ENERGY REQUIREMENTS FOR NUCLEAR TRANSFORMATIONS' BENJAMIN CARROL and PETER F. E. MARAPOD1 Newark Colleges of Rugters University, Newark, New Jersey
Tmm
are several conservation requirements that must be met in nuclear reactions. The familiar Einstein relation E = Me2 is rightfully considered the Rosetta stone for all nuclear transformations. Nevertheless the conservation of charge, angular momentum, i. e., spin, and linear momentum2 are additional factors that should be considered. Most textbooks in chemistry neglect the principle of conservation of linear momentum. In overlooking this principle, textbooks frequently give false estimates of energy requirements for simple reactions. Friedlander and Kennedy's texta is one of the few in which the principle of conservation of momentum is properly considered. Historically this principle was first applied to atomic and nuclear investigations by Rutherford4to derive his famous formula for the scattering of alpha particles. Since that time momentum considerations have figured in many nuclear reactions, particularly those studied in a Wilson Cloud Chamber.s Bethe6 cites the case in which the mass of 2He5is determined by measuring the kinetic energy and momentum of the nuclei in the reaction sLi7
+ ,Hz
-
?He'
+ zHe6
To illustrate the application of momentum conservation we may consider the simple case of a lithium target , which is bombarded by alpha particles to give the reac-
The symbol Q stands for the sum of the kinetic energy of the products minus the sum of the kinetic energy of the reactantsand represents the mass equivalent of the energy released (or absorbed) in the transformation. Using the values for the atomic masses for the substances in equation (1) a value of Q = -2.87 m. e. v. may he obtained.
' Presented
at the 117th Meeting of the American Chemical Society, Philadelphia, April, 1950. a For a. discussion of the validity of Newton's Third Law ss applied to atomic and nuclear phenomena. see GEEIUOY, E., Am. J. Phys., 17,477(1949). a
FRIEDLANDER, G., AND J. W. KENNEDY, "Introduction to
Radioohemistry," John Wiley & Sons, Ino., New York, 1949. ' RUTHERFORD, E., J. CHADWICK, AND C. D. ELLIS,"Radiations from Radioactive Substances," Cambridge University Press, 1o w -.--. T o r example, see POLLARD, E. C., Proe. Roy. Soc., 141, 375 (1933).
BETEE,H. A., "Elementary Nuclear Theory," John Wiley & Sona, Inc., New York, 1947.
The usual impression students get from this information is that if one were to bombard the lithium with alpha particles having a kinetic energy of 2.87 m. e. v. the transformation in equation (1) would be possible. Consideration of the principle of the conservation of linear momentum mould prohibit this reaction. The kinetic energy of the alpha particle must be appreciably greater to permit equation (1) to proceed from left to right. The energy in excess of 2.87 m. e. v. may be calculated by writing equation (1) as
Here P, is the linear momentum of the alpha particle; the lithium nucleus is considered initially a t rest. The symbols m and M represent the mass of the alpha and lithium nuclei, respectively. The momentum P, = mu where w is the velocity of the alpha particle. Both on1 and sB'Oreceive a push in the direction of P vhen they fly apart. The same is true of the compound nucleus (sB1'). Let the kinetic energy of m and 111 m he T, and T ~ + respectively, , ~ h ~ ~ 7
+
Q
=
T , - TM+,
P,,, is the momentum of the compound nucleus. However, because the lithium is initially at rest we have, as a result of the conservation of momentum, P,
=
P.,,+, = P
Thus Tm
=
PP +
Z(d1
or
+ m)
(4)
where m = .lI
Hence or
+m
Q T," = 1 - P
The absolute value of Q is used here.
(5)
NOVEMBER, 1951
587
For the nuclear reaction in question m = 4 and M = 7 so that T, = 11/7Q. Since p < 1 equation (5) may be written as T,
= Q(l+!A
+p2+
For the 'pecia' ease where that T,
GQ
i
