Nuclear and Particle Physics Flashcards
plum-pudding model
atom was uniform distribution of positive charge with negative electrons sprinkled evenly inside
what did Geiger and Marsden expect when firing alpha particles at gold foil
electron too small to deflect so expected alpha particles to pass straight through with very slight deflection
what did Geiger and Marsden actually find
alpha particles sometimes deflected by large angles
led Rutherford to hypothesise the nucleus
foundation for Bohr model
for a nucleus with atomic number Z, the mass is
not just Z times the proton mass
what lead to the discovery of the neutron
needed neutral particle to make up the mass
used to think nucleus also contained electrons but inconsistent with quantum mechanics
number of neutrons N=
A-Z
where Z is atomic number
and A is mass number
we can measure nuclear masses using a
mass spectrometer
nuclei are charged so will bend in a magnetic field
know charge so can work out mass from how much they bend
vary magnetic field strength so only one particular mass will reach detector
nuclear masses are measured in
atomic mass units, u
1u is 1/12 the mass of the carbon-12 atom
atom not nucleus so need mass of electron too
how to convert nuclear mass into energy
E=mc^2
units of MeV/c^2
1eV
the energy an electron gains if accelerated through an electromagnetic potential of 1V
approximately
mp =
mn= 1 GeV/c^2 = 1u
neutron slightly heavier than the proton
nuclear masses and atomic masses are not the same because
atomic masses include the electrons AND atomic binding energy
nuclear mass =
atomic mass - electron mass + atomic binding energy
if we scatter electrons off the nucleus, they form
a diffraction pattern and the position of the first minimum gives us the charge radius of the nucleus
measuring many nuclei with atomic mass A we find their radii obey the rule
R=R0 A^1/3 with R0=1.2fm
makes sense since volume scales like R^3 and volume will scale like mass if nuclei have a constant density
how does nucleus stay together
strong nuclear force binds the protons and neutrons together
mass defect
difference between adding up masses of the protons, neutrons and electrons, and the atomic number
this is the energy that is used to bind the protons and neutrons together in the nucleus
for a general nucleus with atomic number Z and atomic mass number A
mNc^2 = Zmpc^2 + Nmnc^2 - B
=Zmpc^2 + (A-Z)mnc^2 - B
where N is number of neutrons and B is the nuclear binding energy
can rearrange for B
why is atomic binding energy neglected
very small compared to nuclear binding energy
the more binding energy the nucleus has…
the more stable it will be
the nucleus needs more binding energy for
more nucleons
useful to consider the binding energy per nucleon B/A
most stable nucleus
Iron Fe
highest binding energy per nucleon
nuclei heavier than iron
want to break apart to become more stable
fission
nuclei lighter than iron
want to join together to become more stable
fusion
where is everything lighter than iron made
in stars
nuclide
a nucleus with a fixed number of protons and neutrons
heaviest stable nuclide
Pb
heaviest naturally occurring nuclide
uranium 238
unstable but half life of billions of years
neutrons feel the same strong nuclear force as protons but do not
feel electromagnetic repulsion
therefore as nuclides get heavier we need more and more neutrons for stability
isotope
nuclides with same no of protons but different no of neutrons
thus different atomic mass number
isobars
nuclides with the same atomic mass number but different numbers of protons and neutrons
isotones
nuclides with the same number of neutrons but different number of protons
thus different atomic mass number
(name isotone derived from isotope but n instead of p becasue Neutrons stay same)
if we exchange a proton for a neutron or vice versa
resulting nuclide will be unstable
if nuclide has too few protons
it will tend to beta - decay, turning a neutron into a proton
if a nuclide has too many protons
will beta + decay, turning a proton into a neutron instead
valley of beta stability
plot of beta - and beta + decay
parabola
the valley of beta stability is described by
the Bethe-Weizsäcker formula
semi-emirical
semi based on experiment
Bethe-Weizsäcker formula.
each term is inspired by
the liquid drop model, with each term’s coefficient fitted to data
first three terms of Bethe-Weizsäcker formula
volume R^3
surface R^2
coloumb 1/R
final two terms in Bethe-Weizsäcker formula
due to asymmetry and pairing
Bethe-Weizsäcker formula gives a parabola so can find
minimum by differentiating
setting mn=mp gives expected result - need roughly as many neutrons as protons to keep nucleus stable but more for larger nuclei
The nucleons are described by the Schrödinger
Equation with an appropriate potential. We can solve this to find…
the allowed energies of the nuclear states.
nuclear shell model - For the nuclear case, we need to find the
best potential. We want the nucleons to
be almost free but held together by the
potential boundary
nuclear shell model - a first try was
a square well potential but a
better try is the Woods-Saxon potential,
which has a smoother boundary
energy levels of the nucleus tend to
clump together into “shells”
nuclear magic numbers
shells occur at nucleon numbers 2, 8, 20, 28, 50, 82, 128, 184…. and a nuclide
with this number of protons or neutrons will be more stable than naively expected.
doubly magic
If a nucleus has a magic number of protons and neutrons, we say it is “doubly
magic”. For example, the 16O nucleus has 8 protons and 8 neutrons, so is doubly
magic and very stable. Our heaviest stable nuclide, lead-208, is also doubly magic
4 main types of nuclear reactions
α-radiation (4He nuclei)
* β-radiation (electrons and positrons)
* electromagnetic radiation (photons)
* neutron radiation
nuclear radiation is ionising radiation because
It can knock the atomic electrons out of their orbit around the
nucleus, resulting in charged ions and free electrons.
alpha radiation
helium nuclei
two protons, two neutrons
when emitted will change Z by 2 and A by 4
typical energy of alpha particle
3-7MeV
range of alpha particles
big and heavy so have a range of only a few cm in air
do not pass through paper
uses of alpha particles
smoke detectors
energy sources - satellites and space probes
radiotherapy
smoke detectors
Some smoke detectors use Americium-241 as an α-particle
source. The α-particles ionise the air between two charged plates
to create a current in the connected circuit. If smoke gets between
the plates, the α-particles are absorbed by the smoke instead
causing the current to stop and setting off the alarm.
energy sources for remote devices like satellites and space probes
. These convert the heat generated by the radioactive decay
into electricity via the thermoelectric (Seebeck) effect.
warnings for earthquakes
Thermal energy in the Earth’s core comes from radioactive decays of 232Th, 238U, 40K and 235U. 238U may
decay to Radon-222, which is radioactive with a half-life of 3.8235 days. Radon is a gas, so seeps out of
cracks if the molten core is close to the surface and is detected by its α-particle emissions
radiotherapy - alpha
α-particles can deposit targeted doses of energy in radiotherapy, by placing the α-source directly in the
tumour and using their short range to keep the damage localised.
beta radiation - 3 processes
beta + decay
beta - decay
electron capture `
beta + decay
proton –> neutron + positron + electron neutrino
beta - decay
neutron –> proton + electron + electron anti neutrino
electron capture
proton + electron –> neutron + electron neutrino
beta + decay can’t happen outside the nucleus because…
the neutron is heavier than the proton. In fact, the proton is
stable with a half-life > 1034 years!
beta particles can by stopped by
sheet of aluminium
uses for beta decay
positron emission tomography
paper manufacture
positron emission tomography (PET)
A patient is injected with a radioactive material that is taken up in
metabolic processes e.g. fluorodeoxyglucose containing unstable Fluorine-18. This is absorbed by the body
(as a sugar), entering the tissues and accumulating inside tumours. 18F decays to 18O via β+-decay emitting a
positron that annihilates an electron in the surrounding atoms to produce photons (e+e− → ϒϒ). The
photons are detected by the PET scanner, to provide a 3-d image of the body. Other examples are Sodium
Fluoride (again with active 18F), which enters the bones, and 15O, which is used to image blood flow.
paper manufacture
to adjust the width of the paper. Put a βemitter on one side and a Geiger-Müller tube on the
other. The β-particles are absorbed by the paper, so
the amount getting through measures the paper’s
thickness. This can be fed back to adjust the paper
rollers and keep a constant thickness.
photons can be ionising if
high enough energy such as gamma rays
EM radiation - gamma rays
typically MeV energies and are made when nuclei drop from one nuclear energy level to another
gamma rays need what to stop
thick block of lead
gamma rays applications
radiotherapy
neutron activation analysis
neutron activation analysis
uses ϒ-Ray emissions to determine the constituents of matter, similar to atomic
spectra. We bombard a material with neutrons to make unstable isotopes that decay, then use the emission of ϒRays to tell us what was present.
alpha particles have +ve charge so
bend in a magnetic field
beta - particles have -ve charge so
bend in a magnetic field, in opposite direction to alpha (beta + in same direction as alpha)
gamma rays are neutral
don’t bend in magnetic field
neutrons can be ejected when
a nucleus breaks up
neutron radiation like fission is energetically favourable because
lighter nuclei need fewer neutrons to keep them stable
alpha particles lose energy by
ionisation, knocking electrons out of their orbits
lose more energy by passing through dense materials and ionise more when travelling slowly
rate of energy lost with respect to distance travelled (alpha)
-dE/dx prop Z^2/v^2
where Z= atomic number, v=velocity of alpha particle
Bragg peak
most of the energy loss will happen just before the alpha particle stops
processes where beta particles lose energy
ionisation
bremsstrahlung
moller or bhabha scattering
positrons may annihilate with electrons
bremsstrahlung
literally ‘breaking radiation’
radiation emitted by a charged particle undergoing a deceleration
moller or bhabha scattering
electromagnetic scattering off the electrons in the material
energy loss of x rays and gamma rays depends on
energy of the photons and the atomic number
photoelectric effect
photon completely absorbed, giving up all its energy and knocking an electron out of its atomic orbit
compton scattering
photon is not completely absorbed but is deflected with reduced energy
electron is ejected from the atom, ionising process
pair production
in the electric field of the nucleus, gamma ray may have enough energy to split into an electron-positron pair
becquerel
(Bq)
SI unit of radioactive activity
1 decay per second
absorbed dose
energy that a material absorbs (SI unit gray)
equivalent dose
takes into account the biological effect on living tissue
equivalent dose = absorbed dose x RBE
sieverts
effective dose
also takes into account the type of tissue
also measured in Sv
probability of an individual nucleus decaying in time dt is
constant
for N nuclei, dN=-lambda Ndt
where -lambda is the decay constant
half life
time taken for sample to half
given by ln2/lambda
probability of having any nuclei in the sample decay must be proportional to
the sample size
mean life-time
t = 1/lambda see notes for derivation
activity
number of decays per second
A(t)=lambdaN(t)
one step decays
A–>B with B stable
NA+NB=N0
NA(0)=N0 and NB(0)=0
NA(t)=N0e^-lambdat
NB(t)=N0=NA(t)
A to B or C with B and C stable
NA+NB+NC=N0
λA=λB+λC
see notes for full calculation
A to B to C
NB(0)=NC(0)=0
NA(0)=N0
see notes for full calculation
actinide alpha decay chains
thorium,uranium,neptunium,actinium series
chains of alpha and beta decays
alpha decays always reduce
the atomic mass A by 4
beta decay always reduce
unchanged atomic mass
parent daughter relation for alpha and beta decays
Amod4 = A’mod4
A’ is daughter
actinide alpha decay chains - atomic mass numbers differ by
4
thorium series
4n
uranium series
4n+1
neptunium series
4n+2
actinium series
4n+3
thorium decays are useful for
determining the age of the Earth
has a very long half life and then a relatively fast decay chain to stable lead
can work out age of the rocks formed
Q value
difference in kinetic energy between initial and final states
positive Q value
exothermic - energy released
negative Q value
endothermic - energy absorbed
kinetic energy threshold
added energy must be the kinetic energy
minimum kinetic energy the particle needs to make reaction happen is the kinetic energy threshold.
daughter nuclei particle can be produced at rest only if
in the centre of mass frame of the nuclei where all the momentum sum to zero
how to work out kinetic energy threshold
calculate Q in the CM frame and translate to lab frame
fission
breaking apart of heavy nuclei to make more stable, lighter nuclei
often triggered by hitting the nucleus with neutron or proton creating an unstable isoptope which breaks apart
nuclear fission power plants
generate power via controlled fission chain reaction
energetic neutrons provide heat to boil water to make steam which turns turbine and produces electricity
thermal neutrons
energies comparable to the material
want to slow down the neutrons using their extra energy to provide heat energy
do this with a moderator
control rods also used to prevent runaway reaction
nuclear fusion
fuse together light nuclei to form heavier, more stable nuclei
energy source for stars
mechanism for making elements lighter than iron
creating fusion in lab extremely difficult because
need to overcome electromagnetic repulsion of the nuclei to get close enough to fuse
inertial frame of reference
viewpoint for taking measurements that is not accelerating
event
happens at a particular point and time
observers in s and s’ will witness the same events but
assign them different coordinates
event on x-axis
coordinates in s’ will be
x’=x-ut
t’=t
galilean relativity
differentiating terms for x’ and t’ shows acceleration same in both frames
newton’s laws of motion are the same
postulates of SR
- laws of physics the same in all inertial reference frames
- speed of light the same in all inertial reference frames
deriving time dilation
bouncing light off a mirror and measure time taken to come back
t=2d/c
for moving, mirror has moved ut’ so t’=2l/c= 2/c sqrt(d^2+(ut’/2)^2)
rearrange and simplify for equation
proper time
time between two events in the reference frame where they happen at the same place
proper length
length of object at rest
whether events are simultaneous depends on
reference frame
four vector
(ct,x,y,z)
momentum four vector
mV=(gamma mc, gamma mv)
standard model
a relativistic quantum field theory built around the symmetry
SU(3)xSU(2)xU(1)
SU(3) is strong nuclear force or quantum chromodynamics
SU(2) is weak
U(1) is EM or quantum electrodynamics
fermions
quarks and leptons
quarks
up, down
charm, strange
top, bottom
leptons
electron, muon, tau and their neutrinos
bosons
photon
gluon
w boson
z boson
higgs
bosons are
force mediators exchanges between particles to transfer momentum
h bar for bosons
integer multiple
W,Z, photon and gluon have spin 1
H has 0
h bar for fermions
half-integer multiples
all SM fermions have spin 1/2
helicity
component of spin in its direction of motion
right handed particle
spin vector in its direction of motion
left handed particle
spin vector against its direction of motion
proton
uud
neturon
udd
charge of up quark
+2/3
charge of down quark
-1/3
each fermion comes with
antimatter partner
same pass but opposite quantum numbers
if an electron and positron come into contact
they annihilate, turning all their mass into photons
noether’s theorm
any symmetry in physics gives a conservation law
strong force
SU(3)
gluon
conserved colour
electromagnetism
U(1)
photon
conserves electric charge
weak force
SU(2)
W,Z bosons
conserves isospin
gravity
graviton
conserves energy/momentum
not included in SM
quantum gravity
quantum mechanics version of GR where mediator is a spin-2 particle called the graviton
the strong force binds
quarks together to form protons and neutrons
residual force binds protons and neutrons into the nucleus
quantum chromodynamics
red, blue, green
colourless if one of each colour
anti-quarks have anti-colour so quark, anti-quark pair is also colourless
the gluon changes
the colours of the quarks
gluon must also have colour and emit/absorb other gluons
so
QCD force weak at high energies and strong at low energies
asymptotic freedom
quarks in bound states are almost free
difference in W and Z bosons
W bosons have electric charges while Z bosons are neutral
weak force is odd because
it only couples to left handed fermions
baryons
bound states of three quarks, each of different colour
eg: protons
mesons
bound states of quark and antiquark
eg: pion
total spin
J=L+S
angular momentum + quark spin
for lightest states, take L=0 so J=S
two possible configurations of spin:
spins all in the same direction to give J=3/2
one spin in the opposite direction to give J=1/2
J=3/2 baryon decuplet
more u to the right
more s as you go down
upside down trianlge - point is sss
J=1/2 baryon octet
corner states uuu,ddd,sss missing
the uds has two separate states
fermion states must have
an antisymmetric wavefunction under the exchange of quarks
for J=3/2, all the spins
point in the same direction so spin is symmetric in interchange of quarks
(states like sss are also symmetric in flavour so allowed)
for J=1/2 one spin
points the wrong way so spins are not symmetric and symmetric flavour combinations like sss are not allowed
mesons must be
the same colour
eg red and anti-red
pseudoscalar mesons
J=0
vector mesons
J=1
charge conjugation
exchange particles and antiparticles
swaps all charges too eg Q to -Q and red to anti-red
conserved by QED and QCD but violated by the weak interaction
parity
change the sign of all space-coordinates
this will also invert all velocities/momenta
conserved by QED and QCD but violated by the weak interaction
Wu experiment
parity violation in the weak interaction was demonstrated using beta decay of cobalt
magnetic field generated by solenoid to align nuclear spin
if parity conserved, particles emitted in direction of nuclear spin and in opposite direction should be same
time reversal
change the sign on the time coordinate
also changes direction of motion (since v=dx/dt)
conserved by QED and QCD but violated by the weak interaction
CP
weak interaction also violates this combinations
CPT
could be conserved
Luders-Pauli theorem
feynman diagram rules
time flows from left to right
fermions=solid line, photons/bosons=wavy line, gluon=curly, higgs=dashed
fermions have arrow in direction of particle flow, antiparticles have arrow pointing against their direction of motion
if vertical, considering emission and absorption
steps to draw feynman diagram
draw initial particles on left, final particles on right and connect them using only lines and vertices in legal ways
the QED vertex
cannot change the flavour or charge of the fermion
QCD vertex
like QED, cannot change flavour/electric charge
weak interaction vertex
W bosons can change fermion flavour
z bosons cannot change flavour or charge
spectators
when before=after
if fermions are massless, they
travel at c and cannot be overtaken - their helicity becomes fixed and there is no ambiguity
above around 250GeV
gauge symmetries are exact and all particle are massless
below around 250GeV
electroweak symmetry breaks, giving mass to the W/Z bosons and fermions
the higgs mechanism
usually energy in field increases with square of field
system wants to be at minimum energy
particle is an oscillation of the field about this minimum
the second derivative wrt the field at the minimum gives
the mass-squared for a boson or the mass for a fermion
the higgs field has
isospon and hypercharge
the higgs field permeates the
entire universe so everywhere W and B go, will be forced to interact with background higgs field which slows them down
higgs field analogy
higgs field is like universe being filled with treacle
objects move through the treacle will be slowed down depending on how much they stick to the treacle
higgs boson
height of valley Q
if E>Q, particle doesn’t ‘see’ bump and is still symmetric about 0
once particle E<Q, particle falls to minimum and breaks symmetry
LHC
magnets used to accelerate protons
collide at 4 points and 4 experiments used to analyse the collision data
Yukawa couplings
the coupling of the Higgs to fermions needs to be put in by hand
