APPH4010 Intro to Nuclear
Equations and Concepts for Intro to Nuclear
1. Basic Concepts
Name of Concept/Equation  Definition/Equation  Notes  Example 

Isotopes  same atomic number but different neutrons  
Electron Volt  energy equal to the amount gained to accelerate rest electron through a potential difference of 1 volt  
Atomic Mass Unit/Dalton  1/12 of the mass of a neutral atom of $^{12}_6C$  
Magnetic Moment  Magnetic Dipole Moment associated with spin of nucleus  
Spin Quantum Number  nucleons have spin of 1/2 in units of $h / (2\pi)$  
Mass to Energy  $E^2 = p^2c^2 + m_0^2 c^2$  $m_0$ is rest mass, and if at rest, $p=0$  $E_{\mathrm{electron}}=511keV$ 
$E_{\mathrm{amu}}=931.5keV$  
Binding Energy  Energy required to separate its constituent nucleus  using $E=mc^2$  
$B = [Z\cdot m_H + N\cdot m_N  m(A,Z)]c^2$  energy used for binding, so the actual atomic mass $m(A,Z)$ is smaller  consider for carbon $^{12}_6C$ we have $m(A,Z)=12$, but $m_H=1.007825$ and $m_N=1.008665$  
$m_H$ is mass of hydrogen so that it includes weight of electron  
Binding Energy Curve  $B/A$ plotted because binding energy increases just as there are more proton to hold  
the more $A$, much much more $B$ is needed due to short range of internuclear force  
but the above only works until a point where there is only a fixed number of neutrons affecting a proton  
since higher $B$ also means more energy required to break = more energy released when making bonds, we want fission to go from right  
Nuclide Stability Curve  more neutrons is needed for more protons, hence this stability curve in the middle  
so above it is more proton then needed = proton rich  
Properties of Radioactivity  Decay is a random process, but the probability of occurring can be modelled, and macro quantity such as $t_{1/2}$ can be computed  
usually an unstable nuclide “parent” transforms into a more stable nuclide “daughter”  
radioactivity measured in Becquerel = 1 decay per second  
often heavy nuclei gives $\alpha$, and light nuclei $\beta$  
Alpha Emission Properties  $(A,Z) \to (A4, Z2)$  preferred by heavy nuclides  
$\alpha$particle is preferred because it is very stable, hence net effect of decay releases energy  (see below)  
Decay Energy Released  $Q = \Delta m c^2 = (m_{\mathrm{left}}m_{\mathrm{right}})c^2$  if net mass loss = net energy released = RHS more stable = RHS less actual mass  U(238)$\to$ Th(234) + He(4) 
$Q_\alpha = \mathrm{KE}{\mathrm{daughter}}+ \mathrm{KE}{\mathrm{\alpha}}$  for alpha decay  
$Q_\alpha = E_\alpha[1+ \frac{m_\alpha}{m_D}]$  conservation of momentum  
since $Q_\alpha,m_\alpha/m_D$ is known, this means KE energy spectrum for $\alpha$ will be discrete in this case  
Beta Decay Properties  for protonrich nuclides, often see $\beta^+$ decay or electron capture to convert $P \to N$  $p \to n + e^+ + \nu$  
$p+e^ \to n + \nu$  
otherwise, light nuclides often have $\beta^$, which is $N\to P$  $n \to p + e^ + \bar{\nu}$  
KE Spectrum of electron becomes continuous because you have three particles released  
Gamma Decay Property  occurs when excited nucleus lose energy $\Delta E$ as photons  
Rate of Radioactive Decay  $\frac{dN}{dt} = \lambda N$  Given that macroscopically the rate is proportional to number of radioactive nuclei  
$N(t)=N_0e^{\lambda t}$  $\lambda$ is probability per unit time that a nuclide decays  
$A=\lambda N$  since $\lambda$ is prob/time, activity is rate of decay of a sample, measured in Becquerel  
Relativistic Effect  $\Delta T’ = \gamma \Delta T$  high energy particle emitted so that $v \approx c$, then the actual half life becomes $\Delta T’$  $v=0.995c$ when $\nu$ is released with large energy 
$\gamma = \frac{1}{\sqrt{1v^2/c^2}}$  is Lorentz fraction 
5. Interaction of Radiation and Matter
Nuclear radiation normally consists of any particle with energy or photons (particle radiation is the radiation of energy by means of fastmoving subatomic particles). Its interactions with matter gives us opportunity for all experimental work. Therefore, this chapter we consider:
 how charged particle interact with matter
 how uncharged particle interact with matter
 how photons interact with matter
Name of Concept/Equation  Definition/Equation  Notes  Example 

Num Atoms and Atom Density  $N = \frac{m}{M}N_A$  $m$ is mass of the substance you care, $M$ is the atomic weight of it  
$N/V = \frac{m}{V\cdot M}N_A = \frac{\rho}{M}N_A$  $\rho$ is the density of the substance you are  $H_2O$ has $\rho \approx 1g/\mathrm{cm}^3$, and $M=18.0153$  
Heavy Charged Particles Interaction Properties  Two Ways of Interaction: 1) its electric field can ionize atoms in its passage 2) collision (e.g. with electrons) 

Deflect very little from path because its heavy  
Welldefined range, depending on its energy, mass, charge, and the stopping medium  
$\Delta E \approx E_\alpha(\frac{4m_e}{M_\alpha})$  max energy lost per collision  incident alpha particle  
Though short range, as it can produce ionization = excited nuclei produces xrays = need thicker $>R$ protection layer  
Stopping Power  Rate at which a particle loses energy per unit path length  
BetheBloch Formula  
Energy Dependence of Stopping Power  variation of stopping power with energy if incident proton. This general shape works for any charged ion  
$\frac{dE}{dx} \approx \frac{\mathrm{const}}{E^k},k\approx 0.8$  the faster the ion travels, the less chance it interacts with stopping medium, hence negative slope  
$R=\int_{E}^0 dE/(dE/dx) \propto E^{k+1}$  derived from above  
Projectile Dependence of Stopping Power  $dE/dx \propto z^2 f(v)$  depends on particle charge  
$R \propto (m/z^2) F(v)$  range depends on incident particle mass and charge, derived from above  if two charged particle has the same energy, then $R_1/R_2 = (m_1/m_2)(z_2/z_1)^2$ because velocity is the same  
Stopping Medium Dependence of Stopping Power  $\frac{R_1}{R_2} \approx \frac{\rho_2 \sqrt{A_1}}{\rho_1 \sqrt{A_2}}$  depends on stopping medium’s density and mass number  
Bragg Curve  utilize the fact that stopping power increases when energy decreases  
particle travels slower = can produce more ions per unit path  
hence Brag Peak = peak stopping power = peak ionization energy  hit tumor at this distance  
when all energy is lost, it does nothing and stops/halts  
Electrons/$\beta$particle Properties  Two Ways of Interaction: 1) its electric field can ionize atoms in its passage 2) collision (e.g. with electrons) 

a zigzag path  
a much faster speed = less stopping force  
less welldefined range, hence uses meanfreepath  mean free path is the average distance over which a moving particle travels before substantially changing its direction or energy  
you can also use maxrange $R_{\max}$ as a measure, in which case$\rho R_{\max} \propto E_{\max}$  
Stopping Power v.s. Electron Energy  as electrons accelerate by rapidly changing directions, it emits a lot of Bremsstrahlung radiation meaning $\frac{dE}{dx}_{\mathrm{rad}}$ contributes a lot to stopping power  
Reaction Cross Sections  measures area within which if you hit the particle near the target $T$, reaction will occur  can be used for both Neutron and Photon  
hence measures how strongly target $T$ reaction will occur  
Microscopic Cross Section  $\sigma = \sigma(E)$  $E$ is energy of incident particle  
unit is uslaly $\mathrm{cm}^2$ or in Barnes ($10^{24}\mathrm{cm}^2$)  
$\sigma_{\mathrm{total}} = \sigma_{\mathrm{abs}}+\sigma_{\mathrm{sca}}+\sigma_{\mathrm{n,2n}}+…$  
Reaction Rate  $R=N \cdot \sigma \phi = \Sigma \phi$  $N$ is atomic density, $\phi$ is particle flux (particle per cm$^2$ per second)  
$\sigma \phi$ measures number of reactions triggered per second  
Macroscopic Cross Section  $\Sigma \equiv N \sigma$  is the macroscopic cross section  
Gamma Ray Interaction Properties  Primary ways of interaction: 1) photoelectric effect 2) Campton scattering 3) pair production 
because gamma rays are very energetic and they are photons  
we consider attenuation coefficient $\mu_m$ as a measure of probability the photon interact with material = photon vanished  
Photoelectric Effect  photon absorbed by electrons (near nucleus for conservation of momentum) to be ejected  hence that photon disappeared = attenuated  
$T=E_\gamma  B_e = hv  B_e$  $T$ is the emitted photoelectron and $B_e$ is the binding energy of the electron  
secondary reactions might occur if 1) rest of the electrons rearrange themselves for deexcitation and release $\gamma$ray 1) eject low energy electrons for deexcitation and hence Auger electrons to deal with the excess energy 
The Auger effect is a physical phenomenon in which the filling of an innershell vacancy of an atom, by an electron, is accompanied by the emission of another electron from the same atom instead of releasing energy.  
$\sigma_{PE} \propto z^5 / E_\gamma^{3.5}$  free electron = low probability of reaction  
higher electron binding energy = more tightly bound electron has a higher chance (also for conservation of momentum)  
Campton Scattering  $\gamma$ ray scattered off electron and hence we get an electron recoiling and a lower energy photon  
$T = E_\gamma  E_{\gamma ‘} = E_{\text{KE of elec}}$  
$E_{\gamma ‘} = E_{\text{KE of elec}} = E_e  m_ec^2$  for $E_e$ is the total energy of the recoil electron  
$E_{\gamma ‘} = \frac{E_\gamma}{1+(E_\gamma/mc^2)(1\cos \theta)}$  energy of scattered photon from conservation of energy and momentum  
a problem for shielding in real life as $\gamma$ photon didn’t disappear  
Pair Production  creates an electronpositron pair (when heavy nucleus is near to conserve momentum)  
$E_\gamma = 2m_ec^2 + T_ + T_+$  $2m_ec^2$ is the rest mass energy of positron and electron  
has secondary effect when positron recombine with electron $\to$ annihilate and produce two oppositely traveling photon $\to$ can do PE or scattering  
Photon Attenuation  from the three mechanism above, photon either disappear or scattered = not observed by detector = attenuated  
can measure and find out attenuation dependence on the three mechanism  
difficult to measure  
Collimated Photon Attenuation  $dI =  N\sigma I dx$  $x$ is thickness of the material, $N$ is material atomic density, and $\sigma$ is interaction cross section  
$\mu \equiv N\sigma$  linear attenuation coefficient  
represents the probability per unit path length of a photon undergoing an interaction that would remove it from the beam  
$I = I_0 e^{N\sigma x} = I_0 e^{\mu x}$  intensity observed is $I$  
$\mu_m = \mu / \rho$  mass attenuation coefficient  
Photon Attenuation Coefficient Graph  again, there are three phenomenon contributing to $\sigma$ hence $\mu_m$  
Neutron Interactions  A variety of interactions but mostly nuclear reactions: 1) fission if neutrons at few MeV 2) scattering 3) slowed neutrons can give neutron absorption 

Fission Fragments  When $N$ and $_{238}U$ react, it will create two fission fragment and 23 neutrons  hence neutron attenuated  
those neutrons can then start chain reactions  
requires low neutron energy  
Neutron Moderation  if elastic collision, can calculate energy $E$ of neutron with initial energy $E_0$ after colliding with some target nucleus at rest  
$n = (1/\varepsilon) \ln(E_0 / E_n)$  $n$ is the number of collisions of neutrons  
$E_n$ is the energy of neutron after $n$ collisions  
$\varepsilon = (2/A)  (4/(3A^2))$  
if reached low neutron energy = thermal neutron, can do fission  hence attenuated (see above)  
Attenuation in Neutrons  $\sigma_T \approx \sigma_a + \sigma_s$  mostly scattering and absorption  
$I = I_0 e^{N \sigma_T x}$  same equation as photon attenuation  
$I = I_0 e^{\Sigma x} = I_0 e^{ x/\lambda}$  same $\Sigma = N\sigma_T$ as before  
$\lambda = 1 / \Sigma$  mean attenuation length = mean free path  
$1/\lambda = (1/\lambda_a) + (1/\lambda_s)$  since $\Sigma = \Sigma_a + \Sigma_s$ from the first equation 
6. Detectors and Instrumentation
Here, we consider the principle at systems for
 detection radiation
 producing controlled beams of radiation
In general, any detector gets its signal from the interaction of radiations with matter:
 collect charge released by ionization of gas (cased by radiation)
 excitation of electrons in semiconductors (cased by radiation)
 observing fluorescent photons emitted due to deexcitation (cased by radiation)
 making ionization trails visible in film/solid gas (cased by radiation)
and in addition to detecting those radiations, we also want detectors to tell us more information such as energy of the radiation, type of the radiation, dose rate, etc.

dose rate is quantity of radiation absorbed or delivered per unit time

Dose equivalent (or effective dose) combines the amount of radiation absorbed and the medical effects of that type of radiation.
Name of Concept/Equation  Definition/Equation  Notes  Example 

Observe Current by Collecting Charge  $Q = \int_{0}^t i(t)dt$  typically, radiation interact with matter and produces an electric charge (and ion)  
can collect that charge using an electric field = observe a current  
notice the pulse like shape  
In reality, since radiation is random, it is better described by Poisson statistics  
Poisson statistics = probability of a given number of events occurring in a fixed interval of time if the events occur with a known average rate and independently of the time since the last event  
Modes of Detector Operation  Pulse mode: when low radiation flux hence each radiation can be recorded as separate pulse  
Current mode: high particle flux hence average current is recorded  
Detectors Figures of Metric  Things to check for a detector: 1) Energy Resolution 2) Detection Efficiency 3) Dead Time 

Energy Resolution  $E_0$ peak is the source, the Gaussian curve is the fitted energy spectrum measured  
measures FullWidth at Half Maximum (FWHM)  
measured in units of energy or percent of its peak energy $E_0$  
since it is usually first fitted to a Gaussian curve, then FWHM$=2.35\sigma$  
in reality due to all different effects reaching the device  
Detection Efficiency  Measures how much radiation it can capture, and if captured, how much it can record  capture = radiation reaches it record = radiation reached AND recorded 

usually easy to detect charged particles $\alpha$ and $\beta$, but be careful as they have short range  
harder to deal with $\gamma, n$ because they have deposit little energy per unit path  
depends on geometry of the device  if device is liquid so that source submerges in it, then it is $4\pi$ full coverage  
actual performance, dependent on detector material and thickness, and radiation type and energy  
Dead Time  due to physical/electronical problem, there will be deadtime when device becomes unresponsive between events  e.g. takes time for the captured electron to travel to the cathode  
events almost overlap  
$\tau$ = dead time  
so nonparalyzable can recover 4  
paralyzable has the problem of extended dead time  
Dead Time Corrections  Paralyzable detector: $n=me^{n\tau}$  $n=$ true interaction rate, $m$ = recorded interaction rate, $\tau$ = deadtime  
Nonparalyzable detector: $n= m/(1m\tau)$  
when true radiation $n$ is high, paralyzable can mistake it for a low $m$!  
Principles of Gas Detectors  detectors by using gas to be ionized by the radiation $\to$ record those ions  
three types of detectors on this principle 1) ionization chamber 2) proportional counter? 3) GM counter 

Ionization Chamber  works by measuring ionization (of gas molecules) produced solely by incident ionizing particles (e.g. $\alpha$ radiation)  
ionizing particles creates ionized gas $\to$ have a high enough field to prevent recombination $\to$ those ions complete the circuit by having electron goes to anode and positive ion to cathode  
but usually needs an amplifier as current produced could be small  
need a certain amount of electric field applied  see above  
$A$ = #ions per sec produced $\times$ charge per ion  source disposing $1GeV\,s^{1}$ energy per second and air in the chamber ionizes with $34$eV  
#ions per sec produced = $10^9 / 34$  
Pulse Amplitude v.s. Voltage Applied in Gas Detector  the fundamental reason why we have three types of gas detectors  
pulse amplitude can give you information of the energy of ionizing particles!  
region I: increase $V$ means less recombination of ions  
region II: full charge collection = no recombination  Ionization Chamber  
region where output is independent of applied voltage  
Gas Amplification Factor (GAF) = 1  
region III: electrons become more energized $\to$ can cause secondary ionization during collision. Those secondary ionized electrons can further produce ionizations  proportional chamber  
amplification of current = Townsend Avalanche  
GAF up to about $10^5$, but still proportional to the original ionization  
region V: electrons so energized that it can excite inner electrons $\to$ UV radiation from deexcitation $\to$ ionizes other irrelevant atoms in the chamber  GeigerMueller Counter  
can’t distinguish initial input energy of those ionizing particles  
hence can only know the presence of those radiations  
Proportional Chamber  electric field increased beyond region II, so that secondary ionizations occur  
pulse amplitude still tells you energy of input ionizing particles  
GeigerMueller Counter  electric field increased so much that everything is ionized = pulse amplitude does not depend on the energy of input ionizing particles  
hence can only measure the presence of radiation  
Scintillation Detector Mechanics  energy of radiation $\to$ excitation of electrons of scintillation material $\to$ deexcitation/Compton scattering which emits UV/visible light  
those light photons are directed to the photosensitive surface $\to$ emit photoelectrons $\to$ amplified in PMT $\to$ observe pulse of current  
Scintillation Materials  need high effiiency of converting energy to photons  
linear conversion: output proportional to deposited energy from radiation  
short decay time = quick flash = short dead time  
maximize conversion to output fluorescence  
transparent to its own emission (which is photon, which is also energy)  
Scintillation Types  Fluorescence=emit visible radiation with emission time approx. 10 ns  preferred  
Delayed fluorescence=above but loner emission time  
Photofluorescence=longer wavelength and longer emission time  
SemiConductor Detector Mechanics  energy of radiation creates electronhole pair $\to$ electron move in the direction of applied field $\to$ current  works only if those eh pairs do not recombine or get trapped in regions of impurity  
SemiConductor Detector Properties  only requires 34 $eV$ to create eh pair, whereas to create ion pair in gas requires 30 $eV$  
better energy resolution  
faster charge collection = shorter dead time  
SemiConductor Detector Types  diode  
high purity $Ge$  see above  
lithium drifted $Si$ or $Ge$  alternatives to high purity, use $Li$ for drifting as dopant atoms  
Thermoluminscent Detector  operates by accumulating radiation energy and read altogether at the end  
therefore, we want crystals to deexcite as slow as possible when absorbed radiation  opposite of scintillation  
mechanism: electron and holes are elevated but below conduction band, hence “trapped”  
Neutron Detectors Properties  cannot detect neutrons directly, but secondary radiation, such as $(n,p)$  
depending on how fast the neutrons are, there are two types of neutron detectors  
Slow Neutron Detectors Mechanism  for slow, thermal neutrons, nuclear reactions such as $(n,p),(n,\alpha), (n,f)$ have large cross section $\sigma \propto 1/v$  basically can be triggered easily  $(n,p)$ means the reaction of $A+n \to B+p$ 
mechanics: neutron $\to$ nuclear reactions $\to$ charged outputs (e.g. fission fragments) cause ionization $\to$ which can be measured  $^{10} B(n,\alpha)^7 Li$  
Slow Neutron Detector Types  Proportional Counter  $BF_3$ proportional counter performs $^{10} B(n,\alpha)^7 Li$ which a large cross section of 4010 b for thermal neutrons, and $Q=2.79MeV$  
utilizes $(n,\alpha)$  
Fission Counters: coat detector with fissionable material  utilizes $(n,f)$  
Activation Counter: using activation foils composed of material sensitive of neutron of different energies  utilizes neutron capture  
Fast Neutron Detector  use plastic or liquid organic scintillation material instead  $1/v$ means slow neutron detectors become not efficient  
in general, materials of rich hydrogen $\to$ recoiling protons produce energy for scintillation  
Particle Identification  since radiation = any particle with energy, we might also want to know which particle it is  
PI: Counter Telescope  stack two or more detectors, and can measure $\Delta E$ between detectors and $E$ total  
$\Delta E$ can tell you stopping power  
$E\times \Delta E \propto mz^2$  
PI: Time of Flight  measure time between detectors, hence determine $v$  
if radiation is pulsed beam, then you can just measure arrival time at detector and actual beam pulse  
PI: Magnetic Analysis  use spectrometer with magnetic field to measure the deflection of charged particles  
$r = mv/qB$, can measure mass to charge if known velocity  based on Lorentz Force $F=qv\times B$ 
Midterm Concepts
Chapter 1
 atomic structure
 plum pudding model, Rutherford model and its problems, Quantum Theory and its model, Bohr Model and its energy levels
 Rutherford scattering experiment, result
 equivalence of energy and momentum
 massenergy equation
 uncertainty principle
 nuclear models
 liquid drop, shell model, their differences
 electron volt, alternative mass unit
 binding energy, mass excess, and Q calculation
 binding energy curve (reproduce)
 nuclear stability curve (reproduce)
 radioactivity equation and activity
 secular equilibrium
 alpha decay
 energy is discrete, etc
 beta decay
 continuous energy up to a cutoff
 auger electron
 number of atoms and atomic densities calculations
Chapter 5
 stopping power
 two components, use Bremsstrahlung
 BetheBloch that
 stopping power goes $z^2 / v^2$
 stopping power v.s. energy curve
 bragg peak curve
 heavy charged particles
 how it loses energies, ionizing them and/or exciting them
 light charged particles
 zig zag, no welldefined range
 cross sections
 reaction rates, linear attenuation coefficient $\mu$
 attenuation and its differential equation
 photon ways of interacting
 three ways of how they work, but not formula
 neutrons interactions
 three ways, elastic, inelastic, or reaction
 thermal neutrons
Chapter 6

detector two operation mode

detectors figure of merit
 equation of FWHM/$E_0$
 efficiency, especially uncharged particles
 definitions of absolute/intrinsic effiency

dead time
 paralyzable v.s. nonparalyzable
 dead time correction equation
 dead time curves

will need to draw a detector

three types of ionization chambers

gasfield detector v.s voltage
 its five regions and the graph itself

scintillation detector mechanism

semiconductor detector mechanism

TLDs

Neutron Detectoros
 slow.v.s fast needs hydrogen = moderates
 fission counter

particle identification

timeofflight gives velocity + magnetic anlaysis

none of the detector mechanisms except in the particle identification to tell us particle type

Nuclear Structure
Aim:
 understanding what happens inside nucleus
 e.g. understand its properties by understanding what forces are responsible
 no complete theory today fully describes the structure and behavior of complex nuclei
Name of Concept/Equation  Definition/Equation  Notes  Example 

liquid drop model  nucleus regarded as a collection of neutrons and protons forming a droplet of incompressible fluid  
good for systematic behaviors such as nucleon binding energy  
discrepancies in liquid drop model = there is an ORDERED STRUCTURE within the nucleus in which neutrons and protons are arraged in stable quantum states in a potential well  
Shell (Singleparticle Model)  loosely held individual outer nucleons, which account for many of the nucleus’s properties  
very alike the electron shell model and how electrons arrange themselves  
Nuclear Force  binds nucleons in nucleus, nn, pp, pn  extremely complicated, derivation from first principle  
Nuclear Force Properties  short range  
for very small separations, nucleons begin to repel = no clump  
nuclear density approx constant for different sized nuclei = liquid drop  
Nuclear Force Charge Dependency  charge symmetric: same nuclear force for pp as for nn  
Nuclear Force Spin Dependency  Average force for pn > pp or nn by a factor of about 2  both n and p are fermions = obey Pauli Exclusion for spin  
so for pp and nn, you have to have different spin, net $S=0$  
np can have be either antiparallel or paralell. Force in $S = 1$ state is stronger than force in $S = 0$ state. Therefore, avg. force for pn is greater than that for pp or nn (by about factor of 2).  
Explains why can have bound np (deuteron), but not bound nn or pp (i.e., nuclear force not strong enough to bind the latter two configurations)  pp repulsion; nn free particles + not strong enough force  
Nuclear Force SpinOrbit Coupling  $\text{Spinorbit Force}=L \cdot S$  $L,S$ being angular momentum and spin, respectively  
Force is attractive if S and L are parallel, and repulsive if they are antiparallel  
zero on average inside an atom  
SemiEmpirical Mass Formula  estimating binding energy, which can then be used to estimate the actual nuclear masses for unknown nuclei (but known $A$ and $N$)  
based on liquiddrop = estimates collective properties of a nucleus  
SEMF Volume Energy  $a_vA$ term  nucleon feels the force only from its nearest neighbors and the nucleon density is approx constant = force is constant = B/A is approx constant in the interior of the nucleus.  
SEMF Surface Term  $A^{2/3}$ since the first $A\propto \mathrm{Volume}$  reduced by a factor proportional to the surface area of the nucleus  
nucleons on the surface of the nucleus experience the nuclear force from nucleons inside the nucleus, but no force from the outside. hence reduced  
SEMF Coulomb Term  $\propto {Z^2}/{A^{1/3}}$  Coulomb repulsion would further reduce binding energy  
protons repel each other with a longrange Coulomb force; Mean radius of the nucleus is proportional to $R\propto A^{1/3}$ Coulomb force also $\propto Z^2$ 

SEMF Symmetry Term  $\propto (NZ)^2$  nucleus becomes more unstable the greater the difference between Z and N. Has the most effect for light nuclei  
shell model result. The term is zero for $Z = N$ and becomes less important for heavy nuclei (high A), where $N > Z$  
SEMF Pairing Term  $\Delta$  nucleons tend to couple pairwise into more stable configurations  
Δ > 0 if N and Z are both even  from Pauli Exclusion Principle  
Δ < 0 if N and Z are both odd  
Δ = 0 if either N or Z is odd (i.e., A odd)  
Nuclear Fission Energy Barrier  barrier to fission is called the fission barrier or activation energy  
for Heavy Nuclei  As s $A$ increases, the relative importance of the Coulomb repulsion term increases = it becomes energetically possible for the nucleus to split if it becomes deformed enough  
Odd nuclei (($U_{92}^{235}$)) have very low activation energies, since the pairing term is zero, and can fission with low energy neutrons.  
Isobaric Nuclei  same number of total nucleons but swapped #neutrons and #protons  will affect the coulomb term in SEMF  
Vibrational Model  SEMF assumed sphere shape, but in reality can deform  
vibration can actually be modelled with liquiddrop model  
Rotational States  rotation doesn’t produce any change of state  Collective rotational states can only occur in nonspherical nuclei (otherwise how do you know it is rotating?)  
Shell (Independent Particle) Model  aims to model energy level of nucleons  
neutrons and protons fill energy level in the nucleus according to the Pauli Exclusion Principle  similar to electron energy levels  
those structure like properties are not predicted by SEMF, which only deals with collective states  
“islands of stability” corresponding to “closed shells,” also called “magic numbers,”  2 (1s), 8 (2s, 1p), 20, 28, …  
Nuclear Potential Energy  predict energy levels in the nucleus, need to know $V(r)$  arises from interactions with other nucleons.  
infinite well = simplest model  
WoodsSaxon Potential  $V(r) = \frac{V_0}{1+e^{(rR)/a}}$  quite a good approximate  
SpinOrbital Potential  more complicated but accounted for more splittings of energy level  
splittings due to spin $s$ and angular momentum $l$. If aligned, $s\cdot l$ positive hence binding energy is increased 
Nuclear Instability
Aim: Predict/explain why
 some halflives are short and others are long, and
 why certain energy transitions take place and others don’t
In general there will be a) electromagnetic force; b) weak force; c) strong force
Name of Concept/Equation  Definition/Equation  Notes  Example 

Gamma Emission Mechanism  Excited nucleus may deexcite through $\gamma$emission  help understand why certain gamma decay have longer halflives than another  
gamma is “everywhere” in the sense that many radioactive decay accompanies gamma emission (see below, due to excited daughter nucleus)  
Characteristic $\gamma$emission  actually come from the transitions among the energy levels of the daughter nucleus.  
the above means we have two decays: and then 

Selection rules  derived from consideration of conservation of angular momentum and parity, specify the allowable transitions among energy levels  answers: “why are some gamma more likely to happen than others?”  
1. Photon that carries away energy has angular momentum $L>0$  
2. Conservation of Angular Momentum: $\vert l_i  l_f\vert \le L \le \vert l_i + l_f\vert$  $l_i$, $l_f$ is the angular momentum of the initial and final nuclear state  
3. Parity (wavefunction even or odd) is also conserved in EM transitions  whether or not a parity changed decides $\to$  
do not provide information on the probability of its occurrence, only if it can occur  
Competing Process in $\gamma$  as gamma usually comes from deexcitaton, other processes such as deexcite with electron is competing (also comes from deexcitation)  
e.g. Internal Conversion  transfer energy to an orbital electron (Kshell or further out), ejecting it from the nucleus  another way to deexcite = the parent and child would be the same as if performed gamma decay  
Single energy peak for each orbital electron transition unlike continuous β spectrum.  
$\gamma$ Transition Rates  Weisskopf Single Particle γ Transition Rates  
$E_1$ is greatly favored  Selection rules allow E2, M3, E4, M5, and E6 transitions, but the E2 radiation is strongly favored.  
If transitions have high multipolarity, T$_{1/2}$ may be considerable  basically for E6, M6, etc above, the probabiility is low = long half lives  
Mixed $\gamma$ transitions  can substantially raise the probability that a particular energy γray will be emitted  
can’t characterize transition probability using the single particle model  
Beta Decay Equations  
allowed reactions = conservation of energy/momentum/charge/lepton number  
can find the end point of the continous $\beta$ spectrum  recall that $Q$ gives the kinetic energy difference $Q = K_f  K_i = (m_i m_f)c^2$  $T_{end} = E_0  m_e c^2$, the end point of the β spectrum. $E_0$ = total energy of the transition.  
Fermi’s Golden Rule  Probability of beta decay, determine the transition rate between an initial state (i) and a final state (f) for $\beta$ decay  
$\lambda$ measures the transition probability  
Electron Capture  competes with β+decay when both modes are possible  all $\beta$ are due to weak interaction  
Factor affecting EC probability/rate  depends on the overlap of the electron’s and the nucleus’s wave functions  
1. electron is most likely to be captured when it is closer to the nucleus  
2. higher Z = being attracted more = size of $K$orbit electrons orbit is smaller = being captured  EC importance over β+ decay increases with Z  
Alpha Decay  common decay mode for heavy radionuclides  
Shed four units of mass (two neutrons and two protons) in one decay.  
QM “tunneling” in Alpha Decay  heavy nucleus like uranium has barrier of 20MeV, but emitted alpha particle from it can be as low as 5 MeV. now we know it is due to nonzero wave function  
requires $Q>0$  
explanation of QM tunneling  
Preformation Probability  for alpha decay to happen, you need to first form the alpha particle in the nucleus  
Nuclear Reactions
Study a bit further on how/when reaction happens
Name of Concept/Equation  Definition/Equation  Notes  Example 

Types/Classification of Reactions  Elastic scattering: a + A $\to$ a + A  scattering = Incident and outgoing particles are the same  
Inelastic scattering: a + A $\to$ a + $A^*$  some energy goes into exciting internal levels in A, and later will go off $\gamma$ decay  
Knockout: a particle is emitted (“knocked out”) from the nucleus  e.g. stripping of a proton from a carbon nucleus  
Stripping reaction if the transfer is from the projectile to the target.  Transfer reaction: 1 or 2 nucleons are transferred between the projectile and target.  C12 + alpha > C8* + alpha*  
Pickup reaction if the transfer is from the target to the projectile.  Transfer reaction, i.e. target nucleus gained  He3 + p > He4* + gamma  
Direct nuclear reactions: formation of a “new” nucleus typically involve the transfer of just a few nucleons (protons or neutrons) between the colliding nuclei.  without the creation of an intermediate compound nucleus.  e.g. scattering  
Compound nuclear reactions: results in the formation of an intermediate compound nucleus, and can result in the emission of several particles  involve the transfer of many nucleons between the colliding nuclei  
Resonance reaction: incoming particle has right energy to excite an energy level in the target nucleus, greatly increasing the cross section  
Discern what Reaction happened with Energy Spectrum  can distinguish different mechanism because they give rise to outgoing particles have different energy  
Discrete energy peaks at high energies from direct reactions  
At lower energies, peaks correspond to more closely spaced energy levels can’t be resolved  
At still lower energies, compound nuclei are formed, where neutrons and protons share the incoming particle energy and “evaporate” from the nucleus in a continuous spectrum  evaporate: formed compound nucleus $\to$ an equilibrium is reached so the compound nucleus loses its energy slowly over time by emitting particles, mostly protons and neutrons  
Angular Distributions  angle of output particles relative to input particle  
Angular Distributions for Direct Reactions  Direct collisions (few nucleons take part) usually produce forward peaked reaction products  forward peak = products traveling in the same direction as input  
Direct collisions = exhibit oscillations as a function of scattering angle due to the wave nature of the particles  
Angular Distributions for Compound Reactions  Angular spectrum of evaporated particles from a compound nucleus is more isotropic  since the emitted particles “have no memory” of the direction of the incoming particle  
yaxis = prob of observing this  
Reaction rate  each reaction has its own crosssection = own reaction rate  
$R =\sigma N_A \phi=\Sigma \phi$  σ = reaction cross section φ = particle (e.g., neutron) flux $N_A$ = the number of target atoms per unit volume 

DD reaction cross section = higher E better because this is fusion  
Classical Estimate of Reaction Cross Section Assumptions  calculate cross section itself (before we were given this)  
assumptions = reaction happens when come close enough together for the strong nuclear force to act  
Classical Estimate of Uncharged Particles  
$\sigma= \pi (R_1 + R_2)^2 = \pi R^2$  
Classical Estimates of Charged Particles  
Impact parameter, $b$ replaces $R$. $\sigma = \pi b^2 = \pi R^2 (1B/E)$ 
means if $B>E$ reaction cannot occur  
QM Estimate  classical works when particles is more ‘particlelike’  classical approximation of reaction cross section is only decent for particles with de Broglie wavelength less than the size of the nucleus  
also kind of works if we are heavy ions = wavelength often smaller than nuclear dimensions  
but if wave functions coincide, then reaction occur = can occur even if $B>E$  in reality, high energy = looks like particle; low energy = looks like wave  
Usage of Elastic Scattering for Nuclear Structure  force causing scattering depends on the spatial distribution of nucleus => by analyzing the way particles scatterd = know about the size and distribution of force field = know about the nucleus  
Electrons make good probes of the nucleus since they are not absorbed and interact via the wellknow electromagnetic force with the protons  if we take particles such as $\alpha$, then it interacts strongly once inside the nucleus = lose its identity and not reappear in the entrance channel  
Compound Nuclear Reaction Mechanism  Many nuclear reactions proceed in two or more steps  
1. the incoming particle is absorbed by, and excites, the nucleus  
2. the nucleus loses its excitation energy (decays) through one of several different, possible exit channels (decay branches or decay channels)  
Compound Nuclear Reaction Rate  Each decay channel is characterized by a probability and mean lifetime, $\lambda = 1/\tau$  
for compound reaction $A+a\to C\to B+b$ reaction rate is $\sigma_{\alpha, \beta} = \sigma_c (\Gamma_\beta / \Gamma)$ 
$\sigma_c$ is the cross section of forming the compound nucleus $\Gamma_\beta / \Gamma$ is fractional decay width into the final channel $B+b$ 

basically, need $C$ to happen, and then also $\beta$ to happen  
Energy width of Compound Nuclear Reaction  due to the Uncertainty Principle, uncertainties in $\lambda$ and $\tau$ $\to$ uncertainties in energies of the states $\to$ energy spread  
Cross Section Thresholds  each decay mode also has its own ‘excitation energy’ required  
Heavy ion particle accelerator’s goal  Heavy ion particle accelerators are used to cause reactions where the target nuclei are blown apart or to attempt to create new heavy elements through absorption. 
Fission
This section will focus on induced fission from neutron absorption
 Some heavy “transuranic” (near uranium) nuclei can undergo induced fission if supplied with sufficient energy (78 MeV)
 about 6 MeV comes from binding of an extra neutron by the strong force and the rest from external sources
 an extra 12 MeV from pairing (depends on if your $N$ is odd or even)
Name of Concept/Equation  Definition/Equation  Notes  Example 

Why is Fission Energetic Useful?  
Fissile nuclei can fission with low energy, “thermal,” neutrons  
Energy during/Mechanism of Fission  1. normally, the SEMF surface term provides a restoration force, like disturbing surface tension on water  
2. if sufficient energy is supplied, the shape will deform to such an extent that the Coulomb repulsion force will dominate = strong force only works locally = nucleus starts to break  
3. repulsive force will drive the (usually) 2 fission fragments apart and potential energy is converted into kinetic energy  
Fission Activation Energy  The energy required to overcome the fission barrier = deform  
Properties of Fission  Energetically preferred to have one heavy group and one light group as fission fragment  
When a fissionable nucleus absorbs a neutron and forms a compound nucleus, there is a competition between fission and gamma emission to release excitation energy, which doesn’t lead to fission  $\sigma_a = \sigma_f + \sigma_\gamma$  
Discovery of Neutron  James Chadwick found that if the energetic alpha particles emitted from polonium fell on certain light elements, specifically beryllium, an unusually penetrating radiation was produced (not gamma ray)  
this radiation’s range, etc. can be measured by placing paraffin a distance away to perform (n,p) reaction  
the first ‘weird’ event that happened about this was: Walther Bothe and Herbert Becker = discovered that when energetic αparticles from the decay of polonium impinged on certain light materials, they would eject high energy, very penetrating, neutral radiation; they thought they were γrays  
‘Discovery of Fission’  Otto Hahn and Frederick Strassmann  
Observed presence of barium and other middleweight elements in a uranium sample bombarded by neutrons and noted large release of energy.  
12/2/1942 – Enrico Fermi – 1st controlled fission chain reaction in Chicago Pile 1  
Start of Nuclear Weapon  7/16/1945 – 1st nuclear weapon test: Trinity in White Sands Proving Grounds, New Mexico.  
called “The Gadget.” 20 kT (84 TJ). Pu implosion device  
Types of Nuclear Weapons Mechanism  normal plutomium okay, but densely pressed is not = can go over critical mass  
simply have separate halfs of sphere = explosion combine them = increase density = go react now it is uranium 

Why no Pu+Gun Barrel? 1. Because plutonium is less dense than HEU, it would require a larger mass to reach criticality 2. even trace amounts of Pu240 in the plutonium would release enough neutrons from spontaneous fission = wouldn’t explode but fizzle 

Nuclear Power Plant  12/20/1951 – EBR1, Idaho National Lab, Idaho Falls, Idaho. First generation of electric power from nuclear power plant  
Delayed Energy and Delayed Neutrons  nuclear reactor core and spent fuel continue to emit considerable energy even after the nuclear chain reaction process has stopped. This happens because  
1. fission fragments as a byproduct are themselves every radioactive = release energy  
2. delayed neutrons play a critical role in the control of nuclear fission reactions because they are emitted at a slower rate than prompt neutrons, which are emitted immediately following the fission event = helpful to make power change smoother = but still are neutrons  
Neutron Economics  each fission produces 23 neutrons of about 2MeV, but prompt fission neutrons must be thermalized to about 0.025 MeV  
In the slowing down process, must avoid 1. absorption by other nuclei 2. leakage out of the system. 

3. (uncontrollable) absorbed neutron but gave nonfission reaction  
Chain Reaction  
Neutron Multiplication Factor  measure if your chain reaction is sustaining/increasing/decreasing  
recall the section on neutron economics  
Thermal Fission Factor  probability of a fission event occurring when a neutron collides with a nucleus  Not all neutrons absorbed in fuel cause fission  
basically $\sigma_f + \sigma_a = \sigma_T$  if 99.3% U238 and 0.7% U235, then the ratio is $\frac{0.7*\sigma_f(^{235}U)}{(0.7\sigma_T(^{235}U)+99.3\sigma_T(^{238}U))}$ 

because only $^{235}U$ can fission = Most nuclear reactors and all nuclear weapons require higher U235 enrichment  
Methods for U235 Enrichment  Gaseous Diffusion Enrichment: since U238 is a little bit heavier (cannot separate them chemically as they are the same, but physicaly not). Therefore smaller would go through more readily at the top  
and you can repeat this process over and over  
Gas Centrifuge Enrichment: again, U238 is a little bit heavier than U235, so another approach is to spin it very fast  
Enrichment Definitions  LEU – Low Enrichment Uranium: < 20% U235  
HEU – High Enrichment Uranium: > 20% U235  
Weapons Grade Uranium: > 90% U235  
Components in a Nuclear Power Plant  Fuel pellets – typically, sintered UO2 enriched to 35% in 235U.  Fuel pellets and fuel rods are both used to store and transport nuclear fuel in nuclear reactors.  
Fuel rods – typically, clad in Zircalloy to contain fission products and gases  
Control rods – neutron absorbers among fuel assemblies  to regulate the number of neutrons available to sustain the chain reaction  
Reactor core – typically, “square cylinder” shape  
Coolant/moderator – light water most common  to slow down neutrons to thermal  
Structural material – alloys of steel (absorber/reflector)  
PWR and BWR  pressurized, transfer of heat energy  
could have secondary radiation, but more efficient  
PWR Schematic 
Fusion
Why could fusion be useful if we had fission already?
 much less radioactive waste
 but the major problem is it is technically harder
Name of Concept/Equation  Definition/Equation  Notes  Example 

Why is Fusion Energetic Useful?  
Extra Requirement for Fusion  since nuclei are positively charged, so must impart high kinetic energies to the to overcome the repulsive Coulomb barrier = close enough for strong force to form new nucleus  
1. high temperature (i.e., kinetic energy)  
2. high density (close enough)  
3. confinement time: amount of time that the plasma must be confined in order for the reactions to occur  later on will see that density and confinement determines how much energy can be gained  
Magnetic Confinement Basics  confine plasma using magnetic fields = quite complicated setup  because plasma = lot of charged particles in gaseous form  
works based on Lorentz Force  but will lose them if they move in parallel of the field!  
Inertial Confinement  Heat and fuse ions so rapidly that they do not have time to escape.  
typically use laser beams to compress and heat fuel pellets.  
Fusion Reaction Examples  Most uses isotopes of hydrogen to minimize repulsion  
the highlighted ones have high cross section, and nongamma energy release = can be more easily retrained  
Helium as a product is very stable, hence releases a lot of energy!  
Properties of DT Fusion Reactions  high cross section  
produces a lot of energy  
produces an alpha particle, which can be reused to heat up the plasma  
shield against neutrons, which are also very energetic  
tritium (T) don’t occur naturally, so often need to be bred in the reactor factory  can be bred from Lithium  
Why would DT be preferred?  1. requires lower energy 2. produces more energy 

Reaction Rate of Fusion  Reaction prob per unit time = $n_2 \sigma v$  $v$ speed of the particles $n_2$ is density of particle 2 

considers prob of particle 1 interacting with particle 2  
total reaction rate per unit volume: $R=n_1n_2\sigma v$  
$R=n_1n_2 \lang \sigma v \rang$  since plasma = gas particles = velocity is defined by Boltzmann distribution  
Fusion Energy Output and BreakEven  Fusion energy output = $E_f=n_1n_2\lang \sigma v\rang Q \tau$  $\tau$ is confinement time! $Q$ is energy released per reaction 

also includes density $n_1,n_2$  
BreakEven $n\tau > \frac{12kT}{\lang v \sigma \rang Q}$  Lawson Criteria: need to achieve this to get more energy out than input  
Magnetic Bottle Confinement  1. Closedfield geometry (e.g., torus)  
2. Toroidal field (TF) – Coils external and perpendicular to toroidal containment vessel generate TF  
3. Poloidal field (PF) – pass current around axis of torus. Compensates for weakening TF with increasing $r$  
Tokamak is one famous example!  
Inertial Confinement Mechanism  1. the energy from the laser or particle beam pulses is absorbed by the fuel target, heating it to extremely high temperatures and causing it to undergo a rapid expansion.  difficulty is to generate sufficient power to achieve this  
2. This expansion creates a shock wave that compresses the fuel target, increasing the density and temperature of the plasma and creating conditions suitable for fusion reactions.  
therefore, requires lasers and fuel pellets  
National Ignition Facility  NIF: 192 laser beams focused onto small target, uses Indirect target  
Laser pulse vaporizes the heavy metal case, generating intense xrays inside the hohlraum, which compresses and heats the DT fuel  
Hohlraum = the metal cylinder 
Final
 Nuclear sturcture
 spin has an effect on energy
 know every term in SEMF, what it represents
 pairing term
 vibrational harmnoics shapes
 nuclear potential energy
 Nuclear Instability
 gamma emission comes from nucleus, actually comes fro the daughter
 selection rules for gamma emission, including the table
 transition ratess
 Decay schemes
 high polarity and high half life
 internal conversion (eject electron)
 beta decay three possible transitions
 tell by excess protons or neutrons whether if it goes beta plus or beta minus
 alpha decay
 QM consideration that as long as Q is positive it will happen
 Nuclear Reactions
 classification
 scattering, inelastic scattering; knockout
 direct reactions and compoundnucleus decay
 cross section and density equation and reaction rate
 estimate the cross section such as assuming no coulomb interactions with $\pi R^2$; but with intearaction $\pi b^2$
 cross section calculation for charged particles
 no need to know partial waves
 compound nuclear reactions equation
 classification
 Fission
 binding energy curve and why fission is possible
 high cross section with thermal neutrons
 fission activation energy  when you pass the poit of this yuo can fission
 fission history and discovery of neutron
 chain reaction graph of uranium
 neutron multiplication factor
 reactors would want to stay at $k=1$, which is controlled by control rods
 why enrichment is needed
 nuclear reactor basics
 fuel pellets
 Pressurized WR and BWR designs; difference is that water doesn’t boil and goes into heat exchange
 Fusion
 why fusion is possible in binding energy curve
 but the hard part is to have a high temperature, etc.
 magnetic confinement v,s, inertial confinement: heating them so quickly
 reaction rate for fusion equation with $R=n_1n_2 \sigma v$
 triple product and lawson criterion
 toroidal fields, know stuff like this exists
 tokamak
 inertial confinement
 using lasers
 why fusion is possible in binding energy curve

some shared topisc shuch as alpha and beta decay
 Energy calculation, very practical