Formulae Flashcards
deBroglie wavelength
λ = ℏ/p
density
p = M/V
where M = Au
scattering angle
sinθ = 1.22 λ/D
binding energy
B(N,Z) = [(Zmp + Nmn) - M] c^2
where Z is the number of protons
and N is the number of neutrons and M is the mass of the atom
number of fusions
Efusion/Ereactions
fine structure constant
α = 1/4πε e^2/ℏc
α = ke^2/ℏc = 1/137
where ke^2 = 1.44MeV
Energy for a particle in a box
En = ℏ^2π^2/2ma^2 n^2
Electric Parity
P = (-1)^l
The total spin
S = (S1+S2) + (S1+S2-1) + … + |S1-S2|
The total angular momentum
J = (S+L) + (S+L-1) + …+|S-L|
Coloumb Barrier
B = Z1Z2e^2/4πεr
where r = 1.12A^1/3fm
Activity
A0 = Aexp(λt)
where λ = ln2/t(1/2)
Nucleon-nucleon force in terms of the exchange of pions
Yukawa potential
alpha decay
first decay mode.
Beta minus decay
decay involving an e- term on the RHS.
Electron capture decay
The fourth decay mode.
the rate of scattered particles into a given segment of solid angle
N(dot) formula
where L is luminosity and dσ/dΩ is the differential cross section
differential cross section
dσ/dΩ on the formula sheet
where Hint is the interaction hamiltonian
rutherford cross section
(dσ/dΩ) rutherford on the formula sheet
fermi-function
charge density distribution
p(r) on the formula sheet
semi-empirical mass formula
B(N,Z) = aA -bA^2/3 …
first term: volume
second term: surface
third term: symmetry term
fourth term: coloumb term
fifth term: pairing
Q value
Q = (m(initial) - m(final))c^2
the number of possible states a nucleon can occupy in a volume V and momentum interval
dn = 4πp^2dp/(2πℏ)^3 V
Woods-Saxon potential
V_WS on formula sheet
Spin-orbit coupling split in energy
∆Els = 2l+1/2 <Vls(r)>
magnetic moment
µ = g e/2M ℏ/2
number of radioactive nuclei
N(t) = N0 exp(-λt)
where A(t) = λN(t)
N = (mass contained NA)/molar mass (Avogadro’s constant = NA)
dN/dt =
λN
selection rules for multipolarities
|Ji - Jf| ≤ L ≤ |Ji + Jf|
magnetic parity
P = (-1)^(l+1)
Fermi transitions
S = 0, neutrino and electron have anti-parallel spins
∆ J = 0
L = 0
Gamow-Teller transitions
S = 1, neutrino and electron have parallel spins
∆J = 0, ±1
L = 1
how to identify the l (orbital angular momentum quantum number) value
s,p,d,f,g,h etc…
l = 0,1,2,3,4,5…
The most stable nuclei occurs when
dB/dZ = 0
Give the stability of decay modes
Beta minus takes place if Mx > My
Beta plus takes place if Mx - My > 2me
EC takes place if Mx - My > be/c^2
where be is the binding energy of electrons
Global minimum in the liquid drop model
Substitute B(Z,N) into mnuc accounting for me and ignoring b.
Subsitute N = A-Z and collect Z terms.
Giving a second order polynomial
where we can take the derivative with respect to Z and setting equal to zero giving
Z = beta/2 gamma as required.
Geiger Nutall Relationship
log10λ = a + blog10 Rα
where λ is the decay constant and Rα the range of the alpha particles.
Gamov Decay
P = exp(-G)
where G is the Gamov factor and P is the probability of the alpha particle penetrating a barrier.
modern version of the Geiger-Nuttall rule
lnτ = A /√Eα + B
Total lifetime
Γ = ħ/τ
Branching ratio
BR = Γj / Σi Γi
Q value in terms of binding energy
Q = Bfinal - Binitial
Finding the mass of a neutrino - derivation
M(A,Z) = M(A,Z+1) + Em/c^2 + mv
mv = M(3,1 H) - M(3, 2 He) - Em/c^2
if Em is precisely measured.
