Nuclear Chemistry and Mass-Energy Relationships

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Nuclear Chemistry and Nuclear Chemistry and Mass-Energy Mass-Energy Relationships Relationships Chapter 3

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Nuclear Chemistry and Mass-Energy Relationships. Chapter 3. The Nuclear Radius. Nucleus is very small single nucleon ~ 1x10 -15 m or 1 fm fm: femtometer, fermion, fermi nucleus ~ 1 – 10 fm atom ~ 1 Å = 1 x10 -10 m = 100,000 fm All experiments suggest that R = r 0 A 1/3 - PowerPoint PPT Presentation

Transcript of Nuclear Chemistry and Mass-Energy Relationships

Nuclear Chemistry and Nuclear Chemistry and Mass-Energy RelationshipsMass-Energy Relationships

Chapter 3

The Nuclear RadiusThe Nuclear Radius

Nucleus is very small • single nucleon ~ 1x10-15 m or 1 fm• fm: femtometer, fermion, fermi• nucleus ~ 1 – 10 fm• atom ~ 1 Å = 1 x10-10 m = 100,000 fm

All experiments suggest that R = r0A1/3

r0 = constant 1.1-1.6 fm; A-mass number

• Measure scattered radiation from an object; λ = h/p• For nuclei with diameter of about 10 fm λ<10 fm,

corresponding to p >100 MeV/c

Nuclear ShapesNuclear Shapes

2:12:1 3:13:1

R(θ,φ) = R0(1 + βYλμ(θ,φ))

λ=2; β = 0 spherical; β < 0 oblate (disk-like) ; β > 0 prolate (football-like)

λ=3; triaxial, octupole deformed

Nuclear Size and DensityNuclear Size and Density

Density profile of three nuclei.The nuclear radius and volumeas a function of A.

Nuclear PotentialNuclear Potential

Nucleus with radius R

n pO

Center of thenucleus

Potential

Distance

R

Nuclear PropertiesNuclear Properties

Angular momentum and Nuclear Spin• Intrinsic spin +1/2 or -1/2• Orbital angular momentum l• Total angular momentum of a single nucleon is: j = l+s =

l + (+_ 1/2)• The total angular momentum of all nucleons is I = Σj

For all even-A nuclei I = 0 or integralFor all odd-A nuclei I is half integralEven – even nuclei have I = 0

Magnetic MomentMagnetic Moment

• Any moving electrical charged object gives rise to a magnetic moment

• μ = (pole strength) x (distance between poles)

ParityParity• Parity involves a transformation that changes the

algebraic sign of the coordinate system. Parity is an important idea in quantum mechanics because the wavefunctions, Ψ, which represent particles can behave in different ways upon transformation of the coordinate system which describes them. Under the parity transformation:

• The parity transformation changes a right-handed coordinate system into a left-handed one or vice versa. Two applications of the parity transformation restores the coordinate system to its original state.

ParityParity

• The value we measure for the observable quantities depend on

• The we have the following assertion:• If V(r) = V(-r) then

2

22)()( rr

ParityParity

Consequence 1

ψ(r) = ± ψ(-r)

ψ(-r) = + ψ(r) positive (even) parity ψ(-r) = - ψ(-r) negative (odd) parity

ParityParityThe parity of a single particle moving in a fixed potential is (-1)ℓ, whereℓ is the orbital angular momentum.

π(nucleus) = π1π2π3π4… πA

multiply parity of every nucleon to get final parity –

We don’t know the wavefunction (ψ) for every nucleon – but since nucleons pair up, every pair has even parity, π = +• even-even π = +• odd-A π = π of last nucleon, πp or πn

• odd-odd π = πpπn

Just as outer electrons determine atomic, molecular properties, outernucleons determine nuclear properties

• Fermi-Dirac

• Bose-Einstein

• Pauli exclusion principle

Pauli Exclusion Principle Applications

Models of Nuclear StructureModels of Nuclear Structure

• Shell Model (Single Particle Model)

• Fermi Gas Model

• Liquid Drop Model

• Optical Model

• Collective Models

Relative Abundance of The ElementsRelative Abundance of The Elements

Solar system

Magic Numbers and Shell ModelMagic Numbers and Shell Model

Maria Goeppert Mayerand Hans Jensen

Nobel Prize Physics 1963

"for their discoveries concerning nuclear shell structure"

M.G. Mayer, Phys. Rev. 75, 1969 (1949)

• a nucleon moves in a common potential generated by all the other nucleons

2

8

2028

50

82

126

184

Energy required to remove proton or neutron (SP or SN, or a pair S2P, S2N) more difficult for Z,N of certain values

nuclear S2N: atomic ionization energy:

Energy to remove neutron pair Energy to remove electron

(note similar pattern)

large change in nuclear radius when 2 nucleons are added to Z,N of certain values

change when adding 2 neutrons:

Normalized to Rstd = r0A1/3

atomic radii:

In reality

The Pauli principle operates The Pauli principle operates independently for Protons and Neutronsindependently for Protons and Neutrons

Only strong interaction

Fermi Gas ModelFermi Gas Model

• Fermi gas model – also called statistical model• treat nucleus as a statistical assembly of particles – in gas state• calculate their momentum distribution and therefore other nuclear

properties• The nuclear forces are expressed as a nuclear potential• The nucleons are in the possible lowest energy states• The highest filled energy level is called Fermi level• Nuclear excitation are obtain by promoting nucleons above the

Fermi level • Thermodynamic properties of excited nuclei (temperature, entropy,

etc)

r

V

neutrons protons

8 MeV

37 MeV43 MeV

8 MeV

For states – highest occupied level is Fermi level lower states constitute Fermi sea

V

neutrons

43 MeV

8 MeV

Fermi sea

Fermi level

Masses Far From StabilityMasses Far From Stability