Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

1

Nuclear Binding Energy

Btot(A,Z) = [ ZmH + Nmn - m(A,Z) ] c2 Bm

Bave(A,Z) = Btot(A,Z) / A HW 9HW 9 Krane 3.9Atomic masses from: HW 10HW 10 Krane 3.12http://physics.nist.gov/cgi-bin/Compositions/stand_alone.pl?ele=&all=all&ascii=ascii&isotype=all

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

= Btot(A,Z) - Btot(A-1,Z) HW 11HW 11 Show that

HW 12HW 12 Similarly, find Sp and S.

HW 13HW 13 Krane 3.13 HW 14HW 14 Krane 3.14

Magicnumbers

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

2

Nuclear Binding EnergyMagic

numbers

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

3

Nuclear Binding Energy

In generalX Y + aSa(X) = (ma + mY –mX) c2

= BX –BY –Ba The 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

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

4

Nuclear Binding Energy

~200 MeV

Fission

Fusi

on

Coulomb effectSurface effect

HWc 4HWc 4Think of a computer program to

reproduce this graph.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

5

Nuclear Binding Energy

HW 15HW 15A 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.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

6

Nuclear Binding Energy

Is the nucleon bounded equally to everyother nucleon?C ≡ this presumed binding energy.Btot = C(A-1) A ½Bave = ½ C(A-1) Linear ??!!! Directly proportional ??!!! Clearly wrong … ! wrong assumption finite range of strong force, and force saturation.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

7

Nuclear Binding Energy

Lead isotopes Z = 82

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

Remember HW 14 (Krane 3.14).

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

Neutron Number N

Ne

utr

on

Se

pa

ratio

n E

nerg

y S

n (

Me

V)

208 P

b

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

8

Nuclear Binding Energy

Extra Binding between pairs of “identical” nucleons in the same state (Pauli … !) Stability (e.g. -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.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

9

Abundance SystematicsOdd N Even N Total

Odd Z

Even Z

Total

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.

HWc 1HWc 1\\

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

10

Neutron ExcessZ Vs N (For Stable Isotopes)

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100 120 140N

Z

Odd A

Even A

Z = NAsymmetry

AsymmetryRemember HWc 1.

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

11

Neutron Excess

Remember HWc 1.

Asymmetry

Asymmetry

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

12

Abundance Systematics

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

13

Abundance Systematics

NEUTRON NUMBER

MASS NUMBER

AB

UN

DA

NC

EN

EU

TR

ON

CA

PT

UR

E

CR

OS

S S

EC

TIO

N

r s r s

Formation process

Abundance

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

14

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

15

The Semi-empirical Mass Formula

• von Weizsäcker in 1935.• Liquid drop. Shell structure.• 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.…..

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

16

The Semi-empirical Mass Formula

Volume Term Bv = + av ABv 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

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

17

The Semi-empirical Mass Formula

Surface Term Bs = - as A⅔

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

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

32

2

322

0 44

Ar

Ar

o

Nucleons.

31

1

AA

Bs

Remember t/R A-1/3

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

18

The Semi-empirical Mass Formula

Coulomb Term BC = - aC Z(Z-1) / A⅓

• Charge density Z / R3.• W 2 R5. Why ???• W Z2 / R. • Actually: W Z(Z-1) / R. • BC / A = - aC Z(Z-1) / A4/3

Remember HW 8 … ?!

3

3

4r

drr24

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

19

The Semi-empirical Mass Formula

Nuclear and Radiation Physics, BAU, First Semester, 2007-2008 (Saed Dababneh).

20

The Semi-empirical Mass Formula

Quiz 1Quiz 1

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

32

AZZaAaAaMMZAMZAM CSVHnn

From our information so farso far we can write:

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

Is this reasonable…???

So …..!!!!