Download Nuclear and Particle Physics

Nuclear and Particle
Physics
Lecture 2
Dr Daniel Watts
3rd Year Junior Honours
Course
Thursday January 13th
Contact Details
Course Organisers
Franz Muheim (Particle Physics)
f.muheim@ed.ac.uk
JCMB Room 4407
Daniel Watts (Nuclear Physics)
daniel.watts@ed.ac.uk
JCMB Room 5404
Nuclear Part: Course handouts
will be available on course web portal
at end of course (useful as in colour!)
Main points of Lecture 1
Nuclear physics is crucial to many
fields of research
Rutherford: Atoms revealed as consisting of electrons
orbiting an incredibly dense positively charged nucleus
Nucleus comprised of protons (charged) and neutrons
(uncharged) - mn~mp
~10-15m
~10-10m
Typical energy scale in nuclei is MeV - Much higher
than energies associated with atomic electrons (eV)
Nuclei are dense objects: 1cm3 has mass ~ 2.3x1011 kg
(equivalent to 630 empire state buildings!!)
The forces of nature
1. Strong (nuclear) force
• acts on all particles except leptons
• always attractive (on average)
• short range (10-15 m)
ZAP
short range
force
ng,
i
h
not
hin
not
g
10-15 m
STRONGEST FORCE
Fstrong = 1
2. Electromagnetic force
• acts between all particles with charge
• attractive/repulsive
• always present
but inverse square law
~ 1/R2
SECOND STRONGEST FORCE
Fem ~ 10-2 Fstrong
q2
dominant force in atoms,
molecules, solid bodies (binding)
long range
force
q1
3. Weak force
→
+
+ ν
proton electron
(T1/2 ~ 10 min.)
neutrino
• acts on all particles
• very short range (10-18 m)
• e.g. responsible for β decay
SECOND WEAKEST FORCE
Fweak ~ 10-7 Fstrong
4. Gravitational force
• acts between all particles with mass
• always attractive
• always present
but inverse square law
WEAKEST FORCE
~ 1/R2
m1
long range
force
m2
Fgrav ~ 10-40 Fstrong
but dominant force in the
macrocosmos (large masses)
… binds Earth ,solar system
galaxies…
Nucleus – Bound system of protons and neutrons
interacting under the competing influence of attractive
nuclear (strong) and repulsive electromagnetic forces
(between protons)
Nuclear force does not win over em repulsion indefinitely
electromagnetic force (~Z2) wins at Z≥92
Finite number of naturally occurring atoms (Z≤ 92)
Last completely stable nucleus Lead (Z=82)
Weak forces important in unstable nuclei - β decay (see later)
Rutherford scattering revisited
Scattering α particles from gold nuclei
Which force(s) are involved in this process?
At energies available to Rutherford (~5 MeV) α’s cannot
get close enough to nucleus to feel the strong force
significantly – “Coulomb scattering”
See Tutorial questions!
?
197 Au
79
(charge = +79e !)
Only at higher incident energies do we see effects of
strong force – deviation from simple Coulomb scattering
Quantum mechanics
Revision of some basic concepts
• observable = physical, measurable quantity ⇔ operator
examples
total energy
position
linear momentum
angular momentum
⇔ Hamiltonian Ĥ
⇔
x̂
p̂
⇔
⇔
L̂2
• system described by wave function ψ(x,t) obeying
time dependent Schrödinger equation (TDSE)
⎧ h2 ∂2
⎫
∂
−
+
V
(
x
,
t
)
Ψ
(
x
,
t
)
=
i
h
Ψ(x,t)
⎨
⎬
2
∂t
⎩ 2m∂x
⎭
• static potential - stationary states exist. Described by spatial wave
function ψ(x) obeying time independent Schrödinger equation (TISE)
⎧ h2 d2
⎫
−
+
V
(
x
)
⎨
⎬ψE (x) = EψE (x)
2
2
m
dx
⎩
⎭
eigenfunction
eigenvalue
ĤψE (x) = EψE (x)
eigenvalue equation only satisfied by certain values of E
energy is QUANTISED
measurements of total energy can only yield values which are
eigenvalues of Hamiltonian operator
Orbital angular momentum
L̂2 = L̂2x + L̂2y + L̂z2
L̂2
⇔
square of magnitude of angular momentum
In spherical polar coordinates:
1 ∂ ⎛
1
∂2 ⎤
∂ ⎞
L̂ = −h ⎢
⎜ sin θ ⎟ +
⎥
sin
θ
θ
∂
∂θ ⎠ sin2 θ ∂φ2 ⎦
⎝
⎣
2⎡
2
Important property:
L̂2 commutes with any Cartesian component of angular momentum
[L̂ , L̂ ] = 0
2
i
both operators have simultaneous eigenfunctions
m
spherical harmonics Yl (θ, φ)
Choose:
L̂z = −ih
∂
∂φ
z-component of angular momentum vector
• Eigenvalue equation for L̂2
L̂2Ylm (θ, φ) = l(l + 1)h2Ylm (θ, φ)
l = 0,1,2,3,...
orbital angular momentum
quantum number
angular momentum is QUANTISED
• Eigenvalue equation for L̂z
L̂z Ylm (θ, φ) = mhYlm (θ, φ)
−l≤m≤l
in integer steps
magnetic momentum
quantum number
Eigenstates of L̂2 are DEGENERATE ⇔ (2l+1) possible values
experimental evidence: Stern and Gerlach experiment
(spatial quantisation)
N.B. we refer to a particle in a state of angular momentum l meaning
l(l + 1)h
Intrinsic angular momentum
⇔ Ŝ
No good classical analogue
Electrons, protons, neutrons all have half integer spin: FERMIONS
Ŝ
Ŝz
has eigenvalue s = ½
has eigenvalue ms = ± ½
spin up
spin down
Addition of angular momentum vectors in quantum mechanics
when two states of angular momentum quantum numbers j1 and j2
add (or couple) together,
they form a state with definite TOTAL angular momentum j and
definite jz component
⏐j1-j2⏐ ≤ j ≤ j1 + j2
-j ≤ m ≤ j
these states can be expressed as linear combination of states
of the uncoupled basis {⏐j1,m1,j2 ,m2>} with coefficients known
as Clebsch-Gordan coefficients
Total angular momentum J
Sum of orbital angular momentum and spin of nucleon J = L + S
• Eigenvalue equation for Ĵ2
1 3
j = 0, ,1, ,2,...
2 2
Ĵ2 j, m = j( j + 1)h2 j, m
• Eigenvalue equation for Ĵz
Ĵz j, m = mh j, m
−j≤m≤ j
in integer steps
(2j+1) possible projections
set of (2j+1) states
{ j, m }
called MULTIPLET
Central symmetric potential
Potential energy function V=V(r) is function of r only, not of θ and ϕ
Solution of TISE:
u(r,θ, φ) = R(r)Y(θ,φ)
• radial wavefunction R(r) solution of radial TISE:
⎡ h2 1 d ⎛ 2 d ⎞
l(l + 1)h2 ⎤
r
⎟ + V(r) +
⎢−
⎥R(r ) = ER(r )
2 dr ⎜
2
2
µ
dr
r
2
µ
r
⎝
⎠
⎣
⎦
centrifugal potential
(or barrier)
• spherical harmonics Y(θ,ϕ) solution of angular TISE:
⎡ 1 ∂ ⎛
1
∂ ⎞
∂2 ⎤
Y(θ, φ) = λY(θ, φ)
−⎢
⎜ sin θ ⎟ +
2
2⎥
sin
θ
θ
θ
∂
∂
⎠
⎝
sin
θ
φ
∂
⎣
⎦
N.B. potential V(r) does not appear in angular equation
Ylm (θ, φ) always solutions, no matter what form for V(r)
⇒
Very general result following from spherical symmetry of potential
| Ylm (θ , φ ) |2
Parity π
r
r
Parity operator ⇒ inversion in the origin coordinates r → −r
In polar coordinates:
r→r
θ→π−θ
φ→π+φ
Behaviour of eigenfunction under parity transformation determined
by properties of spherical harmonics.
[
]
P̂ unlm (r, θ, φ) = P̂ R(r )Ylm (θ, φ) = R(r )Ylm (π − θ, π + φ) = R(r )(− 1)l Ylm (θ, φ)
P̂ unlm (r, θ, φ) = (− 1)l unlm (r, θ, φ)
Eigenfunctions have definite parity:
EVEN (or positive) for even l
ODD (or negative) for odd l
We will see in later lectures how conservation of
total angular momentum and parity are of
crucial importance in understanding nuclear
structure and reactions