588 BURB I DGE, BURB I DGE, FOWLER, AN D HOYLE A. Path of the r Process We have said that Q„decreases as more and more A nucleus of 6xed Z cannot add neutrons inde6nitely neutrons are added at fixed Z. This statement ignored neutron even in the presence of an intense neutron Qux. The pairing effects. Thus Q„is higher for odd /=A —Z 5 to 3 Mev on than is energy of binding of each successive neutron becomes by 1. (depending A) it for even X=A — This means progressively weaker as more and more neutrons are Z. that instead of a simple monotonic decrease A added, until ultimately the binding falls to zero. This of Q„with increasing (Z fixed) we have two decreasing sequences, one for 1V the sets an upper limit to neutron addition at fixed Z. odd, Moreover, not even this limit will be attained because other for à even, the two sequences being separated about 2 Mev in the range of A of interest in this of the incidence of (y,n) reactions at the temperature by discussion. necessary to produce the neutrons by charged particle Consider the eGect of these two sequences on the interactions. The effect of (y,n) reactions can readily be understood from statistical considerations. Write criterion for neutron addition. Evidently the pairing effect makes it substantially more difficult for to n(A, Z) and n„for the number densities of nuclei A, Z Q„ the when is and of neutrons respectively, and let Q„(A,Z) be the satisfy criterion S even, and hence neutron addition must always be expected to cease when is neutron binding to the nucleus X(A,Z) in the reaction, E even. Thus if a neutron can be added at N even, then X(A,Z)+n X(A+1, Z)+y+Q„(A,Z), a further neutron will always be added, since an addi- tional energy approximately equal to 2 Mev is then such that available for binding the second neutron. On the other Q„(A,Z) =B„(A+1,Z) hand, when a neutron is added' at odd S, the situation is reversed —a second neutron would in that case be less =o'[M(A M Z) (13) Z)+M„—(A+1, j. strongly bound by ~2 Mev —so that, although the first neutron might satisfy our criterion, the second is a good Note that Q for nucleus A, Z is equal to the neutron deal less likely to do and eventually, when à be- binding energy akenpositive) in nucleus A+1, Z. so, 8„(t comes will not in fact do so. Thus we With 7 in units of 10' degrees, Q„in Mev, and neglect- large enough, expect that neutron addition will always cease with ing weight factors of order unity, the statistical balance E at an even value, so that if A is odd Z must be odd, and in this reaction is expressed by if A is even Z must be even. n(A+1, Z) 5,04 These remarks all refer to a fixed value of Z. Now Z ——' log =loge„—34.07 , log Ts+ Q~ (14) must increase by unity from time to time, since the T'9 n(A, Z) neutron-rich nuclei at present under consideration are with all logarithms to the base 10. For neutron addition all unstable against negative beta decay, the lifetimes — to be highly effective we require n(A+ 1,Z)/n (A,Z) ))1. being 0.01 10 sec. When Z thus increases, the energy When n(A+1, Z)/n(A, Z)~1, neutron addition is less of binding of the next neutron increases quite sub- stantially, so that one or more neutrons can again be effective. When n, (A+1, Z)/n(A, Z)((1, neutrons are scarcely added at all. As an approximation, adequate added. Once again, however, neutron addition will cease at some even X. Neutron addition cannot then proceed for the present discussion, we may consider that neu- — trons are added or not added according to whether until Z increases by a further unit and so on. Ke therefore obtain a clear picture of how the nuclei evolve, n(A+1, Z)/n(A, Z) & 1 is satisfied or not, i.e., according with to whether by repeated neutron addition interspersed p T9 processes, the nuclei always being obliged to "wait" for Q„) (34.07+-', logTg —loge„) (15) a P process when neutron addition decreases Q„to the 5.04 limit allowed by our criterion. The process is shown schematically in Fig. V,2. A detail of this figure for or not. With F9~1 and N„~1024as found in is satished 118(A(132 is shown in Fig. V,3. Sec. III F, we have The s and r processes dier in a very crucial way. In &2 Mev. Q the s process, neutrons are made available very slowly neutron is low. In such Larger Q values would be required at higher tempera- so that the density always tures or at lower neutron densities. n(A+ 1, Z)/n(A, Z) circumstances the addition of neutrons to the nuclei is is very sensitive to small changes in Q or T9. Under controlled by their (n,y) cross sections. That is to say, the conditions discussed above 5.04Q„/T9=10.Hence the various nuclei compete among themselves for the a ten percent change in Q„,namely by 0.2 Mev, or a capture of a slow trickle of neutrons, the abundances oI ten percent change in TQ, namely by 10 degrees, will the nuclei being governed by the ffow equation (11) a factor of ten. Con- change n(A+1, Z)/ill'(A, Z), by dn(A)/dt = —)i„(A)n(A)+)„(A—1)n(A —1), versely, for given Q, Tg, and Z the distribution in abun- dance will exhibit a sharp peak where n(A+1, Z) where n(A) is the abundance of that beta-stable isobar =n(A, Z). This peak will be at most two or three units at A which occurs in the s-process capture path. In the in A in width. r process, on the other hand, we are concerned with a SYNTHESIS OF ELK M ENTS IN STARS situation in which there is no neutron shortage, and of A that occur between the waiting points. This treat- neutron addition is limited not by (N,y) cross sections ment is of course approximate, but it is thought ade- but by (p,rt) competition and by "waiting" for P quate for the present purpose, since the present analysis processes to take place. Essentially, we assume that contains other simplifications that are explained at a equilibrium is reached between the rapid (rt,&) and later stage. We arbitrarily take the spread from a given (y,n) processes, viz. , e, ~y, n, and that the slow beta- waiting point at A to the next higher waiting point at decay processes constitute a slow leakage from this A+AA rather than in the reverse direction. Physically, equilibrium. The corresponding Qow equation in the r this corresponds to the assumption that, in freezing, the process, derived from the general equation (10), takes material accumulated at a waiting point with given A, Z the form is spread out, as y radiation dies out, by final neutron captures over an interval in A up to the waiting A for de(Z)/dt= )p(Z) N—(Z)+Xp (Z 1)rt(Z ——1), (16) Z+1. The equation for the equilibrium between (n,p) where e(Z) is the density of that isotope of element Z and (y,n) reactions is much more sensitive to T, (linear at which waiting for negative beta decay occurs, i.e., dependence), than to e„,(logarithmic dependence). the isotope for which Q„2Mev. The quantity Xtt(Z) Thus, as T and e„decrease, the equilibrium is displaced is related to the beta-decay mean lifetime, vp, and half- toward slightly greater A values, representing the cap- life, ts, by Xs= 1/rs=0. 693/tz. The condition for steady ture of the last neutrons. Qow in the r process is The solution of the dynamical problem in the r process falls into two parts. The first part consists in a Xz(Z)e(Z) =Xp(Z —1)n(Z —1)=const determination of the track shown schematically in Ol Fig. V,2, the second consists in estimating the beta- e(Z) ~Xp '(Z)=rp(Z) decay waiting time. The relative abundances given by This is the analog of the steady-Qow condition in the steady Qow then follow immediately from s process, namely, rp(A, Z) dZ &„(A)n(A)=X„(A—1)e(A —1)=const e(A) ~ = rs(A, Z) or hA dA e(A) ~ X„'(A). where rz(A, Z) is now the mean beta-decay lifetime at Just as the abundances of the s nuclei suggest that the waiting point, A, for a given Z. A determination of steady Qow has occurred in the s process, so the abun- the track demands a precise specilcation of the cri- dances of the r nuclei suggest that steady Qow has terion for neutron addition. This criterion can be written occurred in the r process. as Q„)Q„whereQ, is to be a specified quantity, of The rt(Z) given by the flow equations cannot immedi- order 2 Mev, which depends on n„,T in accordance ately be interpreted as atomic abundances. At first sight with the inequality (15).