The Semi-empirical Mass Formulanuclear.dababneh.com/Nuclear-Undergrad/Notes/2012-1/Part-3.pdf · The Semi-empirical Mass Formula • von Weizsäcker invon Weizsäcker in 1935

The Semi-empirical Mass Formulap

• von Weizsäcker in 1935von Weizsäcker in 1935.• Liquid drop. Shell structure.• Main assumptions:Main assumptions:

1. Incompressible matter of the nucleus R ∝ A⅓R ∝ A .

2.Nuclear force saturates.• Binding energy is the sum of terms:Binding energy is the sum of terms:1. Volume term. 4. Asymmetry term.2. Surface term. 5. Pairing term.3. Coulomb term. 6. Closed shell term.…..

Nuclear Physics, BAU, First Semester, 2012-2013(Saed Dababneh).

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The Semi-empirical Mass FormulapVolume Term Bv = + av A

Bv ∝ volume ∝ R3 ∝ A Bv / A is a constant i.e. number of neighbors of each nucleon is independent of the overall size of the nucleus.

B=

ABV constant

The other terms

AThe other terms are “corrections” to this termNuclear Physics, BAU, First Semester, 2012-2013

(Saed Dababneh).2

this term.

The Semi-empirical Mass FormulapSurface Term Bs = - as A⅔

• Binding energy of inner nucleons is higher than that at the surface.• Light nuclei contain largerLight nuclei contain larger number (per total) at the surface.• At the surface there are:

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2

322

0 44 ArAr

=ππ Nucleons.

roπ

1

1ABs ∝

31

AA


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The Semi-empirical Mass FormulapCoulomb Term BC = - aC Z(Z-1) / A⅓

• Charge density ρ ∝ Z / R3.• W ∝ ρ2 R5. Why ???ρ y• W ∝ Z2 / R. • Actually: ρπ drr24yW ∝ Z(Z-1) / R. • BC / A = 4C

- aC Z(Z-1) / A4/3 ρπ 3

34 r

Remember HW 5 ?!Nuclear Physics, BAU, First Semester, 2012-2013

(Saed Dababneh).4

Remember HW 5 … ?!



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Coulomb effectSurface effect Coulomb effectSurface effect


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The Semi-empirical Mass FormulapQuiz Quiz 11From our information so farso far we can write:

...)1()(),( 31

32

+−++−−−=−AZZaAaAaMMZAMZAM CSVHnn

For A = 125, what value of Z makes M(A,Z) a minimum?, ( , )

Is this reasonable…???

So …..!!!!


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The Semi-empirical Mass FormulapAsymmetry Term Ba = - aa (A-2Z)2 / A

• Light nuclei: N = Z = A/2 (preferable).• Deviation from this “symmetry” less BE and stability.• Neutron excess (N-Z) is necessary for heavier nuclei.• Ba / A = - aa (N-Z)2 / A2.• Back to this when we talk about• Back to this when we talk aboutthe shell model.


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The Semi-empirical Mass FormulapPairing Term Bp = δ

Extra Binding between pairs of identical nucleons in the same state (Pauli !) Stability (e.g. α-particle, N=2, Z=2).

even even more stable than even odd or odd even and these are more tightlyeven-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei.Remember HW HW 11\\ ….?!B t d t d ith A ff t f i d l d ithBp expected to decrease with A; effect of unpaired nucleon decrease with total number of nucleons. But empirical evidence show that:

δ ∝ A-¾δ ∝ A .

⎪⎪⎧+ − evenZevenNAap

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⎪⎪⎩

⎪⎪⎨−

=− oddZoddNAa

oddA

p4

30δ


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⎪⎩ p

The Semi-empirical Mass FormulapClosed Shell Term Bshell = ηshell η

• Extra binding energy for magic numbers f N d Zof N and Z.

• Shell model.• 1 – 2 MeV more binding.


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• Fitting to experimental data• Fitting to experimental data. • More than one set of constants av, as …..

I h t t t d ?• In what constants does r0 appear?• Accuracy to ~ 1% of experimental values (BE).• Atomic masses 1 part in 104.• Uncertainties at magic numbers.g• Additional term for deformation.


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)(),(12

−−= MMZAMZAM Hnn

])2()1([ 1231

32

ηδ ++−−−−−− −− AZAaAZZaAaAa aCSV

Variations…….Variations…….Additional physics….Additional physics….Additional physics….Additional physics….Fitting……(Global vs. local)…..Fitting……(Global vs. local)…..


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Work it out …)(),( −−= MMZAMZAM Hnn

)( 2β ZZZAM])2()1([ 123

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2ηδ ++−−−−−− −− AZAaAZZaAaAa aCSV

?),( 2++= γβα ZZZAM

??=

βα

∂M??=

γβ

?0 min =⇒=∂∂ Z

ZM

?=γ ∂Z A


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Mass Parabolas and Stabilityy

2

2

2),( ++= ZZZAM γβα

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−−++−= AaAaAaAM aSVn ηδα

31

4)( −−−−−= aAaMM aCHnβ

3114

)(−− += AaAa

aCHn

γ

β

34 += AaAa Caγβ2

0 min −=⇒=∂∂ ZM


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γ2min∂Z A


Nuclear Physics, BAU, First Semester, 2012-2013 (Saed Dababneh).

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Double β decay! Both Sides!


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Vertical spacing between both parabolas ?

Odd-Odd

p

Even-Even

Nuclear Physics, BAU, First Semester, 2012-2013 (Saed Dababneh).

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Nuclear Spinp• Neutrons and protons have s = ½ (ms = ± ½) so they are fermions and obey the Pauli-Exclusion Principleobey the Pauli-Exclusion Principle.•The Pauli-Exclusion Principle applies to neutrons and protons separately (distinguishable from each other) (IsospinIsospin).• Nucleus seen as single entity with intrinsic angular momentum Ι (orJ).• Associated with each nuclear spin is a nuclear magnetic momentwhich produces magnetic interactions with its environmentwhich produces magnetic interactions with its environment. •The suggestion that the angular momenta of nucleons tend to form pairs is supported by the fact that all nuclei with even Z and even N h l i 0have nuclear spin Ι=0. • Iron isotopes (even-Z), for even-N (even-A) nuclei Ι=0. • Odd-A contribution of odd neutron half-integer spin.Odd co t but o o odd eut o a tege sp• Cobalt (odd-Z), for even-N contribution of odd proton half-integer spin.

Odd N t o npaired n cleons large integer spin


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• Odd-N two unpaired nucleons large integer spin.

Nuclear Spinp

NaturalZ A Spin NaturalAbundance Half-life Decay

26 54 0 0.059 stable ...

26 55 3/2 ... 2.7y EC26 55 3/2 ... 2.7y EC

26 56 0 0.9172 stable ...

26 57 1/2 0.021 stable ...

26 58 0 0 0028 stable26 58 0 0.0028 stable ...

26 60 0 ... 1.5My β-


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Nuclear Spinp

Z A Spin Natural Abundance Half-life Decay

27 56 4 ... 77.7d β+β

27 57 7/2 ... 271d EC

27 59 7/2 1.00 stable ...

27 60 5 ... 5.272y β-


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Nuclear Parityy

• ψ(r) ψ(-r) Even.ψ(r) ψ( r) Even.• ψ(r) -ψ(-r) odd.• For a nucleon ψ is either of even (π = +) or• For a nucleon ψ is either of even (π = +) or odd (π = -) parity.

F th l• For the nucleus π = π1 π2 π3 … πA.• Individually not possible.• Overall π can be determined experimentally.• Overall Ιπ for a nucleus (nuclear state).Overall Ι for a nucleus (nuclear state).• Transitions and multipolarity of transitions (γ-emission)


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emission).

Electromagnetic momentsg• Electromagnetic interaction information about gnuclear structure.• Charge electric; current magnetic.g ; g• Electromagnetic multipole moments.Field∝1/r2 (zeroth, L=0) electric monopole moment.( , ) p

1/r3 (first, L=1) electric dipole moment.1/r4 (second, L=2) quadrupole moment.( , ) q p………1/r2 magnetic monopole (questionable….!).g p (q )Magnetic Dipole (familiar).Higher order magnetic moments.


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g g

Electromagnetic momentsg• Expectation value of the moment. ∫ dvϑψψ *

• Each multipole moment has a parity, determined by ∫ dvϑψψ

the behavior of the multipole operator when r -r.• Parity of ψ does not change the integrand.• Electric moments: parity (-1)L.• Magnetic moments: parity (-1)L+1.• Odd parity vanish.

electric dipole.magnetic quadrupole.electric octupole.


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…………

Electromagnetic momentsg• Electric monopole: net charge Ze.

e e evr

p g• Magnetic dipole: µ = iA

µ = eT

A ; µ = e2π r v

π r2 ; µ = evr2

µ = emvr ; µ = epr ; µ = e Lµ =2m

; µ =2m

; µ =2m

L

• g-factors.


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Nuclear Magnetic MomentgRemember, for electrons Revise: Torque on a current loop.

Gyromagnetic ratio (g-factor)

Z component ?? Experiment applied magnetic field


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Z component ?? Experiment, applied magnetic field.

Nuclear Magnetic MomentgFor Nuclei

For free protons and neutronsProton: g = 5.5856912 ± 0.0000022 ∼ 3.6 Neutron: g = -3.8260837 ± 0.0000018 ∼ 3.8

The proton g-factor is far from the gS = 2 for the electron, and even the uncharged neutron has a sizable magnetic moment!!!

Internal structure (quarks)Nuclear Physics, BAU, First Semester, 2012-2013

(Saed Dababneh).27

Internal structure (quarks).

Nuclear Magnetic Momentg

Magnetic moment µNuclide Nuclear spin Magnetic moment µ(in µN)

n 1/2 1 9130418n 1/2 -1.9130418

p 1/2 +2.79284562H (D) 1 +0.8574376

17O 5/2 -1.8927957Fe 1/2 +0.0906229357C 7/2 +4 73357Co 7/2 +4.73393Nb 9/2 +6.1705


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Electromagnetic momentsg• The nucleus has charge (monopole

t)Classical momentsmoment).

• No dipole moment since it is all positive.B t if th l i t h i ll

moments

• But if the nucleus is not spherically symmetric, it will have a quadrupole

tmoment.


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Electric Quadrupole Momentp• For a point charge e: eQ = e(3z2 - r2).

S h i l t 2 2 2 2/3 Q 0• Spherical symmetry x2 = y2 = z2 = r2/3 Q = 0.• For a proton:

∫I th l Q ⟨ 2⟩

∫ −= dvrzeeQ ψψ )3( 22*

• In the xy-plane: Q ∼ - ⟨r2⟩.• ⟨r2⟩ is the mean square radius of the orbit.

Al Q 2 ⟨ 2⟩• Along z: Q ∼ +2 ⟨r2⟩.• Expected maximum ∼ er0

2A2/3.U t 50 10 30 2• Up to 50x10-30 em2.

• Up to 0.5 eb.


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Electric Quadrupole MomentpNuclide Q (b)Q ( )2H (D) +0.00288

17O 0 0257817O -0.0257859Co +0.4063Cu -0.209133Cs -0 003

• Closed shell Spherically symmetric core Cs 0.003

161Dy +2.4176L 8 0

symmetric core. • Test for shell model• Strongly deformed nuclei ! 176Lu +8.0

209Bi -0.37

nuclei…..!


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Documents

The Semi-empirical Mass Formulanuclear.dababneh.com/Nuclear-Undergrad/Notes/2012-1/Part-3.pdf · The Semi-empirical Mass Formula • von Weizsäcker invon Weizsäcker in 1935