700 (1912).



Nature of X-Rays. 235

Burbidge,* and Burbidge.f These experiments, which have been discussed by Campbellt and Buchwald? did not yield any unambiguous results, amongst other reasons because the random variation of the ionisation effect of one single ray in this case by far exceeded the variation of the number of y-rays passing the diaphragm. Analogous experiments with X-rays (Campbell) were abandoned on account of the great experimental difficulties involved.

? 3. In the laboratory for applied physics at the Helsingfors University detailed investigationslj were carried out during the years 1928-1930 into the variations which are characteristic of the radiation emitted from the anticathode in an X-ray tube. Further, the variations of the intensity of a radiation which had traversed an absorbing layer were investigated, as well as the corresponding variations of a beam formed by interference.

The experimental arrangement was as follows: A Coolidge tubeowas operated by means of a high voltage battery comprising 21,600 cells. The radiation energy within a given solid angle was measured by means of ionisation. The ionisation current was led to a direct current amplifier, worked by means of electron valves, constructed for the purpose. The amplifying factor was 7 75. 106. The amplified current was registered photographically by means of a mirror galvanometer of the Deprez-d'Arsonval type. The film, 50 cm. wide, was carried forward at a speed of 1 cm. per second.

In almost all the experiments a differential method was applied. Two ionisation chambers, filled with a strongly absorbing gas (CH3Br) were subjected to radiation. The results obtained may be summarised as follows:_

The radiation energy absorbed during the time At causes a mean ionisation Q which is subject to random variations AQ. The variations AQ follow the Gauss law of error. The amount of the mean square variation (AQ)2 is, independently of the length of the time interval, given by the equation

(AQ)2/Q - 0 83 X 10-16 Coulombs. (1)

The variations cannot be referred to variations in the intensity of the total radiation nor can they emanate from variations in the position of the focal spot of the cathode rays. Besides, the variations AQ, occurring simultaneously

* T. H. Laby and P. W. Burbidge, 'Nature,' vol. 87, p. 144 (1911); 'Proc. Roy. Soc.,' A, vol. 86, p. 333 (1912)

t P. W. Burbidge, ' Proc. Roy. Soc.,' A, vol. 89, p. 45 (1913). T N. R. Campbell, ' Phys. Z.,' vol. 11, p. 826 (1910); vol. 13, p. 73 (1912). ? E. Buchwald, 'Ann. Physik,' vol. 39, p. 41 (1912). 11 J. A. Wasastjerna, 'Acta. Soc. Sci. fenn. nova Serie,' A, vol. 1, No. 7 (1928), and

vol. 2, No. 1 (1930).




in both ionisation chambers, are quite independent of each other. (The mean square variation is additive.)

If the energy of the radiation absorbed per time unit were constant, these results would either indicate that the ionisation can vary considerably even in the case of constant absorption, or else, that the saturation current during constant ionisation is able to show the variation in question.

As we know, ionisation is a consequence of a secondary 5-radiation. The knowledge we have of the ionising properties of the cc- and 5-radiation as well as of the variations connected with the corpuscular radiation, exclude the possibility of the first-mentioned alternative. The deviation from the mean effect would be of a different kind, and actually of a much smaller order of magnitude than the observed variations AQ.

The second alternative may likewise be excluded. It might, certainly, be possible that the ionisation current should vary during a short interval At even when the ionisation is constant, in consequence of the variations in the course of the ,-particles which certainly occur. If the ionisation is produced in the vicinity of the electrode the ionisation current rises, in the opposite case it falls. This would be a consequence of the limited velocity of the ions. In this case it would necessarily follow that on a variation effect appearing during a given timae interval there would appear an effect of opposite sign during the following interval or intervals. From our investigations it is evident, however, that the mean square variation is characterised by the remarkable property (AQ)2/ At const., which means that the effect observed during a given time interval At in no way determines the nature of the effects which are to be expected during the following time intervals.

The variations observed may, on the other hand, be explained completely by the light quantum hypothesis. According to this hypothesis the mean ionisation Q would be subject to random variations which must obey the Gauss law of errors. According to the theory of probability the magnitude of these variations would, for the actual distribution of spectral energy, be given by the equation

(1/h) . ( (AQ)2/Q) 1- 30 X 1010 Coulomb erg.-L sec.1, (2)

which is independent of the length of the time interval.

The magnitude of an energy quantum is, determined by equations (1) and (2). We then obtain

h - 634 x 10-27 erg. see.




This value of h agrees fairly well with the value of Planck's constant (h = 6 55 x 10-27) theoretically calculated.

Our results may briefly be summarised in the following manner. If the amount of energy E, which during the time At is transmitted by radiation in the frequency range v, v + dv from the focal spot of the cathode in the X-ray tube into the gas present in an ionisation chamber, is divided by the product hv, and if the quotient obtained in this way is denoted by E/hv = n, the variations An appearing, will follow the Gauss law of errors, while the mean square variation (An)2 is given by the variation formula

(An)2 = n,

known from the probability calculus of Poisson. It might appear that the observed variations could be produced by the

absorption mechanism without there necessarily being any variations of the amount of energy passmg the diaphragm per time unit. This is, however, not the case.

We assume that the energy is at every moment localised as to space, and it follows then that through the diaphragm aperture of an ionisation chamber there passes durig each time interval At on an average a given amount of energy which, independently of the structure of the radiation field, may be denoted as no . hv. The mean square variation may be denoted as (Ano)2. (hV)2. The absorption may take place as quanta of the magnitude hv, the absorption process being considered as a chance phenomenon, the probability of which is proportional to the radiation intensity in the neighbourhood of the point considered. Every amount of energy hv which is released by absorption at a point A is definitely abstracted from the primary radiation field, the mean intensity of which has been lowered to a corresponding degree. The. fraction of the incident radiation energy which on an average is absorbed by the gas, may be denoted as p. During the time At an amount of energy pnohv = nhv, will on an average be absorbed by the gas. Its mean square variation may be denoted as (An)2 . (hv)2. It is due partly to the variations of the incident radiation energy (suffix 1) and partly to the variations of the absorbed fraction of the incident radiation energy, consequently to the absorption mechanism (suffix 2). According to Bernouilli's theorem for (An)2 we obtain the equation

(An)2 = [p2 (Ano)2]1 + [nop (1 - P)]2 = [p2 (Ano)2]1 + [n (1-P)]2 (3) If the fraction absorbed is small (p z 0, (1 - p) ~ 1) the equation (3)

changes into Poisson's law / AnA2! -__ [nA- 2, (3A




independently of the value of (\Ano)2. Consequently, if the absorbed fraction is small the variations which are regulated by the mechanism of the absorption process (suffix 2) will dominate. Thus experiments carried out with a low absorption percentage cannot be used as a basis for any considerations concerning the nature of the radiation. This case is encountered in the investigations of E. Meyer, T. H. Laby and P. W. Burbidge on similar variations, brought about by means of radioactive y-radiation.

If, on the contrary, p 1z and (1 - p) 0, the equation (3) changes into the equation (3B)

(An)2 [2 (An 0)2]1 [AnO]12 (3B)

which means that the variations observed in the case of complete absorption are identical with those of the incident radiation energy, and independent of the mechanism of the absorption process. It is perfectly clear that the variations which are due to the discontinuous character of the absorption process disappear in the case of complete absorption, as then every quantity of energy which enters the ionisation chamber will sooner or later be absorbed. Each amount hv can be absorbed at one single point only and not in another as well. Every absorption process which takes place in a given layer of the gas hereby reduces the energy of the radiation field, and thereby also lowers the probability of an absorption in the subsequent layers in such a way that the total variation, due to the mechanism of the absorption process, disappears.

It is, however, evident from our investigations that the variations which correspond to Poisson's law, remain even in the case of complete absorption. As in this case n = no0 and according to the experiments (An)2 = n, it follows from (3B)

(An0)2 _ no. (4)

Thus our experimental results cannot depend on absorption taking place in quanta. As a matter of fact, the experiments mentioned do not throw any light on the mechanism of the absorption process.

When, however, we assume a localisation, as to space and time, of the radiation energy in the radiation field, which involves that the diaphragm aperture of the ionisation chamber during each time interval At is traversed by a certain amount of energy, defined in each single case, then we may conclude that these energy quantities vary according to the probability laws of Gauss and Poisson, in which case a unit of energy s appears which has been found to be identical with a light quantum hv.



Nature of X-Racys. 239

? 4. Let the radiation emitted from the anticathode of an X-ray tube within a given solid angle co traverse an absorbiing layer. This layer, during the time At, will be on the average struck by an amount of energy E. The mean square variation may be denoted as (tXE)2. According to the above-mentioned experimental results (AE)2 may then be calculated according to the formula

(AE)2 (Ano)2. (hV)2 = no(hv)2 = Ehv. Thus, the equation

(AE)2/E -hv (5)

is identical with the variation law of Poisson. The relative mean variation

[AE]/E = /(AE)2/E may be calculated to be

[AE]/E =Vhv/E. (6)

On an average the layer absorbs the fraction p of the incident radiation energy, while on an average the fraction q penetrates the layer. In this case E=- q. E, and its mean square variation (AE')2 may be observed. If the absorption is continuous, so that an identical fraction p is absorbed from each amount of energy, the equation

[AE']/E' = [AE]/E (7)

is valid. If, on the contrary, the absorption is discontinuous, which means that p varies independently of E, we may write

[AE']/E' > [AE]/E.

The following formula applies generally:

(AE')2 - q2 (AE)2 + E2 ( Aq)2 = (1 p)2 (AE)2 + E2 (Ap)2, (8) where (Ap)2 characterises the mechanism of the absorption process. A continuous absorption would give (Aq)2 = (Ap)2 = 0. Consequently, the formula (7) is found to be a special case of formula (8).

Let us now consider the optical phenomena in the strict sense of the word. During the time At the optical system will on an average be hit by the energy amount E. Its mean square variation may be denoted as (AE)2. As an example we may take an interference phenomenon. A beam formed by interference encounters a small surface a, which, during the time At, is on an average -traversed by a fraction g of the radiation E. We are then able to observe




E" - qE and its mean square variation (AEE")2. If the interference is continuous, i.e., in conformity with the classical wave theory, the same equation

[AE"]/E" = [AE]/E (9) is valid here also.

If the phenomenon is discontinuous, we write

[LE"]/E"> [AE]/E, (10) in whlich case agaimn

(AE")2 - q2 (AE)2 + E2 (Aq)2. (11)

Our series of experiments, comprising thousands of At, :have shown that Poisson's law holds good with a great degree of accuracy, both in the case of radiation which has penetrated an absorbing layer, and in the case of a beam produced by interference.

As the variation law of Poisson is valid both before and after the absorption, the values of the mean square variation (AE)2 and (AE')2 satisfy the equation

(AE ')2/E' = (AE)2/E -hv. (12)

According to formula (8) we may write in general

(AE')2 -q2 (AE)2 + E2 (Aq)2. (13)

The absorption mechanism may be characterised by the mean square variation (Ap)2= (Aq)2 derived from these equations (12)-(13)

(Ap)2_ (Aq)2 = pq (hv/E). (14)

Furthermore, as AE and AE' follow Gauss's law, the distribution of Ap - Aq must also obey the same law. The random variations Ap must therefore be assumed to be due to an absorption of some kind of energy elements E/no where no is an unknown though finite number, which process is governed by chance. Thus every incident energy element will either disappear in the absorbing layer (case A), or penetrate the same (case B).

Consequently, one of the cases A, B must necessarily arise so soon as the absorbing layer is encountered by the energy element E/no. The probability p applies to the first case, the probability of q to the second one, so that p + q = 1. This choice between A and B is repeated no times.

Any combination of the cases A, B, comprising no elements and occurring A n' times, B n" times, leads up to a probability product of no factors, of which n' are equal to p and n" equal to q, so that pn' qf" will be the ensuing product.




There are, however, as many combinations of this kind as the permutations given by n0 elements-of which n' are of one kind, and n" of another, i.e.,

no!/n' ! n"

As a probability of pn' qn" applies to every arrangement of this kind,. the total probability that the cases A, B should reoccur in the numbers n' n" is represented by

W (An, Bn") - (no !/n' ! n" !) ptfl q.

The mean value of a magnitude depending on chance is equal to the s'um of its single values multiplied by their corresponding probabilities. Thus, if we write

n nop p nO (p + Ap) nop = noAp, then

0 02 (p)2 (n' - nop)2 (no!/n' n" l) pn" qfl = nopq, noAP

from which

(A p)2-pq (1/no). (15)

If we compare the theoretical formula (15) with the experimentally established formula (14) we find

no E/hv and E/no = hv. (16)

Thus when the absorbing layer is struck by the energy element hv, one of the cases A, B must necessarily occur. The energy element will either be absorbed or transmitted. Absorption is caused by a kind of elementary processes, governed by chance, the primary radiation field in the case of each elementary process of this kind loosing an amount of energy of the magnitude of hv.

As already pointed out above, the mean square variations (lAE")2 and ( AE)2 which occur in interference experiments are likewise governed by Poisson's law, and thus satisfy the equation

(/E")1/E" - (z\E)2/E = hv. (17)

The relation between the energy of the incident, homogeneous radiation and the radiation, which-owing to an interference process-becomes deviated into a given direction is represented by l/q. If we introduce the expression (1 - p) - q the formu-l (13)-(14) still remain valid. The random variations Aq must be assumed to be due to the deviation of direction of each separate energy element E/no in the interference phenomenon being accidental. Every incident quantity of energy E/no wiLl thereby either be deviated in the

VOL. OXXXVI.--A. R



242 Nature of X-Rays.

prescribed direction (case B) or not be so (case A). A probability of q applies in the first case, a probability p in the second. The formulae (15)-(16) are therefore also still valid with respect to interference.

Consequently, an interference phenomenon is composed of elementary processes, governed by chance. In each elementary process of this kind the diredion of propagation of an energy element hv changes in a dcefinite manner.

Summary. If we assume that radiation energy is localised as to space and time, being

propagated from point to point in the radiation field, from which follows that every surface element in the radiation field during each time interval t, t -+ At is traversed by a quantity of energy defined in each case, we are able to conclude, from the experimentally proved variations of the ionisation effects of the radiation-

(1) that the radiation consists of energy elements hv; (2) that the absorption is brouLght about by distinct elementary processes,

whereby the primary radiation field in the case of each elementary process of this kind looses an amount of energy of the magnitude hv;

(3) that the interference phenomena are made up of elementary processes governed by chance. In every elementary process of this kind the direction of propagation of an energy element hv changes in a definite manner.

If, however, with Bohr and Heisenberg, we abandon the claim of a localisation of the energy as to space and time, and instead adopt the view that the energy is in fact not really defined within finite spheres of space and time, but as it were virtually distributed over the entire field of radiation, then the experimental results will imply a quantitative statement of the very variation effects which, according to the above theory, characterise the irrational, non-causal coupling of quantum-processes.



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