Notes 1

Atomic Nuclei

The atomic nucleus is quantum system, which makes up the center of the atom, composed of protons, neutron, and electrons who interact together to form a bound system. The forces acting on this system; Strong force, Weak force, the Coulomb force, and Gravity, act on the system over various length scales that effect the overall distributions of the particle wavefunctions composing the nucleus. The strong force acts on length scales of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-15}} m and below, primarily affecting the quark components of the previously mentioned nucleons. The weak force acts on a scale of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-18}} m, acting on the range of the nucleons as it is the affect of boson exchange. The electric, i.e. Coulomb, force acting on the nuclei also acts at all ranges, modeling the interaction of the charged nucleon present in the system with one another. Finally, the gravitational force acts also acts at all ranges, and is the weakest of the four nuclear forces.

Within a given nucleus, there are multiple levels of organizations. The protons and neutrons, closely bound by the nuclear forces, form a system whose energy levels are quantized, forming a shell structure hierarchy, for both the protons and neutron, whose energy levels often interact between one another. These nucleons are each composed of three individual particles known as quarks, bound together by the strong force.

A nucleus species is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _Z^A X_N} where N denotes the specific species of element X. Z is the proton number which is how an element is defined and A is the Baryon number (Protons + neutrons) and defines the Isotope. For example Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _6^{12} X_N} Is "Carbon 12" which has 6 protons and 6 neutrons. The characteristics of the nuclides follow general trend lines based on both isomer, and isotope number, as seen by the nuclide chart posted here which is a plot of isotopes using proton number vs neutron number. A region of stability is observed, beyond which increases in proton or neutron number outside of the dripline (seen here in black) causes rapid decay of the nucleus in question.

Definitions for Abundances

To facilitate an understanding regarding the elements present in a given cosmological event or object, a framework must be established to properly measure the distribution of particles present. To this end, the nuclear abundance is presented as a measure of the number of a given isotope present in a quantity of measured element. This particle abundance is defined as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i = \frac{n_i}{\sum_{j} n_j} \ , }

where the sum is counted over all isotopes present in the data and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} is the particle's number density. This is often set logarithmically and normalized to the amount of hydrogen present, creating what is known as the relative particle abundance:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_i = \log_{10} X_i + 12 \ , }

Furthermore, the mass fraction is defined to be the fraction of total mass of a sample that is composed of the particular nucleus in question, i.e.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_i = \frac{m_i}{m_{tot}} = \frac{m_i n_i}{\rho} \approx \frac{A_i n_i}{\rho N_A} \ . }

Denoting the mass per baryon as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_i\equiv \frac{X_i}{A_i} \ , }

the particles number density can be defined by this baryon fraction and the density of the sample

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i = Y_i \rho N_A \ . }

The mean molecular weight is determined by

Other important quantities for consideration would be the mean molecular weight, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = \frac{\sum_i{A_i Y_i}}{\sum_i Y_i} = \frac{\sum_i X_i }{\sum_i Y_i} = \frac{1}{\sum_i Y_i} \,} , the electron abundance,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_e = \sum_i Z_i Y_i \ ,} , and the electron number density, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_e = Y_e \rho N_A \ .}

Solar System Abundances

Solar abundances, i.e. the quantity of various species of nuclei present in the solar system, tell us a great deal about how the solar system was formed and provide insight into galactic evolution as a whole. To this end, the nature of studying solar abundances can be broken into three disciplines studying the various classes of data present.

Earth materials

To this end, various materials on earth are examined to see what sort of composition of elements were present during the formation of the solar system. The problem that is encountered, however, is the chemical fractionation strongly hinders this process, changing the apparent concentrations. Isotopic compositions, however, do not exhibit this feature however and are ideal candidates for study.

Solar Spectra

Since the sun formed directly from presolar nebula, examination of the spectra observed from its photosphere, i.e. non-fusion process layer, sheds light on the early composition of the solar system prior to structure formation. Information is gathered by taking a sample of light from the star and putting it through a prism or some other optical element that separates the light as a function of wavelength. Because the gas in the photosphere is cooler than the gas inside the sun it absorbs light from the hotter regions and produces an absorption line. these lines are specific to each element and ion (an element with less electrons) so we can determine the element and even ion concentration in the sun and any other star. The element He was actually observed in the sun before being detected on earth.

Meteorites

Finally, meteorites found on earth, which have not been exposed to extreme temperatures and pressures that would cause the chemical fractionation seen in earthen materials, can be used to sample presolar nebula composition. These meteorites are broken into three classes, stones(93%), stony irons(1.5%), and irons(5.5%). Stones, which are subdivided into chondrites(86%) and achondrites(7%) are by far the largest in abundance, with the chondrites providing some of the best, unfractionated samples of presolar nuclear composition. Most of these are found in the Antarctic where the meteorites are easily spotted on the white snow.

Quantum Mechanics

In the late 19th and early 20th centuries, new physical theories were stretching the limits of classical mechanics as well as classical theories of electricity and magnetism. Physicists needed new theoretical tools for describing quantum systems that would reduce to that of macroscopic systems at the proper boundaries.

The famed double slit experiment offered new, unexplained experimental data which led to the to the understanding of particle-wave duality. This is to say that any particle has wavelike properties and thus can be represented as a wave, and visa versa. This is one of the most important theories in all of quantum mechanics. This data was obtained by firing a beam of electrons through 2 slits, and instead of seeing a bright band on the detector screen an interference pattern, similar to that of light waves, was observed.

Another example of the limitations of classical mechanics is in regard to the stability of the atom. Classical theory would suggest that electrons orbiting a nucleus would lose energy by emitting photons and subsequently crash into the nucleus. Niels Bohr proposed an atomic model in which electrons had discrete energy levels. This theory of quantized energy states was the foundation for modern day quantum mechanics. Once experimental data confirming this theory, or a theory similar, was obtained by James Franck and Gustav Hertz, research into quantum theory greatly increased.

Many Astrophysical systems use Quantum Mechanics such as the cross sections of a nuclear reactions , the probability of a particle or nucleus to decay, or even the force which allows a stable system in a neutron star.

The Schrödinger Equation

Possibly the most important equation in all of quantum mechanics is Schrödinger's wave equation. This partial differential equation represents the change of quantum states over time. The wavefunction represents the probability of a particular quantum state being occupied.

The most general form of the equation is as follows, where E is the energy operator, and H is the Hamiltonian operator.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\Psi = \hat H \Psi}

The equation can be rewritten in a time dependent form and a time independent form.

The time dependent equation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \hbar{\partial \over \partial t} \Psi(x,\,t) = \hat H \Psi(x,\,t)}

The time independent equation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {E}\psi(r) = - {\hbar^2 \over 2m} \nabla^2 \psi(r) + V(r) \psi(r)}

The time independent equation represents a situation in which the energy of the system is constant in time.

Thermodynamics

Thermodynamics is also essential to understanding Astrophysical processes. Thermodynamics, study of energy transport, dominate many Astrophysical processes most notably stars. For this reason included is a brief discussion of variables uses in thermodynamics and the primary equations, pressure, number density, particle density, state density, total number of states, and the equation of state and various other thermodynamic principles where they are needed.

Systems

To begin the discussion of thermodynamics we look at the concept of systems. A system is a region of space that is being studied, this region is defined as the space inside of the boundary. Everything outside of the boundary is known as the environment. The system can exchange energy, work and matter with the environment. There are also closed systems in which only heat and work can be transferred from the system to the environment or vice versa. In a closed system to matter can be exchanged.

Thermodynamical Variables

To begin, we look at the key variables that define a statistical system.

First we define entropy, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\!} , as the energy of a system that is not available to do work. In general, entropy is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = k_{b} \ln{\Omega}\!} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega\!} is the number of states in the system.

The internal energy of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\!} , is the energy necessary to create the system, defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dU = Tds - PdV + \sum_{i}\mu_{i}dN_{i}\!} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\!} is the chemical potential.

The chemical potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\!} , of a system is the change in a characteristic thermodynamic state function per change in the number of molecules.

The Helmholtz free energy, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\!} is a measure the work obtainable from a closed thermodynamic system at a constant temperature and volume, defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = U - TS\!} and the change in Helmholtz free energy as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA = -TdS - PdV + \sum_{i}\mu_{i}dN_{i}\!} .

The Gibbs free energy of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G\!} , measures the useful work obtained from the from an isothermal, isobaric system, and is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = U - TS + PV\!} , where the change in Gibbs free energy is said to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dG = dU - SdT + VdP\!} .

Heat, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q\!} , is defined to be the energy transferred between two thermodynamical systems via a thermal contact and is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = q V \!} with a change in heat given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dQ = T dS \!} .

Finally, enthalpy, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} , is defined as a measure of the total energy of a thermodynamical system, given by the expression Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = U + PV\!} and the change in enthalpy as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dH = TdS + PdV + \sum_{i}\mu_{i}dN_{i}\!} .

Thermodynamical Quantities

Now we move to discuss the various equations that define a thermodynamical system outside of its principle variables. Defining the occupational probability function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p)\!} and the state density function as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega\!} , then we can express the following quantities in compact form:

Pressure

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{1}{3}\int^{\infty}_{0}pv\omega(p)f(p)dp }

Energy Density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = \frac{U}{V} = \int^{\infty}_{0}E\omega(p)f(p)dp }

Particle Density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{N}{V} = \int^{\infty}_{0}\omega(p)f(p)dp }

State Density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(E) = 2\pi\frac{g}{h^{3}}(2m)^{3/2}E^{1/2} }

Number of States

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(E) = \frac{4\pi}{3}\frac{g}{h^{3}}(2mE)^{3/2} }

Occupation Probability

Fermi-Dirac

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p) = \frac{1}{e^{(E(p)-\mu)/k_{b}T} + 1} }

Bose-Einstein

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p) = \frac{1}{e^{(E(p)-\mu)/k_{b}T} - 1} }

Maxwell-Boltzmann

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p) = {e^{-(E(p)-\mu)/k_{b}T}} }

Above, E(p) is the energy of the state, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} represents the chemical potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_b} is the Boltzmann constant, and T is the temperature.

A Simple Application

In stellar structures, the force of gravity acting on the stellar matter must balance with the outward pressure in order to avoid collapse or expansion.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial P_r}{\partial r}=-\rho_r \frac{Gm_r}{r^2}}

Nuclear Physics

Nuclear physics is the field of study pertaining to the interaction of atomic nuclei and the forces that act on them. The applications of nuclear physics range from nuclear defense, to power, to medical technologies and more as new improvements in the field open up new avenues of study.

Basic Ingredients

What one first has to discuss when talking about nuclear physics is what particles and forces are actually playing a roll, and how is their role satisfied in the system. When we talk about nuclear physics, in general, we're talking about the interactions of protons and neutrons inside of a nucleus and how they form together to yield various nuclei.

The forces that govern this interaction are the Strong nuclear force and the electromagnetic force. Due to the interaction with this electromagnetic force, it is possible to probe inside the atom to measure the relative size of the nucleus by use of electron scattering, where electrons are deflected away from the nucleus due to the presence of a Coulomb barrier created by the protons. Examination of the amount of electrons deflected at certain angles gives insight into the charge distribution of the nucleus, which surprisingly does not appear to be uniform. Instead, the charge distribution takes on an effective Woods-Saxon shape defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(r) = \frac{\rho_{0}}{1 + e^{\frac{r - R}{a_{v}}}}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = 1.18\!} x$A^{\frac {1}{3}}-0.048fm$ and $a_{v}=0.055\pm 0.07fm$.

Another factor at play in the nucleus is the binding energy that holds it together, overcoming the Coulomb repulsion at extremely short range. In general, this energy can be related to the difference between the mass of the constituent particles and the actual mass of the system, therefore, letting $c=1$, the binding energy for a nucleus is approxiamtely$B(Z,N)=Zm_{p}+Nm_{n}-M(Z,N)\!$.

Models

In order to better represent the nucleus, models are often used which facilitate the examination of various nuclear phenomena. These models simply the problem and allow for comparison to observed data which dictate how well the model works in its effort to describe the nucleus.

Independent Particle Model

The Independent Particle Model, sometimes referred to as the Fermi model, treats the nucleus as a fermi gas in equilibrium in order to model the energy. It should be noted that this model is only technically correct in the limit of infinite nuclear charge. Here the binding energy can be calculated by use of the number of states, $\Phi \!$, and the fermi energy, $E_{F}\!$ of the nucleus.

Letting $E_{F}={\frac {h^{2}}{2m}}({\frac {3}{8\pi }}{\frac {N}{V}})^{2/3}$, we can define the mean binding energy as

$E_{mean}={\frac {1}{\Phi (E)}}\int _{0}^{E_{F}}EdE={\frac {3}{5}}E_{F}$.

Droplet Model

Since the density of a nucleus was experimentally seen to be roughly constant, the liquid drop model developed as a means of representing this and taking into account the fact that forces acting at the surface of the nucleus would in fact be different then those acting on the internal nucleons. The binding energy can therefore by established by creating a term proportional to the volume of the nucleus, and making corrections to it based on geometry and basic physics. To begin, the volume term, $a_{V}V\!$, must be modified by a correction term to account for the difference in forces on the surface and inside of the nucleus. This surface term is therefore proportional to the surface area of the nucleus, and is a negative to account for the effective over counting of the volume term, therefore given by $-a_{S}A^{\frac {2}{3}}\!$. Furthermore, another term must be added to account for the repulsion between protons, which pushes them apart and lowers the binding energy, therefore making the term negative. This Coulomb term would have to count all interactions of protons inside the nucleus, and therefore would be roughly proportional to $Z^{2}\!$, so that it can be defined as $-a_{C}{\frac {Z^{2}}{A^{\frac {1}{3}}}}\!$. Another term, the asymmetry term is added to account for uneven counts of neutrons and protons, leading to differences in shell energies that bide for control in the nucleus. By use of the fermi exclusion principle, this energy correction term can be written as $-a_{A}{\frac {(Z-{\frac {A}{2}})^{2}}{A}}\!$. A final term is added to account for the Cooper pair interactions for the nucleons, and is different for different types of nuclei. This Term, $B_{pair}(Z,N)\!$, is ${\frac {12}{\sqrt {A}}}\!$ for even-even nuclei, $0\!$ for odd A nuclei, and $-{\frac {12}{\sqrt {A}}}\!$ for odd-odd nuclei. We can then finally write the binding energy for the liquid drop model as

$B(Z,N)=a_{V}V-a_{S}A^{\frac {2}{3}}-a_{C}{\frac {Z^{2}}{A^{\frac {1}{3}}}}-a_{A}{\frac {(Z-{\frac {A}{2}})^{2}}{A}}+B_{pair}(Z,N)\!$

where experimental data can be used to give values for the constants, with $a_{V}=16MeV\!$, $a_{S}=18.5MeV\!$, $a_{C}=0.72MeV\!$, and $a_{A}=23.4MeV\!$.

Shell Model

One of the main problems with the liquid drop model is that it begins to fail at describing nuclei in any consistent manner in the region of very low Z. This this end, the shell model was introduced, effectively accounting for all nuclear interactions regardless of Z number. However, the shell model becomes increasingly complex the higher up in Z one goes, and correlations between neutron and proton shells have to be accounted for, as well as spin interactions which become a predominant factor in heavier nuclei. It is, however, a very effective way of modeling low Z nuclei and eliminates the discrepancies between theoretical and experimental data. This uses the Pauli exclusion principle to define the nucleus in terms of discrete energy levels. This leads to filled "shells" of the nucleus which correspond to certain "magic numbers" for nuclear number Z, corresponding to trends in experimental data where these particular nuclei deviated from trends of their neighbors. This is generally done by taking a harmonic oscillator potential and deforming it into a Woods-Saxon potential, which is very effective at modeling the nucleus, and the corresponding Schrodinger equations are solved, yielding all possible states of the nucleus.

Decays and Stability

Given the wide range of possible isotopes and isotopes and isotones that exist, it becomes important to establish which ones are stable, i.e. long-lived, and which ones have very short life times, leading to the decay of the nucleus into various particles. In order for a particle to decay, two conditions must be satisfied. First, it must loose energy in the process of the decay, so that it can be closer to a ground state, as nuclei do not like being in excited states. Second, a transition must actually exist, via decay chains of alpha and beta decay, that can actually lead to the production of said nucleus. We see, along the middle of the table of nuclides, a finite region of stability known as the nuclear drip line. The equations governing the size of this drip line map out both the isotope stability, i.e. neutron drip line, and the isotone stability, the proton drip line, and are given by:

$S_{n}(Z,A)=M(Z,A-1)+m_{n}-M(Z,A)\!$

$S_{p}(Z,A)=M(Z-1,A-1)+m_{p}-M(Z,A)\!$

Nuclear Reactions and Cross Sections

In order to properly study nuclear reactions, a framework must be presented that details the theoretical values in relation to quantities that are measurable in the lab. To this end, the formalism of both transmission coefficients and nuclear cross sections are discussed below

A basic nuclear reaction is written as

$A(b,c)D\!$

where $A\!$ is the heavy target nucleus, $b\!$ is the projective, that is the particle being accelerated at the target, $c\!$ is the lighter outgoing particle, and $D\!$ is the residual nucleus.

Decays and Transmission Coefficients

In general there are two kinds of reactions. Direct Reactions, where the target nucleus and projectile interact to form a bound state, where the new nucleus is stable. The other is a Resonant Reaction where after the target nucleus and projectile interact an unstable particle is formed that then decays to a stable state.

Transmission probabilities refer to the probability of a nucleus decaying from a state of higher potential to one of lower potential. Within the context of a particle beam and target, the transmission probability can be represented by relating the flux of incident particles to the transmitted particle flux. These values of flux can be elucidated by the quantum mechanical equation of flux, ${\vec {j}}$.

${\vec {j}}={\frac {\hbar }{2mi}}(\phi ^{*}{\vec {\bigtriangledown }}\phi -\phi {\vec {\bigtriangledown }}\phi ^{*})$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T} = \frac{j_{trans}}{j_{inc}} }

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T} } is the transmission coefficient, which relates the intensity of the transmitted wave to that of the incident wave. This value becomes...

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{T}= 4\frac{\frac{2m}{\hbar}\sqrt{(E+V_0)E}}{\left [ \sqrt{\frac{2m}{\hbar}(E+V_0)}+\sqrt{\frac{2m}{\hbar}E} \right ]^2} }

Applied to quantum mechanics, the transmission coefficient acts as a probability of quantum tunneling to occur.

Nuclear Cross Section

Formally, the nuclear cross section is defined to be the effective area of a nucleus for a given reaction channel. Implicitly, this means that the cross section, hereby refereed to as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\!} , is not the actual cross sectional area of a hard sphere-type problem, but rather the area that an incoming particle or photon "sees", such that if the particles path takes it through this area, the particular reaction in study takes place. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\!} is experimentally measured as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{N_R / t}{\left [ N_b/(tA) \right ]N_t} }

Where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_R / t\!} is the number of interactions per time, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_b/(tA) \!} is the number of incident particles per area per time, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_t \!} is the number of non-overlapping target nuclei within the beam.

Relative to the tunneling probability, i.e. the transmission coefficient discussed above, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\!} can be defined, relative to angular momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l\!} by the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{\pi}{k^2}\sum_{l=0}^{\infty}(2l + 1)T_l }

In order to properly define the cross section, the incoming flux as described by a plane wave, is needed, as well as the scattering as a function of angle and the penetration probability of the nucleus. Given an incoming plane wave described as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{in} = \sqrt{4\pi}\sum_{l=0}^{\infty}\sqrt{2l + 1}i^lj_l(kr)Y_{l,0}(\theta) } ,

a transmitted wave described as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_{t} = \frac{\sqrt{\pi}}{kr}\sum_{l=0}^{\infty}\sqrt{2l + 1}i^{l+1}(e^{-i(kr - \frac{l\pi}{2})} - \eta_le^{i(kr - \frac{l\pi}{2})}Y_{l,0}(\theta) } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_l\!} relates to the penetration probability, and an expression for the scattering as a function of angle given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\int\vec{e_r}\vec{j_t}r^2d\Omega = -\frac{\hbar}{2im}\int(\psi^*_t\frac{\partial}{\partial r}\psi_t - \psi_t\frac{\partial}{\partial r}\psi^*_t)r^2d\Omega }

allows us to reach the following expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\!} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{-\int\vec{e_r}\vec{j_t}r^2d\Omega}{|\vec{j_{in}}|}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\vec{j_{in}}| = \frac{\hbar k}{m}} .

From this quantity and the penetration probability and energy, a new variable is introduced known as the astrophysical S-factor, which plays a roll in identifying the type and scale of transitions seen from scattering:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = mc^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(E) = \sigma E e^{2\pi\eta}\!} .

Q Value

Another important quantity in nuclear physics is the Q value, which is the energy released in a nuclear reaction. First we need to remember that some of the mass of an nucleus is in the form of binding energy between the protons and neutrons I.E. the energy it takes to hold the nucleus together because the positively charged protons tend to repulse each other. so the mass of a nucleus the mass of its baryons plus the binding energy. We can then use Einsten's famous equation to relate mass to energy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = mc^2 }

now using this we can apply it to a reaction a(b,c)d

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q = \Delta mc^2 }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta m = m_c +m_d -(m_a+m_b) } this means that the energy released is the difference in final minus initial mass times the speed of light squared. If the Q value is positive then it releases energy and the reaction is favorable. If the Q value is negative it means that the reaction can not take place unless the thermal energy is greater than the Q value for the reaction. This usually only occurs in nova and supernova. Note that this equation also applies for decays.

Thermonuclear Reactions

A thermonuclear reaction refers to the fusion of two light atomic nuclei into a single heavier nucleus by means of collision at extremely high temperatures, releasing a large amount of energy. These reactions can persist in chains, or cycles, that continue the nuclear evolution to higher Z nuclei. These cycles, such as proton-proton, and carbon, account for the majority of energy radiated by the Sun and most other stars in the universe.

Reaction Rates

The reaction rate is intuitively defined as how fast or slow a reaction takes place, and is key in understanding thermonuclear reactions and reaction chains. Relative to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma\!} , the reaction rate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\!} is defined by the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{number\;of\;reactions\;target^{-1}sec^{-1}}{flux\;of\;incoming\;projectiles} = \frac{r/n_i}{n_jv} }

Therefore, in order to discuss the reaction rates properly, one needs to know both the cross section of the reaction and the velocity distribution of the beam and target. To begin, as a relative to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\!} particle, the equation is rearranged to define the reaction rate relative to the integrated mean of the cross sections and velocities:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{i,j} = n_in_j \langle\sigma v\rangle_{i.j} }

This integrated mean is given, in terms of the phase space distributions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_{\vec{v}}\!} as the following: $r_{i,j}=n_{i}n_{j}\int \sigma (|{\vec {v}}_{i}-{\vec {v}}_{j}|)|{\vec {v}}_{i}-{\vec {v}}_{j}|\phi ({\vec {v}}_{i})\phi ({\vec {v}}_{j})d^{3}v_{i}d^{3}v_{j}$ where the distribution functions are defined as $\phi ({\vec {v}}_{i})\phi ({\vec {v}}_{j})={\frac {(m_{i}m_{j})^{3/2}}{(2\pi kT)^{3}}}e^{-{\frac {m_{i}v_{i}^{2}+m_{j}v_{j}^{2}}{2kT}}}\!$.

The next step is to transform the integrated mean into the center of mass, using the substitions $M=m_{i}+m_{j},\mu ={\frac {m_{i}m_{j}}{m_{i}+m_{j}}},{\vec {v}}={\vec {v}}_{i}-{\vec {v}}_{j}\!$ and ${\vec {V}}={\frac {m_{i}{\vec {v}}_{i}-m_{j}{\vec {v}}_{j}}{M}}\!$. Finally, defining $E\!$ as the sum of the new center of mass kinetic energy and relative kinetic energy, the integrated mean can be rewritten as:

$\langle \sigma v\rangle (T)=({\frac {8}{\mu \pi }})^{1/2}{\frac {1}{(kT)^{3/2}}}\int _{0}^{\infty }E\sigma (E)e^{-E/kT}dE$

We now move to applications, and actual types of reactions. First considered are resonance reactions. These reactions have enhanced probabilities at energies corresponding to the energy levels of one of the nuclei involved in the collision. This approach to modeling the integrated mean is often valid at low T, and is approximated by $\langle \sigma v\rangle _{res,n}\approx \hbar ^{2}({\frac {2\pi }{\mu kT}})^{3/2}{\frac {(2J_{n}+1)(1+\delta _{ij})}{(2I_{i}+1)(2I_{j}+1)}}{\frac {\Gamma _{j,n}\Gamma _{o,n}}{\Gamma _{n}}}e^{-{\frac {E_{n}}{kT}}}\!$.

Non-resonant reactions are a bit different to model, and must be handled by type in order to properly define the cross section and velocity.For non-resonant, neutron-type reactions, we return to the 2nd order approximation for $\sigma \!$ near the Gammow peak where $S(0)\!$ is nearly constant. With this substitution and use of the equation defining $\sigma \!$ in terms of the S-factor, the value of $\langle \sigma v\rangle _{nonres}\!$ becomes the following:

$\langle \sigma v\rangle _{nonres}=S(0)(1{\frac {\dot {S(0)}}{S(0)}}{\frac {2}{\sqrt {\pi }}}(KT)^{1/2}+{\frac {\ddot {S(0)}}{S(0)}}{\frac {3}{4}}kT)$

For charged particle-based non-resonant reactions, the use of the S-factor remains, however the correction term, as well as the basic form of $\langle \sigma v\rangle _{nonres}\!$, are changed to compensate for the presence of a Coulomb barrier in the interaction. With this, the integrated mean is now defined in terms of a default, ground-state energy $E_{0}\!$ and the width of the Gamow region, /Delta (see homework for derivations of these factors) which that it is expressed as the following:

$\langle \sigma v\rangle _{nonres}=({\frac {2}{\mu }})^{1/2}S_{eff}(0){\frac {\Delta }{(kT)^{3/2}}}e^{-\tau }$

For charged particle reactions, the relevant cross section that overlaps the Maxwell-Boltzmann distribution function is much larger than kT, which is very different then what is seen in non-resonant neutron reactions. As the energy $E_{0}\!$ gets smaller and smaller, it is clear that the probability that the reaction occurs decreases rapidly due to a smaller and smaller Coulomb tunneling probability.

There are more types of reactions that occur with non-Boltzmann components, and to model them, the concept of reverse reactions must be established. At very high temperatures, i.e. in astrophysical plasma objects, reactions can occur in both the exoergic forward direction and endoergic reverse direction. Assuming a Maxwellian distribution function, the ratio of the reaction cross sections is given by the following:

${\frac {\sigma _{i}(j,o)_{J}}{\sigma _{m}(o,j)_{J}}}={\frac {1+\delta _{ij}}{1+\delta _{om}}}{\frac {g_{o}g_{m}}{g_{i}g_{j}}}{\frac {k_{o}^{2}}{k_{j}^{2}}}$ ;where $g\!$ is the temperature-dependent partition function for the particle.

With use of this, the first reaction process examined is photodisintegration. Photodisintegration is a process where an extremely high energy gamma ray interacts with an atomic nucleus and causes it to enter and excited state which immediately grows through a decay, emitting a proton or neutron. This process is responsible for some of the nucleosynthesis of some of the heavy proton rich elements via the p-process that occurs in supernovae. Here, the wavelength, ergo the energy, of the photon relative to the reaction rate is given by the expression:

$\lambda _{i;\gamma ,o}(T)={\frac {g_{o}G_{m}}{(1+\delta _{om})G_{i}}}({\frac {\mu _{om}kT}{2\pi \hbar ^{2}}})^{3/2}e^{-Q_{o,\gamma }/kT}\langle \sigma v\rangle _{m;o,\gamma }$

A second non-Boltzmann reaction is that of electron capture at high densities. At very high densities, the probability of decay by emitting a positron decreases and thus electron capture again becomes a favored channel of decay. This type of reaction occurs often in the r-process, where the electrons in the supernovae or neutron star can be modeled as a degenerate Fermi gas with energy given by $E_{F}={\frac {\hbar ^{2}}{2m_{e}}}(3\pi ^{2}N_{A})^{2/3}(\rho Y_{e})^{2/3}\!$.

Finally, the last non-Boltzmann reaction discussed is that of inelastic neutrino scattering, which is also prevalent in neutron stars and Type II supernovae. Here the bombardment of a nucleon with a neutrino causes fragmentation, conserving the momentum of the particles by changing the kinetic energy. At very high energies, this fragmentation actually breaks the target nucleus into many other particle, and thus is named "deep" inelastic scattering. Simpler inelastic scattering, like that seen in neutron stars and supernovae, involves the knockout of a proton, neutron, or alpha particle by the weak interaction of the neutrino with the target nucleus.

Nuclear Networks

Within nuclear physics and especially nuclear astrophysics, it is often necessary to look at numerous processes working together as opposed to a single reaction. These "cycles" are extremely important in astrophysics when studying nucleosynthesis, as they offer an explanation as to how the elements were formed. What this means is that after a reaction occurs, the products of the first reaction go on to complete a second, and so on. Some examples include the Hydrogen burning process to form Helium and the CNO cycle.

Whilst determining rates for a secondary or tertiary reaction, it is important to take into account the rates of the prior reactions. In many reaction networks, there will be one process that will take much longer than the others, and therefore determine the rate of the overall reaction. These rate limiting steps are usually processes which involve the weak interaction such as beta decay. These reactions occur on much longer time scales relative to the strong interaction, and thus will limit the overall rate. A general equation for the rate of change of abundance of a nuclei within a nuclear reaction network is as follows.

${\frac {\partial n_{i}}{\partial t}}=\sum _{i}n_{j}^{i}r_{j}+\sum _{j,k}n_{j,k}^{i}r_{j,k}$

The first term represents all of the contributions to the rates from processes that create the particular nucleus, where the second term represents processes that destroy the nucleus being studied.

One simple example of a reaction network is the Hydrogen burning process which is also known as the Proton-Proton chain reaction. The steps in the first part of this reaction are as follows.

$^{1}H(p,e^{+}\nu _{e})^{2}H\!$

$^{2}H(p,\gamma )^{3}He\!$

$^{3}H(^{3}He,2p)^{4}He\!$

One can see due to the products of the reactions that the first step is mediated by the weak interaction; the second by the electromagnetic interaction; and the final step by the strong interaction. Since the weak interaction has the longest time scale of the three, we can deduce that the first reaction will act as a rate determining step. To determine a more exact value for the rate of the reaction, we must determine the rate of change in abundance of each products in the cycle.

${\dot {Y}}_{1}=-{\frac {2}{2}}\rho N_{A}\left\langle 1,1\right\rangle Y_{1}^{2}-\rho N_{A}\left\langle 1,2\right\rangle Y_{1}Y_{2}+{\frac {2}{2}}\rho N_{A}\left\langle 3,3\right\rangle Y_{3}^{2}$

${\dot {Y}}_{2}={\frac {1}{2}}\rho N_{A}\left\langle 1,1\right\rangle Y_{1}^{2}-\rho N_{A}\left\langle 1,2\right\rangle Y_{1}Y_{2}$

${\dot {Y}}_{3}=\rho N_{A}\left\langle 1,2\right\rangle Y_{1}Y_{2}-{\frac {2}{2}}\rho N_{A}\left\langle 3,3\right\rangle Y_{3}^{2}$

${\dot {Y}}_{4}={\frac {1}{2}}\rho N_{A}\left\langle 3,3\right\rangle Y_{3}^{2}$

Above, $Y_{1}\!$ is the abundance of $^{1}H\!$, and $Y_{2}\!$ is the abundance of deuterium, $^{2}H\!$. Similarly, $Y_{3}\!$ is the abundance of $^{3}He\!$ and $Y_{4}\!$ is the abundance of $^{4}He\!$.

The Big Bang and the Early Universe

Thermal History of the Big Bang

The first hint of the existence of the Big Bang came from the observation of what is known as the Hubble Expansion. Galaxies were noted to be moving away from ours at a rate that varied linearly with respect to their distance from us. From this was extract a fit that shows the rate of galaxies propagating away versus distance, such that at some point when all of the galaxies were together, this rate was zero. From the slope of this graph, known as the Hubble constant, an approximate time for this joining of galaxies was calculated, namely $T=1/H\!$. This point in time was termed "the Big Bang", since some sort of massive force would have been required to send the galaxies off at the observed rates.

Theoretical Aspects of the Thermal History of the Big Bang and Microwave Background Radiation

Many new observations have been added to this theory, such as the expansion of the universe due to an increase observed in the acceleration of the galaxies and stronger evidence of the Big Bang in the form of background microwave radiation. The Friedman equation treats this expansion, modeling the expansion of an isotropic gas with a scale parameter $R(t)\!$. This will be covered in greater detail in the mathematics section, however it is important to note that all three versions of expansion, constant, accelerating, and decelerating fit the trend observed, however additional astrophysical data suggests that the galaxies are in fact speeding up as they move away. Another important theory involved in the general understanding of the universe is the Cosmological Principle, stating "when averaged over a large enough volume, the universe appears the same in all locations". This is important in that is imposes the condition that the universe is not outside of the laws of physics so that it can be defined by them, and that effectively the universe has no boundaries. That standing, it is important to understand the progressive change of the thermal history of the galaxy and how that ties into the cosmic microwave background radiation that we observe today. During the initial, sub-planck scale time frame of the universe, all four of the fundamental forces were coupled together, with temperatures in excess of $10^{32}K\!$. By $10^{-35}s\!$ the universe had cooled down to $10^{27}K\!$ and gravity had decoupled from the rest of the forces. At approximately $10^{-12}s\!$, the strong force decoupled and there was a soup of fundamental particles present, excited up to a temperature of approximately $10^{15}K\!$. At $10^{-6}s\!$, protons and neutrons begin to farm as the weak force also decouples, now in a temperature regime of $10^{13}K\!$. Three minutes after the big bang, the universe cooled to approximately $10^{10}K\!$ and nucleosynthesis begins to occur. 300,000 years later, at a temperature of roughly $3000K\!$, atoms begin to form, overcoming the high Coulomb barrier due to excited particles. Roughly 100 million years after the big bang, the universe cooled to approximately $10K\!$ as galaxies began to form. Finally, several billion years after the big bang, the universe has cooled to roughly $3K\!$ and we are able to observe microwave radiation from earlier time periods of the cycle.

As the universe cools through the 300,000 year regime mentioned, when it is at approximately $3000K\!$, an interesting delicate event unfolds. The fundamental particles that formed during the previous epoch, neutrons and protons specifically, start at a ratio of 1:1, however as the universe cools that ratio begins to shift towards protons as the neutrons decay out, being less energetically favored then protons. Nuclei are still unable to form as the photon energy is too great. However, just as the proton:neutron ratio reaches 3:1, the universe cools enough for the fusion of deuteron. This leads to the production of $^{4}He\!$, which is stable to decay, and outpaces the rate of decay of neutron.

Mathematics and Applications of the Thermal History and MWB

When discussing the "Standard" Big Bang theory, there are several assumptions that are made in order to simplify the mathematics: thermodynamical equilibrium of all components, a net neutral charge, domination by radiation, and that particle distributions depend only on temperature.

Thermodynamical equilibrium implies adiabatic expansion represented by this equation of state.

$s={\frac {R^{3}}{T}}\left[\rho _{eq}(T)+P_{eq}(T)\right]=Constant$

Where $\rho _{eq}$ is density at equilibrium, and $P_{eq}$ is pressure at equilibrium.

Conservation of energy in this state can be represented similarly by an equation for the change in pressure with respect to temperature.

${\frac {\mathrm {d} P_{eq}(T)}{\mathrm {d} T}}={\frac {1}{T}}\left\{\rho _{eq}(T)+P_{eq}(T)\right\}$

Equation of State for Extremely Relativistic Particles

Energy Density: $\rho (T)=\int _{0}^{\infty }n(p,T)dp{\sqrt {p^{2}+m^{2}}}$

Pressure: $\rho (T)=\int _{0}^{\infty }n(p,T)dp{\frac {p^{2}}{3{\sqrt {p^{2}+m^{2}}}}}$

Entropy Density: $s(T)={\frac {1}{T}}\int _{0}^{\infty }n(p,T)dp\left[{\sqrt {p^{2}+m^{2}}}+{\frac {p^{2}}{3{\sqrt {p^{2}+m^{2}}}}}\right]$

$\rho (T)=g\int _{0}^{\infty }{\frac {4\pi p^{3}dp}{(2\pi \hbar )^{3}}}\left({\frac {1}{e^{\frac {p}{k_{b}T}}\pm 1}}\right)$

The Time Evolution of Temperature

Acceleration by General Relativity: ${\frac {\dot {a^{2}}}{a^{2}}}={\frac {8\pi G\rho (T)}{3}}$

$t=-\int {\frac {s'(T)dT}{s(T){\sqrt {24\pi G\rho (T)}}}}+constant$

Then, add EOS for entropy S:

$t={\sqrt {\frac {3}{16\pi GNa_{B}}}}{\frac {1}{T^{2}}}+constant$

Generally speaking, only particles with mass>kT are present in any significant quantities. This means when T~1.5e12 K, the universe was dominated by muons, electrons, neutrinos and photons. As adiabatic expansion continues, some particles drop out of thermodynamical equilibrium depending on their masses. This occurs because the temperature decline stops reactions that can only occur at higher temperatures, which results in a freeze-out.

The Main Sequence

The main sequence refers to the stars on the Hertzsprung-Russell diagram whose temperature and and luminosity have a linear relationship. These stars are most often dwarf stars, though as stellar evolution progresses they will move away from the main sequence. The mass of the star is what determines the stellar structure as it evolves. Roughly 90% of stars are in this H-burning stage of stellar evolution. Stars of this nature have radioactive centers and convective outer surfaces, so that the temperature gradient naturally moves energy from the core to the surface. The gradient becomes too steep in stars with M > 15 solar masses, however, where the convective and radioactive zones trade places.

Hydrogen Burning

Hydrogen burning is referred to as the P-P cycle, due to the nuclear interactions of the hydrogen (1 proton) nuclei. This is a chain of 2 body reactions, the participants(particularly the abundances) of which determine the reaction rates. The P-P cycle is as follows:

P-P I

${^{1}H}(p,e^{+})d(p,\gamma ){^{3}He}({^{3}He},2p){^{4}He}\!$

P-P II (14% in Sun)

$^{3}He(^{4}He,\gamma )^{7}Be(e^{-},\nu _{e})^{7}Li(p,^{4}He)^{4}He\!$

P-P III (0.02% in Sun)

$^{3}He(^{4}He,\gamma )^{7}Be(p,\gamma )^{8}B(-,\beta ^{+})^{8}Be(-,\gamma )2\,^{4}He\!$

Due to the relative instability of the deuteron atom, the bottleneck of this process is the slow fusion of $^{1}H(p,\gamma )d\!$, this establishing the rate of the overall process based on $^{2}H\!$ production. As one can see, the P-P I process produces ${\frac {1}{2}}$ $^{4}He\!$ per cycle, but the 3 processes together produces one whole helium per cycle, thus doubling the production.

The CNO Cycle

While hydrogen burning is the primary source of energy in smaller stars, stars with greater than about 1.3 Solar Masses get more energy from the CNO(Carbon, Nitrogen, Oxygen) cycle.

$^{12}C(p,\gamma )^{13}N(-,e^{+}\nu _{e})^{13}C(p,\gamma )^{14}N(p,\gamma )^{15}O(-,e^{+}\nu _{e})^{15}N(p,\alpha )^{12}C\!$

This process will bottleneck at the $^{15}N(p,\alpha )^{12}C\!$ portion of the chain and build up an abundance of $^{15}N\!$ that can be used to determine how long the star has been burning in the CNO cycle. Similar to the occurrence in the PP process mentioned above, the CNO process can also go through several different 'breakout' reactions when burning at high enough temperatures which produce such elements as fluoride, neon, and sodium.

Neutrinos

Neutrinos are particles that are produced as byproducts of nuclear reactions. They are electrically neutral and they were believed to only be influenced by the weak interaction. However, there are 3 flavors of neutrinos, which implies that they have mass and are thus subject to the gravitational force. In addition, the fact that they have mass leads many physicists to believe they might possess a small magnetic moment which would let them interact with the electromagnetic force.

Neutrinos have three distinct mass eigenstates, and three distinct flavor eigenstates. These eigenstates, however, are not the same. It is for this reason that neutrino flavors "oscillate." This means that an electron neutrino produced in a $\beta ^{+}$ decay in the Sun may be detected as a muon neutrino here on Earth. As it would turn out, the rate of these oscillations is determined by the density of the medium. That is to say that the neutrino oscillates more in the time it travels from the core of the Sun through the stellar envelope than it does through empty space to Earth.

Post-Main-Sequence Evolution

As stellar evolution progresses, a star's luminosity changes from changes in burning processes. It is for this reason that stars leave the main sequence. In fact, the main sequence only pertains to relatively young stars, as older stars begin new processes depending on their mass, making up the bands beyond the main sequence.

Low Mass Stars

The timescales for which hydrogen fusion will cease in stars of less than ~0.5 solar masses is greater than the currently accepted age of the universe, so it is unclear as to what will happen once these stars stop burning hydrogen. Larger mass stars would burn helium, but these low mass stars do not create the necessary pressure to fuse helium. Computer models suggest that the electron degeneracy pressure will stop the star from collapsing in on itself, which will create a white dwarf.

The Triple $\alpha$ Process

The triple alpha process refers to the burning of helium inside stars to produce carbon for the CNO cycle. The reaction goes as follows.

$^{4}H(\alpha ,\gamma )^{8}Be(\alpha ,\gamma )^{12}C\!$

The beryllium produced is unstable, and therefore the first reaction is the one which limits the rate. The reaction between beryllium and carbon is resonant which allows the reaction to progress. These two reactions create an equilibrium abundance of Be which is helpful in determining the rate equations. As it would turn out, the rate of formation of carbon is proportional to the abundance of helium cubed.

Sometimes the alpha capture can continue to produce oxygen and rarely neon via the $^{12}C(\alpha ,\gamma )^{16}O(\alpha ,\gamma )^{20}Ne\!$ chain.

Core Collapse Supernovae

A core collapse supernova can occur for massive stars, ($M_{core}>1.35M_{\odot }$), when a star's Ni/Fe core exceeds the Chandrasekhar mass limit and the degeneracy pressure can no longer support the mass of the core. The core then rapidly collapses inward, with the outer wall of the core traveling at over 20% the speed of light. As the collapse occurs, the density increases and the temperature rises drastically. This causes it to become energetically favorable for protons and electrons to fuse, creating neutrons and neutrino's. It is eventually this rise in neutron count that increases the neutron density and stops the collapse via strong force neutron-neutron interactions and neutron degeneracy pressure.

Explosions of Massive Stars

When the core collapse does finally stop, the material of the star is still attempting to fall inward and thereby rebounds off the now stable core, producing a shock front that travels outward through the star, with energy on the order if $10^{51}\!$ergs. This would blow the elemental shell envelopes away, however it stalls prematurely in the iron region due to loss of energy from dissociation of Fe atoms and neutrino losses. At the same time, neutrino's are being emitted from the new superdense core, and become stuck in the surrounding material. These neutrino's 'thermalize' the stalled shock and it continue's its propagation through the star, creating a massive explosion that blows the star's envelope out into space and creating large gamma ray emissions in the process. The actual process is highly asymmetric, and some material from the outer layers collapses inward, while material from the inner layers can get pushed to the surface and thereby emitted faster.

Explosive Nucleosynthesis

Most of the nucleosynthesis that occurs during a supernova is the result of Si and O burning. This burning process produces many different elements in the rare earth region, up to iron and nickel. After the burning occurs, the abundance of neutrons produced left from the core collapse become the major contributors to the production of heavier elements. Initially during the actual supernova event, the primary source of heavier elemental production is through the rapid capture of neutrons, the r-process, which produces unstable, neutron-rich nuclei that begin to decay to stable isotopes. During this decay, the s-process can occur, a slow capture of neutrons that becomes prevalent when the lifetime of the nucleus is on the order if the slow neutron capture rate. Two further processes, the proton capture rp-process, and the photodisintegration p-process are also probable contributors to the heavier element production.

Remnants: Black Holes and Neutron Stars

What is left after the gaseous envelope is expelled depends upon the mass of the star that created the supernova. Typically, for stars heavier then 20 solar masses, the resultant core from the core collapse will be too massive to stop from further decompression and will collapse to a single point, the singularity of the black hole. For lighter stars, who had a core between about 1.35 and 2 solar masses, a neutron star is produced.

Neutron Stars

Neutron stars are very hot, and supported from further collapse by the Pauli exclusion principle which creates an effective neutron pressure. The exact structure of neutron stars is not known, as they are relatively extreme phenomena that are difficult to model since conditions like those present inside of the neutron star cannot, as of yet, be reproduced on Earth. Mathematical models based on current nuclear physics have been used to give some conceptual ideals for the general structure of the neutron. Theoretically, the crust of a neutron star should be a tightly bound lattice, but of Fe or lighter nuclei there is some debate, with a sea of electrons floating between the lattice spacings. Going deeper into the star, one would encounter super neutron-rich nuclei that would quickly decay on Earth, but are held stable due to the intense pressure. Going even further pushes one past the neutron drop line so that free neutrons essentially fall off the nuclei. These nuclei then getting smaller and smaller as the superdense core is reached, and vanish completely there. The physics of the core is highly uncertain, but it is believe to consist of a quark-gluon plasma as well as other forms of strange degenerate matter.

Black Holes

These results of massive star collapses are some of the most exotic stellar objects observed. The concept of a black hole, or rather an object so massive that not even light could escape it, was first hypothesized in John Mitchel and Pierre Laplace. However it was not clear how light, a massless wave, could interact with the gravitational field of an object and therefore the idea was ignored for a long period of time. However with the establishment of Einstein's theory of general relativity, the means through which light interacts with the gravitational field was understood. To this end, Karl Schwarzchild and a student by the name of Johannes Droste independently solved Einstein's field equation for point mass. The behavior of the object, however, became strange at a distance now termed the Schwarzchild radius, where several quantities in Einstein's field equation diverged to infinity. For a sufficiently massive object, this would create and effective event horizon at the given radius, a distance at which it became impossible to travel backwards out of the object's field. Through this, and other mathematical frameworks, the nature of black holes was predicted, however none were observed until 1972 when the first candidate for a black hole, Cygnus X-1, was discovered. Since then, many more candidates for black holes have been discovered, and the concepts of intermediate-mass black holes and supermassive black holes have been introduced, with many candidates found for these as well. Until the gravitational waves from these sources are actually measured, it is not possible to say with 100% certainty that they are in fact black holes, however their existence is still partially validated by the immense extragalactic jets seen coming from the centers of accretion discs and by the massive X-ray radiation seem coming from these sources.

Heavy Nuclei Beyond Fe

The formation of elements beyond Iron has intrigued nuclear physicists ever since the current stellar burning models were presented and generally agreed upon. As it would turn out, new processes needed to be elucidated in order to explain the existence of these heavy elements. These processes happen much slower than the timescales associated with regular stellar burning, which leads to a great difference between the abundances of elements up to Iron and the heavier elements. For example, there is a seven order of magnitude difference between the abundance of Iron and that of Gold.

The S-Process

The S-Process, or slow neutron capture process, occurs in intermediate stars with relatively low neutron densities. These low neutron densities result in longer timescales for reactions to occur. This allows the product nuclei to decay by beta process to stable daughter nuclei before the next neutron capture occurs. This process is believed to occur primarily in intermediate stars among the asymptotic giant branch, and produces around half of the neutron rich nuclei beyond Iron, with peaks corresponding to the magic neutron numbers. These numbers are the numbers in which nuclear shells are filled with neutrons and thus symmetrical.

The R-Process

The R-Process refers to rapid neutron capture. The timescales of these reactions are very short, and therefore multiple neutron captures can occur before a beta-decay into a more stable daughter nucleus. This process likely occurs in core-collapse supernovae and is responsible for the other half of neutron rich heavy elements. As soon as the core collapse occurs, there is a very high neutron flux outward, and it is at the intense temperatures in this environment (100 billion K) that allows the neutron capture timescales to become shorter than that of the beta minus decay. The R-Process begins with a "seed" nucleus, usually Nickel-56. The availability of the seed nuclei will influence the final products of the reactions, with more abundant seed nuclei resulting in lighter nuclei. There exists peaks in R-Process nuclei abundances corresponding to the "Double Magic" numbers, similar to the magic numbers, but referring to when all nuclear shells are filled but with a mixture of protons and neutrons. Both the S and R Processes can only occur in mature stars due to their high metallicity and the need for heavy seed nuclei.

The P-Process

Proton rich nuclei are created by proton capture process which release photons. These reactions also produce the neutrons necessary to fuel the S and R processes. The P-Process refers to any proton capture process, and is less common that the aforementioned neutron capture processes due to the need to overcome the Coulomb barrier. Neutrons, with no net charge, do not have this necessity and can be captured by nuclei at much lower temperatures. In high temperature and high proton density environments, many proton captures can occur before decay to a more stable nucleus, resulting in what is known as the rp-Process, or rapid proton capture process. It is this process which allows for nuclei to break out of the hot CNO cycle.