Principal
On the Scattering of Thermal Neutrons by Bound Protons
On the Scattering of Thermal Neutrons by Bound Protons
Niels Arley
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In 1945, Niels Arley discovered that this seemingly esoteric publication had been used to dimension the Hanford reactor that produced the plutonium for the Nagasaki bomb, and promised never to touch at nuclear physics any more.
Attached to the booklet is a twopage presentation of the research in Danish, presented at the 19. skandinaviska naturforskarmötet, Helsinki 1936.
Attached to the booklet is a twopage presentation of the research in Danish, presented at the 19. skandinaviska naturforskarmötet, Helsinki 1936.
Año:
1938
Editorial:
Levin & Munksgaard
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english
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29
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Det Kgl. Danske Videnskabernes Selskab. Mathematiskfysiske Meddelelser. XVI, l.
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Det Kgl. Danske Videnskabernes Selskab. Mathematiskfysiske Meddelelser. X V I , l. ON THE SCATTERING OF THERMAL NEUTRONS BY BOUND PROTONS NIELS ARLEY K.0BENHAVN L E V IN & M UNKSGAARD HJNAH ML'NKSGAAHf) 1938 IN T R O D U C T IO N 4 fter F e r m i ’s discoveryx) o f the possibility o f producing A jL slow neutrons by surrounding a source o f fast neutrons by hydrogeneous substances such as paraffin wax, the problem of the mechanism o f the collision between neutrons and protons has become important for the study o f the proper ties o f slow neutrons. The problem has already been treated by F ermi himself2), who describes the slowingdpwn pro cess in the following way. Neglecting first the fact that the protons in the paraffin are bound chemically, the fast neutrons which come from the source will make elastic col lisions with the protons giving up on the average half of their kinetic energy at every collision. In this way they will soon reach thermal energies, where they will remain for a relatively long time, because now the chance that a neu tron w ill get by a collision with a proton some o f the ther mal energy of the latter is about the same as that it w ill lose energy by the collision. The neutron w ill therefore diffuse round in the paraffin until it is finally captured by a proton. So long as the neutron energy is large compared with the oscillation energy of the proton it is legitimate to consider the latter as free. As the highest oscillation fre quency o f the proton in paraffin is o f the order 3000 cm 1 0 E. F e rm i , and coll., Proc. Roy. Soc. 149, 522 (1935). Prinled in Denmark. Hianeo Lunos Hogtrvkkeri A/S. 2) E. F e r m i , Ric. scient. VII. 11. 13 (1936). See also H. A. Re t h e , Hev. of Mod. Phys., 9, No. 2 1937. 1* 4 On the Scattering of Thermal Neutrons by Round Protons. Nr. 1. N iel s A k i . k y : corresponding to an energy of 0.37 v o lt1) it will be correct to treat the protons as free for neutron energies down to a crease11 until the energy is small; compared with the energy of the excited state of the deuleron. In this second stage the crosssection will he independent of the energy and it is about one volt. Classically the total crosssection for the scattering should he the same above and below one volt, as the crosssection is classically always the geometrical area of the proton. In a quantum treatment, however, the binding of the proton has a large influence, as first pointed out by F erm i 21, who showed that one may use the Born approximation in cal culating cross sections for the slow neutrons. In this appro ximation the crosssection is proportional to the square of the reduced mass1’1, and as this is equal to the neutron found2) to he about 13 X 10 24 cm2 corresponding to a mean free path o f 1 cm for neutron energies from about 10 000 volts down to resonance energies of the order of some volts, in the third stage when the energy gets below one volt the chemical binding becomes noticeable and the crosssection increases to about 48 X 10 24 cm2 for thermal energies2), so that the mean free path decreases to about 0.3 cm. For the two first stages F ermi has obtained the energy distribution of the neutrons2) which in the second stage, the proton is free, it is seen that the crosssection in the where the mean free path is a constant, turns out to be dE proportional to . In the third stage, neutron energies h below one volt, the problem of the energy distribution has first extreme case will he four times as large as in the second neither as yet been solved theoretically, nor is it known extreme case. For intermediate cases this chemical factor, accurately from experiments.4) mass when the proton is hound strongly compared with the neutron energy but equal to half the neutron mass when as it is called, will lie between one and four. F ermi found For this last problem and for further problems connected by his model for the binding the value 3.3 in the case of with the slowingdown process, such as temperature effects, the Cneutrons. it is of interest to determine theoretically the effect of the Because o f this quantum effect we have therefore diffe chemical binding on the scattering crosssections. Recently rent stages in the slowingdown process. In the first stage, attempts have been made to connect such calculations with fast neutrons with energies of the order some million volts, a still more extended range of problems: it has been pro the crosssection is experimentally found to be of the order posed5) to adopt for the crosssection of free protons — which 1 — 2 x 10 24cm2 4) corresponding to a mean free path in is of considerable importance for the determination of the paraffin of about 5 cm. Owing to the collisions the energy will soon decrease and the crosssection will therefore in1} ^ W>volt 2) 2) M. G oi .dharkr and G. H. B riggs , Proc. Roy. Soe. l(>2, 127 (1937) he 1.5910“ 4) Cf. e. g. H. A. B kthk and R. F. B a c hk r , Rev. of Mod. Pliys., 8, No. 2 (1936) eq. (62). 12 ^ r a ' 1 = 1.233104 (? )ciu  i . Selsk. Skr. Mat.fys. Med. XV, No. 10 (1938). Loc. cit. 3) Loc. cit. 2) cf. eq. (1) p. 12. 4) J. C h a d w i c k , Proc. Roy. Soc. 142, 1 (1933) and J. coll., Phys. Rev. 48, 265 0935). and O. R. F risch , H. v . H aliian jnn. and J. K o c h , Kgl. Danske Vidensk. R. D u n n in g and 4) cf. later p. 9. •’) B kt h k , loc. cit. 7 Nr. 1. N iels A h l e y : On the Scattering of Thermal Neutrons by Bound Protons. neutron and radiation width o f excited nuclear levels1) as the data needed are accurately known. Simpler molecules, well as for the theory o f the deuteron and the discussion like water for instance, have on the other hand so far only of the relation between protonproton and protonneutron been used in the liquid state, and in this the interaction forces2) — instead of the direct experimental value which between the molecules which is o f considerable importance is not very accurate, the quotient o f the thermal crosssec for our problem cannot easily be treated quantitatively. W e tion and a calculated chemical factor. It would, however, shall therefore in the present paper only discuss a very be much preferable for the above purposes to have a more schematic model for the binding. 6 I. Instead of the normal vibrations we assume each pro exact experimental determination o f the free proton crosssection as it is only possible to base such calculations on ton to oscillate independently in a harmonic potential, which very rough models for the binding o f the protons in pa we shall assume to be anisotropic, since it can be deduced raffin and similar hydrogeneous substances. In spite o f this from molecular spectra that the protons oscillate with lar (act it is, as we have seen, of interest to get some rough ger frequencies in the direction o f the valencybond than ideas about the influence of the binding, and we shall in in the perpendicular directions. For the frequencies we this paper treat the problem by help of a model for the shall take v, = 3000 cm1 = 0.37 volts, vx = vy = y vz with binding which we shall discuss in § 1. y = 0.4 so that v_ = v„ = 1200 cm1 = 0.148 volts. II. As we have already mentioned the binding has no influence classically on the scattering. This is also true if § 1. Discussion of a simplified model for the binding of we do not consider the motion as a whole but only the the protons. separate degrees o f freedom. Now we know that the nuclear 1 he scattering crosssection and the energy loss can be motions in the molecules have also in addition to the larger calculated exactly if the proper function for the nuclear mo frequencies which we have accounted for by the assump tion in the molecules concerned is known. Theoretically it tion I, a spectrum extending to quite small frequencies. is possible from an analysis o f the molecular spectra to These small frequencies we w ill take into consideration by obtain the frequencies of the vibrations and the normal assuming that the protons and their potentials can move coordinates which determine the form o f the different normal freely like gas molecules with a vibrations. For the more complicated molecules, however, tion, so that we^ubstitute for the energy exchange between such as paraffin which is mostly used for the purpose of the neutrons and the small frequencies the exchange of slowing down the neutrons, the resulting expressions would kinetic energy between the neutrons and these “ molecules” . indeed be very complicated and unmanageable, quite apart So long as the neutron energy can be considered large com from the fact that for these complicated molecules not all pared with the energies corresponding to these frequencies M axw ell velocity distribu we can namely, as we have just seen, consider these sepa H. A. B e t hk and G. P l ac z e k , Phys. Rev. 51, 450 (1937). 2* G. B r e i t and J. R. St e h n , Phys. Rev. 52, 396 (1937). rate degrees o f freedom as unbound, only the fact that 8 9 Nr. 1. N ihi.s A hi.e y : On the Scattering of Thermal Neutrons by Bound Brotons. they are connected with the other degrees of freedom with elusions regarding the influence o f the binding we can draw the large frequencies must be accounted for. This we do from the model, and next we shall use the results to esti by ascribing an effective mass to the “ molecules” consisting mate the effect o f temperature variation on the mean free path. o f proton and potential, and for this effective mass we choose In order to obtain definite results regarding the last the value 14 times the neutron mass, which is the mass of a problem it is necessary to know the energy ranges of the C//2 group. This figure is rather arbitrary and corresponds neutrons with which we are dealing. We shall assume these to the conception that the energy taken up in the neutron to be the socalled Cneutrons, that is the neutrons which collision by a proton is transferred to a single carbon atom are strongly absorbed in cadmium. The range of strong in the hydrocarbon chain rather than to several of them.1! absorption in Cd extends from 0 to about 0.3 volts.11 Further Our two assumptions are o f course very arbitrary and we must know the energy distribution of the Cneutrons. certainly not fulfilled in nature. No account is taken of This is not exactly known; its theoretical determination is interference effects, and apart from this it is known, for in just one of the aims of the theoretical study of the slowing stance, that the frequency of the CC vibrations in iEthan down process with which we are dealing in the present paper. (C 2Hfi) and other heavy carbon molecules is of the order T w o methods of investigation have been used to determine o f 1000 cm1, which is about five times the energy of the energy distribution of the Cneutrons experimentally. thermal neutrons at room temperature21, so that these v i First the method of the mechanical velocity selector21. By brations cannot at all be considered small. The model de this method it is found that at room temperature the energy scribed is on the other hand the next simplest after that distribution has a maximum for an energy of the order of chosen by F e r m i 31, the isotropic oscillator with infinite mass, kT. Second the method of absorption in B o ro n 31. As the and it is certainly a better approximation than his41. Taking capture crosssection in Boron is assumed to follow the 1 v law 41 it is possible hv absorption experiments in this ele now our model for granted, we shall first see which con1) It must be emphasized that this model is in no way identical with a gas of CHo groups. Firstly, in a C H 2 group the positions of the hydro gen atoms depend on each other; this gives rise to important interference effects which we do not consider in our model; secondly, the slowing ment to compare the mean value of * for different kinds v of neutrons. If for instance the Cneutrons were in thermal equilibrium with the slowingdown medium this mean value down process by free CHo groups would — apart from the slowingdown and hence the Boron absorption should vary with the ab by clastic collisions — take place by energy transfer to the three proper solute temperature of the medium as 7’ ~ 2. W hile between vibrations of the group and the three rotations of the group as a whole, while in our case we have two times three vibrations and no rotation. 2) For T — 290° abs we have k T = 0.025 volts — 203 cm 31 Loc. cit. b After the conclusion of our calculations a discussion of the effect of the anharmonic binding on somewhat similar lines has been published by B k t h e , loc. cit., where, however, the influence of the thermal motions arc not considered (cf. the §§ 4— 6 of the present paper). D Cf. e. g. J. G. H of fman and H. A. B e t h e , Phys. Bev. 51, 1021, (1937). ) J. B. D u nn ing and coll. Phys. Bev. 48, 704 11935). Cf. also B et hi :, loc. cit. 3) For a survey of the literature cf. F iu sc h , H ai .han and K och loc. cit. 1) R. F kisch and G. P i.ac zek , Nature 137, 357 (1936). I). F. W eekks , M. S. L iv in gs t on and H. A. B k t h e , Phys. Bev. 41), 471 (1936). 10 On the Scattering of Thermal Neutrons by Bound Protons. Nr. 1. N iels A r l e y : 400° and room temperature no deviation from this T ~ * law I. The de B roglie wavelength, X,X) for the neutron has been found the increase of the Boron absorption be relative to the proton must be large compared with the tween room and liquid air temperature, and still more range of the neutronproton force, q : between liquid air and liquid hydrogen temperature, is much less than would follow from a X» law. This proves that at least for temperatures of liquid air and downwards the II. The total crosssection, (), must be small compared with the square o f the wavelength: energy distribution of the Cneutrons cannot be represented by a M axw ell Q« distribution with the temperature o f the slowing down medium. The question how far their energy distri bution can be represented by a M axw ell distribution corre sponding to a higher temperature or by a mixture between q. X\ III. For I to be satisfied one can deduce2^ that the di mension o f the proton wave function, a, must be large compared with the range of the neutronproton force: a» a maxwellian and a nonmaxwellian part shall not be dis q. cussed here. In view of these possibilities, however, it re For slow neutrons and protons bound in paraffin all these mains interesting to investigate the energy dependence of conditions are certainly satisfied, as for such neutrons X beam of neutrons. is of the order of 10~ 9cm or more, Q is of the order of W e shall therefore for the purpose of the following calcula 4 8 x l 0 —24cm2 and we further know that q and a are tions assume the Cneutrons to obey the respectively o f the order o f 10 the scattering crosssection for a M ax w ell M axw ell law cm and at least 10 cm. For the differential crosssections per unit solid angle throughout. A consequence o f this assumption together wilh the assumptions made about the binding mechanism is, do, I (0, (p), where I (6,ip)dw is defined as the number however, that we cannot expect a direct comparison o f the of neutrons which are scattered, after having excited the results o f our calculations with experiment to give a quanti proton from its in th into its n th state, into the solid angle tative agreement. dot in the direction per unit time and per scatterer, if there in the incident beam is one neutron crossing unit area per unit time at the place of the scatterer, we have now in the B orn approximation the well known expressions^ § 2 . General theoretical remarks. As first proved by F e r m i^ it is possible to find a “ rectangular hole” potential V with radius q' « X and depth I)', which substituted for the neutronproton potential will give correct crosssections in the P B orn approximation so long as the following conditions are satisfied: — ... h This is for nonrelativistic energies given bv U ) cm — 1 (2 m N E N)  2.85 X 10“ 9 EJ t '  when E s = m N vlel is measured in volts* vrel being the velocity of the neutron relative to the proton. 2) For instance by F o u ri e r analyzing the wave function of the pro ton in respect to velocity. 3) Cf. e. g. M o t t and M assey , “ Theory of Atomic Collisions”, p. 100, eq. (21). (The equation is erroneous, the factor D Loc. cit. Cf. also Re t h e , loc. cit. Part B p. 123. ^ missing). It will be seen that in this approximation I depends on ft only, not on <p. 12 Nr. 1. N ik ns A ki.k y : On the Scattering of Thermal Neutrons by Bound Protons. >...(«) = 2 2M 4 1,1 W u v ( i k l i r ) Y ' ( \ r s  r i>\Wn(r M k"in n = k()0 k~ Q'n E„,))> H < (k 0,> k fiui)/ v U where ifjm and tpn are llie wave functions of the proton before and after the collision, kQ and kmn the initial and final wave vectors l) of the neutron, and A/v , E0 the reduced mass and energy of the neutron. In this expression V' only depends on the distance be tween the neutron and the proton, so taking r y rise to a little confusion we shall briefly give the definitions here, the transformation formulae being derived in Note 2. In the theory for twobody collisions three different coordinate systems are used. O First the system where the one particle is at rest before the col lision, which we shall call the rest system and denote by R 2) (All variables denoted by capital letters). Next the system where 2 MXK0 ti1 ’ k , 0kl mn’ 2 AL 2 ( 7V m( r J the center o f gravity of the two particles is at rest both before and after the collision, which we shall call the center of gravity system and denote by C. (All variables denoted by small letters with an asterisk). Finally the system which has its origin in the center of gravity of the one particle both before and after the collision, which we shall call the relative system and denote by r. (All vari ables denoted by small letters). Let the two particles have masses ni\, m 2 and coordinate vectors/?], /?o, then the center of gravity, Rr , is defined by r p as a new variable in the drN integration we can at once perform ni\R\ } m^R2 = (uh 7;,,n (0) = <1■ r HVn g we get eXP (i kn,nr p) nio) Rc (5) The coordinates referred to the center of gravity are next defined by this and using that the exponential is equal to unitv by this integration due to X » 13 r i* = Ri — Rc, r.,* = R2— /?, (6) Putting ( 6) into (5) we get ,n O'/.) d ' /> ( 2) mi * r* or mi r* i’i llH r2*, mi = a — 02*, 'fi* = <f2* + 71 0 ) if we introduce polar coordinates. Finally the relative coordinates 4 ir2,h4 ^ V dr 4 9  R t W e 'r (3) are defined by r,  Ri Ri = rS — rf, rx = 0 (8) Equation (3 ) we can write in the following way using the the particle with index one being taken as the particle initially resting in the R system. Using (5) we then have, introducing the expression for the total crosssection for scattering between reduced mass a neutron and a free proton2) which we shall denote by Qfl.cc ■t <l = m x • 0 free. (4) We emphasize here that the expression ( 1) or ( 2) is calculated in coordinates relative to the center o f gravity of the system in which the proton is hound and as this fact sometimes gives ' ) The wave vector is just the momentum vector divided by h. ) Cf. Note 1, eq. (N 5). M = mi •m2 mx+ m2 — M To mo ' or *= M  r<> mi “ or i\** = M ri, m2 0O* = M /■•>, 0]* = mi 00, if 2 12 0) U  0o, The following also applies to the case wher two particles are complex, consisting of more parts. In this case the mass is the total mass and the coordinatevector is the one of the center of gravity. 2) It should be noted that this system it not always identical with the coordinatesystem in which we make the observations, cf § 4. 14 Nr. 1. N iels A r l e y : On the Scattering of Thermal Neutrons by Bound Protons. We see from (9) the important fact that the angle of the colliding particle is the same in the C system and in the r system, due to one for a free proton when the proton is strongly bound, 15 which means that the space in which the proper function of the proton is different from zero is very small compared with the wavelength of the neutron. W e can then put the exponential equal to one, so that we get quite independent of the form o f the proper function of the proton L n W = < r  jr  dmn = h0 <?*<*„,„ ( 10) which means that only elastic scattering can occur and that this is spherical symmetric in the relative system just as is the case for scattering by a free proton.1* In the rest system, however, we w ill no longer get the cos 0 law 2* due to the mass o f the scatterer being now larger than the neu tron mass. In F ig . 1 we have plotted in units o f q the curve for ( 10) transformed to the rest system3* for the mass of the scatterer, ms, equal to 14 For the total elastic crosssections we get from (10) ( „ * ; ) «/ve. (m using (4). For the case MN = mN i. e. ms = oc the factor o f Qfree in (11) reduces to the factor 4 first obtained by Fm. 1. A ngular distribution o f scattered neutrons in the rest system cor responding to isotropic distribution in the center o f gravity system, fo r m s = 1 4 mN. which circumstance the formula (1) is often said to be derived in the C system in spite of the fact that it is really derived in the /■ system. From the formulae (2 ) and (4 ) we can at once deduce that, as was already mentioned in the introduction, the total crosssection w ill be nearly four times as large as the F e r m i . 4* W e have in this work taken ms = 14 mN throughout so that (M N\Z 4 —  \mN! ( 14\2 4— ] V15/ 40.871 = 8.48 which makes a considerable difference. o Of. Note 1. 2) Of. Note 2 ecj. (N 19). :5> Of. Note 2 eq. ( N 18). 4> loc. cit. (12) 16 On the Scattering of Thermal Neutrons by Hound Protons. Nr. 1. N iels A heey : § 3. The anisotropic oscillator. V, h «)„ Vo h (oz W We now in ( 2) put the wave functions for our ani e_ m 1+ m F, m NVl 17 ( 15) ' sotropic oscillator and as these are products o f three wave functions for a onedimensional harmonic oscillator, the where VN is the velocity of the neutrons in the rest system, matrix element will be a product of the matrix elements and then we can write (14) in the following form, due to o f the type given by eq. (N 24) in Note 3. Using the for mulae (N 32) and (N 35) in Note 3 we have at once for the 0 9 Kmat = 4 w sin2 2 (by 0 3 )) 0 transition, which is the only one we shall treat here exp (( — ho — (/•exp a„ — Kox + ^no(/) y (/V (\MJp (»_ !)’ ' 9 •) no/t — 9 <1• / . ^nnt j • 4 \\T W sirr t. *4 Ju ; 1 , ( L— t V /22 h'1sin 0 / ~~ \ exp ( t2) dt 2 ( y < 1). (16) *«• (13) For y = l we get the crosssection for the isotropic oscillator1’ 1 ' = i/ ’ iu i> E* loo = 7 ' exp I — 4 W sin ^ (17) Mp being the reduced mass o f the proton, o>„ = 2 tcv,, v„ in the direction o f the In F in. 2 we have plotted in units o f q the curve (1 6 )2) axis and 6 the angle between k0 and k00 i. e. the scat transformed to the rest s y s t e m fo r two different values of the frequency o f the oscillation IT, IT = 0.0697 (full line) and IT = 0.0156 (dotted line) tering angle o f the neutron. Further we must take the mean value of (13) over all which correspond to /uty = 0.37 volts, y = 0.4, ms = 14 mK directions o f the oscillator. This we do by taking the axes and E w equal to the effective energy of neutrons at room of the oscillator as coordinate system and averaging over all respectively at liquid air temperature, i. e. 90° abs.4) It is directions o f k'0'0 in respect to this system, the length o f k'0'0 seen that even at liquid air temperature there is still a being kept constant. In considerable deviation from the spherical symmetry which this way we get, denoting the mean value by /00osc2’ is always assumed in calculations about the diffusion of thermal neutrons.5’ /0 0 „ s c = 9 ' e X P O Cf. F e r m i , loc. cit., and Note 3 eq. (N 34). 2) The function ^ e x p it2) dt is tabulated in Ja h n k e  E mdk “ Tables of W e introduce as new variable the dimensionless quantity 0 Cf. eq. (N 36). 2) A mean value we shall in this paper always denote by this symbol. Functions”, p. 106. •f Cf. Note 2 eq. (N 18) b Cf. § 6 p. 38. Cf. F erm i and He t h k , loc. cit. VidensU.Selsk. Math. fvs. Metkl. XVI.1. 2 18 The curves in F ig . 2 can also be represented by the function oo — 2 r* 4_I ^ 7 r00’ '*00 / 19 On the Scattering of Thermal Neutrons by Bound Protons. Nr. 1. N iels A l l e y : 00, ‘ w ], (18) . o q 4 IF sin2 2 r 1 • 9 8' 4 W sin2 2 y 1 ' gexp •is’' /on = q •exp ! + . ' [ ] + 110 n 2+ ^ n ;' + " ( 11>) i + J n + ; . [ ? + i 18 [ r ,+  " [] = ( * — ljtV V 'sin 2 * so that the two curves have the same starting point and starting tangent and the difference comes first in the second power o f W. From (14) we can now by integrating over 8 and <y get the mean value o f the total crosssection. The result is ^00 ose — (#11— exp 4 U ’ ;/ i (i y )^ (it. ( 20) sin 8 d8 = n q • w  Li — ( i — ;0 h r «  'o For y = 1 we get the wellknown formula for the iso tropic oscillator ^ is Q00 = n q ’ 1 — exp (— 4 W ) W ( 21 ) In F ig . 3 we have in the full curve plotted (20) in units of ()f 2) for y = 0.4 and ms = 1 4 mN. Also we have in the same figure in the dotted curve plotted the curve analog ous to (18), F i g . 2. A ngular distribution in the rest system o f neutrons scattered by anisotropic oscillator. F u ll line corresponds to W = to W = 0.0697, dotted line where /qS0 is given by (17) and I q0 stands for the same function with yoo substituted for w, which we can write as in (18) with <>£ = 3 Oor, + fo iT .. Ooo = Ooo ( ’ w ) , ( 22) 0.0156, W given by (15). W given by (15). The reason why the As we know from (19) the two curves have the same starting point and starting tangent. This can also be seen by direct expanding in powers of W curves (16) and (18) are so like is easily seen analytically 0 Cf. F eiimi , toe. cit. Cf. also Note 4. by expanding in powers of IF. W e then get 2) Cf. eqs. (4) and (12). 2* 20 Nr. 1. N iels A kley 21 On the Scattering of Thermal Neutrons by Bound Protons. (h 1 <1 15 4W ,f q • 4 \ Qis"  1 v 00 15 (2 3 ) 1\" :, c ( n + 1) i) n l 1 3 '' «>/ ,r q •4 Since y < 1 (1 — ;0 /2< 1 (due to y — 0.4 < 1) in the inte gration range of t in ( 20) we can for large IV neglect the exponential and we find then after elementary integration I Qv O 0 r » i ) (24) is" '■ 00 For y — 0.4 the two coefficients are respectively 0.531 and 0.6. That I q0 and (/0S0 are very nearly equal to /^josc and QooLsc a^so physically plausible. l'0s0 and we can namely interprete as the average crosssections for scattering in a substance consisting to one third of oscillators with energy hoK and to two thirds o f oscillators with energy yhoo^, while we by /0()oscand Q00os(, are averaging over all directions of one oscillator with one degree of freedom oscillating with an energy hw, and two degrees of freedom oscillating with an energy yhw„, so that one would think that the two kinds of averaging would give nearly the same result, which is V This series is, as is easily verified, identical with B f.t h e Part B eq (4(>3), if we put ;/i s = by B e t h e . loc. cit. oo , as then our ;t </ —r oq, IP > t\, IT  fo 23 Nr. 1. N iei .s A iiley : On the Scattering of Thermal Neutrons by Bound Protons. in fact found to be the case as we have just seen. Due to only measure the velocity o f the neutrons relative to our the expression ( 20) being far more complicated than the observing system, VN, and not the one relative to the scatterer, expression ( 22), we shall in the following use Q'0S0 instead ©rel, and so we must define an experimental crosssection of QooLe’ tbe error being negligible especially as we shall Qexp by the equation 22 only be interested in that part of ( 20) which belongs to p = (25) qi>n Qexp small values o f W. so that the experimental crosssection is given in terms of § 4. Influence o f the temperature motion of the the usual one bv (26) scattering centers. iY W e must now take the second feature o f our binding model into consideration. At the same time we shall define Now we can take our second assumption about the a new scattering crosssection which can be directly measured. binding model into consideration, the velocity Vs o f the The crosssection is as a rule determined experimentally scatterer by measuring the absorption in varying thicknesses of to some probability law, F ( v s) , the probability for finding paraffin. 0 If now the scatterer does not rest but moves the scatterer with a velocity between Vs and Vs {dV being not being constant, but distributed according with a velocity Vs relative to the coordinate system in which just equal to T (®s) dVs. So on the average we shall find we are measuring, it is clear that another number per unit the scattering probability, which we shall denote by Ps, time o f neutrons w ill be turned out o f the beam and so equal to we shall find another absorption coefficient. This number = j! P E (o s) rf»„. of neutrons expelled from the beam we can easily get by ( F ( v s) ( l v s using the fact that the total crosssection is the same in all G a l i l e i systems2^ and so the total number scattered per unit time and per scatterer or the probability for a scatter and so the average experimental crosssection, () s, will be given by ing process is just \ ° re] Q F ( v s) d V , Q cxpl.S where q is the density of the neutron beam, i. e. number (27) \ F (V .)d V s per unit volume, ©rel the velocity of the neutrons relative to the scatterer and Q the total crosssection calculated in the relative system. M axw ell distribution In an experiment, however, we can Cf. e. g. E. A ma ldi and E. F e r m i , Phys. Rev. 50, 899 (1936). 2) Cf. Note 2. For F ( v s) dVs we have assumed the F ( V S) dVs = (?^) exp (— /*!>“,) dvs, (t = (*28) 24 On the Scattering of Thermal Neutrons by Bound Protons. Nr. 1. N ikls A r l e y : 25 where ms is the mass of the scatterer (which we in this paper have chosen to be equal to 14 times the neutron mass), k is the B oltzm ann constant1), r s is the absolute temper ature, and the constant is chosen so that \ F ( v s)d V s = 1. For Q we ought to take the expression (20), but as we are not interested in temperatures much higher than room temperature, the main part of the integral in (27) will come from that part of Q which belongs to values o f the energy not much higher than ^kT which means that our variable W will be o f the order 0.1 due to the value of h<»_ having been chosen equal to 0.37 volts. For small values of W, however, we have seen that (20) can be approxim ated by ()!0S0 defined in ( 22), so that we can safely put Q1 q0 instead o f the Q from (20) into (27). W e have there j fore first to put (21) into (27) and we get then using (15) ijj!jjlra and (28) rl is Qexpls e * P (  V  ® s® v 2) X ®.vp , _ X exp (— ii up dV s , ““ i r . (29) 'V 2 ho) Taking Vs — VN = V as new variable and choosing a polar coordinate system with VN as polar axis the integration can be worked out and we get3) US <✓4c,J,= T X O iji uv) — exp 4 it r '4. ji ' lliiiiliiiiji l/\.2 X A.8 p 4 //+ ii H v, (30) (4 fi/ “T /*) /. where f/>(.r) is the Gauss error function defined in eq. (N 49). O k = 1.571 X 10 16 erg grad 1 — 8.623 X 10 r’ volts grad F room temperature, T — 290° abs, we have k T = 2> Cf. eq. (15). 2) Cf. Note 5. 0.0250 volts. For .02 W F ig . 4. . 0* .06 OS .12 .It .16 .18 .22 .2* .26 .18 Energy dependence o f scattering crosssection fin units o f 0 fr(,(J fo r scattering centers at room temperature, i. e. 290° abs ( fu ll line). D ol led line corresponds to resting scattering centers. 26 Nr. 1. N iki .s A rl k y : On the Scattering of Thermal Neutrons by Bound Protons. W e introduce now the new dimensionless variables W ]) (32). For x greater than 2.5 <D(x) = 1 and so we get, due and s defined by to s « 1 2 2 m N VN f W  /#■' v% 1+ / /1 k Ts t< h (o m ' 1+ mv £ € 1 is Qexp S m Kr m. 1(31) 27 7r q  W H 1  e x p ( — 4 W )) = ()'\ > 2.5 s and so under the same condition Mn (cf. eq. (N 36); Mp Qis’> Qls ^ exp This result we also get if we take the temperature of Putting (31) into (30) we tinally find, due to ^W the scatterer Ts = 0° which means that the scatterer is resting, and we should therefore as crosssection find just ioft v ex 7Tq • W Q' X 1 For neutrons of room temperature F kin = k T — 0.025 w (32) volts we have W = 0.063 and we see from the curve that is'f ®ll>) X which is in fact the case. the corresponding Q £ kto = For ( ) ‘s i we find the same formula only with j r J exPls y «, is equal to 2.76 •P Jree. If we take we get W = 0.095 and = 2.46 •0 fr(.e.<> W and s substituted for W and s. In F ig . 4 we have in the full § 5. M axwell distribution of the incident neutrons. curve plotted is" Qexp S  xp S (33) JJ v exp in units of ()free2) os a function of W for ms = 14 mN , hwz = 0.37 volts, y = 0.4 and Y’s = 290° abs which makes s — 222’ From the formulae (32) and (33) we can already draw conclusions about the temperature effects. In order to be able, however, to compare the results with experiments, we must take into consideration that the beams of thermal neutrons which can be produced in praxis, e. g. by slowing we ^aye ploded the curve for Qls 3) and it down fast neutrons in paraffin, are never homogeneous is seen that for W > 0.1 the two curves are identical. The but have some energy distribution. As discussed in § 1 reason for this can easily be seen analytically from eq. this is not known quite exactly, but we shall here ap The \V here is formally equal to W in (15) only the E y there is now the kinetic energy in the observation system system as in (15). Only for T s = two systems are identical. 2) Cf. eqs. (4) and (12). 3> Cf. eq. (22). and not in the rest 0°, i. e. resting scattering centers, these proximate it by the M axw ell P As will temperature, M axw ell distribution. If e ( E ) is the distribution for the current, that means that the be seen later, the cf. § 6 especially 2.69•Qfree (cf. also T a b u : effective energy p. 38. 1, Ts = The correct 7\. = 290°). is 1.103 k T at room value is therefore On the Scattering of Thermal Neutrons by Bound Protons. 29 Nr. 1. N iels A h l k y : 28 probability for the neutron which hits the scatterer having an energy between E and E + dE is e ( E ) dE, then the where a is some constant characteristic for the detector used. So we get that crosssection which would be measured should just be the e ( E ) 1( E ) dE  « F (E ) dE, j[ F ( E ) dE = 1 average value of ()exps and as the factor a drops out in (34), what we have to calculate is in fact only the mean value in regard to the M axw ell e (E ) dE distribution for the density In praxis, however, this is not the value measured due to Q exp.s x + ..... the fact that the Boron detector which is mostly used to measure the intensity o f the neutron beam is not equally For ()exps we have now to put Qe L given by (32) sensible for all neutron energies, but absorbs according to and (33), and we must therefore first calculate Q^x])s . If the ^ law. If we then by 1 (E ) denote the sensibility of the we define a new dimensionless quantity, n, by v detector, that means the fraction of the neutrons hitting *Tk the detector which it records, then what is really measured 1) a) Mr mv : is obviously the following average value of the crosssection Oexjs which we shall denote by ()expJv f (cf. eq. (N 36)) (36) 1+ we can write F ( E ) d E = G ( W ) d\V = 2 n v*/i M T’ e x p j  W ) dW jjO a ^ c ( E ) I ( E ) d E 0'•exp (34) s and putting this and (32) into (3o), we get \ e (E )I(E )< iE The M axw ell = /r q •2 n ~ ' 2 0 exp and this we shall now calculate. distribution for the current, e (E ), is pro portional to vx F ( v N) d V s , F ( v n) dVN being given in (28) vi —i ©( vv; (1 + 4 s) X 4W exp(  i + 4 + © if we substitute N for S, or transformed from velocity to X energy, proportional to E v ' 2 it Due to the n "(A7V)' E^exp l r J dE> law we have further that i ( e ) = uf: W exp ( — W )d W Both integrals are here of the same type QC ^ VVr~' exp ( — re2 W ) © (fi w ' ") dW = 2 tT ' :s+ ' Arctg which formula is proved in Note 6. x ( 30 Nr. 1. Niels A isley On the Scattering of Thermal Neutrons by Bound Protons. Putting in the correct values for mental value for the free proton crosssection measured an elementary calculation with resonance neutrons is, however, very inaccurate, as o v eisxp!.s N = ^ q *4 >t , .i Arctg (1 — 4 (n + s)) ms n s * fc — Ts ms already mentioned in § 1. n 1 ' Arctg ^ l + 4(n + s) kTs £ Ti m m ' ‘ 1F— ■ ms 31 n Sj kTx U Ti m (38) ms * ms ml Ms 2 (cf. eq. (N 36) ( Mp ~ ms m% For Ts = 0° we have foundl) that (?eXp<j = ()'s so that we can obtain () isa: by putting s — 0 in (38) lira Qexp rrq2n H i — (1 + 4 ri) k) due to Arctg oc = ^ . For Qexps^we 8et sanie f ° rmula with y substituted for s and n respectively, and so finally q ;exp In F ig . 5 V ^ Qis ''■exp v + o Q.ex p; A and ^ n y (39) JN2 0' e x p ; we have plotted the curve (39) in units of ( ) tree2^ as a function o f T n for various values o f 7’s with nis — 14 mN, Ti m _ = 0.37 volts, and y — 0.4. The values are also given in T able 1. W e see that for Ts = TN = 290° the crosssection is 2.7 times larger than the free crosssection. A m aldi and F e r m i :{) find experimentally for the ratio of the two crosssections the value 3.7. The experi1) Cf. p. 27. 2) Cf. eqs. (4) and (12). •b loc. cit. W e see further that for liquid air temperature the crosssection is 34 °/« higher than for room temperature, the scat 32 33 Nr. 1. Nines A ri .f.y : On the Scattering of Thermal Neutrons by Bound Protons. T able 1. The total elastic scattering crosssection in units calculation o f ()^.JS, the only new formula needed being of Qfrvr as a function o f Ts and 7’v given by eq. (,W) with given in Note 7, ms = U ms i. e. iq = 0.871Qfn,e (cf. p. lb). 20° Ts 90° 1 0 ° ....... 2 0 °....... 90° 29 0 °....... 290° s ^ cons 1 Tx v‘ 7c7’ volts s " V ■2 W rU J\ / K+ 2 W\ , 2 /wx'11 * ) + , " ( . ) { \v\] (40 ) s )\ exp( 1 3.39 ; 3.11 1 2.58 3.59 s 3.17 i 2.58 4.26 3.30 2.61 5.49 3.61 2.69 0 0.00172 0.00776 0.0250 liquid hydrogen temperature liquid air — room — terer being kept at room temperature. The agreement with the experimental value of 26 % found by F i n k 1* where W and s are given by eqs. (31). It is seen that W (40) is only a function o f as it must be, since fiw. does not enter into the problem considered here. Also it is seen /w \ !that, due to <P(oc) = 1, we get for s = 0 or lor ( ) » 1 that, as is physically obvious, is even better than can he expected in view o f the rough assumptions of o. Qexp our model2*. The values for liquid hydrogen temperature ( 20° abs) are only given for the sake of illustration, as for Putting further (40) into (35) we get, after elementary temperatures as low as these our model loses every justi calculations, the only new integral needed being given in fication. In this case, infinite effective mass would be the Note 8, more appropriate approximation. In order to see how much of the variation in our cur ^const Wexp S 02 / V (ns) " + (/i + s) Arctg (41) ves comes from the special form of the crosssection of the anisotropic oscillator and how much from the motion n being given by (36). Also here Hm , drops out, as it must, of the scattering centers the factor —1 in (26)j we have since (41) only depends on Further we get for 7’s — 0° to compare the curves with the curve for Ts = 0, as the lim latter contains only the first influence. W e see that the const Q exp SK Q difference is negligible for room temperature but gets im portant for liquid air temperature. Another way of studying the influence of the motion of the scattering centers con sists in calculating ()exp<Jv for Q equal to a constant. Put ting this into (27) we find, proceeding exactly as in the const, and const as we must get, since lim ()c“ s> 0 exp S const. In F ig . 6 we have plotted the curves (41) as func tion of the two temperatures for ms = 14/nv and Q being taken equal to 1) G. A. F i n k , Pins. Kev. 50. 738 (1936). by F risch , H alban and K o c h , loc. cit. 3) Cf. § 1. A similar value was found f ° r Ts = anc* = 290°, that is Q = 2.58Ofrcj,1^ so that we can directly compare these 1) Cf. T abi.k 1. Vidensk. Selsk. Math.IYs. M edil. XVI. 1. 3 34 N r. 1. N 1k i.s A k lky : On the Scattering of Thermal Neutrons by Bound Protons. curves with the curves in Fid. 5. It is seen that the general 35 Ibis we mean that energy, E, which a homogeneous beam character of the curves is the same, coming from the com of neutrons must have in order that the scattering cross mon integrations, but that the curves in F ig . 5 have another section shall be equal to that of a M axw ell beam of tem perature Ts . For E we therefore have the equation (42) OeJs ( E ) = Qexp S Now our expressions for the ()’s are not given directly as functions of the energies but o f the variables W, n and s. W e have, however,1) IF = a E , a 1 li <o_ ms = 2.52 volts n = u E v, x = ft E s  — 0.180 volts n (o m ' 1+ ~ s ms l , ft ms for hw_ = 0.37 volts and ms = 14/ny where E is the energy o f the homogeneous neutron beam in the coordinate system of the observer and E s , E s are equal to kTN and kTs respectively. So we can solve the equation (42) in terms of W, n and s. This can, however, only he done analytically in a few special cases. Fit., (i. Same as F i g . fo r a scattering crosssection which does not depend on neutron energy in the center o f gravity system. asymptote coming from the special function chosen for Q in (27). I. case n s 1. As the highest temperature we are interested in is less than say 1000° = 0.0862 volts, we see that both n and s § 6. Effective neutron energy. W ith the help o f the curves in F ig . 4 and F ig . a we can now treat the problem of the effective neutron energy. By are small compared to unity and so we can in this case expand everything in (38) with the result that B Cf. eqs. Oil) and (3(i). :s* (43) On the Scattering ol Thermal Neutrons hv Bound Protons. Nr. 1. N iei .s A l l e y : 36 16 n ,s" W‘\p s (44) 7T 37 x being a numerical constant depending on 7\., so that the curve is seen to decrease at the beginning when k'T^ is increased. From this formula we see that Q is large for << 1 II. case: and from the curve for (/tfxps we can conclude that the In 1. corresponding value of ( j is also small. By expanding This we cannot fulfil for all values of ,v if we still want in eq. (32) we then lind both ,s and n to be small compared to unity. If, however, all three conditions are fulfilled, we can put Q and Arctg is'r (45) Q exp S equal to 1 and ^ respectively and we get then from (32) and (38) Putting (44) and (45) into (42) we readily find that the C „.s ”  T < /  4 (i 2 ir ), .Tty4(1 — 3 /i). (48) q exp effective energy is given by ^ 01 Oexp we Ket the same expressions with ^ W and ^ n E = ^ kTs (for Q « 1, /1 « 1, s « 1j (46) substituted for W and n respectively. From (42) we then easily obtain independent of the temperature of the scattering centers. This value is also the effective energy of a M axw ell beam E = 9 kTS for >: 1, n « 1, .s < < 1 (49) in regard to absorption in Boron ^ (while we define the effective energy in regard to scattering) because the crosssections in both cases vary as v This is the classical relation that the mean energy of a M axw ell (cf. eq. (4 5 )). If we take also higher powers in the expansions into beam is equal to also expect to turn out under the conditions stated above. consideration, we are able to get information about the starting tangent of the curve ~E k1v = f ( k T s) . The calcula tions are however lengthy and we shall therefore only k rI\ which result we would n III. case  > 1, n 1. In this extreme case we would find independently of s give the result found, namely that for small values of E rr— t ('•re \,p LS, = 7T(j‘ W i (47) 1) Cf. e. g. H. H. G o l d s m it h and F. H a s k t i , Phys. Hev. 50, 328 (1936). Cf. also B e t h e , loc. cit. Part B p. 136. \’ f .is q ' e x p .r q •l n (50) and so from (42) E = I kTs (for ( ” ) » 1, n » 1 (51) On the Scattering of Thermal Neutrons by Hound Protons. Nr. 1. N iels A u l e y : 38 This case is, however, not of much physical importance, as we cannot neglect the inelastic scattering lor energies which make n » 1. 39 Note 1. For our potential V ' we have assumed 7. > > « ' O, so we know that all the phases will be negligible except the first one, this being given b y  ’ arctg ( ^ tg k ' o ' ) ( Nl ) kn o . Further 1 (exp (2 /i,0) — l ) 2 i A’o since (N 2) < < 1. As in our ease k ' g' and A*0 g' are both smalt, we can expand the tg and arctg in (N 1) with the result v>  1 3 kng'Hk'Z A*n2)  1 3 A',, m x D' . (N 3) Putting (X 3) into (X 2) we find /= /i4 ur (X 4) which shows that I is independent of both the angle and the velocity of the neutrons so that the scattering is spherically symmetric. 0 Since Q = (/dm we finally get for the total crosssection for scattering of slow neutrons by free protons, that this is a con stant given by 4a 9 ( X 5) ( D' g' zj  Note 2. F ig . 7. Effective neutron energy as function o f neutron temperature. As the transformation formulae between different coordinate systems are often used but seldom given in full, we shall here In Fin. 7 we give the curve for ( k 1x as a function of k'l\ for V' — 290° found numerically from the curves in compile them for reference. Firstly let us consider two coordinate systems K and K* so that K* has axes parallel to the axes o f K and further moves along the positive .raxis of K with constant velo F igs . 4 and 5. This “ pipe” like curve we have already used >) Cf. ]). 10. in § 2 !) to obtain that 2) Cf. e. g. M o t t and M a ss e y , “ Theory of atomic collisions” eq. (30/, the effective neutron energy at room and at liquid air temperature is equal to 1.103 kTs and 0.795 kTx respectively. D Cf. p. 17. / p. 21). (T he mass there is equal to the reduced mass, mv\ ' j. •!> M ot t and M asse y , loe. cit. eq. (17), p. 24. b It should he remarked that this means that in the rest system the differential crosssection is proportional to cos <■) cf. Note 2, eq. (N 19). Nr. 1. Xiia.s A iu.k y : On the Scattering of Thermal Neutrons by Hound Protons. city //. A particle is moving with velocity v* in the system K* forming an angle d* with the ,r*axis. In the nonrelativistic ease which we are here considering, u < < e, we have then that in the K system the particle moves with velocity v — u+v*, the angle d between v and the .raxis being determined by during the collision. This energj' can be positive or negative; for m — 0 it is positive for all //. Using (7) and v * = — u, v2* = V)— u, we get from (X 8) due to (N 7) 40 u cos d*ja V* = , COS 0 = u V /. , n'1 , n // 1fj 2 cos d cos d*jV \ V*p* sin d* ty d sin d = b V (NO) t)ni i ( I'n— Rm ) mi /?ii f /n2 17,2 41 (X 9)  nu ( n\\ + nu) (N 6) is now' fully determined by (N 7) and (N 9), but only in the case of elastic scattering, E n— Em — 0, we get simple analytic ex pressions. In this case wTe get, using (7) sin d* . \17 // , .. n = ( ‘” + v*., + 2 p * cos d* !j • To, mi + m2 ' /V* = 77,2n./* = // mi " (X 10) v so that ( X (1) becomes, independently of Vo, using (9) which formulae are at once deduced from fid . 8. , mo tg <)■>= , , m 2 cos dH cos cos do 4  ,, , I . , ms2 , nu Vr14 , + 2 “ cos do = /?ii \ 1+ 01 = For the K and //‘ system we now take the R and C systems') and shall obtain the transformation formulae (N 6) for this case, when the particle with mass nu moves with constant velocity Vo along the .raxis before the collision. From (5) we lind, due to nii (N il) Sill do sin <92 = $*■ X , X ’ nil (n — do), —2 ;  f //?iJ 2 2 cos do nii "1 C Pi = c/2f a, so 0 < ®o < n when nii 4= /?/o, but 0 < 0.2 < 7T wdien nii = nu be cause wTe then simply get = 0 u = V (X7) Wx and as we assume that no outer forces are acting, this velocity is the same before and after the collision. To obtain V\'* and v<'* (the dashes referring to the state after the collision) and so the transformation formulae for the two scattering angles, dj* and d2*, 1 HI1/’!* 1 9 mo m* ( R r EJ 1 9 nnvi'*: 1 9 //Jojo'* ©1 + ©2 — 9 • Solving (X 11) for do we find sin (do — @o) = we only need to use the conservation law for the energy 9 which, combined with (X 11), gives the wellknown relation ( X 8) m~ sin 0o nii (X 12) which for nii > > ni2 reduces to . i mo . mi „ tfa — 0 o“p — sin ©o. where K n— Em is the excitation energy given up by the particle 2 in order to excite the particle 1 from its in'th to its n’th state i) cf. p. i:v  Further we can, using the conservation laws, deduce the formulae for the energies before and after the collision in the rest system: 42 Nr. 1. N iels A h l h y : Ho — ^ in.2 1+ , 4 nii nii H>, } Hi cos2 &i , (nii + m2)2 ,, (nii + nn)2 43 The function g ( o ) we can transform to ft) by (N 11) and we lind then = 0 4 nit ni* E, (1 Ef = On the Scattering of Thermal Neutrons by Bound Protons. ( m2 cos ft) (N 13) 1— \nii //(») 1— cos2 (+ /nr /)+ sin2 ft) ' i (N 20) sin2 ft)’ which for up > > /m just reduces to 1 + 2 /?*2 cos ft). ‘ ‘ By dciinilion we have for the crosssections that I ,nn(®> * ) dP. = I mn (d*, y *) </W* = (d, y) rfr„ (N 14) (X 15) readily seen from the definition. Galling the probability for the scattering process P, that is the number per unit time and per scatterer scattered out of the beam, the crosssection Q is defined as the ratio between P and the number in the incident beam n, we get crossing unit area per unit time at the place o f the scatterer, so that we have (dropping the index 2) which we can write, due to <P = y* = y r ,7 , , . . sin d do ‘„ „ A » . * ) = i ...» ' w ) sin (,) T & For the special case of elastic scattering, m = from (N 11) sin d . . ni‘2~ , , , m o i 1+ „+ 2 cos o sin ft) mi2 nii .< 1 i i}l2 I d  COS nii dQ COS (N 17) mx Putting (N 16) and (N 17) into (N 15) we linally get using (N 11) , ( 1 + ' For mi > > ' COS mi) ' = tive to the scatterer. Since P, o and urel are the same in all Ga l i l e i systems, Q is at once seen to be invariant.) From (N 14) we can also deduce the transformation formulae for the differential crosssections from one Ga l il e i system K to I„„ (»,./ )■ »(») , (X 18) 0 \ (N 21) another K*. The formulae for the angles are, when u is the velo city of the system K * , measured in K 1 + m \ + 2 m cos o' >nn <®> • " ) = / „ „ ( » , . / ) ' P ft + e l where « is the density of the beam, that is the number of par ticles per unit volume, and nr(ll is the velocity of the beam rela do. H+ Q= (N 16) 0 c o s 2 ft) /«i From (N 14) we can at once deduce that the iolal crosssec tions in the two coordinate systems are identical. This is, how ever, only a special case of the more general theorem that the total crosssection is the same in all Ga i .ilki systems.1) This is COS f t ) * + cos ft) II 2 Vi* nii sin ■b we get u * COS Vo* + u  * COS do Vi .cos hi’ * .,+ Vi* v*1 2 V* V2 ll * cos do Vo •(X 22) sin (y,i*— y.2* ) sin do (I  cos' <>)' ■■(1 + in° „ i ,, iiii q (o ) = 1 + 2 — cos o = 1 + 2 — cos ft). /»i n?i *  // 0* + 2 cos »,< /\ ') By a G a l i l e i In the special case /?p = /n2 we find the wellknown formula system is understood a coordinate system which moves with constant velocity along a straight line. > W e have only considered the nonrelativistic case. In tire relativ *7+) cos o 4 cos ft). (N 19) istic case 0 is also invariant so long as the coordinate systems move in the direction of the current. 44 Nr. 1. Nines A r l k y : (with the expression lor eos 0 inserted from the above formula) « 1. 2* = < (® i, 2* , « ) . 0* = < (z»i*, V * ), < &* = < 4.) On the Scattering of Thermal Neutrons by Hound Protons. ([c,*X B ], Bv definition we have1) [v* X a ] ) H jy ) = (— 1)"' exp (if) ( ) exp (— if2). (X 26) ([^ X C o*], [ v f x u ) (and analogous formulae for the quantities without stars) which formulae are obtained from the general formulae of spherical trigonometry, using (N 0), the direction u being taken as the polar axis in a polar coordinate system. Theoretically we can from (N 14) and (N 22) obtain the transformation formulae for the differential crosssections, but in praxis the resulting expressions are so complicated that they are quite unmanageable, with the exception of the special case where u has the same direction as one of the Z)*’s, in which case (N 22) reduces to (X 6). Further, if ti lies in the plane of v * and v * we get <f> = </>* = I), the for mula for © being the same. If we put (X 26) into (X 25) and integrate by parts, we get due to the fact that exp (— z/2) and all its derivatives vanish in + co faster than every power o f y (X27 ) using ll'n = 2«//JI_, n > 1.) Using this recurrence formula / times on itself, we can show' by induction that the result is 11 X T ' S /j_.s)! ^J ^ t>n—l s , w—/ ( X 28) ('i Here l is restricted by the condition that n — l and m   l must both be positive or zero, that is l < min (/??, n). Assuming in < n Note 3. D For the eigenfunction of the onedimensional oscillator we have n !, d~ ■ 2 2(; j!)“ 'I'n (*’) , = ex" (  2 $ " • (X23) _ / , lg _ ( ( H T 9 ) r<(0  1MP („) ’ we can therefore put l = m. By reducing en_ ni.... s () in the same way, wre get M). (  '2) ( N 2!>) due to el0 — ^ exp ( i b y ) exp ( — if2) dy = • — reduced mass of the proton, m the quency of the proton and H n the n’th H ermite polynomial. With these wave functions we shall now calculate the matrix element CO = exp ( — 4 ) \ exI> ( — (11— Y ) ) dU = CXP ( “ Putting now (X20) into (X 28) with l = m, we get exP ( ' C ; 1') I m (X24) l''n* ' exp •\!>m dx = ;i 2 4 ) ;f' "• 2 ( nl in l) V III b( .7  2'" ( i b ) " ~ m nl ml exp jj //n(y) exp ( i b y ) II„,(y) exp ( ~ y 2)d y \y ( i b ) ls 4 1—— 2s si (in — ,s) ! (n mrs)l (m < n). (X 25) b = uk,„„ , n — 1. For n < in we get the same formula with n and m interchanged. (X 30) into (X 24) finally gives us 11 '1 he matrix elements given in this note have been given previously by the author, cf. Nordiska Naturforskarmotet i Helsingfors 1936. The reports, p. 248. b (,f. e. g. H cark and F i i k y : “Atoms, Molecules and Quanta”, p. 533. 1» Cf. Coi k a n t  H i i .is k u t : “ Methoden der p. 78. > Cf. C o c h a n t  H i m i e r t , loc. cit. p. 78. mathematischen Physik”, ( X 30) 46 Nr. 1. N jkls A r i .k y : On the Scattering of Thermal Neutrons by Bound Protons. Uy Mp f71exI) ( ' C ; r) m n hill 2 2 ( nl i ( i />)' n~ m n (  1Y If \ ' exp s 0 /= min (n, m), b = (X 31) 2s si (l — s)l ( \n — m + s ) ! ak'ninx.■ For the one state being just the groundstate we get n exp ( i k'(m x ) 0 ) = ( n !) '2 H i b f exp (N 32) products oi three o f the type (X 23). Due to the states being de generate with the multiplicity gn ^ (/l ” also be obtained by writing the wave functions in polar coordi nates. We only give the formulae for reference1) Xa (21+ 1) (/ — m )! 4a (l + m )!  / n + (+ l\  ^ exp 2 j' n — Z\. V \ 9n \  exp (inup) P\ 2a2/. // I\ i i\ 2 ’ (cos o) X /+ 1 + 1 . ~ 2 ' ’ a2> (XT37) ') I' we must form l < m < 4 /, / n, n —2, n— 1, 1 1 (X30) where mB is the mass of the binding center m B = nis — nip, nip yo mN . For our value of m s = 14 /nv , t has the value 1.0051, which can be safely replaced by unity, so that we apart from the important factor in q are left with F ermi ' s formula. It might be of interest to note that the formula (XT 34) can <P„hlAr,o,t/) For the 3dimensional isotropic oscillator we can at once get the crosssections from (X 32) since the eigenfunctions are only the ( m \ m s \ /nij> + m \ms M m s) \ m v ,n IS 47 l" :l' ( ’K m O  n iijiiK ) j (XT33) o. K. P"' is the ordinary associated L egendre polynomial2) and ^ the confluent hypergeometric function.2) We find [n lm exp ( ik"mr ) 000) This is very complicated unless m — 0 in which case we can at once perform the summations if we only choose the (arbitrary) coordinate system so that k'l'un is along one of the axes h>n ~ '/  V /X •) ‘> kona' (2/41) 4V 2 = 11 = k on 1 i 7 ko ill (( b\n 2 ) CX) M s F, 9 __ Mp h to t n— / (nJcxi) (iA*;;fl.r)0) ( « {/l0)(/i.l0)3 r n + l\ n4/4l (/ ifitl) exP _ ak. ( X 34) Eo F, Putting (X 38) into the formula analogous to (N 33) we just lind the expression in (X 34) for I (m due to the formula proved by the / author1) Fn— FoV “ F (l COS t) ( XT33) ( t ' i ! i (n f /4" 1)! by (1). This is the formula found by F e r m i 1) apart from the factor [n \ in q ^2j (X 38) ‘ (cf. eq. (4)) and trom the factor My in eq. ( X 35) which by F ermi (and by He t h e ) are both put equal to unity: i = n. n—2 ••• >U r 1 2n nl (X 39) ( 't ' ) ! 1> The author, loc. cit. 2> C o u r a n t  H i l b k r t , loc. cit., p. 282. •!> Cf. e. g. M o t t and M assey , loc. cit., p. 38. lots cit. Cf. also B e t h k , loc. cit. Part B eq. (45a.). It should be noted that by the authors quoted m s it put equal to infinity throughout. 4) See Matematisk Tidsskrift, Copenhagen 1937. on his 50th bi rthda y”, p. 42. “ To Prof. H. B ohr 48 Nr. ]. N ikls A hi .k v : In the calculation leading to formula (N 38) we also get the matrix elements lor the fixed rotator with two degrees of freedom whose N l ) l n\' , (a) = — i eigenfunctions are just the first part of (N 37) multiplied by d ( r — ro), r0 being the dimension of the rotator. We find U m exp ( / * " r ) 100)  49 On the Scattering of Thermal Neutrons by Bound Protons. <l0lll ( 2 ) " ( 2 y + 1) ' 2ij (k ”j r 0 )“ ' V . ; , 2(A'"r0) ^ ( n + l ) (— 1)" i f, : ', <4.v) = (X 47) sin 2 i x _ 2n (N 40) c~x— c 2x 4 7i X) 1 Putting (N 42)— ( N 47) into ( N 41) we get, using ,} ko2a2 = W 21 •]j j_i „ being the Bessel function of order 7 f } . I^ n\ 2 I t m ~x m NJ 1— e x p ( — 4 W) . 9 W y Note 4. It may he interesting to note that the formula for the total elastic crosssection for the isotropic oscillator can also be dedu ced by direct calculation o f the B orn phases and their summa tion which is indeed very seldom possible. W e have, since the phases are all small,1) ^0 “ (N 41) + 1) L’n n = which is just eq. (21) remembering eq. (4). Note o. We first prove the formula \ exp ( — « 2(.r — ,t)2) — exp (— « 2(.rj7) 2)j dx — .1 ' ~a 1'f>( « 7) •0 where cP ( x ) is the Gauss error function given by 0 <l>( x ) = CO ‘2 M N k0 f* t* *2 " J, due to o’ « a. n<> Ja*,,,) W X H M r (N 42) h r)rd r. Taking in the first part y — x — /?, in the second y — .r f 7 as new variable we get (N 43) If we put (N 43) into (N 42) using (N 37), we get Ms D 4 1/„ pis ( (— «2 y2) dij so (X 48) follows at once. We can now work out the integral in (29). The two angle integrations being performed, we are left with (N 41) 1 ^00 ! % ( ‘U) exp (— y u2v) i [exp (— a v) — exp ( — ( u + 4 u’) v’2) j X Now 00 \ exp ( (N 49) i o 7’ dt = 1d'o(rY) \(— I ) ’) 4 /' o'3 VL (N 48) 1 P P) Jr ( « 0 Jr (bt) t dt = 2ff_ exp [ — a2f b ' ab \ 2) 4 ' \2p2 “ * ‘T *’o X exp (2 (u vN v) — exp (—2 a u v v)! dv. (N 45) The two integrals here are just of the type (X 48). In the first we I r (x) = exp ( — t' ) ij'j ■Jl,( i x ) (N 46) have a — a, 1 1 = Rat, in the second « 2 = ( « 4 4 u ), i = iv 1 ‘ a Us . ,• a j 4 u Putting in these values in (X 48) we easily find (30). P Cf. M o t t and M assey , loc. cit. eq. (a), p. 138 and eq. (12), p. 90. W a t s o n : “ Bessel functions”, eq. (1). p. 395. 3) W a t so n , loc. cit., eq. (2). p. 77. O W a t s o n , loc. cit. eq. (3). p. 152. 2) Cf. eqs. (1) and (15). Vidensk. Selsk. M:itli.fys. Medd. XVI, 1. 4 50 On the Scattering of Thermal Neutrons by Bound Protons. Nr. 1. Nn:r.s A h f e y : .)! Note <>. ( — i* y2 exp (— « 2y2) d y + 2p \ y exp (  « 2y2)di/ + We prove here the formula •’o •'. X i00 \ w ' exp (,d \ Y ) d‘ ( i W ' ■') (l\v = 2 , r ' V A ret” •o (X 50) •'>? u where P ( x ) is defined in eq. (X 40). We put first t = • (I ,r = ' . (X 51) /* exp ( u x ~~1 [exp ( — « V 2) + a2y2)dy. 2r i + ~i e x p (— « 2;t2) f ' — l rr* •o If we now differentiate the function f ( x ) we gel f'(x ) = «' ■ ) fhe second integral is zero, the two last ones can be performed at once and the first one by integration by parts. The result is I F " as new variai)le and get \ = 2 r l [ exp (  ^ ) <!>(!) dt = 2 4 1/Or), 'o ,+/*> + 4/S V y exp ( — « 2y2) dy + ,t2 1 exp ( '“ y' P ( ,r i) which immediately proves (N 52). “j j [ t P ( t ) } dt. Note S. We prove the formula Integrating by parts we can get the inhomogeneous differential equation for f ( x ) f (•'*') = X ~ l f ( x ) + ;,~J 1 1 ^ \ v ,;s exp ( ~a*W) <!>{? W 'h d W + A rctg'S  7 i ( X 55) ^ where <P(x) is defined in eq. (N 49). We take as new variable / = /? W 12 arid get using (X 49). which hy the ordinary methods can be solved to f ( x )   7T ' ".rArctg .rJ(constant.r). ( — 2 , W i 2 ?f" ' ! ( d/(j d u /2 exp ( — “ J ) exp (  u2). Tlie constant can be determined to be equal to zero by expanding fi’ (0 and integrating term by term. For .r2 < 1 the resultant series is convergent to just n ~ '   x Arctg x. This in (X 51) then proves (X 50). •o l o *° X ow ■x 1 I X dt ( du  *0 X ( dll ( dt *0 * (i #ol ao and so we get, performing the V dt by integrating by parts Note 7. •’ll We prove the formula X \ = 24 \ .r2 1exp ( — <y 2 (.r — ,4)'2) — exp ( — u  (a  f ^)'2) *o •’n <lx (N 52) = ,f> P a + «»] + J 6XP where <T>(x) is defined in eq. (N 49). Putting y — x + ,f we get X 2 a ''" i du exp (— u ) X vn j (43 1 j"« n "21,2 X r « « 2 I X ° X,) i   7  1 + V ( '  * ( aU a t Here all integrations can be performed, using ecp (N 50). The re sult is 4 52 Nr. 1. N ihi .s A ri .k y : s « jS <<2+ ft2 On the Scattering of Thermal Neutrons by Bound Protons. cc 53 ture. In the mathematical notes we have further compiled 2 ~ Arctg y. various formulae for transformation of coordinate systems, which proves (N 53) because we have the elementary identity matrix elements and integrals used in the text. In conclusion I wish to thank Prof. ,y — Arctg  = Arctg x. N ie ls B ohr for his kind interest in this work and to express my appre ciation to Prof. G. P laczek for suggesting the problem to me and for many valuable and helpful discussions in the Summary. course o f the calculations. Further I wish to thank Dr. F. Kalckar, In the present paper we discuss the scattering o f ther mal neutrons in hydrogeneous substances. In § 1 we dis cuss the binding model for the protons. W e assume the protons to be bound independently in an anisotropic os cillator taking the largest oscillation energy equal to 0.37 volts, and the others equal to 0.4 times that. Further we take the lower frequencies into consideration by ascribing an effective mass, which we have chosen equal to four teen times the neutron mass, to the system consisting o f proton plus potential and assuming these “ molecules” to move freely like gas molecules with a M axw ell distribution. In §§ 2 and 3 the crosssections are calculated. In §§ 4 and 5 we discuss the temperature effects. Firstly it is found that when both the neutrons and the scattering substance have room temperatures, the crosssection is 2.7 times larger than the free crosssection. Secondly it is found that the crosssection for neutrons at liquid air temperature i. e. 90° abs is 34 °/o higher than at room temperature. These figures are compared with the experiments. Finally we in £ 6 discuss which effective energy must be attributed to a beam o f M axw ell neutrons in regard to the scattering cross section. It is found that for our model this effective energy lies between 0.7 k T and 1.1 k T depending on the tempera Dr. C. M oller stimulating discussions. and Dr. V. W e is s k o p f for many Pohjoismainen ( 19. skandinaavinen) luonnontutkijain kokous Helsingissa 1936. Nordiska ( 19. skandinaviska) naturforskarmotet i Helsingfors 1936. Eripainos. — Sdrtryck. T A B L E OF C O N T E N T S Page Introduction 3 § 1. Discussion of a simplified model for the binding of the protons § 2. General theoretical remarks 6 § 3. The anisotropic o s c il la t o r ................................................................ 10 § 4. Influence of the temperature motion of the scattering centers 22 §5. M a x w 27 ell distribution of the incident neutrons § 6. Effective neutron e n e r g y .................................................................. Mag. scient. N iees A r e e y , Kobenhavn: 10 Om spredningen a f neutroner med termiske hastigheder ved bundne protoner. 34 Note 1........................................................................................................... 39 Note 2........................................................................................................... 39 Note 3 . ......................................................................................................... 44 Note 4........................................................................................................... 48 Note 5........................................................................................................... 49 Note 6........................................................................................................... 50 Note 7........................................................................................................... 50 Note 8........................................................................................................... 51 S u m m a r y ...................................................................................................... 52 Som det vil blive vist, har Borns approximation 2 gyldighedsomraader: et for hurtige partikler og et for meget langsomme. Man kan derfor i vort tilfaelde benytte denne approximation, og vi finder for det differentielle virkningstvcsrsnit (i relative koordinaterdl,) udtrykket: Tm n W , < P ) = V*n ( r ) e x P ( i k m n r ) f m ( r ) d r % % exP ( i k m nr 2 n %2 ^ dr k0 : neutronernes indfaldsimpuls (i /ienheder) kmn» impuls efter anregung kmn' » impulstab ved stodet (vektorielt) Wn protonens n’te egenfunktion. V (r): protonneutron vexelvirkningsenergien. M: neutronens reducerede masse. q0 er en konstant, der er lig ~n x det totale virkningstvaersnit for neutron Afleveret til Trykkeriet den 28. December 1987. Faerdig fra Trykkeriet den 4. Maj 1988. fri proton (hvilket i relative koordinater betyder fast proton.) Den ubekendte funktion V (r) faar man saaledes ind i en konstant, der kan faas fra experiment, og regningerne reduceres til beregning af matrixelementerne for exp {ik"mn r ) , Disse regninger er gennemfort for tilfaeldene: 1) linear 2) rumlig oscillator og 3) rotator med 2 frihedsgrader. Idet vi nojes med 0—xn overgange, forer regningerne til folgende formler: 1) Dineaer oscillator. "% a 1 k,on ^on d on kn % n! n exp a k (her kan k"on kun antage 2 vserdier svarende til reflexion og gennemgang, idet problemet er 1dimensionalt.) 2) 3dimensional oscillator. j 1 on _ 17 ' 7 1 \/Tl ? r 1\ ^ + x) ( — 2— ) fn —l r ') >2l ' •'till % 1! (n + l + 1)! X 7 k"2 i2 non x 0 2K i f exP ( — (2 betyder summation over alle de til energikvantet n horende tilstande med 1 impulsmomentkvantet l, hvor l kan antage vserdierne n, n 2 , n4, ......... > 0.) Bortset fra de forskellige faktorer er altsaa vinkel og energiafhsengigheden den samme for lineser og for rumlig oscillator. ( a betyder begge steder oscillatorkonstanten a = ( ^ )' 11: protonens masse, on oscillatorens frek\/t CO/ ‘ yens x 2 jt.) 3) Rotator med 2 frihedsgrader. ?on ft I) (K n r) /n + i ( C r) J n r i’ 2 den n ~r ;;~ ’te Besselfunktion. r: rotatorens arm, der faas fra experi ment, idet inertimomentet er u r 2. Helsingfors 19"56. Finska Litteratursiillskapets Tryckeri Al>.