VI. Second Paper on Absorption Phenomena

VI. SECOND PAPER ON ABSORPTION PHENOMENA ON THE DECREASE OF VELOCITY OF SWIFTLY MOVING ELECTRIFIED PARTICLES IN PASSING THROUGH MATTER

(Phil. Mag. 30 (1915) 581*)

*

[See Introduction to Part I, sect. 5.1

ON THE DECREASE OF VELOCITY OF SWIFTLY MOVING ELECTRIFIED PARTICLES I N PASSING

THROUGH MATTER.

RP

5. B O H R . DR. PIIIL, ( O P F N F I A G E N , P.T. RE41)FR IN M A T l I E \ f 4 1 I C A I b'HYSIC& U S I V E R S I T Y OF .\IAIlCIlE871 R

Froin the PHILOSOPHICAL R~AGIAZINIF,vol. xxx. Octo6er lUl5.

On the Decrease of Velocity of Szu;Stly Noz'ing Electrified Particles in passing through Jlatter. B y N. BOHR,Dv. Pl~il.Copenh~~qen j p . t. Reader in Matlieniatical Physics, University of Afanchester *. 1HE object of the present pnper is to continue some calculations on the decrease of' velocity of a and ,B rays published by the writer in a previous paper in this magazine?. This pnper was concerned only with the mean value of the rate of decrenso of velocity of the swiftly moving particles, but from a 'closer comparison with tEe measurements it appears neceEsury, especially for p rays, t o consider the probability distribution of the loss of velocity suffered by the single particles. This problem has been discussed briefly by K. Herzfeld $, but on assumptions as to the mechanism of decrease of velocity essentially cliff erent

1

* Communicated by Sir Ernest Rutherford, F.R.S. t

Phil. Map. xxv. p. 10 (1913). (This pRper will be referred t o as I.)

1 Phya. Zeztschr. 1912, p. 547.

582

Dr. K. Uolir

O)L

the Becrease qj'

from those usrd in the following*. Another question wliich will be considered inore fully in tlie present paper is the effect of tlie velocity of /3 rays being coinparable wit11 t h e velocity of light. These calculations a r c contained in the first three sections. In the two next sect,ions the theory is compared with the ineasureinents. I t will be shown that the al)proxiinate agreement obtained in tlie former paper is improved by tlip c h e r tlieoretical dircussion, :IS well as I)y usiiig tlic recent inore accnriite nieasurciiieirts. Section C; cotititins some conaider:ttions on tile ioiiiz:ttioii produced by a and B rays. A theory for this phenomeuon has been given by S i r J. J . Thoaisont.

6 1. The average value oj'tlie vate ?j' decwaae of relocity. F o r the sake of c1e:irness it is desirable to give :I brief suniinnry of the calcn1:ttions in the fornier inper. Refercnces to the previous literature on the subject will 1)c found in that paper. Following S i r Ernest Rutherford, we shall assume that tho atoin consists of a central nucleus carrying a positive cliarge and surrounded by a cluster of electrons kept together by t h e attractive forces from the nucleus. The n u c l l i s is the seat of practically the entire mass of the atom and has dimensions exceedingly sninll comparcd with the dimensions of the surrounding cluster of electrons. If an tl or @ particle passes through a sheet of matter i t \ d l penetrate through the atoms, and in colliding with the electrons and the nuclei i t will suffer deflexions from its origin:il path and lose part of its original kinetic energy. The deflexions will give rise to the scattering of the rays, and the second effect will produce the decrease in their velocity. The relatiye parts played by the nuclei and tho electrons in these two jihenomenn are very different. On account of the intense field around the nuclei the main part of the scattering will he d u e to collisions of t,he tl or /3 particles with thein ; but on account of the great inass of the nuclei the total kinetic energy lost in such collisions will be negligibly siiiall compared with that lost in collisions with the electrons. I n calculating the decrease of velocity we shall therefore consider only the effect of' the latter collisions.

* Note added in proof. I have only now had an opportunity of seeing recent interesting paper by L. Flamni (Sitzuiiqgsber. d. K. A h ? . d. U ' i ~ s . Wieit, Mat.-itat. h7.cxxiii. 11a, 1914), who has discuesed the problem of the probability variation in the ranges of a particles in air on assuriiptions corresponding with those used in t h e present paper, and has obtained some of t h e results dedriccd i n section 2 (see the note on page 599). t Phil. Illrrg. xxiii. p. 449 (1912). (t

Velocity of Swij'tly hfocing Elect+$ed Particlps.

583

Consider a collision Set\\ een an electrified particle moving with a velocity V and an electron initially a t rest. Let M, E, m, and e be the mass and the electric charge of the particle and the electron respectively, and let the length of the perpendicular froni the electron to the path of the particle before the collision be p . .?f the electron is free, the kinetic energy Q given to the electron during the collision can simply be shown to be

where Clon3ider nest a n a or ,8 particle penetrating t h o u g h a sheet of some substance of thickness Az, and let the number of atoms in unit volume be N, each atom containing 1 ) electrons. The mean value of the number of collisions in which p has a value between p and p + dp is given by

.

dA=2rrNnAxpdp. . . . . (3) If we now could neglect the effect of tlie interatomic forces on the electrons, the average value of the loss of kinetic energy of the swiftly moving particle in penetrating through the sheet of matter would conseqnently be

where the integration is to be performed over all the values for p , from p=O to p = m . The value of this integral, however, is infinite. W e therefore see that in order to obtain agreenient with experiments it is necessary to take the effect of the interatomic forces into consideration. Let us assume, as in the electron theory of dispersion, that the elwtrons norinally are kept in positions of stahle equilibrium and, if slightly displaced, they will execube vibrations around these positions with a frequency v characteristic for the different electrons. I n estimating the effect of the iuteratomic forces it is convenient to introduce tlie conception of the " time of collision," i. e. a time interval of the same order of magnitude as that which the a or ,8 particle will take in travelling through a distance of length 1). If this time interval is very short compared with the time of vibration of the electron, tho interatomic forces will not have time to act before the a or ,8 particle hiis escaped again from

584

Dr. N. Bohr on the Decrease

OJ

the atom, and the energy transferred to the electron will therefore be very nearly the same as if the electron were free. If, on the other hand, the tinie of collision is long compared with the time of vibration, the electron will behave almost US if it were rigidly bound,and the energy transferred will be exceedingly small. The effect of the interritomic forces is therefore equivalent to the introduction of an upper limit for in the integral (4),of the same order of magnitude $he rigorous consideration of the general case would ? i J k e complicated inatheinatical calculations, iind would hardly be adequate in view of our very scanty knowledge as to the mechanism of the forces which keep the electrons in their positions in the atom. Eonever, i t is possible over a considerable range of experimentnl application to introduce great simplifications and to obtain results which to a high degree of approximation are indepeiident of special assumptions as to the action of the interatomic forces. The calculation of the total loss of energy suffered by the u or ,8 particle is very much simplified if we :mume that, for all collisions in which the interatomic forces hare an appreciable influence on the transfer OF energy, tlie displacement of the electron during the collision is sinall compared with p a s well as with the maximum displacement from which it will return to its original position. It can be simply shown that the displacement of the electron during the collision if it were free would be of the same order of magnitude as the above quantity a. The first assumption is therefore equivalent to the condition that V / v is great compared with a. The second assumption is equivalent to the condition that the value for Q which we obtain by putting p = V / v in (1) is small compared with the energy W necessary to remove the electron from the atom. Under these conditions we get by a simple calculation, the detail of which was given in the former paper, that the effective upper limit yv for p in the integral (4) is equal to

where k=1*123. Introducing this, we get for the integral in (4),performing the integration from p=O to p=p, and neglecting a2 in comparison with pv2, log

("r) = log(rE--). a

kV3Mm -7rv e ( M + m )

Velocity qf’ Swiftly illoving Electri~edParticled.

585

From (4)we now get, noticing that v has different values vlv2 , .. Y,, for the dieerent .electrons in the atom,

I n the above we have assumed, as in the ordinary theory of dispersion, that the electrons in the atoms normally are at rest. On the theory of the nucleus atom i t seeins, however, necessary to nssume that, normally the eleclrons rotate in closed orbits round the central nucleus. I n this case it is ti further condition for the validity of the above calculations that the velocity of rotation of the electrons in their orbits is sniall compared with the velocity of the a or /3 p,rrticle and that the dimensions of the orbits :ire sm:ill compared with V / V . I n a previous paper the writer? has attempted to apply the quantum theory of radiation to the theory of‘the nucleus atom. It was pointed out that there appears to be strong evidence for the assumption that for every electron in the atom the energy W will be of the same order of magnitude as hv, where 11 is Planck’s constant. On this assumption it WLLS deduced that in an atom containing n eiectrons the highest characteristic frequeiicy of an electron will be of the same order of magnitude as 2r2e4na v=2; 113

the corresponding values for the \ elocity of rotation, for the diameter of the orbit, and for W will he of the same order of magnitude as

respectively. From these expressions it will be seen that the conditions underljing the :ibove calculation will be the better Eatisfied the smaller the number n of the electrons in the atom. Introducing the numerical values for e, nt, and h, it can be shown that all the coiiditions will be fulfilled, in case of CL particles (v=2.109, E=2e, M=104ux) if n< 10, and in case of B particles (V=2.1010, E=e, M - i n ) if ?E < 100. Now according to Rutherford’s theory the number of electrons in the atom is npproximately equnl to half the atomic weight in terms of the atomic weight of hydrogen as unity. If, therefore, the main assumptions as to the * I. 1.1, 19. t Phil. Nag.. xxvi. p. 476 (1913).

c631

586 1)r. N. Bohr on the Decrease of mec1i:rnism of transfer of energy from the a or p particle to the electrons are correct, we should expect that tlie tbrniula ( 5 ) will hold for absorption of u rays in the lightest elements, , rays illso for the ab5orption in the heiivier elements. and for 9 I n case of ,k? rays it must, however, be remembered that the formula (1) is deduced under the assumption that V is small compared with tho velocity of light. W e shall return to this question iii Section 3, when we have considered the probability variation in the loss of energy suffered by the single particles.

4 2.

l'h probability distribution of the losses qf energy suj'ered by the single a o r partic1e.r. The questions to be discussed in this section are iiitimntely connected with the probability of tho presence of a given number o f particles at a given moment in a small limited part of a l u r p space, in which B large number of the particles are distributed a t random. This problem has been investigated by M. v. Smoluchowslri *, who has shown that the probability for the presence of n particles is given by

where Q is the basis for the natural logarithm and w is the mean value of the number of particles to be expected in the part of the space under consideration. If w is very large this probability distribution is to a high degree of approximation represented by the formula

where s is defined by n = w ( l + s), :ind W(s)tEs denotes the probability that s has a value between s and s+ds. I n the paper cited K. Herzfeld uses the formula (7) in calculating the probability distribution of the distance R which nn u particle of a given initial velocity will penetrate through a gas before it is stopped. Herzfeld inalres the simple assumption that a certain number of collisions with the as molecules is necessary to stop the particle, and he takes t iis number A to be equal to tho total number of ions formed by the particle in the gas. Now the number of collisions suffered by an a particle in penetrating a given distance through the gas is the same as tho number of molecules * Bcltzmann-Festschrift, 1904, p. GPB; see also H. Bnteman, Phil.

P

Mag. xxi. p. 746 (1911).

c641

Velocity of Swijtly Moring Electr9ed Particles.

587 present in a tubular space round the path of the particle. The probability distribution of the number of collisions can therefore be obtained from the above formulae, if for o we introduce the mean value of the number of collisions. Since A is supposed to be very great the variation in the ranges R of the single particle will be very small. The probability that R has a value between R,(l+s) and Ro(l+s+ds), where R, is the mean value oE the ranges, will therefore, on Herzfeld’s assnmption, be simply given by (7) if we put o = A . On the present theory the calculations cannot be performed quite so simply. The total number of collisions is not supposed to be sharply limited, but il is supposed that the amount of energy lost by the a or p particle in collisions with the electrons will depend on the distance of the electron from the path of the particle, and will decrease continuously for an increase of this distance. I n order to apply considerations siinilar to Berzfeld’s, it is therefore necessary to divide the collisions up into groups in such a way that the amount of energy lost by the particles will be very nearly equal for all the collisions inside each group. Consider an a or t3 particle penetrating through a thin sheet of some substance of thickness Act!,and let us divide the number of collisions of the particle with the electrons into a number of groups in such a way that the distance p has a value between p , and p,.+l for the collision in the rth group. Let u s now for the present assume that it is possible in this way to divide the collisions into groups so that the number in each group is large at the same time as the difference between any two values for the energy Q lost by a collision in the same group is small. Let the value for Q corresponding to the rth group be Q,. and let the mean value of the number of collisions in this group be A,., and the actual number of collisions in this group suffered by the given a or 6 particle be A,(1 t s,.). The total energy lost by the particle in passing through the sheet in question is then given by AT= 2 QTAC(l+sr). From this we get, denoting the me:in value of AT by AoT, AT- AoT = Z Q,.A+r. Since the A’s are large numbers, we get from (7) for the probability that s, has a \slue between Pr and sy+dsr, -

W(s,)ds,=2/glr

-+ATSIP

dSr.

588

Dr.

N. Bohr on the Decreare oj'

Now similarly denoting the probability that AT has a value between 4 T and AT+dT by W(T)dT, we get by help of a fundamental theorem in the theory of probability,

W(AI!)dT= (27rPA\c)-t

e

- (AT- A~T)* 2PAx dT,.

.

(8)

where On the above assumptions this can simply be written

PAZ= J QVA. Introducing in this expression the values for Q and d A given by (1) and (3), and integrating for every kind of electron from p=O to p = p v we get

Assuming, 8s in the former section, that p v is large cornpared with a, we get, neglecting the last term under the 2 and introducing in the first the value of a from (2),

It will be noticed that this expression is very simple. It depends only on the t h l number of electrons in unit volume, but neither on the velocity of the BT or @ particle nor on the interatomic forces. From (8) and (9) we can simply deduce the probability distribution of tho thickness of the lttyers of matter through which particles of given initial velocity will penetrate before they have lost all their energy. Putting AT=AoT(l+s), we get for the probability that s has a value between s and (8

+as),

W(s)ds=2/~a-iu"ds,

,

. . .

(10)

where

4

being the nieiln v d u e of

AT --

.

Ax If we now suppose that the straggling of the rays is s m d l -this assumption is already indirectly involved in the assumptions used in the deduction of (8)-the formula (10)

589 will express also the probability that a particle iu order to lose the energy AoT will penetrate through a layer of thickness between Ax=Aox(l +s) and A x + dx=AOx(l+s+rEs), where Aox=AoT/+. I n order to find the probnbility W (R)dR, that a particle iii order to lose all its energy will penetrate through a layer of thickness betweeu R and R+dR, let us now divide the interval froin 0 to T in a great numher of small steps AIT, AzT and let us for the rth step denote the quantities corresponding to Ax, 11, 4, and s by Arz, u,., +,., and 8,. The distniice through which a given particle will penetrate is equal to Velocity of Swiftly Moviny Electrijied Particles.

...

R=XA,.z= 2a.T (l+s,).

6

From this we get, denoting the inem value of the ranges of the particles by Ro, R-&= Z APT -S~

9,.

I n exactly the same manner as that used in obtaining (8) we now get where or siinply

U=PJ:'(J-) (27:

-3

dT,

. . . .

(13)

where the differential coefficient stands for the mean value AT of -. Ax The equations (8) : l i d (9) ant1 consequently also (12) and (13) are deduced under the aammption that the collisions suffered by the swiftly moving particle i n penetrating c1 thin sheet can be divided into groups in such a way tliat the variation of Q for each group is small, while a t the sanie time the number of collisions in the group is large. The condition for this is that the quantity h=dA/$ compared with unity. get

is large

Substituting from (1) and (3) we

X=~.rrhTnA.~(p*+u~).

. . .

,

(14)

c671

5 90

Dr. N. Bohr on the Decrease

OJ

W e see that X is equal to the average number of electrons inside a cylinder of radius Jp2+aa'. Since X decreases for decreasing p , we shall only have to consider its value for p=O. Substituting for a we get

xo= 7re2E2(hf+n ~ ) ~ N n A . r : ~

M2m2V4

If we consider a gas a t ordinary temperature and pressure and introduce the numerical values for e, m, E, M, and N, we obtain both for cc and /3 rays approximately 1037

nAm

v4'

T-his expression varies very rapidly with V, and gives quite different results for a and for ,B particles. For a rays from radium-C: we have V = 1.9 log, this gives X0=1'7 nAm. Now the range of a rays from radium C in hydrogen and helium is about 30 em., and according to Rutherford's theory, the number n of electrons in a molecule of these gases is equal to 2. W e therefore see that Xo will be large compared with unity, provided tlie sheet of matter be not exceedin ly thin compared with the range. F o r other gases Xo wi 1 be even greater, since the product of the number of electrons i n the molecule and the range of rays is greater than for hydrogen and helium. I n case of a rays we may therefore expect that the formula deduced above should give a close approximation. I n order to get un idea of the order of magnitude of the variation to be expected in the loss of energy suffered by a n a particle, consider for instance a beam of a rays penetrating a sheet of hydrogen gas 5 cm. thick. Using the experimental values for the constants, we get from (11)u=3.103 approximately. Introducing this in (10) we see that the probability variation is very small, Thus about half the particles will suffer a loss of energy which differs less than 1per cent. from the mean value, and less than 1per cent. of the particles will suffer rl. loss which differs inore than 5 per cent. I n section 4 wo shall return t o this question and compare the formula (12) with the measurements. For ,B rays of velocity about 2 . lolo, we get for a sheet of aluminium 0.01 gr. per cm.2-a thickness corresponding to that used in the experinients discussed in section 5X0-1.6. Since this is very small compared with unity, it is clear that the assumptions used in deducing the forinuls (8) and (12) are in no way satisfied. Still, it

.

.

f

591

Velocity o t Swgtly Jloving Elect?ijed Puvticles.

appears that it is possible from the calculations to draw some conclusions of importance for the comparison of the theory with the measurements. Consider a p particle passing through a sheet of matter, and let us for a moment assume that ng collision occurs for which is smaller than :L certain vhlue 7 . Let the value for p determined from (14) by putting h.=7 be pr. If T is not small compared with unity the probability distribution of the loss of energ,y will with considerable approximation be given by (8), if in the expression for P the integral is performed from p=pr instead of from p=O. According to the above p r will be great cowpared with a, and we get instead of the expression (9) for P 1 i7rPe4E4N2n2A.v P I = -T -- rn=v4 . (15)

. . .

.

Introducing this in (11) we find for a dieet of aluminium 0.01 gr. per cma2for u approximately ~ t r = 2 5 0 ~ .If T is not small compared with unity, we therefore see .that we obtain a probability distribution of the loss of energy which is oE the same character as that for a rays. The mean value for the loss of energy for the collisions in question is simply obtained from the formula ( 5 ) in the former section by replacing a by p r . This gives

I n the applications the logarithmic term in tliis formula will be large and A r T will depend very little upon the exact value of 7 . Thus for an aluininium sheet ArT will vary only 4 per cent., if T varies from 1 to 2. Let us now consider the probability distribution of tlie loss of energy due to the collisions for which p is smaller than p i . Since p r is large compared with a, it follows from ( l 4 j that the average number of these collisions is very nearly equal to T . If now T is a small number, e. g. ~ = 1it, is evident that the probability distributioii of the loss of energy due to the collisions vill be of a type quite different from that considered above. I n the first place, there is a certain probability that, there will be no loss of energy at all; from (6) we get that this probability is equal to c r . Next, if Qr is the value given by (1) if we put p=pr, no loss of energy greater than zero and smaller than Qr is possible. At Q r the probability enrve suddenly rises and falls off for increasing values of Q approsiiiiately as Q - 2 . For the

c691

592

Dr. N. Bohr on tlw Decrense qf

aluminium sheet considered above we have approximately A,T/Qr= 167. From these considerations it will appear that the probability distribution of the loss of energy suffered by a /3 particle of given initial velocity in penetrating through a thin sheet of matter will show a sharp maximum a t a value very close to &T, if ~ = 1and , fall rapidly off on both Eiides. The value for the decrease of energy measured in the exeriments is evidently this maximum, and not the mean value !or AT given by the formula ( 5 ) , such as was supposed in my former paper. The considerable difference between the two values is due to a very small number of very violent collisions left out in deducing the formula (16) but included in ( 5 ) . Putting ~ = and 1 introducing for p y and p s , we get from (16)

I n section 5 we shall consider the question of the loss of energy suffered by a beam of B rays when penetrating through a sheet of matter of greater thickness.

$ 3. Ffect of the velocity q f f l particles being convarable with the velocity of liyalit. The calculations in the former sections are based on the formula (1)for the energy transferred to an electron by a collision with an a or B particle. I n tho deduction of tliis formula it is assumed that the velocity V is small compared with the velocity of light c. This condition is not fulfilled in case of high speed fl particles. If V is of the same order of magnitude as c, the calculation of the amount of energy transferred by a collision involves complicated considerations for the general caso. The problem, however, with which we are concerned is very much simplified by the circumstance considered in the former section, that the value for the loss of energy of fl particles, measured in the experiments, will depend only on collisions in which the energy transferred is very sniall compared with the total energy of the /3 parti&, i. e. collisions in which LL is small compared with p . Considering such collisions and calculating the force exerted 011 the electron by the fl particle, we can neglect the displacement of the electron during the collision as u ell as its reaction on the @ particle. W e need, therefore, only consider the way in which this force is influenced by the velocity of the /3 particle itself.

Velocity of Swytly Moving Electrijed Particles. 593 I n the electron theory it is shown that the electric force, exerted on an electron at rest by n.particle of charge E and uniform velocity V = & will be directed along the radius vector from the particle to the electron and given by*

1-B2 F = -eTi: r2 (1-p2 sin20 ) Q ’ where r is the distance apart and o the angle between the radius vector and the path of the partible. Let the shortest ?r

distance from the path to the electron be p , and let o = 2 at the time t = O . We have then sin and ra=(Vt)’+p2. For the components of the force perpendicular and parallel to the path of the swiftly moving particle we now get

xt~

F ~ =r

and respectively.

Introducing for

T,

and

and putting (l-,kl2)-i=y,

yVteE F2=((yVt)2+p2)i’

W e see from these expressions that the force at any moment is equal to that calculated on simple electrostatics, if we everywhere replace the velocity V of the swiftly moving particle by yV, and, in calculating the component perpendicular to the path, replace the charge E of the particle by YE, while leaving it unaltered in calculating the component parallel to the path. I n the calculation of the correction due toZhe hi h speed of the 3 , rays we shall, therefore, have to consider t e effects of the two components separately. If the electron is free it will be simply seen that the velocity of the electron, after a collision in which a is small compared with p , will be very nearly perpendicular to the path of the p part.icle. I n calculating the energy transferred i n this c:ise we need therefore consider only the component of the force perpendicular to the path. If V is small cornpared with c we get from (l), neglecting u compared with p,

a

If‘in this expression we introdnce yV for V and y E for E,

*

See, for instance, 0. W. Richardson, ‘The Electron Theory of Matter,’ p. 249, Cambridge 1914.

Pliil. Mug.S. 6. V O ~30. . KO. 178. Oct. 1915.

2Q

Dr. N. Bohr

594

011

the Decrease of

we see that it isunaltered. If the electrons were free there would thus be no correction to introduce in the calculation due to the effect of the velocity of the /3 particle being of the same order as c. If, however, we take the erect of the interatomic forces into account, the problem is a little more complicated. I n this case it is necessary to introduce a.correction in the expression for pv. I n addition the effect of the interatomic forces will involve L: certain transfer of energy due to the component of the force parallel to the path of the p particle ; this is due to a sort of resonance effect which comes into play when the "time of collision " is of the sanie order of magnitude as the time of vihration of the electrons. I n the former paper it was shown that the Contribution to AT due to the component parallel to the path is given by* he2E2NnAz Z= ntv2

*

From (17) it therefore follows that the contribution to ATI, due to the component perpendicular to the path of the /3 particle, is given by

2 (1 (

~ T ~ ~ E * N A X k2V2Nn y = A,T -Z = =08 1nV~

4TV2

,z)

-

If we now in the expression for Y replace V and E by yV and YE, and in the expression for Z replace V by yV but leave E unaltered, we get, by adding the two expressions together rind substituting for 7,the following corrected formula for A,T :

[log(k2V2NnAs 47d )- Iog(l-:)-

AlT= %Te2E2NAx!

* I. p. 17.

:].

The expression deduced in this paper wtis 4xe2E2P\'nAx Z= mvY L,

where

L formed art ot R complicated expression, used in determining p v and evaluated numerical calculation. The value of L, however, can be simply obtained by noticing that f ' ( x )- 1-f(.l.) - f ( 5 ) = 0 .

%y

This gives

Now f ( O ) = l andf'(O)=f(co)=.f'(w )=O; consequently L s i .

c721

(18)

Velocity of Swiftly Mouifig Electrified Particles.

595

It will be seen t,hat tho correction is very small unless V is very near to the velocity of light, since in other cases the

two last terms will approximately cancel each other out.

5 4. Comparison with measui*emeizts on a rays. In the former paper it was shown that the formula ( 5 ) in section 1 gives values which are in closr. agreement with the meamreinents on absorption of a rays for the light elements hydrogen and helium, if we assume that the atoms of these elements contuin 1 and 2 electrons respectively, and if for the characteristic frequencies we introduce the frequencies determined by experiments on dispersion. It was also shown tliat an approximate agreement with the ineasurenients of the absorption in heavier elements could be .obtained by assuming that these elements, in addition to a f e w electrons of optical frequencies, contain a number of electrons more rigidly bound and of frequencies of the same order of magnitude as those determined in esperiinents on.characteristic Riintgen rays ; the values deduced for tho number of electrons were in approximate agreement with those calculated on Sir E. Rutherford's theory of scattering of a rays. I n this section we shall therefore only c o l d e r tho new evidence obtained by 1.,I ter wore accurate measurements. Since the velocity of a particles is small compared with the velocity of light, we have T = & M V . From ( 5 ) we therefore get

where

Thi3 expression depends on two quantities characteristic for t h e different substances, i. e the number of electrons in the molecule it, and the mean value of the logarithm of the cha1 racteristic frequencies of the electrons -2log v. Tho latter n

quautity determines the characteristic differences i n the " velocity curve," i. e. the curve connecting corresponding points in a (3,V) diagram. I n the former paper formula (19) was compared with values for dV/d,z deduced from the measlirements. Since the quantity~directlyobserved is the itis simpler value of V corresponding to different values of a!2., 2Q2

596

Dr. N. Rohr on-tlte Decreose

first to integrate formula (14).

of

This gives

where

A table for the logarithm integral in (20) is given by

*.

Glaisher Considering a gas a t 15’ and 760 mm. presswe we have Ne=1*224 , 10’0. Putting e=4m78.10-10, E=2e, e =5*31 . lo1’, and E/M = 1,448. 1014,we get K1= 1.131. m

and K,= -21.80. I n iiiost measurements rays from radium C are used. This corresponds to Vo= 1,922. l o g t. Assuming that the hydrogen atom contains one electron, we get for the hydrogeii molecule n= 2. If we further assume that the characteristic frequency of both electrons in the hydrogen molecule is equal to the frequency determined by experiments on dispersion in hydrogen, we get $

vl=u2=3%!.

1015 and

1

-2log v=35.78.

n

Using these values and the above values for To,K1and Kp, we get 10gz0=8*75. Introducing this in formula (20) we get for the distance travelled in hydrogen gas by a rays from radium C: before their velocity has decreased to halt’ of its original val~ie,x,=24.0 cm. The first column of the table below contains values for corresponding to different values for V/Vo. No accurate measurements on the velocity curve in hydrogen have been made. Such measurements would form a very desirable test of the theory since the assumptions underlying the calculations may be expected to be closely fulfilled in case of this gas. T. C.Taylor 0 has recently determined the range of a rays from radium C in hydrogen. H e found 30.9 cm. at 15’ and 760 cm. Using the theoretical value 01=24*o cm., we should expect from the iable that the range would be close to 27 cm. This is not far from the range observed. At present it seems difficult to decide whether the small deviation may be ascribed to. experimental errors in the constants involved.

* Phil. Trans. Roy. SOC.CIX.p. 337 (18701.

-f E. Rutherford and H. Robinson, Phil. M R ~ xsviii. . p. 562 (1914).

1 C. & N.Cuthbertaon, Proc. Roy. SUC.A. lxxxviii. p. 166 (1909). 8 Phil, Mag. xxvi. p. 402 (1913).

- --1.o 0.9 0.8 0.7

0 0.338

0x5 0.4 0.3

1~ooo

0.6

0592 W780 0.911

0.2

1.087 1.104

ST;

i

o

l

0.561 0.551

'

0.%6

1 1

111.

IV.

~

v.

n 0,300 0.539 0.730

1.05.5

According to Rutherford's theory the helium atom contains two electrons. Since helium is a nion:~tomic gas this gives n = 2 as for hydrogen. Experiments on dispersion in helium give v = 5 * 9 2 W5. Ilitroduciug these values for 11 and Y in (20) we get values for .c which are a little greater th:in those for hydrogen. The theoretical ratio between the ran es in helium and in hydrogen is 1-09. The measurements of%. P. Adilitis' *, diecussed in the former paper, wei;e in disagreement with the calculation that the range in helium WHS shorter than in hydrogen ; the ratio between the ranges observed being only 0.87.. Taylor's recent measurements, however, give for this ratio 1-05,in close agreement with the theoretical value. For air Mnrsden and Taylor? havs recently made an accurate determination of the velocity curve. They found that a rays froin radium C will travel through 5.95 cm. of air at 15' and 760 mm. pressure before their velocity ia reduced to iVo. If we assume that the nitrogen ritom contains 7 electrons and the oxygen atom 8 electrons, we get for the sir molecule in mean n=14*4. Introducing this in the formula (20) and putting 21=5'!!5 for V=$Vo, we find

.

log q = 5 * 3 i and

-x log u=38*32.

1 11

The values for. .v/.q cor-

responding to this value for logzo w e given in the colunin 11. of the table. Not so many values are given as for liydrogen, since the fulfilment of the conditions mentioned in section 1, on account of the higher frequencies, clriiins greater values for V for air than for hydrogen. Column IV. contains the values for z/zl observed by Marsden and Taylor. The agreement bet\\een the calculated and the observed values is very close. At the same time it will be seen that the values in column IV. differ considerably from those in

* Physical Review, xxiv. p. 113 (1907). t

Prm Roy. SOC.A. lxxxviii. p. 443 (1913).

598

Dr.

iS.Bohr on the Decrease of

columns I. and 111. The values in these columns are c:ilculated by putting logro=8.75 (see above) and 1opz0=4.44 (see below) respectively. If instead of log z0=5% we had used one of the latter values, we should instead of 14.4 h:ive to put n=8.1 or n=22*5 respectively, in order to obtain the observed value for zl. I t will therefore be seen that t h e considerable difference between the values in the columns I., II., and 111. offers a method of determining n, even in caPes

:2 log Y is not known beforehand. n Marsden and Taylor could not observe ang or particle with a velocity smaller tlian 0.42 Vo. When t e velocity had decreased to this value the particles apparently disappenrrd suddenly. Tlris peculiar effect is in striking contrast to \I lint should be expected on the theory. I t nppears, however, that it may possibly be explained by a statistiid effect due to EL small want of iioiiiogeneity in the or-ray pencils used. In t h e firat part of the velocity curve the slope varies gradually, and a possible sinall want of homogencit will have only a very smell effect on the mean value of t e velocity. But near the end of the range the slope of the curve is very steep, and if the pencil for some reason is n o t quite homogeneous, the effect will be that, as we recede from tlie source, more and more of the particles will 90 to speak suddenly fall out of the beam. I n this Hay the velocity will not start to decrease rapidly until almost all the particles are stopped ; but then the beam will contain so few particles that the final descent may be very difficult to detect. The values in column V. correspond to Marsden and Taylor's results for the velocity curve of rays from radium C' in aluniinium. The value for xl corresponding to V = i V 0 measured in gr. per ~ 1 1 1 . ~The d u e for Kl was 9.64. is 9.81. lP6. is niensured in gr. per in aluminium if If for aluminiuiii v e nssu~ne n = 1 3 , and in (20) introduce x1=9.64. lo3 for V=aVo, we get logz0=4'4P and

where

3:

-z log v=39*02.

1 n

As nientioned above this corresponds to

the values in column 111. It will be seen that the values in coluiiin V are much closer to those in 111. t h a n t o those in I. and IT., but the agreement is not nearly so good as for air. This iiiay partly be due to the difficulty in obtaining homogeneous aluminium sheets, but it may also be due to the facat that the assumptions underlying the calculations cannot be expected to be strictly fulfilled for all the electrons in the aluminium atom (see page 586). For elements of

c761

Velocity of Swiftly M o z h g Electrified Particles. 599 higher atomic weight, the assumptions used in the calculations are satisfied to a still sinaller degree than for alurninium, and accurate agreement with the measurements cannot be obtained, although the theory offers an approximate explnnation of the way in which the stopping power of an element and the shape of the velocity curve vary with increasing atomic weight. I n section 2 we considered the probability variation in the ranges of the sin le particles of an initially homogeneous beam of a r:iys. flenoting the inean d u e of the ranges I)y Ro,w e get from (12) and (13) for the probability that the range R has a vnlue between R o ' l +s) nnd Ro(l+ s + d s )

where

This expression is much simplified if w e use anapproximate formula for tET/tl,v. Putting z=CTP, we get

Introducitig this in (22) we get * (23)

* Note added in proqf. For 7*=4, this exprehsion ie equivalent t o the expression deduced by L. Flamm (loo. cit. forniulrt (25)) for the variation in the ranges of u particles due to collieions with the electrons. ThiR author has considered also the collisiolls with the central nuclei m d concluded that, although the eRect of these collisions on the mean value of the rate of decrease of velocity of the CL yrticles is very small compared with that due to the collisions with t e electrons, their effect on the variation in the ranges is not negligible but will be given by an expression of the type (21) for a value of p of the same order of magnitude RS that given by (23). Froin considerations analogous with those applied in section 2 in the case of p rays it appewR, however, that the collisions between the a particles and the noclel will produce R variation in the ranges o f a type different fi*otn(91). In these collisions onlgveryfelv of the particles suffer a considerable diminution of their ranges, while the greater part of the particles suffer diminutions which are very sniall even coiiipsred with the average ditLrexices in the ranges produced by the collisions with the electrons. I t seems therefore that the efect of the collisions with the nuclei iuay he ueglected in R colnprtrison with the measurements.

600

Dr. h'. Bohr on the Decrease of Geiger has shown that we obtain a close approximation to

the velocity curve in air if we put

3

For hydrogen we 5 obtain a similar approximation by putting r = T . The exact d 3r- 2 value for T, however, is of only little importance since f is very nearly constant for values of r between 1 and 2. Putting T=+MV2 and introducing the theoretical expressions ( 5 ) and (9) €or dT/& and P, we get T=

2'

For z rays from radium C we now get, using the same values for log zo as above, for hydrogen and air p =0.86 and p= 1.16. respectively. For a rays from polonium, assuming the initial velocity of the rays to be equal to 0.82 that for radium C, we get €or hydrogen and air p=0*91. and p= 1.20. respectively. Geiger+ and later 'I'aylor 7 have niade experiments in order to measure the distribution of the ranges in hydrogen and air of a rays from polonium and radium C. They counted the number of scintilliitions on a zinc-sulphide screen kept a t a fixed distance from the radioactive source and varied the pressure of the gas between screen and source. The results do not agree with those to be expected from tho theory. The straggling observed was several times larger than that to be expected and did not show the symmetry claimed by the formula (21). These results, if correct, would constitute a serious difficulty for the theory ; they seem, however, inconsistent with the results of some recent experiments by F. FriedmannS. The latter experiments were made in order to test Herzfeld's theory, which also gave a straggling much smaller than that observed by Geiger and Taylor. Friedmann found a distribution of the ranges in air of a rays from polonium which coincides approximately Ae seen, this value with that given by (21),if p = 1.0. is even a little smaller than that calculated from the tlieory. Further experiments on this point would be very desirable.

.

4 5.

Comparison with the measurements

OIL @

vays. The experimental evidence as to the rate of loss of energy by B pirticles in penetrating through matter has until

* Proc. Roy. SOC.Ixxxiii. p. 505 (1910). t

Phil. Mag. xxri. p. 402 (1913). $ Sitzb. d . K,Akad. d. Wiss. Wieti, Mat.-nnt. KI. cxxii. IIa, p. 1169 (1913).

c781

Velocity oj‘ Swqtly illoving Electrfled Particles.

601

recently been very limited on account of’ the great difficulties i n the measurements. Much light, however, is brought upon this question by the study of’ the hoinogetieous groups of /3 rays emitted froih certain radioiictive substances. 0. v. Baeyer. observed that the lines in the ‘ ‘ p ray spectrum,” produced when the rays are bent i n a niagnetic field, were shifted to the side of smaller velocities when the r:idioactive swrce was covered by a thin metal foil. The question has recently been more closely investigated by Danysz 7, who extended the investigation to a great number of the groups of homogeneous rays emitted frotn radium B and C. The Grst two coluitins in the table below headed by Hp:uid A(H ) contain the values given by Danysz for the product of t e magnetic force H and the radius of curvature p lor a number of groups of homogeneous /3 rays, and the corresponding values for the alteration in this product observed when the rays have passed through an aluniiniuin sheet of 0.01gr. per The limit of error in the values for A(Hp) is stated to be about 15 per cent.

L

-

HP. 1391 1681 1748 1918 1983

124 95 90 66 61

2017 .. ~

2224 2275 2939 3227 4789 5830

1

66 57 48 37 48 39 32

w635

0704

0.718 0.750 0,760 0.770 0795 0.802

08G7

0885 (“33 0.960

31 33 33 28 27 26 28 25 24 33 32 28

The values for Hp are connected with the velocity of the /3 pnrticles through the equation

deduced on the espression for the momentum of an electron which follows from the theory of relativity. Denoting V/c by ,8, we get c2m HP---/ql-p)-+.

* Phys. Zeitschr. xiii. p. 485 (1911).

t

Journ. de Pltysiyuc, hi.

11.

. . . .

9.19 (1913).

(24)

602

Dr. N. Bohr on the 1)Pcrease qf

This gives

c2m

A(Hp) = - (l-B')-; A/3. e On the theory of relativity we have further from this we get

T =(.*m( (1-p2)4-1);

AT=c'j)lB(1-P2)-i W e have consequently

A@. .

.

. . .

(25)

. .

(26)

AT=epA(Hp). . . Froni.(18) we thus have, putting E=e a i d V/c=p,

. . . .

(27) Except for very high velocities the variation, of the last factor will be very small, and we shall therefore, accordin? to the theory, expect A(Hp) to be approximately proportiona to p-3. The third column of the table contains the values for 8, and the fourth column the values for ,@A(Hp). It will be seen that the values in this column are constant within the limit of experimental errors. 1 2 log v=39.0 Putting n=13 and using the value

-

7E

calculated from experiments on a rays, we get from (27) €or an aluminium sheet 0.01 gr. per cm.2 /3

= 0.6

p3A(Hp)=

40

0.7 41

0.8 42

0.9 44

0.95 46

Considering the great difficulty in the experiments and the great ditfkrence in mass and velocity for a and p rays, it appears that the approximate agreement mag be considered as satisfactory. The mean values for n(Hp), calculated from the formula (5) in section 1, would be about 1.3 times larger for the slowest velocities and would increase far more rapir!ly with the velocity of the /3 rays, Measurements of the decrease of velocity of p rays in sheets of metals of higher atomic weight are more difficult than with aluminium on account of the ureater effect of the scattering of mys. Danjsz found that %e rate of decrease of velocity was approximately proportional to the weight per cm.2 of the ahsorbing sheet. Since the number of electrons in any substance is approsimately proportional to

Velocity of Swytly Mocing ElectriJetE Particles.

603 the weight, and since the differences in the characteristic frequencies will have a yery much smaller influence for fitst /3 rays than for a mys, results of this kind should be expected on the theory. If we assume that the forinula (18) holds also for the loss of’ energy suffered by /3 rays in penetrating a layer of matter of greater thickness, we obtain for the “runge ” of the /3 particles

wliere I; donotes the last factor in (18) and (27). Considering 2 as const:int, and using tlie :illo~e.formula for AT, we get

Recently R. W. Taider+ Ila5 innde some interesting experiments on tlie absorption of Iioniogeneous B rays. Be measured the vsrintion in the ionization produced by the rays in a shallow ionization clianiber wlren slieets of different thicknesses were introduced in the beam before striking the chanlber. Using aluniiniiuii sheets, he found that the ioiiiention varied very nearly linearly with the tliickness of the sheets, and his cliagrania give a strong indication of the existence of L “ range ” of’tlie /3 porricles. Varder coinpared the ranges observed with the last factor S in the formula (2S), and f‘ouiid that the ratio between the ranges and 5,though nearly independent of the initial velocity of the rays, decreased slowly with this velocity. This should be expected from the above calculations, as 2 will increase slowly with the velocity. Measuring R in gr. per cm,2 Varder found R/S=0*35 for /3=0.8 and R/S=0*30 for /3=0*95. ‘Jhe first factor in the theoretical foriiiuln is equal to 0.42 for /3=0*8 and 0.38 for /3=0.95. W e see thut tlie agreement may be considered as very sntisfwtory. The distribution of the losses of energy, suffered by the individual particles o f a be;uu of initially homogeneous p rnyi in penetrating through :t slieet of iiiatter of considerable thickness, cannot be represented by the fortnula (12) used in the former section, since-see section 2-dready the distribution of the loss of energy suffered in penetrating through a thin sheet differs essentially from that g i w n by formula (a).

* Phil. Mag. sxix. p. 72.i (1915).

604

Dr. N. Bohr on the Decrease of

I n addition, the transverse scattering of the rays due t o deflexions suffered in collisions with the electrons as well as with the positive nuclei must be taken into account. This scattering will cause the nieiin value of the actual. distances travelled by the particles in the niatter to be greater thrin the thickness of the sheet. If, however, we for a moment neglect all collisions in which the particles suffer either abnormally big losses in their energy or big deflexions, we may, as in section 2, espect that the rest of the rays will beh:ive in a similar way to a beam of a rays and that they will show a rsnge of a similar degree of sharpness. Therefore the distribution of the energy of a beam of initially hoiiiogeneous /3 rays emerging froiii a thick la er of matter must, as for a thin sheet, be expected to ex ibit a welldefined peak sharply limited on the side of the greater velocities, but falling more slowly off towards the smaller velocities. The further tbe rays pass rhrough the matter the greater the chance that tho particles will suffer a violent collision, and tbe smal!er will be the number of particles present at the peak oE tho distribution. A simple calculation sliows that by far the greater part of this effect is due to the deflexions suffered in collisions with the positive nuclei. An estimate of the effect of t,hese collisions may be obtained in the followin wa , The orbit o a igh speed /3 particle colliding with R positive nucleus has been discussed by C. G. Darwin E. From his calculations it follows that the angle of deflexion t+ of a /3 p:irticle of velocity V=@e is given by

i

f K

where ne is the charge on the nucleus :tiid p is the length of the perpendicular from the nucleus to the path of the /3 particle before the collision. Let pr be the value of the p correThe probability that a particle will sponding to * = T . pass through a sheet of matter of thickness A x without suffering a collision for which + f > r is equal to 1--WAX, where

Since o A z is small, this probability can be written e--Az,

* Phil. Nag. xxv. p. 201 (1913).

VeZocity of Sic{{tly

Moviit,q

Xlectiaified PaTticles.

605

and the probability that the p particle will penetrate through il. sheet of greater thickness without suffering a single deflexion foi. which + > T is consequantl!- given by W==e-*, where A = [ ~ A . c . Substituting for Ati- from tlie foriiiulu (18) and uiing the same notation as above, we get [ n * ( l -@)AT-

.

~y&:2),,~

*

Considering 2 as a constant we get froin this, using the expression (25) for AT,

where S as above denot,esthe last factor in the expression (28) for the range R. W e hare consequently

where S is approximately proportional to the range of the emergent rays, and K a constant. The formulii (29) gives an estimate of the number of /3 particles left in the peak of the velocity distribution of the emergent rays, and may be compared with the ionization measured in Varder’s experiments. It will be seen that W depends to a very high degree on n, and therefore on tlie atomic weight of the absorbing substance. As mentioned above, C is for these fast rays appyoximately proportional to n, and the exponent in (29) is therefore proportional to 7). It‘ aluininiuiii \I as used i\s absorbing substance Varder found that the ionization was approximately proportional to the range of the emergent rays, while for paper it decreased more slowly, nnd far inore rapidly for silver and platiiiuin. 1 F o r aluininiuiii we have n = 1 3 and - x = l S for /3=O*tr. 72

Putting the exponent i n tlie expression for W equal to 1, w e get in this case 7=0*30 and 9 :ipproxirnately equal to Z O O ; this is an angle of the right order of magnitudo. For paper the exponent in (29) will be halved and for platinum will be more than five times larger than for aluminium, for the same values of T and I n connexion with the calculations in this section, it iiiay be of interest to remark that the approximate agreement ohtainecl between the theory and the measurements seein, to

+.

606

Dr.

N. Bohr

on the Pewease of

give strong support to the espressions used for the momentum and the energy of a hi h speed electron. Let us for a moment suppose that t e ordinnry expressions for the momentum and the energy of slowly moving electrons could be used without alteration. This should not alter the equations (26)and (27), hut the values for V deduced from the values for Hp would be (1-@)-$ tiines greater. Introducing &his in the formula (27) we should have found a value for A(Hp) which for the swiftest rays would be about 30 times s&aller than that observed by Danysz, aud the values in the last column of the table on p. 601 would, instead of being nearly constant, be inore than 20 times sinaller for the slowest rays than for the swiftest. If, on the other hand, we had supposed that the expressions for the momentum were correct, but that the “Ion itudinal” iiiass of the electron was equol to the ‘‘ transversa ” inass, we should hare obtained the same values for V as in the table, but the equations (26) and (27) would have been altered b y a factor (1-B2)-’. I n this case the value calculated for AtHp) for the swiftest rays would ,have been about 15 tiines larger than that observed, and the values in the last column, instead of being nearly constant as observed, should have been expected to be 10 tiines greater for the fastest than for the slowest rays. It thus appears that 1ne:uurements on the decrease of velocity of fl rays in passing through matter m:iy nRord a very effective ineans of testihg the formula for the inoinentum and the energy ot’ a high speed electron.

f

P

€J 6. The ionization produced by a and $ rays.

A theory of the ionization produced by a and B rays in a J. J. Thornson*. I n this theory

gn3 lias been given by Sir

it is assumed that the swiftly moving particles penetrate through the atoms of the gas and suf€er collisions with the electrons contained in them. The number of pairs of ions produced is supposed to be equal to the number of collisions in which the energy transferred from the particle to the electron is greater than a certain energy W necessary to remove the latter from the atom. If we neglect the interatomic forces this number can be simply deduced. By differentiating ( I ) with regard to p and introducing for pdp in (3) we get 2 ~ e ~ E ~ N ucEQ A.t

dA=

mv2

qj-. . . . .

* Phil. Mag. xxiii. p. 449 (1912j.

(30)

Velocity of Swijtly Jloring Electrijed Particles. 607 Denoting by Qo the value for Q obtained by putting p=O in (l), we get, integrating (30) from Q = W to

&=a,

1 Aw= 2?re2EZNnA.f( -mV2

1)

iV-C& ’ *

,

.

(31)

where

If we consider a substance in which t,he different electrons correspond to different values for W, we get instead of (31) simply

Sir J. J. Thoiiison showed that thc formula (31) with close approximation can explain the relative number of ions produced by a and fl rays. If, however, in (31) we introduce the values for \I‘ calculated from the observed ioniziitiou potentials, and the values for the number of electrons in the atoms wllich were found to agree with the calciilutions in section 4, we ohtaiu absolute vdues for Aw which are several times sinaller than the ionization observed. It appears, however, that this disagreeiilent may be explained by considering the secondary ioniz:ition produced by the electrons expelled from the atoins in the direct collisions with the u and ,f3 particles. I n Sir J. J. Thoinson’s paper it is argued that this secondary ionization seems to be very sinall compared with the direct ioniztation, sitlce the tracks of a and /3 particles on C. T. R. Wilson’s photographs show only very few branches. A calculation, Low ever, indicates that the ranges of the great number of the secondary rays able to ionize are so short that they probably would escape observation. The rays in -question will be the electrons expelled with an energy greater than W, and will be due to collisions in which the a or fl particle loses an aniount of energy greater than 2W. The number of such coltisions is given by (31) if IV is replaced by 2W. Let this number be Azw. The total energy lost Iry the particle in the collisions in question is equal to

approximately. The mean value of the energy of the electrons expelled is therefore P=W(2 log (Qo/2W)-1). F o r hydrogen and a rays from radium C this gives approxiniately

608

Dr. N. Boltr

071

tlte Decrease qf

P= 1OW ; corresponding to a velocity of 6 . lo* and a range

of about 1 0 - 4 ~ ~in n .hydrogen a t ordinary pressure. The number of ions produced by the secondary rays cannot be calculated in tlte same simple way us t,he number produced particles, for in tho by the direct collisions with the a or case of the secondary rays we cannot neglect the effect of the interatomic forces. From the considerations in section 1 it is seen that the conditions for the neglect of the interatomic forces is that the value of p corresponding to Q = W i s very small compared with V/v. By help of the expressien (1) for Q and the expressions for W and v on p. 585, it can be siinply shown that this condition is equivalent to the condition that the energy &mV2 of the rays is very great compared with W. This condition is fulfilled for a and fl rays in light gases, but is not fulfilled for rays as slow as the seoondary rays. Recently J. Franck and G. Hertz* have made some very interesting experiments which throw much light on the question of ionization by slowly moving electrons. Experimenting with mercury vapour and helium gas, they found that an electron will rebound froiii the atom without loss of energy if its velocity is less than a certain value. 4 s soon, however, as the velocity is greater than this value the electron will be able to ionize the atom, and it was shown that the probability that ionization will occur at the first impact is considerable. For other gases the results were somewhat different, but in all cases a sharply defined limiting value for the velocity of the ionizing electrons was observed. These experiments indicate that slowly moving electrons are very effective ionizers. We ma*y,therefore, obtain an approximate estimate of the number of ions produced by the secondary rays, if u e siuiply assume that each of these rays will produce s ions if their energy has a v:ilue.between s W and ( s + l ) W . This would give for the total number of ions formed 2re2EZNnAx 1 I=Aw+Aaw+ ... = iuV% 2w * . If Qois very great compared with W this gives approximately

[(&

i)+(

-J; +

This formula applies only to substances for which W has the same value for all the electrons in the atom. For other substances we must take into account that an electron expelled may produce ions, not only in collisions with electrons * Verh. d. Deutsch. Phys. Ges. xvi. p. 467 (1914).

.]

l’elocity of Siviftly U o a i n g Electriped P a d c l e s .

609

corres~ondingto the same value for W, hut also in collisions with other electrons in the atom. Considering, however, the rapid decrease i n the chance of ionizalion with increasing W, we may in a simple wag obtain a n approximate estimate, if we asslime tliat all the ionization produced by tlie secondary rays is due to collisions with electrons corwsponding to the sriiallest value for TV. This vnlue will be the one which is determined in experiments on ionization potentials; let it be denoted by W,. I n tlie same way as above we now get

If Q,, is big compared with all the W’s, we get approximately

On account sf the simplifying aeeumptions used, the forniulm

(34) and (35) can only be expected to indicate an upper limit for the ionization. The minitnuin fall of potential P necessary to produce ionization iii hydrogen, helium, nitrogen, and oxygen was measured by Franck a d Hertz”. They found 11, 20.5, 7.5, and 9 volts respectively.

By help of the relation

Pe W = --, 300

M‘ equal t o 1.75. lo-”, 3.25.10-11, 1.20 lo-’’, and 1-45.l o - ” respectiyelj.. The absolute nuniber of ions produced Ly a rays in air is deteriiiined by H. Geiger t. H e found that every a particle frbm radium C: in passing through 1 cm. of air a t ordinary pressure and teiiiperature proiluced 2.25.10’ pairs of ions. Froin this ~e get, ubing T. S. Taylor’s$ measureinents of the relative ionizations in air, hydrogen, and helium, that the nunnber of pair of ions produced by an a particle from radium C iu passing tlirough 1 cm. of one of the two latter gases is very nearly the same and equal to 4.6 lo3. If now in (31) we introduce the ahove d u e for W for hydrogen and use the same va ues for N, n, e, E, nz, a i d V as io section 4, we get for a rays from radium U we get from this

.

.

* t t

Verb. d. Detctach. Ph!ls. Oea. xv p. 34 (1913). Proc. Hog. SOC.A. luxxii. p. 486 (1009). i’hil, 3hg. xryi. p. 403 (1913).

c871

Dr. N. Buhr 011 t71. Decrease OJ in hydrogen Aw=1*15. lo3. The value given by (34) is 5.9 Aw. The first value is 4 times snialler than tlie iuiiiza-

610

tion obgerved. The latter valiie is of the right order of magnitude, but is a little larger than the experiinent:il value. F o r helium JV is nearly twice ns great as for hydrogen. From (31) and (34) we should therelore espect a Ialue for the ionization only half of that in hydrogen. Taj lor, however, found the same ionimtion in hydrogrii as in heliuni. Since in this caw the value observed is gre:tfer tliaii thnt calculated from t;34)Ftlie disagreeineiit is difficult to exphiii, unless the high value ohserved by Taylor pos4bly iiiay he due to the presence of a siixill nniount of iiiipuriticbs i i i the heliuin used. Thi, wems to be supported hy expei.iirieiits of W. K o s e l * on tlie ioniziition Iiroduced by cathple rays. This autlior found tlirit the ionization in heliuin WAR only half as large as tliat in hydrogen-in agreeinent mitt1 the theory. The cathode rays used liad a velocity of 1.88. lo@, corresponding to a fall of potential of 1000 volts, and the number of ions produced in passing tlirougli 1 cin. of hydrogen a t a presGure of 1 mm. H g was eqwil, to 0.882. This corresponds to G70 pairs of ions at atnioeplieric pressure. Putting V = l * 8 8 . l U 9 and E=c, and usiiig the same value^ for TV, e, m, N, and n as above, we get froin (31) AtV=:300. From (34) we get T = 4 . 5 Aw. If we consider a subst:rnce such as 2ir, which contains a greater number of elpctrons in tlie atonis, 13% do not lrnow the valixe of W for the different electro~s. A suf7ici~iitly close approxiination may, however, be obtaiiied, if in tlie logarithmic term of (35) we put W = 7w, where lz is Planclt’s constant. This gives, it‘ we at the saiiie time introduce the value for Qo from (32),

If now in this formula we introduce the values for n and 1Z log v used in section 4 in cdculating the absorption of

n a rays

.

in air, and put W1=1*25 lo-”, we get I = 3 * 6 . lo4. This is the siime order of magnitude as the value 2.25.10‘ observed by Geiger, but somewhat larger : this would be expected from the nature the calqulation. Tlie value to be Bxpected from the formula (33) cannot be stated accurately on account of the uncertainty as to the magnitude of the W’s,

01

*

Ann. d , Phpik, xxxvii. p, 393 (1912).

V e l o c i 9 qf Sic$ly ;1focing Elect,$ietl Pui ticles.

611

b i i t an eytimate indicates that i t would be 1ecs than a fifth of tlic oI)
which ia siniilar to the variation of AT gireti by the formula 1 (5). Using the same value for 71- 2 log Y as above, we thus get for a rays in air, that the ratio between the values for I, given Iiy (:<(i), for V = l * S . 10° and for V = 1 . 2 109is equal to 1 65. The correspoiidiiig riitio for A T given by (5) is 1.54. Tllis is in ngreemetit with Grigcar's * measurerite~its, utiich gave tliat the ionization produced by :in ?L particle in air, a t any point of the path, was nearly proportional to the 10.s of' energy suffered by tlie particle; both quantities being approximately propoi tioiial to the inverse first power of' t h e velocity. I ' l h e number of ioiys produced hy catliotle rays i n air has been nieasureti by W.Kosselt and .J. L. Glasson$. F o r a velocity of 1 88. l o BKossel found 3.28 pairs of ions per cm. a t I min. pressure. Uiider the same condilions Glasson found 2.01 and 0.99 pairs of ions for t h e velocities 4.08. log a n d ti.12.109 respectively. A t atmospheric pressure this gives for the same three ~elocities2.49. lo3, 1.53 lo9, and 0.75. lo3 pairs of ions respectively ; or 9.0, 14.7, and 30.0 times smaller than Geiger's value for 2 r a j s f'roni radirim C. The v:ilnes calculaled froin (36) for catliocle rays of tlie three velocities in question :ire 7.1, 17.4, and 31.2 times sirialler than the value calculafed for a rays from radiirin C. The calculations in this section cannot be immediately compared with measurements of the ionization produced hy high speed /3 rays, since we have made use of the formula (1) which is valid only if V is small comp:ired with the ~ e l o c i t y of light. I n a nianner analogous to t h e conqitferations in section 3, i t can, however, be simply shown that the correction to be introduced in the foriiiula (36) is very small and will only affect the logarithmic term. F o r liigh speed /3 rays, the variation of this term with the velocity V will further-as in the calculations i n section 5-be very small compared with the variation of the first factor. F r o m (36) we shall therefore expect that t h e ionization produced by these ra,ys will be approximately proportiorial to the inverse square of the kelocity. This is in agreement withW.Rilsoii's$ measurements. ' Proc. Roy. SOC.A. lxxxiii p. 505 (1910). t I S O C . cit. 1 Phil. Map xxii. p. 1347 (1911). 4 Pios. Roy. soc. -1.Ixxxr. p, 240 (1U11).

.

.

Xwn )navy, Accortling. to t h e tlieory' discussed in this paper, the decrease of velocity of a :ind p r:iys i n pwiiug tlii-ough matter dol~entl.;es,5enti:illj on t,he cliitimtrihtic freqiiencies of tlio electron in tlie iltolns, in a siinilar way as the phenomena of refraction and (lispcrsioii. I n a previous paper i t was shown tlint the tlieory leads to results wliioli :ire in close a.greeiiient with experinients on a1,sorption of a rays in Iijdrogcn and helium, i f we assuinc) tliat tlie atoins O F tliese elcincats coiitiiiii'l and 2 elcctroiis respect>ively,and it' the freqnciicirs of thcse e1ec:trons are init equal to t h o f'rchqiieiicies c:ilciilatetl froill espariinents on disliersion. It was also sliown tliat an apl)i.oxiiiinte espliunatioq oE tlie aLso,prion of' rilj,s in lienvier siil)stiinccs c:in lie ol)tainetl, if we :i+siiine that tlie atoiiis of sncli eI(siii(wt4,i n ntltlitioii to a f e w electrons of oplical fiwliieiiciv, coiit:iiii :I niinil)er of elwtrons inore rigi ily Lorunt1 : u i t I , of freqiiencir~s o f tlie wine order of niagnitude as charnctcridic l{iint,gPti lays. rl ,l i e nii.iil)ar of elcctrona deduoctl was in npprosiiii:ite agrceinent nirh tliosc cdciilnted in Sir E . Il~utlit~rforcl's tlicory of scattering of a rays. T l i e ~ cconclusions have h e n verified Ly using tlie later inore :iccornte nieasureinenf:'. I n illy forincr p p r , very few data were :iy:iil:il)le on {lie decrease of velocity of p r a ~ in s triiver,sing matter and the agrceincnt betv-ecn theory and &x1)eriineiitw a s not vcry c1o.e. Tlic agreenient 1)etween tlieory a n d e s p r i i i i r n t is iiiiproved m:itci*i:illy, 1):iitlx Ly using n e w nicasrireineiits and 1)artly 1)y taking the pim1ial)ility wri:tt.ions i n the l o ~ sof energy siifferrtl by the int1ividu:il ,i3 particles into :icBcouiit. 111 this connesion it is pointetl out tliat it :ii)peai*st1i:it nie:ixureuients on tlie deerewe of vclocity of p r a j s :itl'orcl an diective test oE the forniulre for tho enei'gy and nioiiimturii of n high spped electron tl(diicct1 on tlie theory of' rehiivity. I n coniiexion with tlio calculations of the absorption of a nntl f rnys, tlie ionimtion prudncecl by such rays is cQiiaiderocl. I t is shown tlint tlie tlieory of S i r J. J. Thomson gives results in approximnte agreenient with the nieasureilitmts it' the second;iry ionization procluctd hy tlie electron expelled Ly ttio direct iiiipact of t,he a and ,d pays is t:ilien into account.

I wish to express my best th:inl