Formulae

Question 1

deBroglie wavelength

Answer 1

λ = ℏ/p

Question 2

density

Answer 2

p = M/V

where M = Au

Question 3

scattering angle

Answer 3

sinθ = 1.22 λ/D

Question 4

binding energy

Answer 4

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

Question 5

number of fusions

Answer 5

Efusion/Ereactions

Question 6

fine structure constant

Answer 6

α = 1/4πε e^2/ℏc

α = ke^2/ℏc = 1/137

where ke^2 = 1.44MeV

Question 7

Energy for a particle in a box

Answer 7

En = ℏ^2π^2/2ma^2 n^2

Question 8

Electric Parity

Answer 8

P = (-1)^l

Question 9

The total spin

Answer 9

S = (S1+S2) + (S1+S2-1) + … + |S1-S2|

Question 10

The total angular momentum

Answer 10

J = (S+L) + (S+L-1) + …+|S-L|

Question 11

Coloumb Barrier

Answer 11

B = Z1Z2e^2/4πεr

where r = 1.12A^1/3fm

Question 12

Activity

Answer 12

A0 = Aexp(λt)

where λ = ln2/t(1/2)

Question 13

Nucleon-nucleon force in terms of the exchange of pions

Answer 13

Yukawa potential

Question 14

alpha decay

Answer 14

first decay mode.

Question 15

Beta minus decay

Answer 15

decay involving an e- term on the RHS.

Question 16

Electron capture decay

Answer 16

The fourth decay mode.

Question 17

the rate of scattered particles into a given segment of solid angle

Answer 17

N(dot) formula

where L is luminosity and dσ/dΩ is the differential cross section

Question 18

differential cross section

Answer 18

dσ/dΩ on the formula sheet

where Hint is the interaction hamiltonian

Question 19

rutherford cross section

Answer 19

(dσ/dΩ) rutherford on the formula sheet

Question 20

fermi-function

Answer 20

charge density distribution

p(r) on the formula sheet

Question 21

semi-empirical mass formula

Answer 21

B(N,Z) = aA -bA^2/3 …

first term: volume
second term: surface
third term: symmetry term
fourth term: coloumb term
fifth term: pairing

Question 22

Q value

Answer 22

Q = (m(initial) - m(final))c^2

Question 23

the number of possible states a nucleon can occupy in a volume V and momentum interval

Answer 23

dn = 4πp^2dp/(2πℏ)^3 V

Question 24

Woods-Saxon potential

Answer 24

V_WS on formula sheet

Question 25

Spin-orbit coupling split in energy

Answer 25

∆Els = 2l+1/2 <Vls(r)>

Question 26

magnetic moment

Answer 26

µ = g e/2M ℏ/2

Question 27

number of radioactive nuclei

Answer 27

N(t) = N0 exp(-λt)

where A(t) = λN(t)

N = (mass contained NA)/molar mass (Avogadro’s constant = NA)

Question 28

dN/dt =

Answer 28

λN

Question 29

selection rules for multipolarities

Answer 29

|Ji - Jf| ≤ L ≤ |Ji + Jf|

Question 30

magnetic parity

Answer 30

P = (-1)^(l+1)

Question 31

Fermi transitions

Answer 31

S = 0, neutrino and electron have anti-parallel spins

∆ J = 0
L = 0

Question 32

Gamow-Teller transitions

Answer 32

S = 1, neutrino and electron have parallel spins

∆J = 0, ±1
L = 1

Question 33

how to identify the l (orbital angular momentum quantum number) value

Answer 33

s,p,d,f,g,h etc…

l = 0,1,2,3,4,5…

Question 34

The most stable nuclei occurs when

Answer 34

dB/dZ = 0

Question 35

Give the stability of decay modes

Answer 35

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

Question 36

Global minimum in the liquid drop model

Answer 36

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.

Question 37

Geiger Nutall Relationship

Answer 37

log10λ = a + blog10 Rα

where λ is the decay constant and Rα the range of the alpha particles.

Question 38

Gamov Decay

Answer 38

P = exp(-G)

where G is the Gamov factor and P is the probability of the alpha particle penetrating a barrier.

Question 39

modern version of the Geiger-Nuttall rule

Answer 39

lnτ = A /√Eα + B

Question 40

Total lifetime

Answer 40

Γ = ħ/τ

Question 41

Branching ratio

Answer 41

BR = Γj / Σi Γi

Question 42

Q value in terms of binding energy

Answer 42

Q = Bfinal - Binitial

Question 43

Finding the mass of a neutrino - derivation

Answer 43

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.