Notes 2

Nuclear Astrophysics

Nuclear Astrophysics is a combination of nuclear physics and astrophysics. Nuclear physics is the study of atomic nuclei, their composition and their interactions, while astrophysics aims at studying galactic objects such as stars and galaxies. Nuclear astrophysics delves into the questions of where and when the elements were created and how nuclear reactions drive cosmic events.

The Nucleus and the forces that govern it (and some that don't)

Atomic nuclei sizes are on the 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^{-14} } meters and form the (relatively) small centers of atoms while carrying most of the mass. The basic constituents of atomic nuclei are protons (Z) and neutrons (N). Protons and neutrons are fermions (particles with half-integral spin, obey Fermi-Dirac statistics, obey the pauli exclusion princile) comprised of three quarks each, fundamental particles on a size 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^{-19} m } . The proton consist of 2 up quarks and one down quark, the neutron consist of two down quarks and one up quark. The up and down quarks have charges +2/3e and -1/3e respectively. The charges of the proton and nuetron (+1, 0) are just the sums of their constituent quarks charges. The relevant interactions on this size/mass scale are the electro-weak and strong forces.

Elements are arranged in the periodic table by order of the atomic number (Z, the number of protons in the nucleus). The atomic mass of an element is A, where A=Z+N. The radius of the nucleus is proportional to the cube root of the mass number

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=R_{0}A^{\frac{1}{3}}}

The strong force (SF) (also known as the Nutclear force) is a short range force that dies off after a few femto-meters (3 x 10^-15 m) (in other words, the strong force is only relevant in the nucleus of an atom) and acts on color and charge. It is 100 times stonger than EM but many many orders of magintude stronger than the other 2 forces, gravity and the weak force. The SF mediates interaction between quarks, gluons, and anti-quarks. The gluons are the exchange particles, gluons being spin 1 particles existing only inside hadrons. Hadrons feel the strong force indirectly (or they feel the residual effect of the SF). Hadrons are baryons (made of three quarks, i.e. the proton and neutron) and mesons (a quark and anti-quark pair). The strong force binds nucleons (protons and neutrons) together in the nucleus. It is charge independent, but dependent on spin orientation. Basically, the strong force glues (via gluons) protons and neutrons together.+

The range of the strong force determined by the uncertainty principle is 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=\frac{\hbar}{mc} } where m is the mass of the virtual pion.

The weak force acts on flavor. Quarks and leptons experience this force. The W+, W-, and Z0 bosons are the mediating particles. They are all very massive which means their range is very small, as are their lifetimes. Like the strong force, the stregnth dies off rapidly with distance. The weak force in a nutshell is the force which governs nuclear decay.

The electromagnetic force acts on charged particles with the exchange particle being the photon. The Gravitational force works on all mass and energy and is mediated by the graviton.

For every matter particle there is an antimatter particle, denoted with the same symbol but with a bar over it. Antiparticles have the same mass, but have opposite values for electric charge, baryon number, and strangeness.

Definitions for Abundance

The particle abundance of isotope 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 } 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 X_{i} = \frac{n_i}{\sum_j n_j} \, }

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_i } is the the number density of particle 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 the sum is taken over all isotopes present. It is also useful to define a relative particle abundance, which is set logarithmically and normalized to the abundance of hydrogen:

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 \ . }

The mass fraction is the fraction of total mass in the sample constituted by species 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 } :

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} } ( Note that 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 m_i \approx A_i m_u \ } 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 m_u = \frac{m_{12C}}{12} = \frac{1}{N_A} } )

Denoting 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\,} 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} , }

allows us to define the particle's number density by the density of the sample and this baryon fraction,

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 \ . }

Another useful quantity is the average mass number, or mean molecular weight, 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 \mu_i = \frac{\sum_i{A_i Y_i}}{\sum_i{Y_i}} = \frac{\sum_i{N_i}}{\sum_iY_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 \ } , which can also be written 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_e = \frac{\sum_i{Z_i Y_i}}{\sum_i{A_i Y_i}} ,}

is the ratio of protons to nucleons in the sample, and similarly to nuclei, the electron number density is found 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 n_e = Y_e \rho N_A \ . }

Determining Solar Abundances

It is widely held that the solar system was formed partly from the gravitational collapse of interstellar matter billions of years ago. If so, then a study of the elemental abundances in our solar system should give us a window to what comprised our region of the galaxy at that time. There are several methods of determining the elemental abundances in our solar system.

Terrestrial methods

Examining the materials that we find on earth can yield the desired abundances. However when studying “native” rocks one must be careful in interpreting the data. Through time chemical fractionation would have separated out some of the elements and affected their distribution. However, chemical processes are not dependent on which isotope of the element is involved. Therefore on Earth the distribution of the isotopes within each element should be the same as when the solar system came into existence.

An exception to the chemical fractionation problem is found in the study of a certain class of meteorites called carbonaceous chondrites. About 6% of all meteorites fall into this category. What is significant here is that they appear to have experienced very little heating; therefore they have undergone minimal chemical process and should reflect what the accurate solar system abundances are.

Extra Terrestrial methods

By observing electromagnetic radiation from the sun we may also determine elemental distributions. This follows the basic premise that each element and chemical compound has its own energy levels and therefore its own unique pattern of absorption and emission lines. One must simply examine the received solar spectra and resolve these patterns to detect these elements and compounds. A complication of this method is that one must have a precise statistical knowledge of how the atoms are ionized and how they interact with photons and other particles like atoms and electrons.

Properties of Astrophysical Plasmas

Quantum Mechanics

A knowledge of quantum mechanics is necessary for many astrophysical calculations. This is Schrödinger's 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 \hat{H}\left |\Psi\right \rangle = i\hbar\frac{\partial }{\partial t}\left |\Psi\right \rangle}

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{H}} is the Hamiltonian operator 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 \left |\Psi\right \rangle} is the complex state vector describing the quantum mechanical state of the system. In this notation the right-pointing vector is called the ket while left pointing vectors are called bra. The inner product of a bra and a ket yields an inner product which is a scalar: 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 \left\langle \Phi | \Psi \right\rangle} . In the position representation we define the wave function 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(\vec{x},t) = \left \langle x | \Psi \right \rangle} The probability density of this system is 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 |\psi|^2 = \psi*\psi} which must be normalized when integrated over all (relevant) space:

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 \left \langle \Psi|\Psi \right \rangle = \int_{}^{} |\psi|^2 d^3x = 1}

In the presence of conservative forces the Hamiltonian operator can be expressed in a the classically equivalent form (for one particle):

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{H} = \frac{\hat{p}^2}{2m} + V = -\frac{\hbar^2}{2m}\nabla^2 + V(\vec{x},t)}

where the position representation of the momentum operator is used: 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{p} = \frac{\hbar}{i}\vec{\nabla}}

A notable case for us is that where the potential is time-independent: 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 V = V(\vec{x})} . In such a case the separation of variables method can be applied and the time-dependent part of Schrödinger's equation can be separated from the space-dependent part. This will yield the time-independent Schrödinger's equation (TISE):

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{-\hbar^2}{2m}\nabla^2\phi(\vec{x}) + V(\vec{x})\phi(\vec{x}) = E\phi(\vec{x})}

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 \phi} is the position dependent part (stationary state) of the wave function 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 E} is defined as the energy of the state.

For a collection of n particles we may write the TISE 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 \left(\sum_{i=1}^{n}\frac{-\hbar^2}{2m_i}\nabla_i^2\right)\phi_{1,2,...,n} + \left (\sum_{i=1}^{n}\sum_{j=i}^{n}V_{ij}\right )\phi_{1,2,...,n} = E\phi_{1,2,...,n}}

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 \nabla_i^2} is the Laplacian operator acting on 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{x}_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 V_{ij}} is the potential energy between particles i and j. For a non-interacting 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 V_{ij} = 0} 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 i \neq j} and we may use separation of variables again to express the stationary state 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 \phi_{1,2,...,n} = \phi_1\phi_2...\phi_n } and the TISE for the entire system as a set of independent TISE's for each particle.

The TISE for a system of free (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 V = 0} ), non-interacting particles in a 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 d\times d\times d} volume yields single particle eigenenergies 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 E = \frac{\pi^2 \hbar^2}{2md^2}(n_x^2 + n_y^2 + n_z^2)}

Fermi Gases

Every quantum particle in nature can either be classified as a fermion or a boson. Identical fermions are known to always have wave functions that are anti-symmetric, 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 \psi_{1,2} = -\psi_{2,1}} while identical bosons are known to have wave functions that are symmetric, 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 \psi_{1,2} = \psi_{2,1}} . This knowledge is necessary to formulate a thermodynamic theory for gases composed of these kinds of particles.

For Fermions some detailed calculation in statistical mechanics yields that their occupation distribution is

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^{\frac{(E(p)-\mu)}{kT} + 1}} }

Using the non-relativistic density of states 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 \omega(E) = \frac{2\pi g}{h^3}(2m)^{3/2}E^{1/2}} we can find the 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 = \int_{0}^{\infty} \omega (E) f(E) dE = \int_{0}^{\infty} \frac{2\pi g}{h^3}(2m)^{3/2}E^{1/2} \frac{1}{e^{\frac{(E-\mu)}{kT} + 1}} dE = \frac{4\pi (2m)^{3/2}}{h^3}\int_{0}^{\infty}\frac{E^{1/2}}{e^{\frac{(E(p)-\mu)}{kT} + 1}}dE}

where we substituted g = 2, the spin degeneracy factor for spin-1/2 fermions. Now we make a coordinate transformation 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 = \frac{E}{kT}} :

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 = 4\pi \frac{(2mkT)^{3/2}}{h^3}F_{1/2}\left (\frac{-\mu}{kT}\right )}

where the Fermi function $F_{n}(z)=\int _{0}^{\infty }{\frac {x^{n}}{1+e^{x+z}}}dx$

Similarly,

P =