Second Series February sgzg Pol. 33,¹. z THE PHYSCAL REVEW QUANTUM MECHANCS AND RADOACTVE DSNTEGRATON' BY R. %. GURNEY AND E. U. CQNDoN ABSTRACT Application of quantum mechanics to a simple model of the

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Second Series February sgzg Pol. 33,¹. z THE PHYSCAL REVEW QUANTUM MECHANCS AND RADOACTVE DSNTEGRATON' BY R. %. GURNEY AND E. U. CQNDoN ABSTRACT Application of quantum mechanics to a simple model of the nucleus gives the phenomenon of radioactive disintegration. The statistical nature of the quantum mechanics gives directly disintegration as a chance phenomenon without any special hypothesis. )i contains a presentation of those features of quantum mechanics which are here used and gives a simple calculation of the disintegration constant. $2 discusses the qualitative application of the model to the nucleus. $3 presents quantitative calculations amounting to a theoretical interpretation of the Geiger-Nuttall relation between the rate of disintegration and the energy of the emitted a-particle. n getting this relation one arrives at the rather remarkable conclusion that the law of force between emitted e-particle and the rest of the nucleus is substantially the same in all the atoms even where the decay rates stand in the ratio 10 . $4 calls attention to the natural way in which the paradoxical results of Rutherford and Chadwick on the scattering of fast O.-particles by uranium receive explanation with the model here used. (5 discusses certain limitations inherent in the methods employed. HE study of radioacitivity itself together with the application of it as a working source of high speed helium nuclei and electrons has played a fundamental role in the development of quantum physics. The scattering experiments of Rutherford and his associates gave the picture of the nuclear atom on which all of the success of modern atomic theory depends. Bohr's formulation of quantum postulates to be applied to such a model was a great step in the extension of knowledge of atomic structure and finally culminated in 1925 in the discovery by Heisenberg and by Schrodinger of a reformulation of mechanical laws which has subsequently proved extremely powerful in handling atomic structure problems. n this development of the last fifteen years little advance has been made on the problem of the structure of the nucleus. t seems, however, that the new quantum mechanics has had sufhcient success to justify the hope that it is competent to carry out an effective attack on the problem. The quantum mechanics has in it just those statistical elements which would seem appropriate to an explanation of the phenomenon ' An account of this work was first published in Nature for September 22, n a number of the Zeitschrift fur Physik (51, 204, 1928) received here two weeks ago there appears a paper by Gamow who has arrived quite independently at the same basic idea as was presented in our letter and which is here treated in detail. Reports of this paper were also given at the Schenectady meeting of the National Academy of Sciences on November 20, 1928 and at the Minneapolis meeting of the American Physical Society on December 1, 128 R. V. GURNEY AND E. U. CONDON of radioactive decay. This is the feature of the general problem with which we are concerned in this paper. We believe that the results provide at last an interpretation of nuclear disintegration which in its fundamental points is very close to the truth although it is necessarily quite incomplete. The outstanding difficulties in the way of a good theoretical treatment of nuclear structure at present are mainly bound up with our lack of understanding of the quantum mechanics of the magnetism of the fundamental particles. This question has been much advanced this year by Dirac's extension of Pauli's theory of the spinning electron' but this remains essentially a theory of the behavior of one electron in an electromagnetic field. Not only is it apparently still unsatisfactory as such but this limitation must necessarily be disposed of in principle before the many body nuclear problem can be approached. And with that done there will remain the inevitable analytical difficulties. Enough is known, however, to teach us that probably the magnetic interaction is not to be handled simply by an alteration of a potential energy function depending solely on the coordinates of the several interacting particles. This tends to detract from the value of arguments based simply on the use of quantum mechanics with the positional coordinates of the nuclear constituents. Nevertheless we shall restrict ourselves to the use of such methods in the discussion of the instability or capacity for spontaneous disintegration of a very much simplified nuclear model. The simplification to be made will consist in supposing that we can discuss the behavior of any one constituent by applying the quantum mechanics to it as a single body moving in a force field due to the rest of the nucleus. The difference between quantum mechanics and classical mechanics which is here made responsible for the disintegration process is easily stated. n classical mechanics the orbit of a particle is entirely confined to those points in space at which its potential energy is less than its total energy. This is not true in quantum mechanics. Classically if a particle be moving in a basin of low potential energy and have not as much total energy as the maximum of potential energy surrounding the basin, it must cerfairlly remain there for all time, unless it acquires the deficiency in energy somehow. But in quantum mechanics most statements of certainty are replaced by statements of probability. And the above statement must now be altered to read it may remain there for a long time but as time goes on the probability that it has escaped, even without change in its total energy, increases toward unity. n $1 of this paper the detailed development of the argument leading to the conclusion of the preceding paragraph is given. n f2 we discuss its qualitative application to the nuclear disintegration problem. f3 is devoted to semi-quantitative estimates of the rates of decay. CQUPLNG QF JUQTtoNs of EQUAL ENERGY Consider a particle of mass p, t is sufficient to consider one degree of freedom; let the coordinate of the particle be x and let the forces be measured by the potential energy function V(x). ~ Dirac, Proc. Roy. Soc. A117, 610; A118, 351 {1928). QUANTUM MECHANCS: RADOACTVE DSNTEGRATON 129 n classical mechanics the equations of motion possess the energy integral V(x) = W (p =momentum) which, for values of x such that W V(x) &0, can only be satisfied by P pure imaginary. Therefore, classically, one had the result that a particle could only be where W V(x) &0. An important consequence of this was that if there were several ranges of x for which W V(x) &0 separated by ranges where W V(x) & 0, then there were several different motions possible with the energy level 8', each of which was wholly confined to one of these separate ranges. Thus in Fig. 1 for the energy level indicated there mould be two distinct types of motion of the same energy 8 ;one is a libration in the range, and the other a libration in the range. These results are modified considerably by the new quantum mechanics. il V the first place, Fq. (1) loses its validity and is replaced by an integral theorem, as Born' has shown, in which there is no longer a definite correlation between simultaneous value of position and momentum as (1) implies. The quantum Fig, 1. mechanical form of (1) is, if lf (x) is Schroedi'nger's wave function f'+ h' dp df tr= ++. +V(*)04)d* (& ) J 8x'p dx dx The lack of a precise correlation has been much emphasized by Heisenberg and by Bohr, 4 and is a general characteristic of quantum mechanics. From the new standpoint, one has to consider the behavior of Schrodinger's equation for the problem d'p Sm-'p + (W V(x))iP =0. (2) dx' h' As is well known, in some problems there are solutions f(w, x) for certain values of 8' which are finite and continuous everywhere. These are the allowed values of quantum theory. For the ip(w, x) which comes out of (2) as a by-product, Born has shown that its square may be satisfactorily interpreted as giving the probability that the particle lies between x and x+dx when it is in the state of energy B. This is really the ground for requiring that ip remain finite. For an energy level, such that P(W, x), does not remain finite as x~+ 00, the probability that it is not at infinity is vanishingly small, and therefore these states do not exist physically. Adopting the probability interpretation of ip(w, x) one has at once the result that there is a finite probability of being outside the range of the classical motion of that energy. ' Born, Zeits. f. Physik 38, 806 (1926). 4 Heisenberg, Zeits. f. Physik 43, 172 (1927); Bohr, Nature, April 14, 1928, W x 130 R. lk GURNEY AND E. U. CONDON A simple case is the lowest state of the harmonic oscillator, which has the energy hv/2. The P(W, x) for this state is e r'&*&'&&' so P=e &*'&' where u is the classical amplitude of motion associated with this energy. The probability of being outside the classical range is therefore a =0.157 or more than 15 percent. When one studies the behavior of f(w, x) from (2) for a V(x) somewhat like the one in Fig. 1, he finds that, if the W is one for which P is finite everywhere, then P approaches zero very rapidly (exponential decrease) as x~+ ~. n the neighborhood in which W V(x) is small, the function takes on appreciable values and has oscillatory character where W V(x) (0, and non-oscillatory character elsewhere. Cases like that of Fig. 1 have been discussed by Hund' in connection with his studies of molecular spectra. An important case is that in which the potential energy curve consists of a single obstacle or barrier as in Fig. 2, and the motion is one of insuscient energy, 8', to clear the obstacle. n such cases there are taro finite solutions lf &( W, x), and tp2(w, x) associated with each energy level, W, and so an arbitrary linear combi- V nation of them is also a solution of (2). Born has Fig. 2. shown that there is always a combination of them which depends on x as e+' and represents a pure left-to-right progressive wave motion as x~+ ~. Such a solution for x large and negative can then be said to represent an incident left-to-right wave coming from the left side and a reflected wave which is not as strong as the incident wave. The interpretation is that the incident beam of particles is partly reflected and partly transmitted. n the range where (W V) (0 the de Broglie wave-length h/p becomes imaginary, and so gives rise to an exponential behavior of P whose nearest analogue is, perhaps, in optics in the slight penetration of a refracted ray into a rarer medium even beyond the angle of total rehection where the refracted angle is imaginary. n this way, one can find the probability that a particle comi'ng up from the left will get through the wall and escape to the right. The case illustrated in Fig. 3a for which Fig. 3a. V(x) =0 V(x) =V V(x) =0 x& a, a&x&0, and for 0& 8 & V, is a simple one with which to illustrate the nature of the calculation. For a given energy level, W, there are two g functions satisfying ~ Hund, Zeits. f. Physik 40, 742 (1927); Kentzel, Zeits. f. Physik 38, 518 (1926). QUANTUM MECHANCS: RADOACTVE DSNTEGRA TON 13i the requirements of finiteness everywhere and of continuity for the ordinates and slopes at the discontinuities in V(x). These are readily found to be ( cosh o,a cos o,(x+a) (o, /o. q)sinh o,a sin uq(x+a) (x& a) fg(w, x) = ~ cosh o,x ( a&x&0) COS 0 yx (o&*) ' (3) (o&/a2)sinh 02a cos o.z(x+a)+cosh 02a sin o.q(x+u) (x& a) Pg(W, x) = (o ~/u, ) sinh u2x ( a&x&o) Sln 0'yX (0&*) where a&=(2s./b)(2pw)'~' and o, = (2s/h) [2p(V W) j'~' To find the lp function corresponding to a beam of particles incident from the left which is partly transmitted and partly rehected, one has to add these together in such a way that to the right of the obstacle there is only the pure left-to-right How, i.e., one must take Pg(lV, x)+i/2(w, x) To the left of the obstacle, the P function represents the superposition of a left-to-right, or incident beam and a rehected The transmitted Z oy og cosh O. ~a sinh cr2a e' 1(*+' 'y beam beam is simply tt y 0'2 + sinh o ~ae ' 1( +') 2 0'g 0 y ego'1a (4) These expressions have, of course, the conservation property (4'4') *-= (44) +(AW i' The probability that a particle coming up to the wall shall get through to the other side is simply (fp)~ '. (PP);, which for e'~ &&1 is clearly equal to ' (W) =16(W/V)(1 W/V)s ~'. (6) The controlling factor is the exponential term except when W/V is very near to Oor i. For application to a theory of the pulling of electrons out of metals by electric fields Fowler and Nordheim' have derived the probability expression by similar methods for Fx the curve of Fig. 3b, i.e. V(x) =0 x&0 V(x) =C Fig. 3b. 6 Fow1er and Nordheim, Proc. Roy. Soc. A119, 1 (1928). 132 R. O'. GURNEY AND E. U. CONDON The probability that a particle of energy S get through the wall they find to be P i,(w) = 4 [(W/C) (1 W/C) ] 'exp( 4k(C W)' /3F) from their Eq. (18) p The term in the exponent can be written 4k(C W) ' /3F = 4ee/3a (k' = 8ir'p/k') (7) to exhibit the similarity with the case of the square wall. Here a is the positive value of x for which V(x) = W, and o ~ is defined as ee = (2x/k) [2ti(C W) ]'t' The exponents in each of these cases can be written in the form (4x/k) ~f [2&(V W)'t'dx the integration extending across the barrier, the limits being the two places where V(x) W=O. Application of the method of approximate integration of Schrodinger s wave equation which was first used in quantum mechanics by Wentze15 indicates that such a result is quite general. The probability of getting through the wall at a single approach is governed essentially by the factor exp 4' h 2p V W ' 'dx (8) being equal to it except for a factor of the order of magnitude of unity. We have next to consider the case of a potential energy curve of the type shown in Fig. 4. According to classical mechanics there are two modes of motion associated with energy levels below the maximum such as 8' in the figure. One is a periodic motion in the range while the other is an aperiodic motion in the range. By the Bohr-Sommerfeld rule the periodic motions would give a discrete spectrum of allowed energy levels which would overlie the continuous spectrum associated with the aperiodic motions. On the quantum mechanics every energy level is allowed with the essential difference that there are no energy T1 levels with which two types of motion are asso Fig. 4. ciated. With each energy level there is associated just one wave function f(w, x) whose square gives the relative probability of being at different parts of the possible range of x. The P(W, x) functions do show traces of the discreteness of the energy levels which the Bohr-Sommerfeld rule associates with the perodic motions in, in an interesting way. The f(w, x) for every W show sinusoidal oscillations as x~~ and also oscillate in the range. For most energies the amplitude of the oscillations in the range is overwhelmingly large compared to that in range, the ratio being of the order of exp[(2ir/k) f[2ti(v- +] 'dx] the integration extending across the barrier. This situation is just reversed however for little ranges of S' values near those given by the old quantization rules. For these the amplitude in is large compared to that in in the same ratio. These then are the allowed QUANT UM MECHANCS: RADOA CTVE DSNTEGRA TON 133 energy levels. t is not a stationary state for the particle to be in range and remain there. But for certain energy levels there is an extraordinarily large probability of being in unit length of range relative to unit length of range. We have to find the mean time which a particle remains in the range before leaking through to the outer range. This can be obtained from the following simple consideration. When the particle is at a place of x large and positive, V(x) =0 (Fig. 4) so the energy is all kinetic and the speed is therefore (2W/]M)'/'. The amount of time which the particle spends in unit length for x large is therefore (p/2 W)'/'. The time spent in a range of length a is therefore o (]s/2w) '. Now according to the wave-functions the probability of being in unit length of range for one of the quasi-discrete energy values relative to the probability of being in unit length of range is of the order exp (4m./h) f[2p( V W) j'/'dx. Therefore since the motion is aperiodic and the particle escaping from range mill in the mean only go through unit length of once, the time T which must be spent in range before getting through to range is of the order of 1' (p/2w) ~ p (]4 /h) Jf [2l (V W)] '1 where a is of the order of the breadth of range. Like all of the results of quantum mechanics this is to be interpreted as a probability result. So that if we start with a number of particles in the same allowed energy level in identical regions similar to range, the number which leak out in time dt is governed by dã= EXdt which gives the usual exponential law of decay X(]') =Roe ' where X= 1/T. (9) The expression for 1may be arrived at in a somewhat different way. One can think of the particle as executing its classical motion in range, but as having at each approach to the barrier the probability of escaping to range given by expression (8) above. The frequency of the periodic motion in, which represents the number of approaches to the barrier in unit time, is of the order a(]s/2 W)' so the mean time of remaining in range before excape comes out as the quotient of these two quantities as before. The reader will find it of interest to examine Oppenheimer's formula' for the pulling of electrons out of hydrogen atoms by an electric field. His formula for the mean time required for dissociation of the atom by a steady electric field splits naturally into a factor which is the classical frequency of motion in the Bohr orbit multiplied by an exponential probability factor of the type of expression (8) used in this paper. 2. APPLCATON TO RADOACTVE DSNTEGRATON After the exponential law in radioactive decay had been discovered in 1902, it soon became clear that the time of disintegration of an atom was ~ Oppenheimer, Proc. Nat. Acad. Sci. 14, 363 (1928). 134 R. tv. GURNEY AND E. U. CONDON as independent of the previous history of the atom as it was of its physical condition. One could not for example suppose that an atom at its birth begins to lose energy by radiation and that its instability is the result of the drain of energy from the nucleus. On such a view it would be expected that the rate of decay would increase with the age of the atoms. When later it was observed that the number of atoms breaking up per second showed the fuctuations demanded by the laws of probability it became clear that the disintegrating depended solely on chance. This has been very puzzling so long as we have accepted a dynamics by which the behaviour of particles is definitely fixed by the conditions. We have had to consider the disintegration as due to the extraordinary conjunction of scores of independent events in the orbital motions of nuclear particles. Now, however, we throw the whole responsibility on to the laws of quantum mechanics, recognizing that the behaviour of particles everywhere is equally governed by probability. From what was said in the preceding section it is clear that the property of the nucleus which we need to know in order to apply the theory is its potential energy curve; and this happens to be a property which we know fairly definitely. Outside a nucleus whose net charge is given by the atomic number we should expect to find a Coulomb inverse-square field of the appro- 0 l 2 0 g 8 X F'ig. 5. &he unit of abscissas is 10 cm. The horizontal line gives the energy of the a-particle emitted by uranium, 0.5 &(10 ' ergs. priate strength. And it is well known that in experiments on the scattering of alpha particles from heavy nuclei the proper inverse-square fiel
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