The Scattering o f Slow Alpha Particles hy Helium.

Transcription

1 380 The Scattering o f Slow Alpha Particles hy Helium. By P. M. S. B lackett, M.A., and F. C. Champion, B.A., Hutchinson Research Student, St. John s College, Cambridge. (Communicated by Sir Ernest Rutherford, P.R.S. Received November 3, 1930.) [P late 6.] 1. Gordon* has proved that the scattering of particles by an inverse square field is the same on the wave mechanics as on the classical theory. Mottf has however shown, from consideration of the symmetry of the wave functions, that the scattering is quite different from the classical when the scattering and scattered particles are identical. The scattering of alpha particles by helium is typical of the case of the collision of identical particles that have no spin and obey the Einstein-Bose statistics. For this case, the expression given by Mott for the number of alpha particles scattered between angles 0, and 0 + d%,in going a distance dr in a gas containing n ato where N db dr = {cosec4 0 + sec cosec2 0 sec2 0 cos u) c?0 (1) m2v u = - log cot 0, (2) 137 v and N0 is the total number of incident particles, v, m and e, their initial velocity, mass and charge and c the velocity of light. On classical mechanics the number of alpha particles scattered through the angle 0 is given by the first term in the bracket, while the number of helium atoms projected at this angle is given by the second term. The last term, which arises from the interference in co-ordinate space of the waves representing these two streams of particles, is an oscillating function of 0 and v, and so gives rise to an interference pattern which varies in scale with the velocity. The intensity of the scattering at 0 = 45 is double the classical for all values of the velocity. The oscillations of intensity with varying 0 are more marked and closer together, the lower the velocity. In fig. 1, curves (a), (b) and (c), is shown the variation of scattered intensity with 0 for velocities of 108, 2 X 108 and 8 X 108 cm. per second, corresponding to ranges in air at * Gordon, Z. Physik, vol. 48, p. 180 (1928). t Mott, Proc. Roy. Soc., vol. 126, p. 259 (1930).

2 Scattering o f Slow Alpha Particles. 381 N.T.P. of 0-4, 1*0 and 7*7 mm. The dotted line shows the classical intensity. In the first case there are three minima, in the next, one, and in the last, none Angle of scattering 8 F ig. 1. The relative amplitude of the oscillations decreases rapidly as 0 is reduced. It is interesting to note that for small velocities the angular width of these fringes is given approximately by 0 X/d, where X is the De Broglie wavelength of the alpha particle with its initial velocity, and d is half the closest distance of approach. The angular width of the fringes is therefore just that of the pattern due to the diffraction of a wave of wave-length X by an aperture or screen of dimensions d. As the velocity is reduced X increases as 1/v but d increases as l/v2 leading to the contraction in the width of the fringes. Mott s theory has been tested by Chadwick* using the scintillation method. The number of alpha particles scattered by helium atoms at an average angle of 45 were determined as a function of their velocity. The earlier work of Rutherford*)* and Chadwick had shown that the scattering of fast particles at 45 was greatly in excess of the classical value. This was attributed to the structure of the particles ; that is the assumption was made that the forces * Chadwick, 4Proc. Roy. Soc., A, vol. 128, p. 114 (1930). f Rutherford and Chadwick, 4Phil. Mag., vol. 4, p. 605 (1927).

3 382 P. M. S. Blackett and F. C. Champion. between them when at the small distances reached in a fast collision, were no longer those of the inverse square. This law of force must, however, be valid at sufficiently large distances. Thus, as the velocity of the particles is reduced, the ratio of the observed scattering to that calculated classically for the law of the inverse square should-tend to unity, while according to Mott s theory it should tend to 2. Chadwick in fact found that this ratio did approach the value 2 at the lowest velocity (8 5 X 108 cm. per second) used. Slower particles could not be utilised owing to the difficulty of counting the scintillations. The calculation of the absolute magnitude of the scattered intensity was made by comparison with the scattering by argon, which was assumed, on the basis of some earlier measurements of Rutherford and Chadwick, to scatter classically. By using a Wilson chamber the scattering can, however, be followed down to a velocity of 108 cm. per second, corresponding to a range of mm. in air at N.T.P., that is to velocities eight smaller than is possible with the scintillation method. At such low velocities the maxima and minima predicted by Mott s theory are very marked, so that a very definite test of the theory is possible. The further advantage that the absolute values of the scattered intensity can be directly determined is offset by the large chance variation of the intensity due to the small number of measured collisions. For these slow particles the closest distance of approach is so large that there is no possibility of any deviation from the inverse square law complicating the scattering. The actual apsidal distance for a particle of \ mm. range which makes a 40 collision in about 5 X 10~n cm. 2. The Experimental Method. Photographs of about 50,000 alpha rays from polonium were made using an automatic Wilson chamber,* containing a mixture of helium and oxygen,, and measurements were made of the forks due to the collision of alpha particles with helium atoms. From these measurements were calculated the lengths R# and R^ of the two arms of the fork and the angles 0 and </>between them and the stem of the track. For an elastic collision between two identical particles 0 -[-< = tc/2, and Ve/v# tan 0, (3) where ve and v$ are the velocities of the two particles after the collision. The test of the first relation, together with that for the coplanarity of the fork, serves to identify a collision as an elastic one with a helium atom. But when * Blackett, 4Proc. Roy. Soc.,5A, vol. 123, p. 613 (1929).

4 Blackett and Champion. Proc. Roy. Soc., A, 130, PI. Collisions of Alpha Particles with Helium Atoms. 1he photographs show the tracks 3*7 as large as they would be in air at N.T.P. (Facing p- 383.)

5 Scattering o f Slow Alpha Particles. 383 (One arm is very short, its direction cannot be determined with sufficient accuracy to make this test feasible. In this case use was made of the known relation between the range and velocity of the particles (see below) to calculate Vejv<t>from the ranges, and to compare it with the ratio as calculated from the observed value of 0. This method allows collisions with helium atoms to be distinguished from those with oxygen atoms, even when </> is not measurable. In every case the initial velocity of the particle is deduced from the relation v2 = ve2 + v ^ and the corresponding initial range R determined. Since the two particles colliding are identical, it is in principle impossible to determine whether the collision corresponds to a deflection of 0 or </>. If the convention is adopted of calling the smaller of the two angles 0, then no values of 0 greater than 45 occur. The result of the measurements is the pair of values of 0 and R for each of the measured helium collisions. The values of the latter are all reduced to the equivalent ranges in air at 15 C. and 760 mm. Since collisions of sufficiently small angle are not observable, it is necessary to have a criterion to govern the decision to measure a fork. The one adopted is that all forks are counted if the length of the shorter arm is equal or greater than some definite value ; actually the criterion used was R^ > 0* 20 mm. in air at N.T.P. Further no tracks w^ere counted for which the initial range R, was less than 0*5 mm. or greater than 8*0 mm. To calculate the expected number of scattered particles, it is necessary to know accurately how the velocity of a particle varies along the last few millimetres of its track. This can be determined up to about 4 mm. from the end by using Briggs results for air,* corrected for the change of relative stopping power of the helium-oxygen mixture relative with air. The data for this are available from the measurements of Gurneyf and Bates.* To extend the curve up to the extreme end of the track, use is made of a method described in a former paper. Measurements of 0, R0 and R^ are made for a fork for which R^ > 4 mm. and R^ < 4 mm. (photographs Nos. 3 and 4). From the value of R^, the velocity ve is obtained using the known relation between the two quantities, and then v(f) is calculated from (3), thus giving, with the observed value of R^, a new point on the range velocity curve. By this means an accurate determination of the range velocity relation right up to the end of the range has been made. From these new results, which will shortly be described in detail, * * Briggs, Proc. Roy. Soc., A, vol. 114, p. 341 (1927). t Gurney, Proc. Roy. Soc., A, vol. 107, p. 340 (1925). t Bates, 4Proc. Roy. Soc., A, vol. 106, p. 622 (1924). Blackett, 4Proc. Roy. Soc., A, vol. 103, p. 62 (1923). VOL. CXXX. A. 2 C

6 384 P. M. S. Blackett and F. C. Champion. the following table gives the results relevant to the present discussion. probable error of these velocity determinations is a few per cent. Table I. The Range in air at 760 mm., 15 C. in - millimetres Velocity in centimetres per second X 10" The Results. For each of the 428 measured forks found among the 56,820 photographed tracks, a dot, corresponding to the observed values of 0 and R, was placed on a diagram with these quantities as co-ordinates (fig. 2). The density of dots is then a measure of the scattered intensity.* Velocity of alpha particles, cms. per sec. x R Range in mms. in air at 760rnms., 15 C. F ig. 2. * Note. The diagram should contain a dot for all those and only those forks which satisfy the conditions, 0*5 < R < 8 mm., R^, >0*2 mm. It will, however, be noticed that some dots fall below the line = 0 2 mm. This is due to the fact that though the criterion that R^ 0 2 mm. was used to decide whether a fork should be counted, the actual dot was plotted at the measured value of 0 and R. If the measurement of the three quantities 0, R and R<j>were exact, a fork for which R^, > 0 2, would, when plotted as the point (0, R), lie above the line R^ = 0-2 mm. But as the measurements are not exact some of such points will lie below the line. It is legitimate to count these points as belonging to the nearest cells above the line, and these dots have in fact been included in

7 Scattering o f Slow Alpha Particles. 385 It is immediately evident that the observed distribution is incompatible with classical scattering, which predicts a steady decrease in the number of collisions with increasing 6. On the other hand the distribution is seen at once to be in complete qualitative agreement with Mott s theory, which predicts a maximum of double the classical for 0 = 45, and a minimum along the line (shown dotted), defined by u tz, that is, cos u = 1. As R increases from mm. the deep minimum of nearly zero intensity at about 0 = 40, moves to smaller values of 0, and becomes shallower. The obvious general agreement of the experimental results with the theory is very striking. The second minimum cannot be observed as it exists only for ranges less than 1 mm. and also lies very close to the limit of observation given by the line R^ = 0 2 mm. The apparent slight minimum about the point R = 1*2, 0 20 is probably to be attributed to chance variations. A quantitative comparison with the theory is obtained by dividing the diagram into cells and comparing the observed and calculated numbers of dots in each cell. The boundaries of the cells were chosen in the following way. The range of R from 0-5 to 8*0 mm. was divided more or less arbitrarily by the ordinates at 1 5 and 3 0 mm., to give suitably sized cells, containing a convenient number of dots. The limits were thus 0*5, 1-5, 3*0 and 8 0 mm. The limiting values of 0 were determined by putting cos u = 0, that is, the horizontal divisions between the cells were chosen as the lines along which the quantum theory scattering is equal to the classical. These lines are shown full on the diagram. Between 0 = 45 and the line given by u = tz/2, the theoretical intensity is everywhere greater than the classical, since here cos u is positive ; between the latter line and that given by u = 3tc/2, it is everywhere less, since here cos u is negative ; between the last and the line defined by R^ = 0-2 mm. which marks the adopted lower limit of the diagram, it is partly greater, and partly smaller, but the calculated intensity variations are too small to be detected with so few observations. The expected number in any cell is with the integration limits given by the cell boundary. the numbers given in Table II. It would be certainly theoretically preferable to use instead the criterion that the point corresponding to the measured values of 0 and R should lie above the curve R</, =0-2 mm. Though the former criterion was adopted owing to its greater convenience, the experience of this work has emphasised the advantage of the latter (see conclusion of section 3). 2 c 2

8 386 P. M. S. Blackett and F. C. Champion. This gives where and N, - K j K 2m; (4) Kx = 4rcN0w0e /m2, (5) K2 /y T 0L/TR0 (6) w = j J [cosec4 h + sec cosec2 0 sec2 0 cos u\ d(), (7) where as before lo7 v u = tl7log cot 0. In these equations n0 is the number of molecules in 1 c.c. at 760 mm. 15 C., R is the range in air under the same conditions, V = X 108, and / fraction of helium in gas mixture, p = final pressure in chamber in length of tracks in chamber, R0 range of alpha particles from polonium in air at 760 mm. 15 C. Using the accepted values of e and mf or the a the following quantities determined experimentally, X0 = number of tracks = 56,820,/= 0-65, L = 10-0 cm., R0 = 3-87 cm., = 0-80, T = 15 C., we get - Kx 3500 so that K2 = 1 34, Ng = 4700 w. Note. The number of helium molecules in 1 c.c. of the gas in the chamber is n=fpt 0n0/T, and the equivalent range dr in air at N.T.P. is d.r = R0dr/L, so that ndr = K,re0dR. The integrations to give w have to be carried out graphically, using in the integration with respect to R the experimental relation between v and R given in Table I. The limits 0Xand 02, both of which, in general, depend on R, are taken from the cell boundaries in the diagram. The numbers Nc expected on the classical theory were calculated by putting cos 0 in (7). In this case the first integral is obtained directly. The final results are shown in fig. 3, which is a diagrammatic representation of the cells. The observed numbers are shown in block figures, the calculated numbers on the quantum theory in small figures, and the calculated classical numbers in brackets. Neglecting for the moment the lower three cells, excellent agreement is found between the observed numbers and those calculated on the quantum theory. The actual observed numbers will, of course, be liable to chance

9 Scattering o f Slow Alpha Particles. 387 Range in mms. in air, N.T.P. Fig. 3. variations. If Nr is the observed number of dots in the rth cell of a set of n cells and if Nr is the mean number, over a large number of experiments, then the mean value of the quantity r 1 V (Nf - N l)2 n - 1 Nn taken over a large number of experiments, is unity. If in this expression Nr is given the calculated theoretical value for each cell, a value of F in the neighbourhood of unity will indicate a distribution quite consistent with the calculated average. For the upper six cells, F has the value 1 04 when the average values predicted by the quantum theory are used and a value 15 3 for the classical figures. The observed distribution is therefore a probable one if the quantum theory of the scattering is correct, but a most improbable one if the classical theory is right. If the lower boundary of the lowest row of cells is taken as the line = 0 2 mm., the numbers expected on the quantum and classical theories are nearly the same, but over double the observed numbers. The numbers for the three lower cells are : Table II. N observed Nri Nc It is very improbable that this is to be taken as indicating a real reduction in scattering below that calculated, as the effect of the electronic screening only enters at much smaller angles. It is probably to be attributed to the unexpected and disappointing failure to count more than a fraction of the forks

10 388 Scattering o f Slow Alpha Particles. lying just above the criterion line. It is possible that a large number of such small angle collisions with helium atoms were wrongly attributed to collisions with oxygen atom and so were not counted. It must be remembered that the limiting spur length of 0-2 mm. is only about double the breadth of the tracks so that exact length measurements are difficult, and that it is through the length measurements that the possibility of distinguishing between small angle helium and oxygen collisions depends. The procedure adopted was to raise arbitrarily the lower limit of the lower right-hand two cells from the line R^ = 0 2 to the line R,;, = 0-3 and the lower limit of the left-hand lower call to the line joining the points R = 0 5, 0 = 25 to R = 1 5, 0 = 15. The calculated and observed numbers for the reduced cells so formed are given on the diagram. The agreement with the left-hand cell is good, but the discrepancy for the others still marked, so that it must be assumed that many collisions above this new line were still being missed. Taking the nine cells as a whole the agreement obtained with Mott s theory is very satisfactory. Since the scattered intensity is approximately proportional to ljvi, the numerical agreement obtained gives strong confirmation of the accuracy of the velocity determination results given in Table I. Summary. A Wilson chamber was used to investigate the scattering of slow alpha particles in helium. The results are in excellent agreement with Mott s theory. We are indebted to the Department of Scientific and Industrial Research for a grant to one of us (F. C. C.) and for a grant to meet some of the cost of the apparatus. Description of Photographs. 1 No P R, mm The ranges are in millimetres of air at N.T.P. Only one of the two photographs of each track is reproduced All the photographs reproduced are of tracks whose planes are roughly at right angles to the axis of the camera, so that the apparent angles do not differ very much from the real angles. It will be noticed that the ends of many of the tracks (e.g., Nos. 10 and 11) are sharply curved, so that an exact determination of the angles becomes difficult. The origin of this curvature is under investigation.