RICE UNIVERSITY
NUCLEAR STATISTICAL EQUILIBRIUM
AND THE IRON PEAK
by
Kem Hainebach
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
Thesis Director's signature:
Houston, Texas ABSTRACT
NUCLEAR STATISTICAL EQUILIBRIUM AND THE IRON PEAK
Kem Lawrence Hainebach
The equations governing the abundance distribution of the nuclear species (atomic nuclei) under conditions of statistical equilibrium of the strong nuclear interaction are rederived and discussed. An algorithm for their solu¬ tion on a computer is presented, giving abundance distribu¬ tions as a function of temperature, density and overall neutron-proton ratio.
A computer search of nuclear species abundance dis¬ tributions and mixtures of two distributions is carried out in an attempt to match equilibrium distributions, after weak interaction decay, of certain "iron peak" species to their observed solar system distributions. A mixture is found which reproduces the solar system abundance distribution
54 56 of Fe ' '57,58> Ninety-nine percent of the Fe~*^ is produced in nuclear statistical equilibrium as Ni 5 6 , lending hope to the proposed observation by gamma ray astronomy of the decay of Ni 56 to Fe 56 following a super¬ nova event. 266 C5llm, SBuubcvfi bu bid) atfcufaftC OB bcr fommcnbctt (9cfcf;id)lcn: 2)aC ©c[c(3 beC Scltenaflc ftomiten voir ?mei itid)t bcrnicBten. ©Wjt«. 9?cmicn foil i$, fagt i()v, tncitte knight im SiebcrreicBl Khoren, bie il)r feib, id) hunt ©te am hjcnigficti t>ott end). gragt mid) nad) ber Shtgcn garBe, Rragt mid) uad) ber ©tintmc £ott, ftragt itad) <$attg imb $£att} unb Raiding, 21$, unb maG meift i$ bafcon. 3ft bte ©onne nid)t bie DttcITe Me8 2eBcnC, atlcC SicBt'e! Unb mac miffett fcott berfelBefc 3$ unb mtb atfe? — nic^ts. gttbrlegiidj. 5Dit meiftt cC too^l, id) BlciB’ biv fent Unb matty' biv feitten Summer, Gin £rauiitBilb aBer jog' id) gent 2ftand)mat burd) bcinen ©glummer. Gin latter SBinb Bemegt baC $)iol)r, Unb BlaftBlau BliiBt ber glicbcr, 3d) fiiffe bir auf 2ttunb unb Dfyr 2)aC fd^iJnftc tnciner 2icber. 2)tt fpricbft mit ntir 311111 erflcitmal, 2D?eiit Gott, nic^t fo gcfcBnrittbe, ©0 jaud)$t ber 23ad) I;iuaB itiC ©0 rauftyi eC in ber £itibe. SCBir merbett bieC oBtt’ Uttterlaft 3m £i»tmcl cittntal BaBcn, SBcrbriefjlid) i(l baBci tiur, bag .©ie fritter unc BegraBett. TABLE OF CONTENTS
Page I. INTRODUCTION 1
II. THE EQUATIONS OF NUCLEAR STATISTICAL EQUILIBRIUM 3
Existence and Uniqueness of Solutions 14 Numerical Method 16 Application of the Method to the Problem.... 18
III. NUCLEAR STATISTICAL EQUILIBRIUM AND THE IRON PEAK NUCLEI 20 Fitting of Abundance Ratios: A Norm 22 NSE as an Approximation to Freeze Out 25 One and Two Zone Models: Mixing 27 Results 39
IV. CONCLUSIONS 55
ACKNOWLEDGEMENTS 66
REFERENCES 67 LIST OF TABLES
Page
1. 1964 Atomic Mass Table 32
2. Table of Two Zone Mixture of NSE Distributions 7 3 at Tg = 3, p = 10 g/cm , = 0.0837, = 0.0037, a = 0.013 40
6 3 3. NSE Distribution at Tg = 3.8, p = 3.1 x 10 g/cm , r\ = 0.069 45
4. NSE Distribution at T = 3.0, p = 3.1 x 10^ a/cm3, T) = 0.0 69 50 LIST OF FIGURES
Page
1. (none)
2. Graph of Two Zone Mixture of NSE Distributions 7 3 at Tg = 3, p = 10 g/cm , nx = 0.0837,
T|2 = 0.0037, a = 0.013 40
6 3 3. NSE Distribution at Tg = 3.8, p = 3.1 x 10 g/cm , T) = 0.069 45
6 3 4. NSE Distribution at Tg = 3.0, p = 3.1 x 10 g/cm , Ti = 0.069 50
NSE Abundances (Mass Fractions) vs. T), for 7 Tg = 3, p = 10 ; Stable Isotopes of the Elements:
5. Titanium (Ti) . 56
6. Vanadium (V) 56
7. Chromium (Cr) 56
8. Manganese (Mn) and Cobalt (Co) 56
9. Iron (Fe) 56
10. Nickel (Ni) 56
11. Copper (Cu) 56
12. Zinc (Zn) 56 I. INTRODUCTION
This study of nuclear statistical equilibrium extends the work of many previous investigators, one of the more recent studies being the work of Clifford and Tayler (1965)
The present study uses the experimental values for nuclear binding energies given in the 1964 Atomic Mass Table by
Mattauch, Thiele and Wapstra (1965) whereas Clifford and Tayler used the 1960 Nuclear Mass Table by Everling, Koenig
Mattauch and Wapstra (1960).
We have developed a computer code which solves the equations of nuclear statistical equilibrium, giving nuclei abundance distributions as a function of the three indepen¬ dent parameters: temperature, nucleon density, and neutron excess. The code is also part of another code which was developed to search for linear combinations of equilibrium distributions which approximate the solar system distribu¬ tion of certain nuclei in the iron peak, specifically in' the range of elements between titanium and zinc (atomic numbers 22 to 30). This search was undertaken as the next reasonable step following the failure of any single equi¬ librium distribution to produce a good fit to the solar system iron peak distribution. 2
Attempts are made to fit the solar abundances of the four stable iron isotopes with linear combinations of NSE abundance distributions. The possible consequences for gamma ray astronomy are discussed. 3
II. THE EQUATIONS OF NUCLEAR STATISTICAL EQUILIBRIUM
We repeat, with slight change in notation, the sketch of the basic equations given by Clifford and Tayler (1965).
In nuclear statistical equilibrium (NSE), the number density of any species is governed by the equation
( (Eir“Mi)//kT Vn. E (1) 1 r where the particles are contained in volume V, n^ is the number density of the particles of the i kind. The sum is over all (relativistic) energy states E^r, internal and kinetic, of species i, and includes the 2j + 1 spin states of any-internal energy level (J = spin). Mi is the chemical potential of species i, defined by
Mi = (Hr) S-V'N <2> where E is the average relativistic internal energy of all particles in volume V; Ni = Vni ; the subscripts S,V,N denote entropy, volume and numbers of all other species held constant. The plus and minus signs refer to Fermi-
Dirac and Bose-Einstein particles, respectively. 4
If species A^ react to form species (and vice versa) as
E a.A. E 8 .B . (3a) 1 1 1 j 3 D then, from the definition of chemical equilibrium (see e.g., Reif, Ch. 8), their chemical potentials obey the relation
2 aiMi = 2 (3b)
At the temperature and densities of interest in this study, the nuclear particles are non-degenerate and non- relativistic. This results in several simplifications.
One can write. E. =“e. +m.c 2 ;e. = pT/2 2m. + lr lr I ' lr *1' 1 internal energy terms where ITU is the mass of a nucleus of species i. 2 M. N, , ( ) denotes 1 3N. (< >) 1 energy average
2 “ aST [< sNj(^7 + mjc )>]
= < + m. c 2m^ 1 5
The term ) can l>e looked on as a non-relativistic 1 2 chemical potential. Denote it by U. . Then M. = u. + m.c . l ill Then equation (3b) becomes
2 2 S or. u. + £ a.m.c = £ p.|i. + £ p.m.c i 1 1 i 1 1 3 3 3 3 3 3 or S = £ P-U. - Q (4) i j J J where 2 2 Q £ or.m.c - £ p.m.c (5) i 3 3 3
Another simplification is that equation (1) goes over to the Maxwell distribution
1 ^n. = £ exp[-(Eir-M.)AT] = £ exp^e^-u^AT]
= expAT^ £ exp^-eirAT) . (6)
The sums of kinetic and internal energies can be separated and the sum over kinetic energies converted to an integral by allowing one state per volume h in phase space. Then 6
U.AT e T Vn. e 1 S (2J +l) - int,lA 1 r
-p2/2m.kT 4nV 2 3 p e dp (7) h where p = momentum and we have multiplied by the density of states in phase space and integrated over all of phase space contained in the volume V.
The integral in equation (7) can be done, giving
li.-AT t ^ \3/2 n^ = e (ju^[2TimJcT/n j (8) where e " iintnt rr^ a)± = S (2Jr+l) e ' y (9) r
tTi Jr = spin of the r excited state.
In the present study, temperatures are encountered at which the excited states of some nuclei become impor¬ tant, and these states are included in the statistical weights UK . The temperature dependent partition func¬ tions (partition function = statistical weight) of
N. Bahcall and W. Fowler (1970) were used, viz. the approximate curve fittings of the form (in their notation) 7
GL = g? + K± exp(-a±/T9) , (10)
r where , g? is the ground state partition function, and a^ are fitting constants. (See Table 1.) Equations (4), (5) and (8) relate the abundances of all species to the abundances of free neutrons and free protons, or, through these, to the abundances of any two chosen species. From equation (4) one has
Vp + Niun = ui - Qi (ID
Qi " (Zimp + Nimn - mi) °2 (12) and Z^ = atomic number of species i, ISL = neutron number, m = proton mass, m = neutron mass, and equation (12) Jr ** defines Q^, the binding energy of species i.
Exponentiating equation (11), one obtains
z.|i AT N.U AT U.AT -Q.AT e 1 p e 1 n = e 1 e 1 (13)
From equation (8) one obtains expressions for the U±AT e , which when substituted in equation (13) yields 8
N. n z. n n
[ 2 3/2 J L 2 3/2 U) (2™ kTA ) rr u>n(2nmnkT/h ) r
n. -Q.AT x (14) 2 rimikTA 2)372
Since JQ = for both free proton and neutron, and there are no excited states, one has from equation (9) u) = tu =2, P n giving
m. 3/2 Q.AT Z „ 2/ 2 /2nkT\ ^ ) / i \ 6 n X n 0 z N —r n n i (_m . im_ i) A. p n P n (15)
If one chooses temperature (T) as an independent parameter, then equation (15) gives an expression for n^ as a function of two unknowns, n and n . Two equations p n in these two unknowns are needed to determine them. Once determined, they determine all the other n^ through equation (15).
One of these two equations is usually chosen to be an expression of the mass density in terms of equation
(15). Mass however is not a conserved quantity, so this choice can lead to problems when one later wishes to cal¬ culate energy per some conserved quantity. We have 9
therefore chosen to use an equation expressing the conser¬ vation of nucleons (baryons). Let NN = total nucleon number density, N1SL = number density of nucleons bound th up in nuclei of the i kind.
One has
NN. = A.n. (16) .1X1
NN = E NN. = S A.n. . (17) x. X l.11
The sum is over all species, including free protons and neutrons.
We give this quantity units of mass density and obtain an approximate expression for the mass density by multiplying by the atomic mass unit Mu = 1/a, a = Avogadro's number. The approximate mass density, p* , is
pi* = (NN^) Mu (18)
A n p* = (NN) Mu = S i iMu (19)
The actual mass density is p = S n^m^ • P* approximates i it to the extent that nu =* A^MU • The error is less than 1% for all species.
p* is the second free parameter, and equation (19) with equation (15) is one of the two equations in n^ and nn« One assumes that strong reactions are occurring so fast that weak reactions do not have time to occur signi¬ ficantly. The number of neutrons relative to protons are therefore fixed. We define the third free parameter to be the neutron excess, r| 7
T) s (N - Z)/(N + Z) = (N - Z)/A (20) where N, Z and A are total (bound and free) neutron, proton and nucleon number densities, respectively, t] ranges from -1, corresponding to all protons, to +1, corresponding to all neutrons. So
E(N.-Z.)n. 1 1 -L r\ (21)
Equation (21) with equation (15) gives the second equation in n and n p n Before solving the system (19), (21), we shall make change of variables, from the number densities n^ to dimensionless variables. Equations (18) and (19) give 11
Dividing by p* ,
2 (p±*/p*) = 1 . (22)
Now define the nucleon fraction as
xi = p.Vp* (23)
This is the fraction of nucleons bound up in nuclei of til. the i kind. Then equation (22) becomes
EX. = 1 (24) i
Now define another quantity, , called the nucleon fraction per nucleon:
Y. = X./A. = yp.*/pM*A. = A.n.M /p*A.M = n.M /p*M i'- 11 i 1 l l u 1 l u
= atomic mass unit = 1/a, a = Avogadro's number ; (25) Equation (24) becomes
SAY = 1 (26) 1 and equation (25) gives
n. = p*Y./M = p*aYy . (27) i w x' u l 12
Consequently equation (21) becomes
Z(N.-Z.)Y.p*a N z )Y .1 x f i- i i _x X] = S AiYip*a | AiYi which by equation (26) becomes
i - ?
Using equations (27) and (15) we obtain
Yi - !
Q / kT Z/Z 3/2 i ’ z. N. ^2trkT^ m.m. v - e- -j.Y a Y ~ip a Yjp.a = (—zv Nj ) -A—-( p»* ) ( n * ) m 1 m 1 P n or,
Qj/kT ,2T,kT 3/a .“1 Ai_1 Z. N. Y. = a 1 l “i (—) (• Z?NJ A—A. (»* ) * WP m lm 1 2 1 P n (29)
Define:
3 2 (1 A J 3/2 xA. —1 ^2rrkT^ ^ “ i ( m_.m± v^ -' e f \ Gj^tT.p*) (T) a - »i (—) (—-hr)l l -K— (p* ) m m P n (30) 13
Equation (29) becomes
(31)
Equation (31) with equations (26) and (28) give the system
Z S A.Y. = 1 where Y. Y . l l i l P (32) ?
This is a non-linear system of two equations in two unknowns, Y and Y . P n 14
Existence and Uniqueness of Solutions
The solution by Newton's method of y = f(x) = 0, where x and y are real numbers and f is a real-valued
function of a single variable, is familiar. One iterates
to a solution x* by the formula
xn+l = xn ' f*'131!!1’’1 f
If the method works, it converges to x* such that
x* = x* - [f /(x*)]“1f (x*) so f(x*) = 0 .
Newton's method is easily generalized to finding the solution of y = f(x) = 0, where x and y are vectors and f is a vector function of a vector. Here we assume that x and y have the same dimension. The derivative of 15
a vector is just the matrix: f'(x) = (df^/Sx^) and the inverse [f/(x)] ^ is just the matrix inverse.
If f and f' satisfy certain conditions, then there exists a unique solution to y = f(x) = 0, and Newton's method converges to that solution with known accuracy.
This has been proved by Kantorovich (1952). A simpler proof by Tapia (1971) has recently appeared. The Kantoro¬ vich theorem is difficult to apply in practice, and in this case not really necessary, since our problem is phy¬ sically founded. That is, given any temperature, density, and neutron-proton ratio, there does exist some unique nuclei abundance distribution such that the nuclear reac¬ tions going on do not change that distribution as time goes on.
-Thus encouraged, I have used Newton's method to find solutions to the equations of nuclear statistical equili¬ brium without bothering to show in each case that such a solution exists and is unique. 16
Numerical Method
Rewrite the system (32):
Z. A. A.Y XY 1 1 E i p n i (33) Z. A.-Z. E(A 2Z.) Y XY 1 1G- i 1 p n l r\ i
Given T, o* and r) , we solve the system (33) by applying a modified Newton-Raphson method, a description of which follows.
Suppose one wishes to solve the system
cp(x,y) = 0
♦(x,y) = 0 (34)
Expending cp and f in a Taylor series about a point
a (xc / YQ) > nd letting cpQ = cp(x0#y0), etc., one has
2 0 = cp = cpQ + |£) Ax + ||) Ay + 0 o * o
2 0 = * = *Q + |~) Ax + f£) Ay + 0 (35) o x o
2 where Ax = x-xQ , Ay = y-yQ , and 0 indicates terms of order 2 and higher in Ax and Ay. If (xQ,yo) is sufficiently 17
2 2 near the solution (x,y), then 0 =** 0. Setting O =0, the system (35) becomes linear in Ax and Ay and may be easily solved for these quantities in terms of cp , i|r and their derivatives, all evaluated at (x , y ). An o 2o approximate solution (x^,y^) results, where x^ = XQ + Ax, y^ = yQ + Ay . The process is then repeated using
ace terat ns P^ (VY0) * ^ i° continue x until an approximate solution ( n'Yn) is obtained such that cp(xn,yn) 0 and \]r(x ,yn) “ 0 to desired accuracy. 18
Application of the Method to the Problem
Writing
"VV - l AiGiYpZiYnAl ^ - 1
Z. A.-Z. 1 1 1 S(A.-2Z.)G.Y Y - 1 * the system (33) becomes oVV - 0 (37) ♦ Differentiating the system (36) gives 5cp 5Y_ - acp “- S A. (A.-Z.)Y. 3Y Y . l l l l n n l at — E(A -2Z )Z Y 5Y i i i i P 1 — E(A.-2Z. ) (A -Z )Y dY. i i i (38) n n l 19 Thus the values of cp , \|r and their derivatives are known explicitly for any point (Y , Y ) - P ^ In practice it was necessary to take each new approximate solution to be: X = X ,, + e(Ax) where new old 0 < e < 1. eis decreased whenever the process shows signs of not converging. For these calculations, the iterations are continued until |cp| < 10~8 and |i|f/r)| < 10~8. Once Yp and Y^ are found for a given T, p* and ri , Z. A.-Z. 1 1 (31): Y± = Yp *Yn G±(T,p*), and (25): X± = A±Y± give the abundances of all other species. The 207 nuclei (0 £ Z £ 30) in the 1964 Atomic Mass Table (Mattauch, Thiele and Wapstra (1965) were used in the calculation. 20 ' III. NUCLEAR STATISTICAL EQUILIBRIUM AND THE IRON PEAK NUCLEI The theory of stellar nucleosynthesis postulates that the observed abundances of the nuclear species (the elements and their isotopes) are primarily the result of the nuclear processes occurring in stars. One assumes that the matter in a star undergoes a certain amount of nuclear processing and is then dispersed, perhaps by a supernova explosion which causes further processing. The processed matter mixes with the interstellar gas and dust and eventually becomes part of another accumulation of matter, e.g., a planet or star. The stellar evolution approach to the problem is to follow the nuclear and hydrodynamic life of a star to obtain"the resultant distribution of nuclear species, and then to compare this with the distribution observed in the solar system and in stellar atmospheres. Another approach is to examine the results of certain nuclear processes in an attempt to find a nucleosynthesis process which gives the observed abundances and, if successful, find a possible stellar setting for the con¬ ditions which produced this success. This is the approach I have chosen. In general, one does not expect any single proces to produce all the nuclear species in their observed abundance ratios. 22 Fitting of Abundance Ratios: A Norm To compare the calculated and observed distributions, one usually looks at tables or makes graphs, plotting the logarithm of the species' abundances (calculated and observed) vs. atomic weight. One discerns by eye how good a fit has been obtained. While this is a valuable tool, it is also desirable to have some quantitave measure of the fit. Given any two sets of abundance ratios of the same set of nuclides, one wishes to produce a non-negative real number which is a measure of the closeness of the two vectors. One wants a norm. A well known set of norms on finite dimension vector spaces is the ln(p) norms (pronounced "1-p"): 1 n (P)^ for a difference vector ; n and p are finite positive integers; n = dimension of the vector space. I have chosen p = 2, giving the ordinary Euclidean distance. Also I have used the equally good norm i=l Clifford and Tayler (1964) used such a measure of the fit of equilibrium distributions of elemental rations to observed ratios in certain metal deficient red giants. 23 Let be the calculated abundance of species i (abundance = nucleon fraction = = n^A^/S^n^A^ , where n^ = number density of species i, A^ = atomic weight of species i, and the sum is over all species.) Let be the observed (e.g., solar system) abundance of species i. Then where the double sum is over all chosen species, is a quantitative measure of the fit. Each term of the sum is non-negative, and the sum is zero if and only if all the abundance ratios exactly match the observed ratios. As the observed abundance ratios of species which are isotopes of a single element are known with less error than the ratios of species belonging to different elements, it is desirable that the "error function", i.e., the above sum, depend more strongly on isotopic ratios than on elemen¬ tal ratios. To this effect we have introduced another term, a "weighting factor", into the error function EF where EF = error function and the are the weighting factors. The weighting factors might be set to 1.0 for 24 isotopic ratios and to 0.1 for elemental ratios. They can be altered to place stress or lack of stress on any ratio; e.g., certain W.. may be set to zero to eliminate those ratios from the Siam. 25 NSE as an Approximation to Freeze Out We have used this technique to search for equilibrium abundance distributions, and linear combinations of such distributions, which approximate the observed distributions of certain species in the iron peak, viz. certain stable isotopes of elements 22 (titanium) through 30 (zinc). The unstable species which exist under conditions of nuclear statistical equilibrium (NSE) are assigned to the stable isobars to which they decay. (See Table 1.) The search is carried out at temperatures (Tn = 3) at which at least some of the reactions have fallen out of n equilibrium. (The notation Tn is defined as Tn = T/10 °K.) These distributions are taken as approximations to the actual distributions resulting from a "freeze out" of material initially at much higher temperatures (T_ ^5) at which true NSE does prevail. Doing an actual freeze out calculation involves following the strong and weak nuclear reactions and abundances as stellar material cools and expands (as in a supernova explosion). As this happens, the reactions, whose rates go as high powers of temperature, freeze out, i.e., they slow and then cease to occur. The abundances are thereafter no longer characteristic of NSE at the ensuing temperatures, but may be approximated by NSE abundances at higher temperatures, viz. at some "freeze 26 out" temperature. The crude validity of this approximation has been demonstrated by performing freeze out calculations. See e.g., Arnett, Truran and Woosley (1971). They show that Tg = 3 is a typical freeze out temperature. We will come back shortly to how well the approximation holds. 27 One and Two Zone Models: Mixing The distribution of the nuclear species in NSE is determined by three independent parameters, which may be chosen to be temperature, nucleon density and neutron excess [r| = (N - A)/(N + Z), where N and Z are neutron and proton number densities, both bound and free.] A simple thing to do is to search over temperature, nucleon density and neutron excess for a single NSE abun¬ dance distribution which fits the solar system distribution of iron peak nuclei. By a fit, we mean that isotopic abun¬ dance ratios are correct to within 10-15% and elemental ratios to within a factor of about 2. This would have to be where Fe^ or one of the isobars which decays to it (Co56 or Ni^) is dominant, since Fe^ is the most abundant of the iron peak nuclei in the solar system. Figures 5 through 12 are graphs of NSE distributions of the stable iron peak nuclei, after all isobars have undergone weak interaction decay. The common logarithm of each species' nucleon (or mass) fraction, i.e., = p^*/p*, is plotted vs. r| (neutron excess) for 0 £ t] S 0.15 , for the temperature Tg = 3 and density p* = 10 7g/cm 3 . As these figures, especially Figure 9, show, Fe~*^ dominates (X5(- =“l)at r] 0 ± 0.01 , where NSE produces 28 Ni^®, and at r| =“ 0.071 ±0.01 , where NSE produces Fe^. By consulting the solar system ratios to Fe^ of the various iron peak species listed in Tables 2, 3 or 4, one can see in Figure 9, with the help of a slide rule, that KA KC. c;7 at t) 0.002, Fe ' ' are approximately in their proper ratios, but the isotopic ratio Fe^^/Fe^^, which should be approximately. 3.7-3 x 10 is many orders of magnitude lower. Another approximate fit comes at r) =* 0.069. Here all four iron isotopes are made in approximately the correct ratios. However Figure 10 shows that Ni^, which should be approximately 1.7 x 10 as abundant as Fe° , is about 10 as abundant, a bad overproduction. Figure 5 shows that Ti 48 , which should be about 1.8 x 10”_3 as abundant as Fe 56 , __ g is about 3 x 10 as abundant, a bad underproduction. Clearly no good fit of NSE to solar distributions can be . 7 3 obtained at Tg = 3, p* = 10 g/cm . A fit at Tg. = 3.8, p* = 3.1 x 10^ g/cm^ is described later in this paper. It too suffers from various over- and underproductions when the iron isotopes are in their correct ratios. Extensive searches over T, p* and r) by many inves¬ tigators have shown that no single NSE distribution can fit the solar distribution of the iron peak. As different stars and different parts of a single star may be expected to have different temperature, density 29 and nuclear processing histories during an explosion, it is logical to try to mix the abundance distributions resulting from such different histories in attempting to derive the solar distribution. The simplest mixture is of two distributions, and this is the first mixture we have tried. These combined, or mixed abundance distributions are of the form X. = a.X'!' + a_X? 1 1 i 2 I where the superscripts refer to different zones, and al'a2 ^ 0 (one may not subtract material, only add it). By "zone" we mean any set of T, p*, r) . Since for any abundance distribution, X^ , it is true that E X. all species i. i 3 3 then for the mixed distribution, one has that + a 2 ' or a 2 1 30 Therefore Xi = ax* + (l-a)X? , where the subscript of "a", the mixing parameter, has been suppressed. We have used the error function, developed above, to search for mixed abundance distributions which approximate the observed solar system distribution. We have used, for this purpose, the abundance distri¬ bution of solar, terrestrial, and chondritic meteorite material given by Cameron (1967) with later revisions from Cameron. For these values, as ratios to the abundance of C CL Fe , see Table 2, 3 or 4, under the column labeled "SS RATIO". Note that these are abundances by mass. . ^Because of chemical fractionation, the relative abun¬ dances of the elements vary greatly from place to place. The ratio of any two species of different atomic number ("elemental" ratio) is somewhat uncertain because of this, and is generally taken to be known only to within a factor of about two. m Chemical fractionation has little effect on the iso¬ topes of a single element. Isotopic abundance ratios are therefore much better known, on the order of 10% accuracy. 31 For the above search, we chose a fixed temperature and nucleon density for both zones and allowed only the neutron excess of each zone to vary. The mixing parameter "a" is also allowed to vary. The search is carried out on a computer, using a function minimization algorithm. The solution, i.e., the neutron excesses of the two zones, the mixing parameter, and the resultant mixed dis¬ tribution, varies with the species chosen to be included in the error function, and with the weighting factors chosen. 32 TABLE 1. Table 1 lists 205 of the nuclei in the 1964 Atomic Mass Table [Mattauch, Thiele and Wapstra (1965)], their binding energies, and their temperature dependent partition func¬ tions when available [Bahcall and Fowler (1970)]. The neutron and.proton binding energies (zero) and partition functions (one, no temperature dependence) are omitted. The nuclei are grouped by element. The isotopes of each element, given by atomic number, atomic weight, and bind¬ ing energy in MeV as defined by equation (12) are listed three to a line. Beneath all of the isotopes of any one element, temperature dependent partition functions: g? , and a^ [see expression (10) in the text] are listed four to a line, corresponding to the order of the isotopes. The chemical symbols of the elements are given to the right of each group. If any and a^ is blank, is taken to be g? . For example: o 8. 16. 127.6200 8. 17. 131.7625 8. 18. 139.8087 1. 6. 2. 10.11 1. 33 Interpretation: >16 8 has binding energy 127.6200 MeV >17 8 has binding energy 131.7625 MeV 0 18 8 has binding energy 139.8087 MeV 16 = 1/ K and a are not given. g?(8° ) 4<8°17) 6, K = 2, a = 10.11. o. ~18> = 1, K and a are not given. 9i<8° > The stable nuclei having 22 s Z s 30 are doubly underlined. Isobars of these nuclei which are converted to them by weak interactions are underscored with a broken line with an arrow. A right (left) arrow indicates that the species decays to the stable isobar of next higher (lower) atomic 48 4ft number. E.g., Cr (24. 48.) decays to Ti (22. 48.) 28.2961 s : o2 j *H ! ro 1 0r-i > C 1 ^ 1 vO•h- i • i i ! o i ' 0 ! o ; »H i ^ 1 t i 1 i j rH rH VO : OJ OJ 0 OJ O v£> 0 0 • ! 1 0 ! i r-H ! 0 i ^ 1 0 i v£> ! in 1 O' ! ro ■ 0 h- pH PH rH in in rH rH 4 o 4 6 0 • ro OJ 0 0 00 1 > ; «H ; o- ; rH ! OJ ' in | CO ! €» i 1 1 i rH : C i rH 1 O' ! co 1 co i o ' co j j j j j N rH pH OJ 0 00 vD vO 0 OJ 00 rH h- ro f- o e 0 0 0 6 0 0 0 rH rH vO OJ j O j rH < rH j PH ] OJ | ro in O 0 0 9 d ' • ! c ! 0 j ; j i ! , 0 rH O' i rH CC CO rH CO o- inOJ O (Vj4 4* 0 OJ 0 CVJ 0 F Oj C 9 ! I O l r e ! rH : 0*) ojin | rH 1 | i *-* 10 j • | 1 j 1 i ! ojrvj j on 1 * | F rH rH rH CVJ 00 rH rH OJ rH rH O' (Vi r- n r*- O OJ H rH O 4 0- in vo 00 f- CO rH rH vO CO vO CO O OJ co 4 vO UJ LU 0 0 • 0 • c » e • o N rH O rH v£> rH O vO O 0 « 0 C ! 1 i j ! ; : N rH O 0J rH rH 00 O' O' 4- CO UJ -yr e • 0 rH 0J OJ rH CO ON o CO in rH CO rH O co 4: vO 4 co 2: UJ 0 • c • c 0 • 0 « 0 • i rH in »H CO 4“ rH rH 0J 4- vD O' rH 4 2: UJ < 0 © 0 e © 6 e rH rH OJ \S) O rH rH 4' i 15. 28. 221.920 15. 29. 239.280 15. 30. 250.603 15. 31. 262.9155 15. 32. 270.8520 15. 33. 280.9553 15. 34. . 287.530 i 4 " © rH O r-H CM o rH r-H CO r-H CO CM CO 4 o LT) 4 VO © • 0 • O © © • ■ j LD CO in CO CM o 4 4 00 : e 0 s r • e 1 i 1 j i i 1 1 1 © 1 i O 1 © ! CM : 4 i co : r-H » vO ! CO ! -* 1 • 1 i in vO i * 1 vO j CO I co i ! -* ! vo ! ft i i ! CO j CM i • 1 | I CM ! I- i “H 1 • 1 ^ ! i O' in h- vO r-H v£ 4 r-t • e 00 CM CO rH ro in o 00 CO CO r-H CVJ CM CO CM O O' r-H O' ao 4 vO rH ro CM 00 O 4 vO vO CM CO 4 c © © • 0 e c 6 • i ; | o CO o CO o h- *-H r-H r-H 00 CO CM r-H CO CO vO o r-H ro O 00 co h- H ro vO vO O sO • © © e 0 0 e • e O' co GO rH r-H ] 4 r-H 4 00 CM 4 r-H rH 4* C • 0 0| 0 oo o oj CM (V r5n CM OJ (VJ £V| CM CM a^fM 0 4 Ch 4 4 •—« CM O' ro ro j ; i : Is H ^ o 4 4 O a t-i o 1 a A- ro fO O'h O' O O' 4 4 A* O' A- i l *-* 4 vO 4 © n o' aj *-< co O ro A- O' j HAW • •91 0 0 0 0 0 0 • l • 0 0 43 r-t 03 LA O' co 43 a ro 4 rH | , a 43 o AJ 43 CO •-« LA O' a LA o ro 43 | a LA CO n ro ro 4 n CO 4 4 4 4 4 i .444 • • o o 0 0, St di CM LA OO ro '0 a 4 fl444 : 4 4 L/1 A ■ •I • •! • ii o o o o • • a a 0 ro ro ... CM CM CM CM 4* GO co 4> co a a4a|oo oo c^\|ja|I'C 4 i ro co do ZZZZZZbJUJUJIijUiljJUJOOOOOOOwMHHH ZZZZZrLLU.U.lLlLli.UUUUUUUZZZZZ 2: 2 © c vo n CO CO O - rT, yv1s C\j| IT 4'; I 1 1 * mlm LO O <5 rS cc cv oJ CM (VjfvS cv CM{CV ro h- r-i | O' r-H ! H r-i H H : | H X O i *"< (\) H j c- in CM ; ; c- co co CM i ! 4* o : 1 O' ro ! O' O' X o o m CM o o in 0 ino DZDrDDIDIDIDZZZZZZlZ UUUUUUUNNNNNNN rH d O ♦I *| ( c ooi^l Results Special attention was paid to the stable isotopes of iron: Fe^'^6#57,58. ^ s ^ ^ .005 say) Ni^ is 7 3 the dominant species for NSE at TQ = 3, p = 10 g/cm , typical freeze out conditions (see Figure 9). After dis- 5 6 persal this species beta decays to Co with a half life of 6.2 days, and then to Fe^® with a half life of 77.3 days: Ni (p v)Co (p v)Fe Each beta decay is accompanied by line gamma ray emissions. If much of the Fe^® we observe is made in this manner, then these gamma ray lines might be observed immediately follow¬ ing a supernova. Clayton and Silk (1969) have discussed this possibility. If, however, Fe^® were produced as itself at TI “.071, then the synthesis of this abundant species would not be so observable. It is this possibility that I explore in the remainder of this paper. Our calculations show that the observed abundances of the stable iron isotopes can be reproduced very well by mixtures of low r](T) “ .004) and high T)(r) “ .07 - .08) NSE material, the low r| component always dominating. Figure 2 shows an attempt to match the four stable iron isotopes abundances. This can be accomplished by 40 TABLE 2. This table shows in column 3 the abundances (mass fractions) of 29 species relative to Fe^, resulting from a mixture of two NSE distributions, both at Tg = 3, p = 3 x 10 7g/cm 3 . 0.013 of a zone at r|^ = 0.0837 and 0.987 of a zone at = 0.0037 produces the mixture. Column 1 gives the rela¬ tive abundances of the species for n = and column 2 for r| = T)2 • The abundances are normalized such that columns 1 and 2 sum to column 3, allowing one to tell from which zone each species is derived. Column 4 gives Cameron's (1967) solar system mass fraction ratios to Fe^. Column 5 is log^Q(column 3/column 4). FIGURE 2. Figure 2 is a plot of the solar system (circles asterisks) and NSE (letters = first letter of chemical symbols) abun- dances relative to Fe 5 6 of the 29 iron peak species listed in Table 2. Atomic weight (46 to 70) runs the length of the page; mass fraction relative to Fe^® is the other axis, _7 running from 1 to 10 on a semilog scale. Isotopes of each element are connected by solid lines (solar system values) or dashed lines (NSE values). 41 CN W& £ 42 ---I 1*""/ I HI IM to*3cdo*i IH 10*35*6*9 /K IH 10*3056*9 IH 10*3526*9 IH 10*3006*9 IH I0*35i8*9 IH 10*3058*9 IH 10*3528*9 IH 10*3008 t IH 10*35/.i* i IH 10>305**9 ‘ IH_I0*352**9 IH I0*300i.*9 Weight IH I0*3S*9*9 IH 10*3QS9*9 IH 10*3529*9 l H 10 ♦ 3 0 0 9 * 9 IH I0*35*S*9 TH'10*30SS*9 ' IH 10*352S*9 IH !C**30d5*9 IH 10*35*9*9 IH' 10*3059*9 IH_10*3529*9 IH 10*3009*9* IH I0*35*C*9 IH I0*305C*9 JH 1 0*352C*9 IH* T0*300t*9 1H I 0*3S*2*9 IH 10*30S2 *9 IH 10*3S22_*_? C\! w PH o H fa IH 10*30SI*S IH 10*3S21*S IH"10*3001*9 IH 10*3510*5 IH 10*3050*5 10*3520*5 TH '(O'*3000*5“ IH 10♦35*6 *9 IH 10 *30S6 * 9 IH 10*3526*9 IH 10*3006*9 IH 10*35*8*9 IH 10*3658* IH 10*3528*9 IH 10*3008*9 IH 10*35** *9 IH 10*305**9 IH 10*3521*9 loOooZ*9 ' IH 10*35*9* 9 IH'10*3059*9 IH 10*3529*9 SM "10*3009*9 I 1 I 1 1 10-3000* 1 I C0-32**9" 90-3951 50-3000*l I *0-3299*9 ‘ I I 20-3951*2 I C0-3000*l I 50-3299*9 90-3951*2 I *0-3000*1 310S *WDIia3A 9011*35 10-35C5 *l * UNO TV 1H0Z I ttOH * 33VdS 1VHI N QJllUTd SI lNlOd 3N0 NvHl 3dO* S31VOlON I s= S3DNV0Nnav (•) W31SAS avlUS ONV (SIObrtAS) HUIddlimOJ 43 mixing 0.013 of a zone at r| = 0.083 with 0.987 of a zone at TJ = 0.0037. Of the four stable iron isotopes, Pe54,56,57 come from the low t) zone, while Fe^® comes from the high t) zone (see Table 2). The worst error in 57 the iron isotopes was in Fe , which was about 0.7 of its solar system abundance ratio to Fe^. As lagniappe, this mixture produces, as well as it does the iron isotopes, _ 54 „.58 - , . 62 Cr , Ni and NiT 55 Mn , primarily from the low r| zone, and made as Co 55 (see Figure 8), is the only species to be seriously overproduced, and it by only a factor of about 3. This is important, because any underproduced species can be hoped to be produced elsewhere, while an overproduction in any zone of any species, relative to Fe^, say by a factor of A, limits the fraction of solar system Fe^ than can come from such a zone to 1/A. Of the nuclei Ti , Cr , Fe and Ni which are not produced in explosive burning models, Cr^ and Fe^® are produced here.. Both are made as themselves in the high t) zone. Ti^ and Ni^ are badly underproduced. As Figure 9 shows, the Fe^^ produced in the low r| zone, which comprises most of the mixture, was made in NSE as Ni56. 44 One may now ask whether it is possible to make the four stable iron isotopes in their correct abundances by a NSE distribution at an r) where Fe^ is made as itself, 2 or by a mixture composed primarily of such a zone. B FH (1957) found such a zone (their Figure IV,1, page 579). Fowler and Hoyle (1964) found essentially the same zone, using Clifford and Tayler's NSE tables (1965). The zone C. O was Tg=3.8, p = 3.1 x 10 g/cm , Z/N = 0.872 (corres¬ ponding to n = 0.068). I have repeated this calculation at the same temperature and density, performing a computer search over r) to find the best fit to solar system iron ratios. The results are shown in Figure 3 and Table 3. The best fit to all the iron ratios, weighting all six ratios equally, occurred at t) = 0.069, virtually the same as Fowler and Hoyle's result. As can be seen, the iron group was fit excellently. However, severe overproductions, ranging from 0.5 to 1.5 orders of magnitude resulted for V , Cr , Mn , Co and Ni One may now ask how these overabundances fare in an actual freeze out calculation. Arnett, Truran and Woosley (1971) and Woosley in his Ph.D. dissertation (1971) showed a seesaw effect, depending on the details of the freeze out and the resulting free particle (neutrons, protons and alphas) density. If the freeze out is "normal", free par¬ ticle abundances decrease rapidly with time, and become 45 TABLE 3. This table lists in column 1 the NSE mass fractions 5 6 6 relative to Fe for the zone T^ = 3.8, p = 3.1 x 10 3 g/cm ,11= 0.069. Column 2 lists Cameron's (1967) solar system mass fraction ratios to Fe^®. Column 3 is log^Q (column 1/column 2). FIGURE 3. Same type of diagram as Figure 2, for the abundances given in Table 3. 46 1 2 3 ETA SS-RAHH ALOG10 < CLMN--1/CLMN- 2) TI46/FE56 6.6S8E-06 1.815E-04 1.435621 - - -Tl-47 /F-E-56 2.576E-06 l-^713E-04 -1.822622 -TI4B/FE56— 1.213E-04 1.770E-03 1.164081 -TI49/FE56— 3.881E-05 1.349E-04 -.541192 -T1-50/FE-56 2-^934E—05 1,332E~04 -.657205 -V—50/FE56 — 1-.349E-05 7.874E-07-— 1.233817 -V—51/FE56 1,682E-03 ——3.277E-04 .710270 -CR50/FE56 7-.4 96E-04 ■ 5-.918E-04 .102698 -GR52/FE56 1.S68E-01 1.197E-02 1.117284 -CR53/FE56 2.487E-02 1.384E-Q3 —1.254580 -ER54^Ft56 2-.G06E-03 3.S48E-04 ,752329 -MN55/FE56- ——5.665E-02 1.113E-02 .708362 'FE54/FE56 5.600E^02 6.-139E.-02 -.039885... ‘FE56/FE56 1.0 0 0E+0 0_ __1.. 000E.+ 0 0 0.0 000 00 LFE57/FES6 3.87-6E=02 : 2^34 E= 02 1016 03 'FE58/FE56 3.675E-03 3.684E-03 -.001021 •C059/FE56 8.857E-03 2.825E-03 .496229 -NI58/-FE56 8^721E^-03 4.17.2E-D2 _-_.67-97.94 NI60/FE56 1 .086E-01— 1.671E-02 .812823 — NI61/FE56 1 •127E-03 7.701E-04 .165446 -NI62/F-E-56 3.331E-03 2.4Q3E--83 J .14133 0 N164/FE56 — 3.854E-07 7.337E-04 -3.279535 — CU63/FE56 6.536E-05 5.551E^--04 -.929018 XU6S/FE56 5_4.42£=03 2.S62E=04 .-2.673908 ZN64/FE56 9.358E-06 1.016E-03 >2.035699 - -ZN66/FE56 1.790E-06 5.969E-04- -2.522957 - 2N622EE5 6 _2.S73E-0 9 8.97.8E^05. -4.5.42733 2N68/FE56 1.546E-09 4.100E-04 -5.423592 ZN70/FE56 1.148E-14 - - _ -1.412E-05 -.9.089892 TABLE 3 47 IH 10*3000*/. TT?T0r*'3S£6*7' IH 100056*9 IH tO *3926*9 IH 10*3006*9 IH 10*35/8*9 IH 10*3058*9 /s ItrT0^O538 9T IH t00008*9 IK 10*3SZZ*9 IH 10*30SZ*9 IH I0*3S2Z*9 IH 10*200Z*9 Tin0V3Sf9rT IH 10*3059*9 ’IH 10*3529*9 IH 10*3009*9 ’IH'T0*3SZS*9 Weight IH 10*3055*9 TH~T3*3S25*9' o IH 10*3005*9 •r-i IH 10* 3SZ6*9 IH 10*30S*7*9 g IH 10*3S2*?*9 o IH 10*300*7*9 lH"T0*3Sie-9' IH 10*305C*9 IH I0*3S2C*9 IH 10*3002*9 IH 10*35/2*9 IH I0*30S2*9 IH~10*3522*9“ IH 10*3002*9 IH *lO*3SZt *9 IH t0*30SI*9 IH t0*352l*9 IH 10*3001*9 IH 10*35/0*9 iH 10*3050*9 ---IH- 10*3S20 *9 - IH 10*3000*9 IM-tO*3S/6'S- IH 10*3056*5 l*-l 0*3526 *5* IH !0*3006*S IH 10*3S/8*S IH t0*3058*S CO IH-10*3S28*S- IH I0*3008*S W IH l0*3SZZ*S~ Pi IH I0*305/*S IH 10*352/* 5 & IH I0*300Z*5 0 IH 10*35/9*5- H IH IO*30S9*S P4 FH-l0*3525*5- !H t0*3009*S IH io*35/S’S IH 10*3055*5 IH I0*3S2S*S- IH 10*3005*S IH-I0*3S/*7*S- IH 10*30S*7*S IH -10*352*7*5 ‘ IH 10*3006*S —IH10*35/C‘S IH io*3ose*s -lH~-tO<3S2C*S- IH 10*3005*5 IH 10*3S/2*S IH 10*30S2*S IH“I0*3S22*5 IH 10*3002*S -lM-ttt*3SZt"S- IH I0*30St *S lHtO*3S2t*S IH 10*300 t *S IH 10*3S/0*S - IH 10*3050*5 IH-10*-3S20*S— IH I0*3000*S IH l0*35/6**7 - IH I0*30S6**7 IH I0*3S26**7 IH I0*3006**7 lH-I0*3S/8**7- IH 10*30S8**7 IH 10*3529**7 IH 10*3008* *7 IH 10*35 ZZ*«7 IH 10*30 5/**7 IH—I0*3S?/**7 IH l0*300/**r IH l0*35/9**r IH I0*30S9**7 lM 10*3S29.^ SH 10*3009**7 T- 1 10-3000-1 I Z0-32*>9**7 | I 20~3*/SI I ZO-3COO*l ?X1*3A 5031*35 3Dv«;s ivHi Ni ainoid si IN IOd 3N0 NVHl 3d0n S31V3IQNI = S3vMV0NDtiV ini HllSZS ttV 30s osv (s"loanAS> wniadinnoa 48 lower than their NSE abundances at the given temperatures. This causes species of A < 56 to have higher abundances r /• c O than predicted by NSE. E.g., one has X(Fe )/X(Cr ) a X(ct), where X denotes mass fraction. If X(a) is lower than NSE predicts, then X(Cr )/x(Fe ) must be higher. On the other hand, species of A > 56 have lower abundances than NSE predicts. E.g., X(Ni60)/X(Fe56) <*X(a). If the details of the freeze out are altered, i.e., temperature and density as functions of time are altered, an "alpha rich" freeze out can occur. X(of) instead of plunging, remains steady below Tg ^ 3, and just the oppo- 52 site effect from that noted above occurs. Cr is lower 60 and Ni is higher in abundance than NSE predicts. The same thing occurs for other species on corresponding sides of A --56. Thus the seesaw effect. 6 3 As at T. = 3.8, p = 3.1 x 10 g/cm , and r) = 0.069, there are overproductions on both sides of A = 56, we can¬ not be saved by doing the freeze out correctly instead of approximating it by NSE. Let us ignore for the moment the actual behavior of the abundances during freeze out. It is a general feature of NSE distributions that as the temperature is lowered at any given density and ri , more of the material present goes into the dominant species. That is, the ratio of every other species relative to the dominant species drops. 49 One might suppose it possible to put together the solar system abundance distribution by mixing material from zones where in any one zone only one species is dominant and the temperature is low enough so that no other species is overproduced relative to it. This is similar to adding up Dirac delta distributions to obtain any desired distribution. Unfortunately this will not work. The reason is that some species never dominate any zone. Whatever the neutron excess (r|), some other species is more abundant. This is the problem for Fe 57(see Figure 9).57 If Fe is made in NSE, it must be made in a zone where its ratio . 56 to the dominant species is correct. If Fe is made as itself in the region around ri = 0.069, then the tempera¬ ture must be lowered below T_ = 3.8, where the iron ratios y are correct, in order to eliminate overabundances of V, Cr 54 57 58 and Ni. When the temperature is thus lowered, Fe ' ' are then too low and must be made elsewhere. Figure 4 and Table 4 show this effect beginning to occur at T_ = 3.0, y same density and n . (No search over ri was performed.) 58 Fe may be made by mixing in material from a zone where it is dominant or nearly so, say t) S: .08 . As 57 58 Figure 9 shows, Fe is less abundant there than Fe , not 54 more abundant as it is in the solar system. Fe may be made in a zone near r) = 0.037, where it dominates. But in 50 TABLE 4. Same type of table as Table 3, for the zone T^ = 3.0, 6 3 p = 3.1 x 10 g/cm , t] = 0.069. FIGURE 4. Same type of diagram as Figures 2 and 3, for the abundances in Table 4. 51 1 _ 2 3 ETA SS RATIO ALOGIO(CLMN 1/CLMN 2) TI46/FE56 2.009E-07 1.G15E-04 -2.955906 TI47/FE56 2•606E-08 1.713E-04 -3.817696 TI48/FE56 3.935E-06 1.770E-03 -2.652966 TI49/FE56 8•074E-07 1.349E-04 -2.223061 TI50/FE56 3.239E-07 1.332E-04 -2.614233 V 50/FE56 1•586E-07 7•874E-07 -.695759 V 51/FE56 . 1.142E-04 3.277E-04 -.457718 CR50/FE56 1.325E-04 5.918E-04 -.650026 CR52/FE56 5•722E-02 1•197E-02 .679625 CR53/FE56 5.305E-03 1.384E-03 •58365A CR54/FE56 1.433E-04 3.548E-04 -.393767 MN55/FE56 2.339E-02 1.1I3E-02 .322690 FE54/FE56 5.215E-02 6.139E-02 -.070816 FE56/FE56 1•OQOE+OO 1.000E+00 0.000000 FE57/FE56 1.164E-U2 2•434E-02 , -.320409 FE58/FE56 3.969E-04 3.684E-03 -.967671 C059/FE56 1.895E-03 2.825E-03 -.173386 NI58/FE56 8.281E-03 4«172E-02 -.702267 NI60/FE56 1 .007E-01 1.671E-02 .780296 NI61/FE56 1.498E-04 7.701E-04 -.710948 NI62/FE56 6.089E-04 2.403E-03 -.596166 NI64/FE56 3.385E-09 7•337E-04 -5.335988 CU63/FE56 5.085E-06 5.551E-04 -2.038093 CU65/FE56 . 7.943E-09 2.562E-04 -4.508689 ZN64/FE56 1.195E-06 1.016E-03 -2.929510 ZN66/FE56 7.408E-08 5.969E-04 -3.906210 ZN67/FE56 8.116E-12 8• 978E-05 -7.043833 ZN68/FE56 4.B76E-12 4.100E-04 -7.924716 2N70/FE56 7.768E-19 1.412E-05 *3.259470 TABLE 4 52 I 1— I T i In to*3ooo*i I i IH 10*3 5*6*9 I I I IH 10*3056* 9 I I I IH 10*3526*9 I I I IH 10*3 00b * 9~ I. . I IH 10*35*9*9 I I I IH 10*3059*9 l I IH 10*3529*9 I I i IH"l0*3009*9 I i _J IH I0*35***9 T IH t0*3~05*~*9 IH 10*352* *9 IH T0*3Q0*."9~ IH 10♦3SA9 *9 I H~ IC ♦ 3 0 5 9 * 9 IH 10*3529*9 Atomic Weight W &D O H &4 THTO*‘3059*,7 IH 10*3529*9 SH '10*3009*9 I 10-3000* l I C0-32*?9* 9 ' I ‘ *;0-3*>5I*2 I ' 50-3000*1 " I *0-3299* V ~ I I 20-395l*2 I E0-3000M I S0-32*>9**> I 90-3951*2 I *0-3000*1 ,r _ _ ■~"WDl f«3A~90niH3S ~lO-35C5 T =" I IH(3 WIfiOZIdOH *1DV55 IYHF ^r ailTOlcT^I' I'TrcrcrjNTT ?:VHr*l>iGH-53IYTn3ra ' =' S3DNVQN09V l«) H3J.SAS dV30S QNtf (S30a«AS) wnId«IHli03 53 c 7 c A —O this zone, X(Fe )/X(Fe ) =** 10_ , not 0.4 as required. This ratio might be brought up by increasing the tempera¬ ture, but this would cause overproductions of other species 54 50 52 relative to Fe . At Tg = 4.5, both Cr and Cr are 54 overproduced relative to Fe by a factor of 2 and still 57 Fe is underproduced by a factor of 4. (This calcula¬ tion is not shown.) Also, the NSE distribution at these higher temperatures are no longer even crudely close to the correct freeze out abundances. There does exist the possibility of an alpha rich 57 54 freeze out, where the underproduction of Fe to Fe is alleviated. It is one I plan to investigate. However, if this is not the case, we are unable to 57 56 make FeD correctly if Fe3 is made as itself at high r) (*“* 0.069) . The only solution is the one found by the mixing search program (see Figure 2 and Table 2). Fe^ is made primarily as Ni^ at low r) (r| = 0.0037). This zone produces Fe^'^'^ correctly as well as Ni^, with 55 an overproduction of Mn by a factor of 3. This overpro¬ duction is troublesome, but not disastrous (dis-aster = evil star) since it is the only stable isotope of Mn, and its ratios to other species are therefore necessarily elemental ratios and uncertain vaguely by a factor of 2. 54 5 8 The lack of Fe in this low r) zone was remedied by the inclusion of 1.3% of a zone at t| = 0.0837. Fe^ still dominates in this zone, so a small but noticeable amount of Fe 5657and Fe as well as the58 required amount of Fe were added to the mixture. 55 IV. CONCLUSIONS In conclusion, if NSE is an adequate approximation to what really happens to nuclear abundances during a stellar explosion, and if the four stable iron isotopes are to be made in such a process, then no single zone does this without seriously overproducing other species, and a mixture of zones is necessary. The only mixture that seems to work requires that about 99% of the Fe^ be made in a low rj zone as Ni^. The possibility of detecting the synthesis of the important species Fe^ by gamma ray astronomy therefore remains. 56 FIGURES 5 — 12. These figures are plots of logarithmic mass fractions of 29 species as a function of ri , 0 ^ r) ^ 0.15, all at 7 3 T = 3, p = 10 g/cm . These are abundances after weak interaction decay. Explanation of the plotting symbols: 46 „.58 FIGURE 5. A = Ti FIGURE 10. R = Nl 47 60 B = Ti S = Ni 48 61 C — Ti T = Ni 62 D = Ti49 U = Nl XT-64 E = Ti50 V = Nl 50 „ 63 FIGURE 6. F — V FIGURE 11. W = Cu „ 65 G = v51 X s= Cu 50 ^ 64 FIGURE 7. H — cr FIGURE 12. Y- =: Zn I = cr52 Z = Zn66 „ 53 J = Cr 1 = Zn67 D54 68 K = Cr * 2 = Zn 3 = Zn70 FIGURE 8. L __ Mn55 59 Q = CO „ 54 FIGURE 9. M Fe 56 N = Fe 57 0 == Fe P __ Fe58 57 The various parts of the abundance curve for each stable species are labeled with the dominant isobar in NSE, before weak interaction decay, which produces that species. The stable isobars are underlined. 58 -or *-| D_3_ -&± -Hi in FIGURE -»—% fv-l-• ia\_ J.st qr Tjrr >60— -U_ JLL •**»•• I»-"r"~ *>0t< W «HlOC ClilO*W Si • *iJ *13VrfS 1*H1 Kl OliJOt* SI lNIGrf iHQ trtMi ivcm S 11*310*1 • 59 FIGURE »-*l «r*O0 CjliCW Cl iHjMOlkNl tfOclHW • *1? 'yjtn i»-u MI cmovi ti .1*104 >MO MTr«i >*©« iaif3iONt • 60 mmcn • t- 2- c- v- V- %• l- s- fr~ •!- I x r •til* • • r * r x* 1 r - . r x 1 r x l r * x CM i r •ovi* « i A j . r x . I r * y .jj i r —n FH^ \ C * r’ ! ffO • ; ;on » !r > • OCI* II » * V ri -C71V » r i . V r l • ... w r i X r i • •21* V r l * r l f l z H X* J ... X 1 1 *SIl* r l ^ : r i in » r i r t HL r i •on* ' r \l * r i ^lx ri * X ri X I r ir H .SOI* .. V I r N « * * r X 1 r T * M * * i 2 r M *8#l* i t r- . . ._ »i i r xv r N * x > r X * ill**** r' « x t t r\ r n ”%! r • H •{«* Tx _q[ X K * .**- I M * ix r N * M *«l* ax r N * ix r M * i x r M • i v r * * C^l i x r H *0*0* [> I A i x r « u J i x r M * i x r M * -r-tSi I x r M * U J *~C- : u ($JLU a if M * I r X « * i r I r X N i r I f Ml FIGURE M X I Iro r H X r _ M X • MI* J-O r M X t f r*-r< X * £3 r V X ' v V r tf) X M . r i’c\\ •e«o* s r M k—1 X - - i • f M . - Ji . - . . . - - ? • j J! U V ■ CCS* i r M X ? r . M X I r * M X • I f i r M X •esc* M « M X • 4- X -4 X o r j J V ri N X r 1 r i M X • 01#* r • X • n* i X * « • I x • r« X * i- X ltd* x • in X i r x x • i r x X » i r N a «•<•’ t r M X j r M X • ...... M X !ro r x x • cNi im r 2 X .Itf in ; ( r M * 1 r M vttf* II It ICO I * 1- «• C- - *- V- C- 4- c. t* *1- •J9H »*t «►*.» CJHO\d tl fcO«in>i • *13 4»»U KI OJUCW tt 1*IC* JwO **lHi 3*.0« C)ir3!0N| • 61 mmol t 1- 2- C- V- V- »- 1- »- f 1 • r* "» TO i?— • J ^ ei r «r» \n ti ? n,ri SI U r • 2 O e n Bn... i*— • n O' s T ; i* * ■» ki 5 4 # i ui— * ■» O "1 fl ^ O 1 r 1 • !• -io -| o 1 o I* n o i e 9* 1 V LO i o . n 0 8* 45 in _ I! :r o 16 (J 1 c . “10 S* ^ f» co -1 • n * TB »* - TO -1 • n fi *1 0 •• i e 1 A FIGURE 1 to n a ... . . i:_ n o 1 o 1 B #• in - - in T 5 . - - -- - . - 0) 4 5 f— fa : cn s i in • n *H * •• l T2Tt T 0 fl* T » 1 • 1 • f n B 1 0 n fi ■t . > *» e i e •• *i » 1 5 ^ t - C* "i i LO } CTi ® in : in . •• On *rl B X-)n |2T #t n B T_ B n •• n B i # n _ • ^ n • 1X111(91 • 1- 2- t~ V- fc- i- »- *■ '101* Xt b*C8 OJiiO'W SJ Ol*>0 MOalOX ■ »ll JZ1*t if*i Ml 02110X4 il 1M104 XO MTM1 3»3« «>l»0i meifoi • l- J- t- s- *- X- #- _ 1 A i n • *S1* SA n : . ! • TH * • n - in , *S1» n m * . AV At n • pr 1 * AS 1 n •»M* A i A t i n • 1 A l in A S in • A ift • set*"’ A IS ru A n j - -v 0 II X A n i . ♦«!’ * rVj .. n "■ i " • A s n • r* \ \D ] A s n • A Cr* •sir*— s n 1 Vr) X A ns 1 - 1 A n s 1 ♦an* "A n i 1 Aft S 1 • S 1 n A S 1 • S X Trtir n A S 1 a n A S' J CO n A S X y * . n A V 1 —trr n A Si *r*t a V. N n ASX T I 8 in n AS X 8 n AS 1 52} 1 4 n ■ X a " - VA 4 •SIT*— n S A X a n 8 A 4 —c n * A 1 a ■ ^ n S AX n S AX a • *fc** 1 11" «r n E AX a • n ^E~ » z • ft % t 1A 8 n ■ s a— • Wt* " n s 1A a n s m\£7 I 1C 0 n s A 8 • —n- 5- »r-{|* n s A a •tM* n s - Sli - A n s A a • (i s 4 A n s 1 t i a “ •Uf* " sr. X a A ■s _n * 1 8 A s n i 8 A . n « O s X * A A s 8 n X A •er.«* VL/ ~a n 1~ * • Jl* 1« n x A n * * n x ' A FIGURE 10 hy\i a n x A • f I 1 V ■ II 1 »svr t V n i A 5 I TIT A s a ru A • s a (U s a ru ^ J A .(«* sa xn VO A* 4ft p as in r \ VJ a s I "n a s *X ’ ft • A a t i n ♦•so* 4 ft a s i n - a s —X n a s rH x n - a t KD *1 "ft a *s _r\ i ft • 8' " 1 S 8 t "ft •*•** ■s a 1 s *x "ft : ■I * ‘l • a 1 * s -1° CO- X jj VO 1 in a *s 4 ft ]q • •H a ’s lx ft 1 •*c** s i n 13 a t 4 n • 1 4 4 ft a S 4 n • m •srr* 8 s X ft 1 " 8 *1 xn • a xrt a m s 1ft •iri* a c in - * a • VO € • • 8 •H \ • 8 " t 8 ru 8 a. 8 s ru • " 8 ' i 8 % ft *x •*i** S' 4 a 1i i ft 4 • t , ft 4 a € n i • n ’ "4 8 A ft 4 ■ 1 tJ 1 af t ft X • a * x 8 i n «•*•*• 01*1*01 * l- X- c- s- *- X- •- X- il* •aow * cntttol • I* I- c- t- |- ft. ft- «I- i « ' ' »tSf t" R * * Ico l : UO * * : in « o ? KO T o S •»»i* ■ -rU : • la; • :sc * * "i « • • * • • eel* X: - — ft * H • -K ft X « ..... _ - - - ‘ * fMl*- * -ft X ft - V R •m* • -ut « 9 ‘O * * : -^1 •* * ^ * * • ' ft X K X •*u« • R X ft X • W X R If * A * •vei A * • ftR X 1 ft X •HI* • ~ ■ R" !0“ 1 *■ • « 1.0 * \ oh • X 1 R X • Oft** ft x R X • At A V • „ - H 1 •ssx* R X A A A V • - " R » R X • 0S9* - R ;—x R X • A 1 A V • R X • R I R X * R X •czx* R x • FIGURE 11 R t • R i R X * : M? : • A V0 ^ ft- X •«» \ CO ft X * ID "* *x »eK* ft X • 3 Ol R « *_ "R « • R X * " R «•%•* R * ~co V TD " -tSJ- i »«e— •ixtxttoi • |»_ i- c- •30*4 3*1 ***00 C3WOVJ SI iN)t«01VN3 • *13 •30*4i l»*4l MI OlliOV St 1MIM 3*0 ***l 3*0* S3031 OKI • 65 mottol • I- x- r- •- V- «- X- v- •i- - ^^ !iu* *12 z Z UJ i 1 l ^in ? • VO * -! 1 tw I—1*- Z 6 n^ 52 •*H 1 |lc £ 2 • z ♦HI* Sa1 l 1 S N z 2 ■ 1 X Z 1 c Z a A Z —{—E— Z z 2 A • z VO I r z —-i- »icr- z . z A * Z- -pi c A z c z A" Z— r\ c z 1 c z A* *CCI* z 1 - -£ l z t c z A 1 ~r *• A g 1 -• - Z * , — t • A •«r w z I c z A ~z l c “ 2 z t 1 : z t l • z 1— 1 . i A • Ml* z r .1 rz * i^TZC A • f t 1 z i ,-<1 f? C A • * t • A • £tt ! C A *z ’.tf1—1 t A Z 1 Z C A - . z 1 z c A • *eu* "Z 1 Z ( : A z • C . A • 2 * Zl C A • Z‘ A » i * •Mr 2 Z 1 C A Z Z 1 AC l C 1 At 2 2 1 A C • 001* 2* 1 A V z2 2x tr A A 2 : * A * z •ese* A ‘z • : • to z •_ .. VD . z •" "A K I ‘z • TNf| A * Itctf A • A • »c# A ; A • A ♦#c«* A * A r »«rt '»m' ,T • nr* ...-..,..1....--,... , **“• mmoi • 1- i- c- _ v- fr- *■> X- *- k- «!• •>*>»* >n **oo CJuo-u il AKZ'oolirHl • «u *23TA il ACKNOWLEDGEMENTS I wish to thank my advisors Donald D. Clayton and W. David Arnett, committee member Raymond J. Talbot, Jr., and fellow student Stanford E. Woosley for their help, encouragement, patience and friendship, without which I still would not believe in myself. Particular thanks to Ray Talbot for many long discussions when I was beginning my apprenticeship in alchemy two years ago. I also wish to acknowledge the financial support of the National Science Foundation and the Fannie and John Hertz Foundation while this work was being done. 67 REFERENCES Arnett, W. D., J. W. Truran and S. E. Woosley 1971, Ap. J. 165, 87. Bahcall, N. A. and W. A. Fowler 1970, Ap. J. 161, 119. Burbidge, E. M., G. R. Burbidge, W. A. Fowler and F. Hoyle 1957, Rev. Mod. Phys. 29, 54. Cameron, A.G.W. 1967 (published 1968), Origin and Distri¬ bution of the Elements, ed. L. H. Ahrens (Oxford: Pergamon Press), p. 125. Clayton, D. D. and J. Silk 1969, Ap. J. 158. L43. Clifford, F. E. and R. J. Tayler 1964, M.N.R.A.S. 127, 185. Clifford, F. E. and R. J. Tayler 1965, Mem. R.A.S. 69, part 2, 21. Everling, F., L. A. Koenig, J.H.E. Mattauch and A. H. Wapstra 1960, Nuclear Physics 18, 529. Fowler, W. A. and F. Hoyle 1964, Ap. J. Supp. IX, supp. #91. Kantorovich, L. V. 1952, Functional Analysis and Applied Mathematics, translated from the Russian by C. D. Benster, National Bureau of Standards Report 1509. Mattauch, J.H.E., W. Thiele and A. H. Wapstra 1965, Nuclear Physics 67, 1. Reif, F. 1965, Fundamentals of Statistical and Thermal Physics, (New York: McGraw-Hill), p. 317. Tapia, R. A. 1971, Am. Math Monthly 78, 389. 68 Truran, J. W. 1972, On the Synthesis of Neutron-Rich Iron Peak Nuclei (paper to be published). Woosley, S. E. 1971, Nucleosynthesis During Advanced Burning Stages of Stars, Ph.D. Dissertation, Rice University, (unpublished).