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	$(A,Z) \to (A-4, Z-2)$	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 proton-rich 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
	$\gamma = \frac{1}{\sqrt{1-v^2/c^2}}$	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 fast-moving 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
	Well-defined 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 x-rays = need thicker $>R$ protection layer
Stopping Power	Rate at which a particle loses energy per unit path length
		Bethe-Bloch 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 zig-zag path
	a much faster speed = less stopping force
	less well-defined range, hence uses mean-free-path	mean free path is the average distance over which a moving particle travels before substantially changing its direction or energy
	you can also use max-range $R_{\max}$ 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) photo-electric 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 de-excitation and release $\gamma$-ray 1) eject low energy electrons for de-excitation and hence Auger electrons to deal with the excess energy	The Auger effect is a physical phenomenon in which the filling of an inner-shell 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 electron-positron 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 2-3 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 semi-conductors (cased by radiation)
• observing fluorescent photons emitted due to de-excitation (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 Full-Width 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 dead-time 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 non-paralyzable 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: $n= m/(1-m\tau)$
		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 de-excitation $\to$ ionizes other irrelevant atoms in the chamber	Geiger-Mueller 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
Geiger-Mueller 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 $\to$ de-excitation/Compton scattering which emits UV/visible light
	those light photons are directed to the photosensitive surface $\to$ emit photo-electrons $\to$ 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
Semi-Conductor Detector Mechanics	energy of radiation creates electron-hole pair $\to$ electron move in the direction of applied field $\to$ current	works only if those e-h pairs do not recombine or get trapped in regions of impurity
Semi-Conductor Detector Properties	only requires 3-4 $eV$ to create e-h pair, whereas to create ion pair in gas requires 30 $eV$
	better energy resolution
	faster charge collection = shorter dead time
Semi-Conductor 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 de-excite 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
  • mass-energy 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 cut-off
• auger electron
• number of atoms and atomic densities calculations

Chapter 5

• stopping power
  • two components, use Bremsstrahlung
• Bethe-Bloch 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 well-defined 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. non-paralyzable
  • dead time correction equation
  • dead time curves

• will need to draw a detector

• three types of ionization chambers

• gas-field detector v.s voltage

  • its five regions and the graph itself

• scintillation detector mechanism

• semi-conductor detector mechanism

• TLDs

• Neutron Detectoros

  • slow.v.s fast needs hydrogen = moderates
  • fission counter

• particle identification

  • time-of-flight 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 (Single-particle 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, n-n, p-p, p-n	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 p-p as for n-n
Nuclear Force Spin Dependency	Average force for p-n > p-p or n-n by a factor of about 2	both n and p are fermions = obey Pauli Exclusion for spin
		so for p-p and n-n, you have to have different spin, net $S=0$
		n-p can have be either anti-parallel or paralell. Force in $S = 1$ state is stronger than force in $S = 0$ state. Therefore, avg. force for p-n is greater than that for p-p or n-n (by about factor of 2).
		Explains why can have bound n-p (deuteron), but not bound n-n or p-p (i.e., nuclear force not strong enough to bind the latter two configurations)	p-p repulsion; n-n free particles + not strong enough force
Nuclear Force Spin-Orbit Coupling	$\text{Spin-orbit Force}=L \cdot S$	$L,S$ being angular momentum and spin, respectively
		Force is attractive if S and L are parallel, and repulsive if they are anti-parallel
		zero on average inside an atom
Semi-Empirical 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 liquid-drop = 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 long-range Coulomb force; Mean radius of the nucleus is proportional to $R\propto A^{1/3}$ Coulomb force also $\propto Z^2$
SEMF Symmetry Term	$\propto (N-Z)^2$	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 liquid-drop model
Rotational States	rotation doesn’t produce any change of state	Collective rotational states can only occur in non-spherical 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
Woods-Saxon Potential	$V(r) = -\frac{V_0}{1+e^{(r-R)/a}}$	quite a good approximate
Spin-Orbital 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

1. some half-lives are short and others are long, and
2. 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 de-excite through $\gamma$-emission	help understand why certain gamma decay have longer half-lives 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 de-excitaton, other processes such as de-excite with electron is competing (also comes from de-excitation)
	e.g. Internal Conversion	transfer energy to an orbital electron (K-shell or further out), ejecting it from the nucleus	another way to de-excite = 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 γ-ray 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 non-zero 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.	C-12 + alpha -> C-8* + alpha*
	Pickup reaction if the transfer is from the target to the projectile.	Transfer reaction, i.e. target nucleus gained	He-3 + p -> He-4* + 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
		y-axis = prob of observing this
Reaction rate	each reaction has its own cross-section = 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 $R$. $\sigma = \pi b^2 = \pi R^2 (1-B/E)$	means if $B>E$ reaction cannot occur
QM Estimate	classical works when particles is more ‘particle-like’	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 well-know 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 (7-8 MeV)
• about 6 MeV comes from binding of an extra neutron by the strong force and the rest from external sources
• an extra 1-2 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 α-particles from the decay of polonium impinged on certain light materials, they would eject high energy, very penetrating, neutral radiation; they thought they were γ-rays
‘Discovery of Fission’	Otto Hahn and Frederick Strassmann
	Observed presence of barium and other middle-weight 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 Pu-240 in the plutonium would release enough neutrons from spontaneous fission = wouldn’t explode but fizzle
Nuclear Power Plant	12/20/1951 – EBR-1, 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 by-product are themselves every radio-active = 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 2-3 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 non-fission 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% U-238 and 0.7% U-235, 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 U-235 enrichment
Methods for U-235 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% U-235
	HEU – High Enrichment Uranium: > 20% U-235
	Weapons Grade Uranium: > 90% U-235
Components in a Nuclear Power Plant	Fuel pellets – typically, sintered UO2 enriched to 3-5% 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 non-gamma 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 re-used 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 Break-Even	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 $n_1,n_2$
	Break-Even $n\tau > \frac{12kT}{\lang v \sigma \rang Q}$	Lawson Criteria: need to achieve this to get more energy out than input
Magnetic Bottle Confinement	1. Closed-field 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 x-rays 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 compound-nucleus 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
• 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

• some shared topisc shuch as alpha and beta decay

• Energy calculation, very practical