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The dynamics of nuclear reactions
T H E D Y N A M I C S OF N U C L E A R R E A C T I O N S . BY C. E. M A N D E V I L L E Bartol Research Foundation of the Franklin Institute. ABSTRACT. T h e d y n a m i c s of n u c l e a r r e a c t i o n s are s u r v e y e d on a n e l e m e n t a l basis w i t h e m p h a s i s u p o n t h e c.g. s y s t e m of coordinate axes. C o r r e c t i o n s a r e g i v e n for a n g u l a r d i s t r i b u t i o n s , a n g u l a r straggling, a n d t h i n t a r g e t straggling. I. INTRODUCTION.
T h e elements of nuclear dynamics have been discussed by a ' n u m b e r of writers2-3 This paper makes no a t t e m p t to present any new facts b u t rather to summarize the ideas of the previous articles and to give some additional information in w h a t is hoped to be a clear and lucid manner. In general, the final expressions have been given' in the form which has been standardized by reference (2). When any result has been different from t h a t of reference (2), a footnote has been given. In order to aid visualization of the mechanics of nuclear reactions, free use has been made of simple geometry. T h e t r e a t m e n t is entirely non-relativistic, and great emphasis has been placed upon thinking in terms of the set of coordinate axes in which the center of gravity of the reaction is at rest. n . BASIC EQUATIONS.
A nuclear reaction is usually considered to take place as in the selfexplanatory diagram of Fig. 1. Let M1, M2, M3, and M4 refer to the masses of the incident particle, the initial nucleus, the residual nucleus, and the emitted particle. Let El, E2, Es, and E4 and Pl, P~,,P3, and p, have the same meaning in kinetic energies and momenta. Remembering t h e equivalence of mass and energy, the "energy release" of the reaction is defined as Q = (MI+M~)
-
(Ms+M4)
= E3+E4--E1.
(1)
In practice, it is generally true t h a t E2 = 0. If m o m e n t u m is to be conserved, the m o m e n t a of the various particles will form the closed triangle of Fig. 2. 4 It follows immediately 1 H. W . N e w s o n , Phys. Rev., 4 8 : 7 9 0 (1935). 2 M . S . L i v i n g s t o n a n d H. A. Bethe, Rev. Mod. Phys., 9 : 2 4 5 (1937). 8 N. N. D a s G u p t a a n d S. K. G h o s h , Rev. Mod. Phys., 1 8 : 2 2 5 (1946). • 4 T h e e q u a t i o n s d e d u c e d f r o m Fig.. 2 a n d t h e succeeding e q u a t i o n s are valid for m a s s n u m b e r s a n d e n e r g i e s in M e v . O n l y w h e n c o m p u t i n g Q directly from m a s s differences m u s t a c c u r a t e m a s s e s in m a s s u n i t s be used. 385
[J. F. I.
C. E. MANDEVILLE.
386 that
2p~p4 cos ~,,
p ~ = p ? + p~" -
M3E~ = M~E1 -J- M~E4 -
2(MIM4E1E4)I cos ~,
M 3 ( Q + E~ - E O = M~E~ + M4E4 -
2 ( M I M ~ E ~ E 4 ) ~ cos ¢.
INCE)ENT PARTICLE
, ~
~
(2) (3) (4)
)
FIa. 1. The nuclear reaction.
L e t t i n g Ms + M4 = M1 -t- M2 -- M, (4) b e c o m e s E , + E l ( M 1 -- M a ) / M
-
MaQ/M -- [ 2 ( M I M 4 E I E 4 ) ~ cos so-]/M = 0.
(5)
T h e solution of this q u a d r a t i c in E4~ is E4 ~ = [ M I M 4 E , I-]~ [-cos , p ] / M 4-
.E,½ - - - , ( ~ )
( M a M , E ~ cos ~ ~ -
-~
1
M3Q])~,
(6)
M ~ M 4 E ~ sin ~ ~ + E ~ M , M 2 ) ~ .
(7)
M F E ~ M 1 - EaM3 -
[ - M i M ~ l - ] ½ [ c o s (p]
-4-
~
(MMsQ -
FI6. 2. If momentum is conserved, the momenta of the incident particle, the emi.tted particle, and the residual nucleus will form a closed triangle from which the energy of the emitted particle, a function of the angle of observation, is deduced.
Nov., I947.]
387
T H E DYNAMICS OF NUCLEAR FRACTIONS.
Thus if E4 is measured at some given value of ~o, Q may be obtained. The significance of the fact that E4~ is a double value function of will be discussed w i t h reference to the system of coordinates in which the center of gravity of the interacting particles is at rest (c.g. system). If subscripts (3) and (4) are interchanged, the recoil energy of the residual nucleus is obtained. Ez½ = ( ~ )
1
[M1MvEI']½(cos~b)
4-
_~
(MM4Q - M1M3E1 sin s ~ + EIM4M2)L . (8)
Referring the reaction to the e.g. system of coordinate axes, conservation of momentum and energy require that the following equations hold: 5 (a) M1 V1°* = Ms V2~*, (b) M3V3 o~ = M4V, o*, (c) (E~0, + E,o,) - (E~o, + Ee*) = Q,
(9)
Also, (a) V1
= ~=c*+ V1c*, (VIM2)/M, (c) V2o* = ( V 1 M 1 ) / M = (2EIMx)~/M.' (b)
(10)
V l cg =
Since the initial nucleus is at rest in laboratory coordinates, (d). V~°~ = Vows where V~,~ is the velocity of the origin of the c'g" system in laboratory coordinates. From the equations above,
(e) Ex °* = EIM~2/M 2;
E2 ~, =
E1M1M2 M~
;
E z °g =
M,E, _ M.~
_
Substitution of these values in II(9c) leads to
(f) E,°*(1 +
M4/M3) -
E1M~/M
= Q
or
(g) E4cg =
(
E1M~) M3 Q q- M M '
the energy of the emitted particle in the c.g. system. Interchanging subscripts (3) and (4), the recoil energy of the residual nucleus in the c.g. system is found to be (h) Es cg =
(
E1M2)M4 Q+ M M
5 Equations I I (9) and I I (10) relate to the
magnitudes
of the velocities concerned.
388
C.E.
MANDEVILLE.
[J. F. I.
From equations II(10g) and II(10h), the magnitudes of V~cg and V4~" are immediately obtained. Suppose now t h a t the velocity of the emitted particle in the c.g. system is less than the velocity of the origin of the c.g. system in laboratory coordinates. From equations II(10c), II(10g), and II(10h), it is clear that
M4
Q+
M3
Q+
--~ 2
< ELM1,
Vg~/Vo,., < 1,
(11)
< EIM~,
V3~/Vcg~< 1.
(12)
FIG. 3. The reaction may be visualized as taking place a t the center of Circle I which is moving in laboratory coordinates with the velocity of the e.g. system. The radius of Circle I is the velocity of the residual nucleus in the c.g. system. When V.jc*/Vog~ < 1, two velocities of the residual nucleus such as OP and OQ will be found in laboratory coordinates a t the angle ~b. The separation of the centers of the two circles is V~g,. In making the diagram, it has been assumed t h a t any angle of emission in the e.g. system is possible.
It is obvious that this condition arises more frequently when the residual nucleus is under consideration. The vector diagram of Fig. 3 applies to that situation. The radius of circle I of Fig. 3 is V3% the velocity of the recoil nucleus in the e.g. system. To this randomly oriented radius vector is added Vo~s. The distance of separation of the centers of circles I and II is obviously Vega.. The resultants, which will terminate on the circumference of circle II, are the various values of V3, the velocity of the residual nucleus in laboratory coordinates. Two of the resultants, OP and OQ, are drawn in Fig. 3. T h u s two resultant velocities of differing magnitudes are found at the same angle ~b in laboratory coordinates. OQ corresponds to the plus sign before the radical of equation II(8), whereas OP corresponds to the minus sign. Consider the radius vectors OM and ON in Fig. 4, where the two circles are redrawn. OM and ON rotating in clockwise and counter-
"Nov., I947.]
THE
DYNAMICS
389
OF N U C L E A R FRACTIONS.
clockwise directions, respectively, will coincide at some position OD where ~ takes on its m a x i m u m value ¢~. . . . and the discriminant of equation I I(8) vanishes. From Fig. 4, sin ~max = V3°*/Vo~.
(13)
N o recoil nuclei will be found beyond ~ . . . . and all of them, corresponding to both plus and minus sign before the radical of I [ (8), will be found in a cone about the forward direction having a half-angle ~b. . . . Assuming an isotropic distribution of recoil nuclei in the c.g. system (which m a y or m a y not be the case), the fraction of the particles in the cohe corresponding to the minus sign before the radical would be. 4--~
2~ sin • d~3
where
sin (~r - ~0) -
Vae, sin ~bm~
. ¢illcll
I
ClltCll
II
FI6. 4. When the vectors OM and ON of Circle I coincide at OD, the discriminant of equation II (8) vanishes, and the maximum value of q~ is obtained. The circles of figures 3 and 4 may also be applied to the "emitted particle," since subscripts (3) and (4) of the text are interchangeable.
Now suppose t h a t the velocity of the emitted particle in the c.g. system is greater than or e q u a l to the velocity of the c.g. system in laboratory coordinates. This condition does not often apply to the residual nucleus, unless Q is large and E1 is small. In Fig. 5, Vo,~ corresponds to the separation of the two points O and P in the circle. The vectors giving the various magnitudes of V4 and the related values of ~ now e m a n a t e from _P and terminate on the circumference of the circle. The velocity of the emitted particle is thus seen to be a single value function of ~, and the energy of the emitted particle is obtained by taking only the plus sign of equation II(7). T h e preceding discus¢~ is taken positive counterclockwise from 00'. The radial vectors corresponding to the minus sign before the radical then lie in a cone of half-angle ~r -- d0 in the c.g. system.
390
C.E.
MANDEVILLE.
[J. [;. I.
sions obviously apply to the e m i t t e d particle and the residual nucleus interchangeably. F r o m Fig. 5 it is also clear t h a t only when V4c~ >/ Vc~ is emission at 90° to the incident beam possible. W h e n II (7) is squared, .E4 =M--~2 [ - M I M 4 E 1 c o s 2
+2[(EIQMM~M3M4+E1M1M2M3M4 -- (MIM4E1) 2sin -° ~) ~ cos ~o] + M M 3 Q + E 1 M 3 M ~ - M I M 4 E ~ sin s ¢].
(14)
FIG. 5. When V4¢g/Vcg~> 1 only the plus sign before the radical of equation I I ~7) applies, and the velocity of the emitted particle in laboratory coordinates is a single valued function of ,p. The reaction may be represented as taking place at P in laboratory coordinates. The distance OP is the velocity of the c.g. system in laboratory coordinates.
W h e n ~ = 90 °, when the p a t h of the e m i t t e d particle is at 90 ° to the p a t h of the incident particle, 1 E , 9° = M2 [MMaQ + E~(M3M~ - MaM4)]
M~Q -
E1
M
(15)
For ~o differing only slightly from 90 °. E , - 2 c o s ~ (MIMdE~E49o) + E490" M
(16)
Denoting the angle between the perpendicular to the path of the inci7r d e n t particle and the path of the e m i t t e d particle by 0 so t h a t 0 = ~ - ~, for small values of 0, 20 E4 = -~ (M,M4E,E49°) + E4 °°.
(17)
Nov., I947.]
THE
DYNAMICS
OF N U C L E A R
FRACTIONS.
391
EXAMPLE : Suppose t h a t Li s is produced b y t h e reaction LF + D --~ Li s + H x - 0.26 Mev at a b o m b a r d i n g energy of 1 Mev. F r o m II (13), 21 sine, . . . . = ( g ~ [ - - 0 . 2 6
+;])'
9 X (2 X 2 ) '
0.54,
q¢..... = 32.7 °. A t ¢~ = 20 °, the two energies of the recoiling" Li s are found from I[ (8) to be 0.053 M e v and 0.36 Mev.
E x p e r i m e n t s to d e t e r m i n e the angular distribution of the disintegration products are frequently performed. T h e d a t a m u s t then be transformed to the c.g. s y s t e m for theoretical interpretation. Let the n u m b e r of particles e m i t t e d per unit solid angle in laborat o r y coordinates be N(9). T h e n N(~o) sin 9 d 9 = N ( 3 ) sin fl d3,
(18)
where 3 and 9 are clearly defined b y Fig. 6 as the angles of emission in the c.g. s y s t e m and the l a b o r a t o r y system. A relation exists between
V,~,
FIG. 6.
Vector diagram relating velocities in the c.g. and laboratory systems of coordinates.
the expressions sin 9 d9 and sin/3 dfl which will now be derived. Fig. 6, cot 9 --
cot~=
Vcg8 V4°* sin 3
+ cot 3,
r + cos/3 ~ / l _ c o s 2/3'
From (19)
r = V~gs/V4 cg.
(20)
This gives a q u a d r a t i c in cos/~. csc 2 9cos2/3 + 2r cos/3 + r 2 -- cot s 9 = 0, -
(a) cos/3 =
-
r + 4r 2 -
(b) cos/3 = - r s i n
csc 2 9 ( r 2 -
cot 29)-
C8C 2 ~p
294-cos9
~/1 - r 2sin 2 ~o.
(21) (22)
392
C.E.
MANDEVILLE.
[J. F. L ,
C o n s i d e r the case of V~°"/Vc,. /> 1. T a k i n g the plus sign before the radical, since cos/5 is a single v a l u e d f u n c t i o n of ¢ in this case, we h a v e on d i f f e r e n t i a t i n g sin /5 d/5 = 2r cos 9 + x/1 - r 2 sin ~ 9 + ~/1 sin ~ d ¢
r'~ cos 2 ~ r 2 sin 2 ¢ -
(23)7
-
T h e n if N ( 9 ) is observed, N(/5) m a y be calculated. N o w suppose t h a t V4°./Vo,. < 1. In this case, cos/5 is a d o u b l e v a l u e d f u n c t i o n of ~. D e n o t i n g all v a l u e s of/5 lying w i t h i n t h e cone of half-angle ~r - /50 by/sa a n d all v a l u e s of/5 exterior to the cone by/52, N(~)
=
N(/5)
sin ~d~
N(~o) = 2 N ( / 5 ) [ 1
slrn ~ d ~
'
-- r 2 sin 2 ~ + r 2 cos 2 ¢~ 41 -
(24) (25) ~
r ~ s i n ~ ~v
T h e m i n u s sign before the radical of e q u a t i o n II(22b) relates to /5:, the phis sign to 152. EXAraPLE: Concerning the angular distribution of the neutrons from the reaction D + D He s + n + Q. V4~. is always greater than V~,, and the positive sign of equation II(22b) applies. r 2 = ( V ~ , . ) 2 / ( V , ~ ) 2 = E ~ / 6 ( Q + ½E~).
Substitution of this value of r in equation II (23) leads to sin fl dO E~ i cos so[-6Q + 3E1 - E~ sin 2 so]~ + (3Q + 2E1 - E~ sin~ so) sin so dso [](Q + ½E1)]~[6Q + 3E: - E1 sin2 ~o]t An alternate method of approach is to employ a vector diagram similar to that of Fig. 6. For the D + D reaction, V~ = V. = velocity of the neutron in laboratory coordinates. From Fig. 6, V. sin B = ~ sin so,
d0 sin # d¢~ sin sodso
V. cos so dso +
V 0. cost~
sin so V.~-cosO
0 V. dso Oso
'
[ 11. ~z cos so V. sin so OV,, \ V , . " q cosO + i V . ° t ) 2 c o s ~ 0so '
where
cos t~ =
(cos so)24g~. - ½EO [](Q + ~E.)]~
V. and ~ are obtained from equation II(7). oq~
III. ~
sck~rggtl~O.
The collision of a neutron with a proton is depicted in the vector diagram of Fig. 7. For this reaction, the magnitudes of the velocities 7 Compare reference (2), equation 784a. s Compare reference (2), equation 784b.
Nov., I947.]
T H E DYNAMICS OF NUCLEAR FRACTIONS.
393
satisfy the e q u a t i o n V,°* = V, °* = Vc,.
and
/3 = 2~,
(1)
If it is assumed t h a t the scattering is spherically s y m m e t r i c a l in the c.g. system, sin t3 d/~ = sin 2~ 2d9 (2) N(¢) = sin ~ d e sin ~ d9 T h e total n u m b e r of particles to be found in l a b o r a t o r y coordinates between ~ a n d ~ + d¢, would be Nr(~o) d ~ = N(~o) 2~ sin ~ d e = 87r sin ¢ cos ~od~.
(3)
i
FIG. 7.
Vector diagram relating velocities of the recoil proton in the c.g. and" laboratory systems of coordinates.
For this reaction, M1 = M~ = Ms = M4 = 1 m.u., M = 2 m.u., Q = 0. B y e q u a t i o n II(7), 2E,i = E 0 cos ~ + E1~41 - sin ~ ~, E , -- E1 cos~ ~,
dE, = -- 2E~ sin ~ cos ~ d¢, N(Es) d E , = - NT(9) d 9 = 47r/E,.
(4) (5) (6) (7)
T h e n u m b e r of scattered particles per u n i t range of energy in the l a b o r a t o r y s y s t e m is constant. IV. GEOMETRICCORRECTIONS.
T h e r e are several t y p e s of straggling which m u s t be considered when the e m i t t e d particles of nuclear reactions are observed. A m o n g these
394
(7. E .
MAN1)I~;VILLE.
[I. ['. l,
are : (a) Range straggling, (b) Angular straggling, (c) Straggling arising from target thickness. It has long been known from experiment that range straggling obeys the well known Gaussian distribution; that is, N(R)dR
OL
(1)
= -~-:- e .... ~1~ ~o)"dR, ~Tr
where N ( R ) d R is the probability that the charged particle come to rest at a distance between R and R -[- d R from the starting point. R0 is the range exceeded by half the particles and is known as the .mean range.
Ro [~ex Fw,. 8.
Differential a n d integral curves giving extrapolated and m e a n ranges.
The average value of the mean square deviation (R -- R0) 2 of the range of the particles from the mean range is given by (R
-
R
=
e - , ~ ( R - n o ) ' ( R -- Ro)~-dR
1
= 2a-~;.
(2)
When experiments are performed, the mean range of the emitted particle is the sought after quantity. This mean range is usually obtained by subtracting an appropriate correction from the extrapolated range. The extrapolated range is determined by extrapolation of the
Nov., 1947.]
THE
DYNAMICS OF NUCLEAR FRACTIONS,
395
point of m a x i m u m slope of the integral curve. Differential curves are often converted to integral curves for the purpose of this convenient extrapolation. These curves are illustrated in Fig. 8. T h e o t h e r types of straggling simply increase the distance between R0 and Rex already introduced by normal range straggling. At R = R0, the height of the integral curve is one-half its initial value. From Fig. 8, 2(R0 - Rox) = t a n x = - 4~/~, Rex -
Ro =
4-~/2a
= zx.
From IV(2), I
zX2 = ~r (R -- R0) ~.
(3)
It is clear that, for Gaussian straggling, a q u a n t i t y ~ m a y be c o m p u t e d which when subtracted from the extrapolated range will give the mean range. For other types of straggling, ~ m a y be computed in the same manner, t h o u g h an approximation is introduced when the result is taken to be Rex - R 0 , The several types of straggling will now be considered.
NCE
Fie. 9.
N(E4) as a function of E4 when the particles are emitted from a "thin" target. THIN TARGET CORRECTION.
When a thin target is bombarded, the emitted particle will emerge from various d e p t h s in the target material. Assuming t h a t the yield function of the reaction under s t u d y is constant over an energy interval as large as the thickness of the targgt, the n u m b e r of particles produced at any given d e p t h and thus the n u m b e r of emitted particles produced per unit range interval will be constant. For small intervals of energy, for intervals of energy comparable to the thickness of the thin target, the range-energy relation m a y be regarded as linear so t h a t a rectangular
396
C.E.
MANDEVILLE.
[ J - F . 1.
function similar to N(R) will describe N(E). The function N(E) is plotted inFig. 9. Let (R - Ro)~ equal the average square value of the fluctuation of R about R0 arising from the finite thickness of the thin target. Then,
fn
Rm*x(R _
(R -
Ro)~
=
Ro)2N(R)dR
~"°
(R
Ro)2KdR
=
K = constant, ](_R (R -
Ro)~
Ro)~
=
]""" -IR.i,,
(Rm.x -- Rmi.) 2
=
Rm,~ -- Rmi.
12
remembering that R0 = (Rmi, + R .... )/2.
Then
71-
AT = ~
(4)
(Rmax -- Rmin)
and Z~r is the contribution to t h e difference Roz - R0 made by the finite thickness of the thin target.
By Geiger's law, g n =
2d(log R)
d(log E)
Then, z~R = n AE R 2E'
a r =.
(146)
RO "~E'---~ =
R0 2
E0
(5)
Em,z is the energy of a particle emitted from the surface of the target. Emln is the energy of a particle emitted from the back of the target. 1° A rough estimate of R0 and corresponding E0 is necessary to determine z~r. The bombarding energy is taken to be the energy of the incident particle at the center of the thin target. ANGULAR STRAGGLING.
Observation of the emitted particle is usually made at 90 ° to the incident bombarding beam in the laboratory system. However, a 0 For values of n for this differential form'of Geiger's law, see reference 2. t ° E ~ u - Emi. is usually determined by the thickness which the thin target presents to the emitted particle. However, if the energy of the emitted particle varies rapidly with bombarding energy, and if the thin target is ~sufficiently thick to make that variation appreciable, a n additional correction for the difference between the bombarding energy at the surface and back of the target must be made.
Nov., x947.]
T H E DYNAMICS OF NUCLEAR FRACTIONS.
397
finite solid angle is subtended at the target by the detecting apparatus, and a spread in energy a n d thus in range of the emitted particle is intercepted by the detector. Since the experiment is not performed at exactly 90 °, a correction must be made. Most experiments are now performed only with "good geometry" as defined by Livingston and Bethe. 2 The angular straggling of emitted particles, collimated by tyro square apertures of side 2a and separation b, will now be computed for
111111
I
-
b Fio. 10.
good geometry; that is, a << b. From Fig. 10, the number of particles detected at the angle O will be proportional to (2a'- b tan O). From equation II(17), E 4 - - E , 90
and by Geiger's law,
R , - R,,o
20 (M~M~E~Ep°) t = _~_
(6)
. (E,-
(7)
Then, E,9--~ (R, -
R,'o)'
]
(2a - b tan O)dO
(8)
=
o*~j~ (2a -- b tan O)dO
(R, - R,,o),
2 ~ n~ = 3 b* M* (R~)2
Aa = ~
(M1M,el o----~ E,
n a Rpo ( M~MoE' ) '
] I '
(9) (ao)
398
C . E . MANDEVILLE.
IJ. F. I.
RECOIL P R O T O N S .
As has long been known, the most precise method of observing neutron energies is simply the measurement of the proton's recoil energy arising from a head-on collision with the neutron. In order to obtain su.fficient intensity, proton recoils making an angle as great as some arbitrary limit 70 are usually accepted. The straggling of the neutrons is then increased. By II[(7), it is seen t h a t assuming spherical symm e t r y in the c.g. system, the number of recoil protons per unit interval of energy is constant in the laboratory system. A rectangular function similar to that of the thin target correction then applies. By IV(5),
146 R o ~n~E - ~ o where by III(5)
AAn = }
AE
= E 4 9 ° - - E4no = E49°
AA n =
(11)
sin 2 r/. = E49°r/0 ~,
(12)-
Rondo2"
After all other corrections, the mean energy of the group of recoil protons will differ from the energy of the neutrons emitted at exactly 90 ° to the incident beam by an a m o u n t ½(E4 - E4 cos ~ r/o) = ½E,~oL This amount must be added to give the correct energy. • Were a single straggling correction involved, the mean range would be obtained by subtracting that correction from the extrapolated range. However, when more than one correction is made, such as range straggling, straggling arising from target thickness, and angular straggling, the corrections are combined so that =
+ A.42 +
+
.-
(13)
For corrections for normal range straggling and the range energy relation, see references (2) and (12). EXAMPLE: The observed extrapolated range of the alpha particles from the reaction Flg + H 1 --- 0 TM+ He 4 was found to be 6.03 cm. of standard air at a bombarding energy of 0.85 Mev. The target thickness was estimated to be 30 Key for the incident protons and 67 Key for 7 Mev alpha particles. The beam was defined by two rectangular slits having a width of 0.3 cm. and a separation of 3 cm. ; n = 3.18 (Fig. 35, reference 2), the range correction, 1.12 per cent. (Fig. 37, reference 2). From equation IV(10), the angular straggling correction is given by R,~---~=
~ ~r
E49o
/
= 1.02 \ ~ - 6 7 1 \ - - f - / \
7.1
= 0.56%.
By equation IV(5), R, 9°A---Z'r= 21 /rn62 Emax EO--~---,min= 0.36 (3A8)(0.067__ +?(0.03))__ = 0.6%. n Compare reference (2), equation 809.
Nov,, 1947.]
THE
D Y N A M I C S OF N U C L E A R FRACTIONS.
399
The final percentage to be subtracted from the observed extrapolated range is q(1.12)' + (0.56) 5 + (0,6) ~ = 1.38%, The mean range of the alpha particles is then 5.95 cm. equation" II(15), Q is readily determined.
From the range-energy relation ~.12and
ACKNOWLEDGMENTS.
The writer wishes to t h a n k professor W. E. Bennett of the Illinois, Institute of Technology for m a n y discussions and suggestions with regard to the text. He wishes also to acknowledge the interest and advice of Dr. W. F. G. Swarm, Director of the Bartol Research Foundation. ~ M. G. Holloway and M. S. Livingston, Phys Rev. 54, 18 (1938).
T h e elements of nuclear dynamics have been discussed by a ' n u m b e r of writers2-3 This paper makes no a t t e m p t to present any new facts b u t rather to summarize the ideas of the previous articles and to give some additional information in w h a t is hoped to be a clear and lucid manner. In general, the final expressions have been given' in the form which has been standardized by reference (2). When any result has been different from t h a t of reference (2), a footnote has been given. In order to aid visualization of the mechanics of nuclear reactions, free use has been made of simple geometry. T h e t r e a t m e n t is entirely non-relativistic, and great emphasis has been placed upon thinking in terms of the set of coordinate axes in which the center of gravity of the reaction is at rest. n . BASIC EQUATIONS.
A nuclear reaction is usually considered to take place as in the selfexplanatory diagram of Fig. 1. Let M1, M2, M3, and M4 refer to the masses of the incident particle, the initial nucleus, the residual nucleus, and the emitted particle. Let El, E2, Es, and E4 and Pl, P~,,P3, and p, have the same meaning in kinetic energies and momenta. Remembering t h e equivalence of mass and energy, the "energy release" of the reaction is defined as Q = (MI+M~)
-
(Ms+M4)
= E3+E4--E1.
(1)
In practice, it is generally true t h a t E2 = 0. If m o m e n t u m is to be conserved, the m o m e n t a of the various particles will form the closed triangle of Fig. 2. 4 It follows immediately 1 H. W . N e w s o n , Phys. Rev., 4 8 : 7 9 0 (1935). 2 M . S . L i v i n g s t o n a n d H. A. Bethe, Rev. Mod. Phys., 9 : 2 4 5 (1937). 8 N. N. D a s G u p t a a n d S. K. G h o s h , Rev. Mod. Phys., 1 8 : 2 2 5 (1946). • 4 T h e e q u a t i o n s d e d u c e d f r o m Fig.. 2 a n d t h e succeeding e q u a t i o n s are valid for m a s s n u m b e r s a n d e n e r g i e s in M e v . O n l y w h e n c o m p u t i n g Q directly from m a s s differences m u s t a c c u r a t e m a s s e s in m a s s u n i t s be used. 385
[J. F. I.
C. E. MANDEVILLE.
386 that
2p~p4 cos ~,,
p ~ = p ? + p~" -
M3E~ = M~E1 -J- M~E4 -
2(MIM4E1E4)I cos ~,
M 3 ( Q + E~ - E O = M~E~ + M4E4 -
2 ( M I M ~ E ~ E 4 ) ~ cos ¢.
INCE)ENT PARTICLE
, ~
~
(2) (3) (4)
)
FIa. 1. The nuclear reaction.
L e t t i n g Ms + M4 = M1 -t- M2 -- M, (4) b e c o m e s E , + E l ( M 1 -- M a ) / M
-
MaQ/M -- [ 2 ( M I M 4 E I E 4 ) ~ cos so-]/M = 0.
(5)
T h e solution of this q u a d r a t i c in E4~ is E4 ~ = [ M I M 4 E , I-]~ [-cos , p ] / M 4-
.E,½ - - - , ( ~ )
( M a M , E ~ cos ~ ~ -
-~
1
M3Q])~,
(6)
M ~ M 4 E ~ sin ~ ~ + E ~ M , M 2 ) ~ .
(7)
M F E ~ M 1 - EaM3 -
[ - M i M ~ l - ] ½ [ c o s (p]
-4-
~
(MMsQ -
FI6. 2. If momentum is conserved, the momenta of the incident particle, the emi.tted particle, and the residual nucleus will form a closed triangle from which the energy of the emitted particle, a function of the angle of observation, is deduced.
Nov., I947.]
387
T H E DYNAMICS OF NUCLEAR FRACTIONS.
Thus if E4 is measured at some given value of ~o, Q may be obtained. The significance of the fact that E4~ is a double value function of will be discussed w i t h reference to the system of coordinates in which the center of gravity of the interacting particles is at rest (c.g. system). If subscripts (3) and (4) are interchanged, the recoil energy of the residual nucleus is obtained. Ez½ = ( ~ )
1
[M1MvEI']½(cos~b)
4-
_~
(MM4Q - M1M3E1 sin s ~ + EIM4M2)L . (8)
Referring the reaction to the e.g. system of coordinate axes, conservation of momentum and energy require that the following equations hold: 5 (a) M1 V1°* = Ms V2~*, (b) M3V3 o~ = M4V, o*, (c) (E~0, + E,o,) - (E~o, + Ee*) = Q,
(9)
Also, (a) V1
= ~=c*+ V1c*, (VIM2)/M, (c) V2o* = ( V 1 M 1 ) / M = (2EIMx)~/M.' (b)
(10)
V l cg =
Since the initial nucleus is at rest in laboratory coordinates, (d). V~°~ = Vows where V~,~ is the velocity of the origin of the c'g" system in laboratory coordinates. From the equations above,
(e) Ex °* = EIM~2/M 2;
E2 ~, =
E1M1M2 M~
;
E z °g =
M,E, _ M.~
_
Substitution of these values in II(9c) leads to
(f) E,°*(1 +
M4/M3) -
E1M~/M
= Q
or
(g) E4cg =
(
E1M~) M3 Q q- M M '
the energy of the emitted particle in the c.g. system. Interchanging subscripts (3) and (4), the recoil energy of the residual nucleus in the c.g. system is found to be (h) Es cg =
(
E1M2)M4 Q+ M M
5 Equations I I (9) and I I (10) relate to the
magnitudes
of the velocities concerned.
388
C.E.
MANDEVILLE.
[J. F. I.
From equations II(10g) and II(10h), the magnitudes of V~cg and V4~" are immediately obtained. Suppose now t h a t the velocity of the emitted particle in the c.g. system is less than the velocity of the origin of the c.g. system in laboratory coordinates. From equations II(10c), II(10g), and II(10h), it is clear that
M4
Q+
M3
Q+
--~ 2
< ELM1,
Vg~/Vo,., < 1,
(11)
< EIM~,
V3~/Vcg~< 1.
(12)
FIG. 3. The reaction may be visualized as taking place a t the center of Circle I which is moving in laboratory coordinates with the velocity of the e.g. system. The radius of Circle I is the velocity of the residual nucleus in the c.g. system. When V.jc*/Vog~ < 1, two velocities of the residual nucleus such as OP and OQ will be found in laboratory coordinates a t the angle ~b. The separation of the centers of the two circles is V~g,. In making the diagram, it has been assumed t h a t any angle of emission in the e.g. system is possible.
It is obvious that this condition arises more frequently when the residual nucleus is under consideration. The vector diagram of Fig. 3 applies to that situation. The radius of circle I of Fig. 3 is V3% the velocity of the recoil nucleus in the e.g. system. To this randomly oriented radius vector is added Vo~s. The distance of separation of the centers of circles I and II is obviously Vega.. The resultants, which will terminate on the circumference of circle II, are the various values of V3, the velocity of the residual nucleus in laboratory coordinates. Two of the resultants, OP and OQ, are drawn in Fig. 3. T h u s two resultant velocities of differing magnitudes are found at the same angle ~b in laboratory coordinates. OQ corresponds to the plus sign before the radical of equation II(8), whereas OP corresponds to the minus sign. Consider the radius vectors OM and ON in Fig. 4, where the two circles are redrawn. OM and ON rotating in clockwise and counter-
"Nov., I947.]
THE
DYNAMICS
389
OF N U C L E A R FRACTIONS.
clockwise directions, respectively, will coincide at some position OD where ~ takes on its m a x i m u m value ¢~. . . . and the discriminant of equation I I(8) vanishes. From Fig. 4, sin ~max = V3°*/Vo~.
(13)
N o recoil nuclei will be found beyond ~ . . . . and all of them, corresponding to both plus and minus sign before the radical of I [ (8), will be found in a cone about the forward direction having a half-angle ~b. . . . Assuming an isotropic distribution of recoil nuclei in the c.g. system (which m a y or m a y not be the case), the fraction of the particles in the cohe corresponding to the minus sign before the radical would be. 4--~
2~ sin • d~3
where
sin (~r - ~0) -
Vae, sin ~bm~
. ¢illcll
I
ClltCll
II
FI6. 4. When the vectors OM and ON of Circle I coincide at OD, the discriminant of equation II (8) vanishes, and the maximum value of q~ is obtained. The circles of figures 3 and 4 may also be applied to the "emitted particle," since subscripts (3) and (4) of the text are interchangeable.
Now suppose t h a t the velocity of the emitted particle in the c.g. system is greater than or e q u a l to the velocity of the c.g. system in laboratory coordinates. This condition does not often apply to the residual nucleus, unless Q is large and E1 is small. In Fig. 5, Vo,~ corresponds to the separation of the two points O and P in the circle. The vectors giving the various magnitudes of V4 and the related values of ~ now e m a n a t e from _P and terminate on the circumference of the circle. The velocity of the emitted particle is thus seen to be a single value function of ~, and the energy of the emitted particle is obtained by taking only the plus sign of equation II(7). T h e preceding discus¢~ is taken positive counterclockwise from 00'. The radial vectors corresponding to the minus sign before the radical then lie in a cone of half-angle ~r -- d0 in the c.g. system.
390
C.E.
MANDEVILLE.
[J. [;. I.
sions obviously apply to the e m i t t e d particle and the residual nucleus interchangeably. F r o m Fig. 5 it is also clear t h a t only when V4c~ >/ Vc~ is emission at 90° to the incident beam possible. W h e n II (7) is squared, .E4 =M--~2 [ - M I M 4 E 1 c o s 2
+2[(EIQMM~M3M4+E1M1M2M3M4 -- (MIM4E1) 2sin -° ~) ~ cos ~o] + M M 3 Q + E 1 M 3 M ~ - M I M 4 E ~ sin s ¢].
(14)
FIG. 5. When V4¢g/Vcg~> 1 only the plus sign before the radical of equation I I ~7) applies, and the velocity of the emitted particle in laboratory coordinates is a single valued function of ,p. The reaction may be represented as taking place at P in laboratory coordinates. The distance OP is the velocity of the c.g. system in laboratory coordinates.
W h e n ~ = 90 °, when the p a t h of the e m i t t e d particle is at 90 ° to the p a t h of the incident particle, 1 E , 9° = M2 [MMaQ + E~(M3M~ - MaM4)]
M~Q -
E1
M
(15)
For ~o differing only slightly from 90 °. E , - 2 c o s ~ (MIMdE~E49o) + E490" M
(16)
Denoting the angle between the perpendicular to the path of the inci7r d e n t particle and the path of the e m i t t e d particle by 0 so t h a t 0 = ~ - ~, for small values of 0, 20 E4 = -~ (M,M4E,E49°) + E4 °°.
(17)
Nov., I947.]
THE
DYNAMICS
OF N U C L E A R
FRACTIONS.
391
EXAMPLE : Suppose t h a t Li s is produced b y t h e reaction LF + D --~ Li s + H x - 0.26 Mev at a b o m b a r d i n g energy of 1 Mev. F r o m II (13), 21 sine, . . . . = ( g ~ [ - - 0 . 2 6
+;])'
9 X (2 X 2 ) '
0.54,
q¢..... = 32.7 °. A t ¢~ = 20 °, the two energies of the recoiling" Li s are found from I[ (8) to be 0.053 M e v and 0.36 Mev.
E x p e r i m e n t s to d e t e r m i n e the angular distribution of the disintegration products are frequently performed. T h e d a t a m u s t then be transformed to the c.g. s y s t e m for theoretical interpretation. Let the n u m b e r of particles e m i t t e d per unit solid angle in laborat o r y coordinates be N(9). T h e n N(~o) sin 9 d 9 = N ( 3 ) sin fl d3,
(18)
where 3 and 9 are clearly defined b y Fig. 6 as the angles of emission in the c.g. s y s t e m and the l a b o r a t o r y system. A relation exists between
V,~,
FIG. 6.
Vector diagram relating velocities in the c.g. and laboratory systems of coordinates.
the expressions sin 9 d9 and sin/3 dfl which will now be derived. Fig. 6, cot 9 --
cot~=
Vcg8 V4°* sin 3
+ cot 3,
r + cos/3 ~ / l _ c o s 2/3'
From (19)
r = V~gs/V4 cg.
(20)
This gives a q u a d r a t i c in cos/~. csc 2 9cos2/3 + 2r cos/3 + r 2 -- cot s 9 = 0, -
(a) cos/3 =
-
r + 4r 2 -
(b) cos/3 = - r s i n
csc 2 9 ( r 2 -
cot 29)-
C8C 2 ~p
294-cos9
~/1 - r 2sin 2 ~o.
(21) (22)
392
C.E.
MANDEVILLE.
[J. F. L ,
C o n s i d e r the case of V~°"/Vc,. /> 1. T a k i n g the plus sign before the radical, since cos/5 is a single v a l u e d f u n c t i o n of ¢ in this case, we h a v e on d i f f e r e n t i a t i n g sin /5 d/5 = 2r cos 9 + x/1 - r 2 sin ~ 9 + ~/1 sin ~ d ¢
r'~ cos 2 ~ r 2 sin 2 ¢ -
(23)7
-
T h e n if N ( 9 ) is observed, N(/5) m a y be calculated. N o w suppose t h a t V4°./Vo,. < 1. In this case, cos/5 is a d o u b l e v a l u e d f u n c t i o n of ~. D e n o t i n g all v a l u e s of/5 lying w i t h i n t h e cone of half-angle ~r - /50 by/sa a n d all v a l u e s of/5 exterior to the cone by/52, N(~)
=
N(/5)
sin ~d~
N(~o) = 2 N ( / 5 ) [ 1
slrn ~ d ~
'
-- r 2 sin 2 ~ + r 2 cos 2 ¢~ 41 -
(24) (25) ~
r ~ s i n ~ ~v
T h e m i n u s sign before the radical of e q u a t i o n II(22b) relates to /5:, the phis sign to 152. EXAraPLE: Concerning the angular distribution of the neutrons from the reaction D + D He s + n + Q. V4~. is always greater than V~,, and the positive sign of equation II(22b) applies. r 2 = ( V ~ , . ) 2 / ( V , ~ ) 2 = E ~ / 6 ( Q + ½E~).
Substitution of this value of r in equation II (23) leads to sin fl dO E~ i cos so[-6Q + 3E1 - E~ sin 2 so]~ + (3Q + 2E1 - E~ sin~ so) sin so dso [](Q + ½E1)]~[6Q + 3E: - E1 sin2 ~o]t An alternate method of approach is to employ a vector diagram similar to that of Fig. 6. For the D + D reaction, V~ = V. = velocity of the neutron in laboratory coordinates. From Fig. 6, V. sin B = ~ sin so,
d0 sin # d¢~ sin sodso
V. cos so dso +
V 0. cost~
sin so V.~-cosO
0 V. dso Oso
'
[ 11. ~z cos so V. sin so OV,, \ V , . " q cosO + i V . ° t ) 2 c o s ~ 0so '
where
cos t~ =
(cos so)24g~. - ½EO [](Q + ~E.)]~
V. and ~ are obtained from equation II(7). oq~
III. ~
sck~rggtl~O.
The collision of a neutron with a proton is depicted in the vector diagram of Fig. 7. For this reaction, the magnitudes of the velocities 7 Compare reference (2), equation 784a. s Compare reference (2), equation 784b.
Nov., I947.]
T H E DYNAMICS OF NUCLEAR FRACTIONS.
393
satisfy the e q u a t i o n V,°* = V, °* = Vc,.
and
/3 = 2~,
(1)
If it is assumed t h a t the scattering is spherically s y m m e t r i c a l in the c.g. system, sin t3 d/~ = sin 2~ 2d9 (2) N(¢) = sin ~ d e sin ~ d9 T h e total n u m b e r of particles to be found in l a b o r a t o r y coordinates between ~ a n d ~ + d¢, would be Nr(~o) d ~ = N(~o) 2~ sin ~ d e = 87r sin ¢ cos ~od~.
(3)
i
FIG. 7.
Vector diagram relating velocities of the recoil proton in the c.g. and" laboratory systems of coordinates.
For this reaction, M1 = M~ = Ms = M4 = 1 m.u., M = 2 m.u., Q = 0. B y e q u a t i o n II(7), 2E,i = E 0 cos ~ + E1~41 - sin ~ ~, E , -- E1 cos~ ~,
dE, = -- 2E~ sin ~ cos ~ d¢, N(Es) d E , = - NT(9) d 9 = 47r/E,.
(4) (5) (6) (7)
T h e n u m b e r of scattered particles per u n i t range of energy in the l a b o r a t o r y s y s t e m is constant. IV. GEOMETRICCORRECTIONS.
T h e r e are several t y p e s of straggling which m u s t be considered when the e m i t t e d particles of nuclear reactions are observed. A m o n g these
394
(7. E .
MAN1)I~;VILLE.
[I. ['. l,
are : (a) Range straggling, (b) Angular straggling, (c) Straggling arising from target thickness. It has long been known from experiment that range straggling obeys the well known Gaussian distribution; that is, N(R)dR
OL
(1)
= -~-:- e .... ~1~ ~o)"dR, ~Tr
where N ( R ) d R is the probability that the charged particle come to rest at a distance between R and R -[- d R from the starting point. R0 is the range exceeded by half the particles and is known as the .mean range.
Ro [~ex Fw,. 8.
Differential a n d integral curves giving extrapolated and m e a n ranges.
The average value of the mean square deviation (R -- R0) 2 of the range of the particles from the mean range is given by (R
-
R
=
e - , ~ ( R - n o ) ' ( R -- Ro)~-dR
1
= 2a-~;.
(2)
When experiments are performed, the mean range of the emitted particle is the sought after quantity. This mean range is usually obtained by subtracting an appropriate correction from the extrapolated range. The extrapolated range is determined by extrapolation of the
Nov., 1947.]
THE
DYNAMICS OF NUCLEAR FRACTIONS,
395
point of m a x i m u m slope of the integral curve. Differential curves are often converted to integral curves for the purpose of this convenient extrapolation. These curves are illustrated in Fig. 8. T h e o t h e r types of straggling simply increase the distance between R0 and Rex already introduced by normal range straggling. At R = R0, the height of the integral curve is one-half its initial value. From Fig. 8, 2(R0 - Rox) = t a n x = - 4~/~, Rex -
Ro =
4-~/2a
= zx.
From IV(2), I
zX2 = ~r (R -- R0) ~.
(3)
It is clear that, for Gaussian straggling, a q u a n t i t y ~ m a y be c o m p u t e d which when subtracted from the extrapolated range will give the mean range. For other types of straggling, ~ m a y be computed in the same manner, t h o u g h an approximation is introduced when the result is taken to be Rex - R 0 , The several types of straggling will now be considered.
NCE
Fie. 9.
N(E4) as a function of E4 when the particles are emitted from a "thin" target. THIN TARGET CORRECTION.
When a thin target is bombarded, the emitted particle will emerge from various d e p t h s in the target material. Assuming t h a t the yield function of the reaction under s t u d y is constant over an energy interval as large as the thickness of the targgt, the n u m b e r of particles produced at any given d e p t h and thus the n u m b e r of emitted particles produced per unit range interval will be constant. For small intervals of energy, for intervals of energy comparable to the thickness of the thin target, the range-energy relation m a y be regarded as linear so t h a t a rectangular
396
C.E.
MANDEVILLE.
[ J - F . 1.
function similar to N(R) will describe N(E). The function N(E) is plotted inFig. 9. Let (R - Ro)~ equal the average square value of the fluctuation of R about R0 arising from the finite thickness of the thin target. Then,
fn
Rm*x(R _
(R -
Ro)~
=
Ro)2N(R)dR
~"°
(R
Ro)2KdR
=
K = constant, ](_R (R -
Ro)~
Ro)~
=
]""" -IR.i,,
(Rm.x -- Rmi.) 2
=
Rm,~ -- Rmi.
12
remembering that R0 = (Rmi, + R .... )/2.
Then
71-
AT = ~
(4)
(Rmax -- Rmin)
and Z~r is the contribution to t h e difference Roz - R0 made by the finite thickness of the thin target.
By Geiger's law, g n =
2d(log R)
d(log E)
Then, z~R = n AE R 2E'
a r =.
(146)
RO "~E'---~ =
R0 2
E0
(5)
Em,z is the energy of a particle emitted from the surface of the target. Emln is the energy of a particle emitted from the back of the target. 1° A rough estimate of R0 and corresponding E0 is necessary to determine z~r. The bombarding energy is taken to be the energy of the incident particle at the center of the thin target. ANGULAR STRAGGLING.
Observation of the emitted particle is usually made at 90 ° to the incident bombarding beam in the laboratory system. However, a 0 For values of n for this differential form'of Geiger's law, see reference 2. t ° E ~ u - Emi. is usually determined by the thickness which the thin target presents to the emitted particle. However, if the energy of the emitted particle varies rapidly with bombarding energy, and if the thin target is ~sufficiently thick to make that variation appreciable, a n additional correction for the difference between the bombarding energy at the surface and back of the target must be made.
Nov., x947.]
T H E DYNAMICS OF NUCLEAR FRACTIONS.
397
finite solid angle is subtended at the target by the detecting apparatus, and a spread in energy a n d thus in range of the emitted particle is intercepted by the detector. Since the experiment is not performed at exactly 90 °, a correction must be made. Most experiments are now performed only with "good geometry" as defined by Livingston and Bethe. 2 The angular straggling of emitted particles, collimated by tyro square apertures of side 2a and separation b, will now be computed for
111111
I
-
b Fio. 10.
good geometry; that is, a << b. From Fig. 10, the number of particles detected at the angle O will be proportional to (2a'- b tan O). From equation II(17), E 4 - - E , 90
and by Geiger's law,
R , - R,,o
20 (M~M~E~Ep°) t = _~_
(6)
. (E,-
(7)
Then, E,9--~ (R, -
R,'o)'
]
(2a - b tan O)dO
(8)
=
o*~j~ (2a -- b tan O)dO
(R, - R,,o),
2 ~ n~ = 3 b* M* (R~)2
Aa = ~
(M1M,el o----~ E,
n a Rpo ( M~MoE' ) '
] I '
(9) (ao)
398
C . E . MANDEVILLE.
IJ. F. I.
RECOIL P R O T O N S .
As has long been known, the most precise method of observing neutron energies is simply the measurement of the proton's recoil energy arising from a head-on collision with the neutron. In order to obtain su.fficient intensity, proton recoils making an angle as great as some arbitrary limit 70 are usually accepted. The straggling of the neutrons is then increased. By II[(7), it is seen t h a t assuming spherical symm e t r y in the c.g. system, the number of recoil protons per unit interval of energy is constant in the laboratory system. A rectangular function similar to that of the thin target correction then applies. By IV(5),
146 R o ~n~E - ~ o where by III(5)
AAn = }
AE
= E 4 9 ° - - E4no = E49°
AA n =
(11)
sin 2 r/. = E49°r/0 ~,
(12)-
Rondo2"
After all other corrections, the mean energy of the group of recoil protons will differ from the energy of the neutrons emitted at exactly 90 ° to the incident beam by an a m o u n t ½(E4 - E4 cos ~ r/o) = ½E,~oL This amount must be added to give the correct energy. • Were a single straggling correction involved, the mean range would be obtained by subtracting that correction from the extrapolated range. However, when more than one correction is made, such as range straggling, straggling arising from target thickness, and angular straggling, the corrections are combined so that =
+ A.42 +
+
.-
(13)
For corrections for normal range straggling and the range energy relation, see references (2) and (12). EXAMPLE: The observed extrapolated range of the alpha particles from the reaction Flg + H 1 --- 0 TM+ He 4 was found to be 6.03 cm. of standard air at a bombarding energy of 0.85 Mev. The target thickness was estimated to be 30 Key for the incident protons and 67 Key for 7 Mev alpha particles. The beam was defined by two rectangular slits having a width of 0.3 cm. and a separation of 3 cm. ; n = 3.18 (Fig. 35, reference 2), the range correction, 1.12 per cent. (Fig. 37, reference 2). From equation IV(10), the angular straggling correction is given by R,~---~=
~ ~r
E49o
/
= 1.02 \ ~ - 6 7 1 \ - - f - / \
7.1
= 0.56%.
By equation IV(5), R, 9°A---Z'r= 21 /rn62 Emax EO--~---,min= 0.36 (3A8)(0.067__ +?(0.03))__ = 0.6%. n Compare reference (2), equation 809.
Nov,, 1947.]
THE
D Y N A M I C S OF N U C L E A R FRACTIONS.
399
The final percentage to be subtracted from the observed extrapolated range is q(1.12)' + (0.56) 5 + (0,6) ~ = 1.38%, The mean range of the alpha particles is then 5.95 cm. equation" II(15), Q is readily determined.
From the range-energy relation ~.12and
ACKNOWLEDGMENTS.
The writer wishes to t h a n k professor W. E. Bennett of the Illinois, Institute of Technology for m a n y discussions and suggestions with regard to the text. He wishes also to acknowledge the interest and advice of Dr. W. F. G. Swarm, Director of the Bartol Research Foundation. ~ M. G. Holloway and M. S. Livingston, Phys Rev. 54, 18 (1938).