## Fme

LnNA(av)T lnc- -InT-c'y (3.115) _ am na( v)t _3 _ c _ 3 (3116) Na av)T NA oV)T0 (T To)c'Er T-3 2 NA av)To (T Tq )11.605 Er T-3 2 where Er in the last term is given in units of MeV. Figure 3.22 shows the exponent n versus temperature for several values of the resonance energy Er. Clearly, the temperature sensitivity of narrow-resonance reaction rates increases for decreasing temperatures and increasing resonance energies. Depending on the values of T and Er, narrow-resonance reaction rates may...

## V

Fig. 4.10 Representations of the (a) Compton effect, and (b) pair production. Note that in part (b) neither the electron nor the positron existed before the interaction. The atom is a spectator only and facilitates the simultaneous conservation of energy and linear momentum. in Fig. 4.10b. The photon must have at least an energy of twice the electron rest energy (2 511 keV 1022 keV) for this process to occur. Also, pair production must involve a third body (usually the nucleus of an absorber...

## Initial mass of M 11 Mq

The evolution of stars in this mass range is in many ways fundamentally different compared to our earlier discussion. Schematic evolutionary tracks for 13 Mq, 15 Mq, 20 Mq, and 25 Mq stars are shown in Fig. 1.5b. The case of a 25 Mq star with initial solar composition will be discussed in the following as an example (Chieffi, Limongi and Straniero 1998 Limongi, Straniero and Chieffi 2000 Woosley, Heger and Weaver 2002). The total life of such a massive stars is relatively short and amounts only...

## Info

Fig. 5.77 (a) Section of the chart of the nuclides in the region of neutron-deficient Sn, Sb, and Te isotopes. Stable nuclides are shown as shaded squares. The letters s, r or p refer to their productions in the s, r- or p-process, respectively. (b) Decay constants of neutron-deficient tellurium iso topes, calculated for a temperature of T 2.5 GK (from Rauscher and Thielemann 2000). At 120Te, the (7, ) reaction dominates over the competing (Y,n) reaction. As a consequence, the abundance flow...

## Ppo

(a) AE 10 keV, r 0, Astragg 0, Abeal 0, 1, 3, 5, 8 keV (b) r 0, Abeam 0, const 1.2, AE 1, 3, 5, 10 keV (c) the target thickness amounts to AE 10 keV for each curve r 0, Abeam 0 r 0.5 keV, Abeam 0, As 0.5 keV, Abe 0.5 keV, Abeam 1.5 keV, const 1.2. The areas under all curves shown in part (c) are identical. find Abeam 4.0 keV. Note that the Doppler effect due to the thermal motion of the target atoms also contributes to the slopes of yield curve edges (Rolfs and Rodney 1988). The beam spread...

## Silicon Burning

Near the conclusion of core oxygen burning, when the 16O fuel is depleted, the most abundant nuclei are 28Si and 32S (Fig. 5.50). The stellar core contracts and the temperature increases. Fusion reactions such as 28Si + 28Si or 28Si + 32S are too unlikely to occur because of Coulomb barrier considerations, even at the elevated temperatures achieved at the end of the evolution of a massive star. Instead, the nucleosynthesis proceeds via photodisintegrations of less tightly bound nuclei and the...

## A20Nea7a 16 x 104 s A23Naapa 57 x 103 s1 and A24Mga7a

4.1 x 102 s Hence, some of the a-particles will be captured by 16O, synthesizing again 20Ne. But there is also a good chance that the liberated a-particles will be consumed by reactions such as 20Ne(a,Y)24Mg, 23Na(a,p)26Mg, or 24Mg(a,Y)28Si. A number of other a-particle-induced reactions will occur that release protons and neutrons, and these light particles will also participate in the nucleosynthesis. Details will be discussed below. To summarize, the network of reactions consisting of the...

## Fep

Fig. 4.28 Response of a 7-ray detector to monoenergetic incident radiation. (a) Representation of different photon histories. (b) Pulse height spectrum the meanings of the labels are full-energy peak (FEP), multiple-site events (MSE), Compton edge (CE), single-escape peak (SEP), double escape peak (DEP), Compton continuum (CC) and back-scattering peak (BSP). ferred to the recoil electron depends on the scattering angle. All scattering angles are possible and, therefore, the energy distribution...

## M

With rnsample and M the mass and relative mass of the sample, respectively. If the sample consists of a compound, then msample, M, and the mass fraction X refer to the active sample nuclei, that is, the nuclei participating in the reaction of interest. Masses of self-supporting samples are frequently determined by weighing, whereas masses of deposited samples can be found from the weight difference between the backing and the combined sample-plus-backing. For compounds or samples consisting of...

## Energy generation

The rp- and ap-processes generate energy in a completely different manner compared to the HCNO cycles. The former processes consist of sequences of capture reactions and j8+-decays. Note that an (a,p) reaction followed by (p,Y) has the same product as a single (a,7) reaction. Reaction cycles play only a minor role, and, therefore, none of the nuclides involved in the nucleosynthesis will act as catalysts. Energy is generated not by the fusion of four protons to one 4He nucleus, but by using...

## Hydrostatic Hydrogen Burning

Hydrogen is the most abundant isotope in the Universe. The fusion of four nuclei to the tightly bound 4He nucleus is called hydrogen burning. Indepen- dently from the details of this transformation, the process releases an energy (Section 1.5.3) of Q 4(M.E.)h - (M.E.)4He 4 (7288.97 keV) - (2424.92 keV) The obvious question arises as to precisely how this fusion process takes place. Early estimates showed that the probability for the simultaneous interaction of four protons in the stellar plasma...

## Solar System Abundances

It is commonly accepted that the solar system formed from the collapse of a gaseous nebula that had an almost uniform chemical and isotopic abundance distribution. Abundances in the solar system are also similar to those found in many stars, in the interstellar medium of the Sun's neighborhood and in parts of other galaxies. Therefore, it was hoped for a long time that a careful study of solar system abundances would provide a cosmic or universal abundance distribution, that is, an average...

## Xiso Mu0 rp150 176 s

uOje Xli0Je Mis0J Tp(140) 102s L7 (5-82) in agreement with the numerical results displayed in Fig. 5.23b. This situation prevails almost until the end of the calculation. Only at times very close to hydrogen exhaustion do the mean proton-capture lifetimes sufficiently increase to cause a slight drop in the 14O and 15O abundances, with a corresponding rise in 12C, 13N, and 14N abundances. Nevertheless, even at hydrogen exhaustion (XH 0.001) we have E tcno1 (18 + 77 + 102 + 8 + 176) s 381 s...

## MAXN mz41 Xn c2 mAXN mZXN1 2me

Positron emission (1.43) electron capture (1.44) where me and Eb denote the electron mass and the atomic binding energy of the captured electron, respectively. The released energy is almost entirely transferred to the emitted leptons. For example, in ft--decay we have Qft- Ke + Ev, where Ke and Ev denote the kinetic electron energy and the total neutrino energy, respectively. Since there are three particles after the interaction, the electron and neutrino energy distributions must be...

## Neon Burning

At the end of core carbon burning, when most of the 12C nuclei have been consumed, the core consists mainly of 16O, 20Ne, 23Na, and 24Mg. Other nuclides will be present as well, but with much smaller abundances (X < 5 x 10 3 see Fig. 5.44). The core contracts gravitationally and the temperature and density both increase (Fig. 5.1a). It is reasonable to assume that the next nuclear fuel to ignite is oxygen via the 16O + 16O fusion reaction. However, before this happens the temperature has...

## C

(2003), and reflects our incomplete knowledge regarding the issues mentioned above. Above mass A 60 the situation is much clearer since for the overwhelming number of nuclides the relative contributions of the s-, r-, and p-process can be estimated in a straightforward manner (Section 5.6). These are shown in Fig. 5.76. For more information on the origin of the nuclides in nature, see Woosley, Heger and Weaver (2002) and Clayton (2003). 5.1 Calculate the lifetime of (i) a proton against...

## The 56Ni bottleneck

To understand why the abundance flow is significantly delayed at 56Ni, we need to consider the unique nuclear properties of this waiting point nucleus. It has a relatively long half-life of T1 2 6.1 d in the laboratory and decays with 100 probability by electron capture. At elevated temperatures and densities, the j8-decay half-life will change somewhat (Section 1.8.4 see also Fuller, Fowler and Newman 1982). Values of Qp7 and T1 2 for nuclei in the vicinity of 56Ni are given in Fig. 5.38a. The...

## Breakout from the HCNO Cycles

For stellar temperatures of T < 0.4 GK, very little material is lost from the operation of either the cold or the hot CNO cycles (Sections 5.1.2 and 5.2.1, respectively). This is explained by the fact that the heaviest nuclei synthesized in the CNO and the HCNO cycles are 19F and 18F, respectively. As can be seen from Figs. 5.9d and 5.22a, the branching ratios BpX p7 of these two isotopes amount to factors of 103-104 in the temperature ranges of the CNO and HCNO cycles (T < 0.1 GK and T...

## Cno1

Tp (12 C) + Tp (13C) + Tp(14 N) + Tp (15 N) Hence, the energy generation rate at equilibrium can be written as CND1 P Tp(l2C)+Tp(l3C) + Tp(l4iV) + Tp(15iV) The sum of the lifetimes in the denominator is called the cycle time, and is almost entirely dominated by the long 14N lifetime. Hence e 25.030 MeV E CNO1 25.030 MeV , eCN01 --- (14N) ---(ECN01)HM14N(p,7) 25.030NaM14N(p,7) (MeVg_ls_1) (5-69) where the sum is over all CNO1 isotopes. Clearly, the energy generation rate in the CNO1 cycle at...

## Effective Cross Section

If the incident neutrons are not monoenergetic, then an effective cross section is sometimes introduced which is defined in terms of the neutron current density, or neutron flux, instead of the number density of neutrons. If we divide the neutron energy distribution into thin slices, then the number of reactions per volume and per time from each slice is given by Eq. (3.1), with Nt V and NVii V the target density and neutron density, respectively. Integrating over all energies, we find The...

## Atomic Mass and Mass Excess

Direct measurements of nuclear masses are complicated by the presence of the atomic electrons. Atomic masses, on the other hand, can be measured with very high precision. For this reason, experimental mass evaluations tabulate atomic rather than nuclear masses. These quantities are related by matom (A, Z) mnuc(A, Z) + Zme - Be(Z) (1.8) where me and Be denote the electron mass and the electron binding energy in the atom, respectively. Nuclear reactions conserve the total charge. Therefore, one...

## Emm V425

The cross section for Compton scattering is given by the Klein-Nishina formula (see, for example, Leo 1987). A polar plot of the angular distribution of scattered photons is shown in Fig. 4.11a for different energies of incident photons that approach the scattering center from the left. It can be seen that the distribution is symmetric around 9 90 for small photon energies (E7 < 1 keV), whereas scattering into the forward direction is strongly preferred for large 7-ray energies. The...

## Pp2 and pp3 chains

So far we have neglected reactions other than 3He(3He,2p)4He that destroy 3He. Figure 5.5a also compares the quantity T3He(3He) e with the lifetime of 3He against destruction via the 3He(a,7)7Be reaction, Ta(3He). It can be seen that the 3He(a,Y)7Be reaction becomes the dominant destruction mechanism for 3He if the temperature and the 4He abundance are sufficiently large. The 4He may either be produced during hydrogen burning or may be of primordial origin. Following the 3He(a,Y)7Be reaction,...

## D

Fig. 3.30 Cross sections for neutron capture on 7Li,31P, and 90Zr versus energy. The curve in the upper panel shows a 1 v behavior, while resonances are visible in the middle and lower panels. overlapping contributions. The S-factor rather than the cross section is shown in Fig. 3.29 since the latter quantity drops rapidly for decreasing energy. Total reaction rates NA & v) of charged-particle-induced reactions depend strongly on temperature, as shown in Sections 3.2.1 and 3.2.4. The...

## Network Calculations at Constant Temperature and Density

We will now turn our attention to the nucleosynthesis that results after breakout from the HCNO cycles has occurred. A representative value of p 104 g cm3 is again chosen for the density. Numerical network calculations are performed for three different temperatures (T 0.5,1.0, and 1.5 GK) and the results will be discussed below. In order to account properly for the nuclear activity at such high temperatures, the network has to be expanded dramatically in size compared to our earlier...

## Fe12 JWpJ J7

Fig. 3.5 Comparison of the exact expression E2 (ee r kt 1) (solid line) and the approximation Ey eey kt (dashed line) at a stellar temperature of T 5 GK. For a sufficiently large threshold energy (in this case, for Et > 1.5 MeV), the difference between both expressions is negligible. Fig. 3.5 Comparison of the exact expression E2 (ee r kt 1) (solid line) and the approximation Ey eey kt (dashed line) at a stellar temperature of T 5 GK. For a sufficiently large threshold energy (in this case,...

## TinuQ Ti215Q574

P(Xh Mh ) Na < v) N(p,7) ln2 ln2 All curves are calculated for a solar value of Xh Mh 0.70. On the left-hand side of each curve, the p+-decays are more likely to occur than the proton-induced reaction, while the opposite situation prevails on the right-hand side. The CNO1 cycle operates in region 1 where 13N p+-decays Tp (13N) < Tp(13N) and where 14N(p,Y)15O is the slowest link in Tp(15O) < Tp(14N) . Suppose we start out in region 1 and slowly increase the temperature by keeping the...

## Decay Constant Mean Lifetime and Half Life

The time evolution of the number density N (or of the absolute number of nuclei N) of an unstable nuclide is given by the differential equation The quantity A represents the probability of decay per nucleus per time. Since it is constant for a given nuclide under specific conditions (constant temperature and density), it is referred to as decay constant. Integration of the above expression immediately yields the radioactive decay law for the number density of undecayed nuclei remaining after a...

## Transition Probabilities

A detailed discussion of the quantum theory for the interaction of nuclei with electromagnetic radiation is beyond the scope of this book. We will instead summarize the most important steps in the derivation of the transition probability. For more information, see Blatt and Weisskopf (1952). The decay constant (that is, the probability per unit time) for the emission of electromagnetic radiation of a given character (for example, E1 or M2) in a transition connecting two given nuclear levels can...

## One Dimensional Square Well Potential

In region III, we have again E V > 0, and the general solution is given by miii Feikr + Ge ikr, F' sin(kr + 50) The solution is the same as the one obtained in the study of the square-well potential (see Eq. (2.59)). First, we are interested in the transmission probability through the potential barrier. it is convenient to start from the wave function solutions in terms of complex exponentials (see Eqs. (2.84)-(2.86)). We must again perform the calculation for the one-dimensional case....

## Approach to steady state in the CNO cycles

We have so far considered only the steady state operation of the CNO1 cycle. We will now investigate nonequilibrium situations. Two aspects are of special interest (i) the approach to steady state in the CNO1 cycle, and (ii) the simultaneous operation of all CNO cycles. The system of coupled differential equations describing the abundance changes of all CNO nuclei is similar in structure to Eqs. (5.48)-(5.53), but it is more complex because of the inclusion of oxygen and fluorine isotopes. Such...

## References

Lett. 93, 161103 (2004). Abramowitz, M., Stegun, I. A., Handbook of Mathematical Functions (New York Dover, 1965). Adelberger, E. G., et al., Rev. Mod. Phys. 70, 1265 (1998). Ajzenberg-Selove, F., Nucl. Phys. A506,1 Ajzenberg-Selove, F., Nucl. Phys. A523,1 Almen, O., Bruce, G., Nucl. Instr. Meth. 11, 257 (1961). Anders, E., Grevesse, N., Geochim. Cosmochim. Acta 53, 197 (1989). Anderson, M. R., et al., Nucl. Phys. A349, 154 (1980). Angulo, C., et al., Nucl....

## Dc

103 102 101 10 101 102 AE T Fig. 4.58 Ratios Ymax Ymax,AE , FWHM AE and (Er Eof50 ) T as a function of AE T for a Breit-Wigner resonance with energy-independent partial widths. where g(E0, Ei) dEi is the probability that a particle in the incident beam of mean energy E0 has an energy between Ei and Ei + dEi f (Ei, E, E') dE is the probability that a particle incident on the target at an energy Ei has an energy between E and E + dE at a depth inside the target corresponding to the energy E'...

## Q

Fig. 2.23 Total cross section for neutrons incident on a target consisting of a natural isotopic mixture of cadmium. The data are fitted by a one-level Breit-Wigner formula, superimposed on a 1 v background, The deduced resonance parameters are Ea 0.176 eV, r 0.115 eV, and amax 7.2x10-21 cm2. The Breit-Wigner formula Fig. 2.23 Total cross section for neutrons incident on a target consisting of a natural isotopic mixture of cadmium. The data are fitted by a one-level Breit-Wigner formula,...