Nuclear Binding Energy

Nuclear Binding EnergyBtot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2 B mBave(A,Z) = Btot(A,Z) / A HW 8HW 8 Krane 3.9Atomic masses from:http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all

Separation EnergyNeutron separation energy: (BE of last neutron)Sn = [ m(A-1,Z) + mn – m(A,Z) ] c2

= Btot(A,Z) - Btot(A-1,Z) HW 9HW 9 Show that HW 10HW 10 Similarly, find Sp and Sα.HW 11HW 11 Krane 3.13 HW 12HW 12 Krane 3.14

Nuclear and Radiation Physics, BAU, 1st Semester, 2006-2007 (Saed Dababneh).

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Nuclear Binding EnergyIn generalX Y + aSa(X) = (ma + mY –mX) c2

= BX –BY –BaThe energy needed to remove a nucleon from a nucleus ~ 8 MeV ≅ average binding energy per nucleon (Exceptions???).

Mass spectroscopy B.Nuclear reactions S.Nuclear reactions Q-value


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Nuclear Binding Energy

~200 MeV

Fission

Fusio

n

Coulomb effectSurface effect


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HW 13HW 13A typical research reactor has power on the

order of 10 MW.

a) Estimate the number of 235U fission events that occur in the reactor per second.

b) Estimate the fuel-burning rate in g/s.


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Is the nucleon bounded equally to everyother nucleon?C ≡ this presumed binding energy.Btot = ½ CA(A-1)Bave = ½ C(A-1) Linear ??!!! Directly proportional ??!!!Clearly wrong … ! wrong assumption

finite range of strong force,and force saturation.


Lead isotopes Z = 82

For constant ZSn (even N) > Sn (odd N)For constant NSp (even Z) > Sp (odd Z)

Remember HW 12 (Krane 3.14).

208Pb (doubly magic) can then easily remove the “extra” neutron in 209Pb.

Neutron Number N

Neu

tron

Sep

arat

ion

Ene

rgy

S n(M

eV)

208 Pb


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

Sn (A, Z, even N) – Sn (A-1, Z, N-1)This is the neutron pairing energy.

even-even more stable than even-odd or odd-even and these are more tightly bound than odd-odd nuclei.


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Neutron Excess

Remember HWc 1.

Asymmetry

Asymmetry


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Abundance SystematicsOdd N Even N Total

Odd ZEven ZTotal


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HWc 1HWc 1\\

Compare:• even Z to odd Z.• even N to odd N.• even A to odd A.• even-even to even-odd to odd-even to odd-odd.


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Abundance Systematics

Abundance Systematics

NEUTRON NUMBERNEU

TRO

N C

APT

UR

E C

RO

SS S

ECTI

ON

Formation process

Abundance


11MASS NUMBER

AB

UN

DA

NC

E

r s r s


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The Semi-empirical Mass Formula

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

1. Incompressible matter of the nucleus R ∝ A⅓.

2.Nuclear force saturates.• 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.


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The Semi-empirical Mass FormulaVolume 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.

The other terms are “corrections” to this term.

=A

BV constant


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

• Light nuclei contain larger number (per total) at the surface.• At the surface there are:

• Binding energy of inner nucleons is higher than that at the surface.

32

2

322

0 44 ArAr

o

=ππ Nucleons.

31

1

AABs ∝

Remember t/R ∝ A-1/3


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

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

- aC Z(Z-1) / A4/3

Remember HW 7 … ?!

ρπ 3

34 r

ρπ drr24



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The Semi-empirical Mass FormulaHW 14HW 14Show that from 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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• Light nuclei: N = Z = A/2 (preferable).• Deviation from this “symmetry” less BE and stability.• Neutron excess (N-Z) is necessary for heavier nuclei.• Fraction affected = |N-Z| / A• Total decrease in BE ∝ fraction x excess.• Ba / A = - aa (N-Z)2 / A2.• Back to this when we talk aboutthe shell model.

Asymmetry Term Ba = - aa (A-2Z)2 / A


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The Semi-empirical Mass FormulaPairing 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 tightly bound than odd-odd nuclei.Remember HWc 1HWc 1\\ ….?!Bp expected to decrease with A; effect of unpaired nucleon decrease with total number of nucleons. But empirical evidence show that:

δ ∝ A-¾ .

⎪⎪⎩

⎪⎪⎨

⎧

−

+=

−

−

oddZoddNAaoddA

evenZevenNAa

p

p

43

43

0δ

Effect on:• Fission.• Magnetic moment.Effect of high angular momentum.


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The Semi-empirical Mass FormulaClosed Shell Term Bshell = η

• Extra binding energy for magic numbers of N and Z.• Shell model.• 1 – 2 MeV more binding.


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• Fitting to experimental data. • More than one set of constants av, as ….. • In what constants does r0 appear?• Accuracy to ~ 1% of experimental values (BE).• Atomic masses 1 part in 104.• Uncertainties at magic numbers.• Additional term for deformation.


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Nuclear Binding Energy