Nuclear and ParticlePhysics

Lecture 2

SubAtomic PhysicsNuclear Physics

Dr Daniel Watts

Main points of Lecture 1As well as being an active current area of researchnuclear physics techniques are used extensively in otherdisciplines and in industrial applications

Rutherfords α scattering experiments- atoms contain an incredibly densepositively charged nucleus

α

Au foil

Mass spectrograph - nucleus must contain unchargedneutral component – “neutron”. Subsequently discoveredby Chadwick - mn~mp

Typical energy scale in nuclei is MeV - million times larger than energies associated with atomic systems (eV)

Nucleons in the nucleus rarely collide with sufficient energy to excite the nucleons - nucleon good d.o.f

Nuclei are dense objects: 1cm3 has mass ~ 2.3x1011 kg (equivalent to 630 empire state buildings!!)

Notes Notes

The forces of nature1. Strong (nuclear) force

• acts on all particles except leptons• always attractive (on average)• short range (10-15 m)

STRONGEST FORCE Fstrong = 1

ZAP

10-15 mshort range

force

2. Electromagnetic force

• acts between all particles with charge• attractive/repulsive• always present

but inverse square law• always present

but inverse square law

~ 1/R2

q1

q2

long range force

SECOND STRONGEST FORCE Fem ~ 10-2 Fstrong

dominant force in atoms, molecules, solid bodies (binding)

3. Weak force

• acts on all particles• very short range (10-18 m)• e.g. responsible for β decay

SECOND WEAKEST FORCE

Fweak ~ 10-7 Fstrong

→ + + νproton electron neutrino

(T1/2 ~ 15 min.)

4. Gravitational force

• acts between all particles with mass• always attractive• always present

but inverse square law

WEAKEST FORCE Fgrav ~ 10-40 Fstrong

but dominant force in the macrocosmos (large masses)… binds Earth ,solar system

galaxies…

~ 1/R2

m1

m2

long range force

Which forces are important to understanding the nucleus??

electromagnetic force (~Z2) wins at Z≥92

Finite number of naturally occurring atoms (Z≤ 92)Last completely stable nucleus Lead (Z=82)

Bound system of protons and neutronsInteracting under the competing influence ofAttractive nuclear (strong) and repulsiveElectromagnetic forces (between protons)

Weak forces are negligible in terms of binding the nucleus, but important in that they can change neutrons ↔ protons β decay (see later)

Nuclear force does not win over EM repulsion indefinitely

Notes Notes

Role of the different forces depends on the distance scale

Example: Which forces dominate in Rutherford’s α particle

Strength of forces at different distance scales

?

Need to get within few fm of thenucleus for strong force to contributesignificantly. In Rutherford’sexperiment (~7 MeV) α’s cannot getwithin this range of the nucleus →

therefore Coulomb force dominates.

See Tutorial question !

19779Au

(charge = +79e !)

Would need higher incidentα energies to see effectsof the strong force in thescattering → large deviationsfrom simple Coulombscattering

Example: Which forces dominate in Rutherford’s α particle scattering measurement?

Revision of some basic concepts to be used later in the course

Quantum mechanics

• system described by wave function ψ(x,t) obeying time dependent Schrödinger equation (TDSE)

),(),(),(2

2

22

txt

itxtxVxm

Ψ∂∂

=Ψ

+∂∂

− hh

• observable = physical, measurable quantity ⇔ operator

examplestotal energy ⇔ Hamiltonian

position ⇔linear momentum ⇔

angular momentum ⇔

Hxp

2L

• static potential - stationary states exist. Described by spatial wavefunction ψ(x) obeying time independent Schrödinger equation (TISE)

)x(ψE)x(ψH

)x(ψE)x(ψ)x(Vdxd

m2

EE

EE2

22

=

=

+−h

eigenvalue equation only satisfied by certain values of E

energy is QUANTISED

measurements of total energy can only yield values which areeigenvalues of Hamiltonian operator

eigenfunction eigenvalue

Notes Notes

Orbital angular momentum ⇔ 2L

2z

2y

2x

2 LLLL ++= square of magnitude of angular momentum

In spherical polar coordinates:

∂∂

+

∂∂

∂∂

−= 2

2

222

φθsin1

θθsin

θθsin1L h

[ ] 0L,L i2 = both operators have simultaneous eigenfunctions

commutes with any Cartesian component of angular momentum

Important property:

2L

spherical harmonics )φ,θ(Yml

Choose:φ

iLz ∂∂

−= h z-component of angular momentum vector

• Eigenvalue equation for 2L

)φ,θ(Y)1()φ,θ(YL m2m2ll hll += ,...3,2,1,0=l

• Eigenvalue equation for zL

)φ,θ(Ym)φ,θ(YL mmz ll h= ll ≤≤− m in integer steps

Eigenstates of are DEGENERATE ⇔ (2l+1) possible values2L

angular momentum is QUANTISED

experimental evidence: Stern and Gerlach experiment (spatial quantisation)

orbital angular momentumquantum number

magnetic momentumquantum number

N.B. we refer to a particle in a state of angular momentum l meaning

hll )1( +

Intrinsic angular momentum ⇔ S

No good classical analogue

Electrons, protons, neutrons all have half integer spin: FERMIONS

has eigenvalue s = ½ S

zS has eigenvalue ms = ± ½ spin upspin 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-j ≤ m ≤ j

these states can be expressed as linear combination of statesof the uncoupled basis {j1,m1,j2 ,m2>} with coefficients known

as Clebsch-Gordan coefficients

Total angular momentum J

m,j)1j(jm,jJ 22h+=

• Eigenvalue equation for 2J

,...2,23,1,

21,0j =

m,jmm,jJz h= jmj ≤≤− in integer steps

(2j+1) possible projections

set of (2j+1) states called MULTIPLET{ }m,j

• Eigenvalue equation for zJ

Sum of orbital angular momentum and spin of nucleon J = L + S

Notes Notes

Central symmetric potential

Potential energy function V=V(r) is function of r only, not of θ and ϕ

Solution of TISE:

• radial wavefunction R(r) solution of radial TISE:

( ) ( ) ( )rERrRrµ21)r(V

drdr

drd

r1

µ2 2

22

2

2=

+++

−hllh

centrifugal potential(or barrier)

• spherical harmonics Y(θ,ϕ) solution of angular TISE:

( ) ( )φ,θYλφ,θYφθsin

1θ

θsinθθsin

12

2

2 =

∂∂

+

∂∂

∂∂

−

u(r,θ, φθ, φθ, φθ, φ) = R(r)Y(θ,φθ,φθ,φθ,φ)

N.B. potential V(r) does not appear in angular equation⇒ always solutions, no matter what form for V(r) ( )φ,θYm

l

Very general result following from spherical symmetry of potential

2|),(| φθm

Yl

Distance of eachpoint from the origin proportional to the magnitude in that direction

Parity π

Parity operator ⇒ inversion in the origin coordinates rrrr

−→

In polar coordinates:

φπφθπθ

rr

+→−→

→

( ) ( ) ( )[ ] ( ) ( ) ( )( ) ( )φ,θY1rRφπ,θπYrRφ,θYrRPφ,θ,ruP mmmmn l

l

lll −=+−==

( ) ( ) ( )φ,θ,ru1φ,θ,ruP mnmn l

l

l −=

EVEN (or positive) for even lEigenfunctions have definite parity:

ODD (or negative) for odd l

Behaviour of eigenfunction under parity transformation determined by properties of spherical harmonics.

We will see in later lectures how conservation of total angular momentum and parity are of crucial importance in understanding nuclear structure and reactions This overview deals with the QM of stationary states produced in a static potential. To calculate the probability of transitions between these states we need more advanced QM. (We will touch a little on this later when we look at electron scattering -Fermi’s golden rule)

zPolar plots of sections at y=0 through spherical harmonics

z

y y++

-20

0|),(| φθY

20

1|),(| φθY

+

+--

Even parity Odd parityEven parity

20

2|),(| φθY

+, - sign offunction in each region

Notes Notes

The nucleus and its properties

Nucleus = central part of an atomit contains A nucleons (nucleon = proton or neutron)

A = Z (protons) + N (neutrons)

mass numberatomic number

A bit of nomenclature…

NUCLIDE element with given N and ZISOTOPES elements with same Z but different NISOTONES elements with same N but different ZISOBARS elements with same AISOMERS elements in metastable (i.e. very long-lived) state

A

Z NX

Z ≡ X chemical symbol)

ISOMERS elements in metastable (i.e. very long-lived) state

Chart of nuclides

isotopes

isotones

isobars

Time evolution of the nuclear chart

Prot

on n

mbe

r (

Z)

Neutron number (N)

Chart still being added to as experimental facilities improve

We are still quite some way from reaching the limits of nuclear existence – particularly on the neutron rich side.

New facilities coming online in the near future like FAIR in Germany (http://www.gsi.de/fair/) should remedy this!

Notes Notes

Mass M: nuclear and atomic masses are expressed in ATOMIC MASS UNITS (u)definition: 1/12 of mass of neutral 12C ⇒ M(12C) = 12 u

1u = 1.6605x10-27 kg or 931.494 MeV/c2 (E=mc2)

M(A,Z) < Zmp + Nmn

difference:

External nuclear properties

Charge Ze: protons have +ve charge e = 1.6022x10-19 Cneutrons have zero charge

neutral atom: (A,Z) contains Z electrons orbiting around nucleussymbolically: (Z ≡ X chemical symbol)A

Z NX

difference: ∆M = Zmp + Nmn - M(A,Z) ⇒ mass defect (excess)

accounts for BINDING ENERGY of nuclei (see later)

N.B. we typically use ATOMIC and not NUCLEAR masses⇒ mass of electrons also included

Size R: nuclear radii are expressed in fermis (fm) 1 fm = 10-15 mcompare with atomic dimensions 1 Å = 10-10 m

matter essentially

EMPTY SPACE!

Notes Notes

… more on nuclear sizeWhat do we define as nuclear size?

Consider the following:• the nucleus has a net positive charge Ze (Z protons)• take into account Coulomb + nuclear force

extends to ∞as 1/R2

has short(~10-15 m) range

Resulting potential

Define:barrier height B at a distance from centre R:

V

Coulomb repulsiveB

B = Zze2

4πε RrR

0

-V0 nuclear attractive

B = 4πε0R

for incident charge ze

R = POTENTIAL RADIUS ⇒ related to distribution of protons & neutrons AND the range of nuclear force

potential radius > charge (or mass) radius

CHARGE RADIUS ⇒ related to charge (proton) distribution

Notes

Notes