Nuclear Physics equations and more Flashcards
What is the atomic length and energy scale?
What about nuclear?
Atomic length: ~10pm (10^-10), energy: ~1-100eV
Nuclear length: ~1fm (10^-15), energy: ~1MeV (10^6eV)
What is the approximate value of hbar c in MeVfm?
~200MeVfm
What is the de broglie wave equation?
lambda = h/p
What are the heisenberg relations for position-momentum and energy-time?
energy width * lifetime ~ hbar
momentum width * position width ~ hbar/2
Derive the equation for half life in terms of the decay constant lambda.
N is total # nuclei.
time derivative of N is -lambda N
Integrate to give N(t)
N(t1/2) = N0 / 2
What are the spectroscopic notation terms for angular momentum values?
0: s
1: p
2: d
3: f
g, h, i…
How can we extract nuclear mass from atomic mass?
What have we neglected?
Nuclear mass = Atomic mass - Z * electron mass
**we have neglected the electron binding energy (small)
What is the definition of the mass excess?
Mass excess = Atomic mass - A * atomic mass unit
How can we extract the nuclear mass from the mass excess?
Nuclear mass = Mass excess + Au - Zm_e
How can we extract the binding energy using atomic masses?
Binding energy = [Z*(atomic mass of hydrogen) + (#neutrons * neutron mass) - Atomic mass] * c^2
(the sum of constituent parts - the mass)
positive for stable nuclei
How should we interpret binding energy?
Usually binding energy per nucleon is considered.
Positive binding energy implies that a bound state is energetically favourable!
What is a direct method of measuring nuclear masses?
Use a uniform magnetic field and measure circular motion radius. Then use the lorentz force law to calculate the mass.
How is the Q-value defined for a reaction?
Initial rest mass energy - final rest mass energy.
IN TERMS OF NUCLEAR MASSES
**i.e: final KE - initial KE
How should the Q value be interpreted?
Positive: exothermic, energetically favourable
Negative: endothermic, unfavourable
What is Gamow’s saturation hypothesis?
What is the value of the constant as found by matching the model to empirical data?
The nuclear force is “saturated” within a nucleus, i.e: as A increases, the volume per nucleon remains the same:
R = R_0 * A^(1/3)
R_0 = ~1.2fm
What is the definition of “skin thickness”?
What is its usual value?
Skin thickness is the radial distance over which the density of the nucleus falls from 90% to 10% of the central value.
~2.3fm for all nuclei
What distribution do we often use to approximate the charge density distribution in a nucleus?
The fermi-dirac distribution:
rho_0 / [1+ e^((r-R)/a)]
How can we investigate the matter density of a nucleus given information about its charge density?
The matter density is linearly related to the charge density.
rho_mass ~ A/Z * rho_charge
What are the two key observations from measurement of nuclear size?
Density at the nuclear core is ~constant, regardless of A. This suggests saturation of the nuclear force and that the nuclear force is repulsive at very short range.
Skin thickness t is ~constant, regardless of A. This suggests that the nuclear force is attractive at short range.
What is the meaning of charge symmetry and independence?
What is some evidence for this?
The nuclear force acting between two protons ~ that between two neutrons ~ between a proton and a neutron.
Approximately equal mass of the three pions (the exchange particles for nucleon interactions.
-> scattering
The similar properties of mirror nuclei (charge symmetry) and isobaric multiplets (charge independence).
What are the 5 terms in the SEMF (semi-empirical mass formula)?
Volume term, Surface term, Coulomb term, Asymmetry term, nucleon pairing term.
What are the three classical SEMF terms, and their physical interpretation?
Volume term: assumes binding energy per nucleon constant so energy proportional to A.
Surface term: accounts for nucleons at the surface being bound less tightly.
Coulomb term: accounts for the electromagnetic repulsion between protons. (assuming homogenous nucleus)
For what range of mass number is the SEMF sucessful?
A > 20
What is the role of the asymmetry term in the SEMF?
Nuclei prefer N = Z. This effect decreases for heavier nuclei.
What is the role of the pairing term in the SEMF?
It is energetically favourable for nucleon spins to be paired. i.e: for there to be an even number of protons, an even number of neutrons.
Even-even is the most energetically favourable, and odd-odd is the least
**this allows for multiple minima in the binding energy-Z space.
What is the Q value for beta- decay?
[atomic mass (Z, A) {mother atom} - atomic mass (Z+1, A)]c^2
What is the Q-value for beta+ decay?
[atomic mass (Z, A) {mother atom} - atomic mass (Z-1, A) - 2m_e]c^2
What process happens in electron capture?
What is its Q-value?
In proton-rich nuclei, a proton and an electron combine to form a neutron and an electron neutrino.
Q = [atomic mass (Z, A) {mother atom} - atomic mass (Z-1, A) - m_e]c^2
Over what range is fusion of nuclei energetically favourable? Why?
Binding energy per nucleon increases with A up until A=56. I.e Iron is the maximally stable nucleus.
What are the similarities and differences between S and R processes?
They both describe neutron capture by nuclei.
They allow for nuclei heavier than iron to exist.
The intensity of neutron flux determines S or R.
S (slow) absorption means that the nucleus is more likely to beta- decay before absorbing another neutron.
R (rapid) absorption means that the nucleus is more likely to absorb more neutrons before decaying. This allows the “fission gap” to be traversed, producing elements like uranium. (likely to only happen in neutron star mergers.
What is alpha-decay?
What is the Q value for an alpha-decay?
Nuclear fission producing a helium nucleus.
Q = [atomic mass(Z, A) - atomic mass(Z-2, A-4) - atomic mass(2, 4)]c^2
What failure of the SEMF does the shell model account for?
We expect nucleons to occupy a shell-like structure with energy levels inside the nucleus, as they are fermions confined by a potential.
The SEMF fails to predict the existence of “magic numbers” (full nucleon shells)
What mean field theory of the nucleus is successful in reproducing the magic numbers?
The Woods - Saxon potential, including the spin-orbit interaction.
i.e: V = V_ws - epsilon/hbar^2 [L dot S]
**epsilon is a constant determined experimentally to be positive -> states with larger j are lower in energy
How many sates are there for each j value?
2j + 1
What is the expectation value for [L dot S] for:
j = l + 1/2
j = l - 1/2
hbar^2 *
l/2 for j = l + 1/2
-(l+1)/2 for j = l - 1/2
How does angular momentum combine in a nucleus?
Azimuthal components (m_j) can be summed to give m_J, which can then be set as J.
How does parity combine in a nucleus?
The total parity is the product of (-1)^l for each nucleon.
What is the J^P for even-even nuclei?
0+
What are “spin-pairs” and why do they form in a nucleus?
A “spin-pair” is a pair of like nucleons in the same subshell that have equal parity and equal and opposite m_j.
As nucleons are fermions, the overall wavefunction for a pair must be antisymmetric. However due to the attractive nature of the nuclear force, nucleons occupy symmetric spatial states, so must have an antisymmetric spin wave function.
What are the approximate energies required for a single nucleon excitation to the next subshell?
The next shell?
Next subshell ~ 1MeV
Next shell ~10MeV
Are the constants in the SEMF positive or negative?
What ~energy magnitude are they?
Positive
On the order of 10MeV, apart from the coulomb term constant, which is ~1MeV
How is spectroscopic notation written?
What is the form of the more common notation?
(2S + 1) superscript on the left
L main script
J subscript on the right
More commonly
nL_j with 2j+1 states for each j
How should the Gamow energy be calculated in the Geiger-Nuttall relation?
Roughly, use atomic mass units and A to calculate the reduced mass (product of daughter nuclear masses / sum)
How can the Geiger-Nuttall relation be related to nuclear decay?
The decay constant lambda is proportional to the transmission coefficient.
