By: Sushil Kumar

HMV Jalandhar

2 convenient units

Atomic mass unit, u thmass of a neutral C12 atom 1

12

by definition (of a mole) 1 mole, C12 = 12 grams

1u = 1 gram

6.022521023 = 1.6604310-24gram = 1.6604310-27 kg

the electron volt, eV Energy acquired by a particle with 1e charge when

accelerated through a potential difference of 1 volt.

1eV = 1.60210-19 Coul-volt = 1.60210-19 Joule

1u c 2 = 1.6604310-27 kg

1eV = 1.60210-19 Coul-volt = 1.60210-19 joule

c 2

1.60210-19 joule/eV

= 1.6604310-27 kg (2.998108 m/sec) 2

1.60210-19 joule/eV

= 9.315825108 eV = 931.58 MeV

Rest mass energy can be measure in terms of eV or MeV.

1u c 2

= 931.58 MeV/c 2 1u

Sometime mass is measure in the convenient unit of MeV/c2.

1 keV = 103 eV

1 MeV = 106 eV

1 GeV = 109 eV

1 TeV = 1012 eV

• 1012 tera (trillion)

• 109 giga (billion, “80 Gigabyte hard drive”)

• 106 mega (million, “128 Megabytes of RAM”)

• 103 kilo (thousand, 1 kilogram = 2.2 pounds)

• 10– 2 centi (hundredth, 1 in. = 2.54 cm)

• 10– 3 milli (thousandth, “milliVolt”)

• 10– 6 micro (millionth, micron=micrometer=10-6m)

• 10– 9 nano (billionth, “nanosecond”)

• 10– 12 pico (trillionth)

• 10– 15 femto

Particle symbol rest energy in MeV

electron e 0.5109989

muon 105.658357

neutral pion 0 134.9766

charged pion 139.57018

proton p 938.27200

neutron n 939.56533

deuteron H2 1875.580

triton H3 2808.873

alpha (He4) 3727.315

or mass in MeV/c2

Rest energies

of a proton + neutron = 938.27200+939.56533 = 1877.83733 MeV

while of a deuteron H2 = 1875.61269 MeV

difference = 2.22464 MeV

A deuteron cannot spontaneously decay into a proton and a neutron.

To break up a deuteron 2.22464 MeV must be added by

bombarding deuterons with an energetic beam of particles or

electromagnetic radiation.

For the alpha particle Dm= 0.0304 u which gives

28.3 MeV binding energy!

For “bound states” the binding energy, B>0

representing the energy required to disintegrate the

nucleus (into individual neutrons and protons)

),()(2

ZAMmZAZmc

Bnucleusnp

),()(2

ZAMmZAZmc

BatomneutralnH

Working in atomic mass units, u

the difference between atomic rest mass energy M(A,Z)

and atomic mass number A u

is called the MASS DEFECT (or “mass excess”)

AuZAM D ),(

Notice with Au Zmp + (A-Z)mn

this is essentially the same as the Binding Energy

Binding Energy Per Nucleon The binding energy per nucleon of a nucleus is the binding

energy divided by the total number of nucleons in the nucleus

Binding energy per nucleon = total binding energy of nucleus _____________________ number of nucleons in the nucleus

In the case of

helium

28.3 7.1 MeV ________ = 4

The great of nuclides have a value of around 8MeV per

nucleon

Peaks at ~8.795 MeV near A=60

for A>50

~constant

8-9 MeV

Bin

din

g e

ne

rgy p

er

nu

cle

ar

pa

rtic

le (

nu

cle

on

) in

Me

V

The key to this release is the chain reaction

produced by the free neutrons realesed

during fission when the mass of U-235

exceeds a critical level (about 52kg- a

sphere17cm in diameter

This confirms the short range ascribed to the nuclear force…

it must involve only nearest neighbor nuclei

If the binding involved contributions from all nucleons

the total number of pair-wise bonds

2)1( AAA

then AA

B (not constant!)

The binding energy curve is obtained by dividing the total nuclear

binding energy by the number of nucleons. The fact peak in the

binding energy curve near iron means that either the breakup of

heavier nuclei (fission) or the combining of lighter nuclei (fusion)

will yield nuclei which are more tightly bound (less mass per

nucleon).

The binding energies of nucleons are in the range of millions of

electron volts compared to tens of eV for atomic electrons. Whereas

an atomic transition might emit a photon in the range of a few

electron volts, perhaps in the visible light region, nuclear transitions

can emit gamma-rays with quantum energies in the MeV range.

Nuclear Binding Energy Curve

The buildup of heavy elements by the nuclear fusion

processes in stars is limited to elements below iron, since the

fusion of iron would subtract energy rather than provide it.

Iron-56 is abundant in stellar processes, and with a binding

energy per nucleon of 8.8 MeV, it is the third most tightly

bound of the nuclides. Its average binding energy per nucleon

is exceeded only by 58Fe and 62Ni, the nickel isotope being

the most tightly bound of the nuclides.

The Iron Limit

The Most Tightly Bound Nuclei 62Ni .

The most tightly bound nuclides are all even-even nuclei.

The high binding energy of the “iron group” of around A=60 is significant in

the understanding of the synthesis of heavy elements in the stars.

B/A (keV/A)

62Ni 8794.60 +/- 0.03

58Fe 8792.23 +/- 0.03

56Fe 8790.36 +/- 0.03

60Ni 8780.79 +/- 0.03

The Semi-emperical Mass Formula

is an approximate fit to all this data

np mZAZmZAM )(),(

The 1st correction to this (the “mass defect” or binding energy)

AaE vV

~proportional to volume

A

B/A

(M

eV)

true enough for

very small A !

Volume term

But the nuclei at the surface have fewer nearest-neighbor bonds!

by something the nucleus’ surface area

3/2AaE ss

With these effects alone,all isobars might be

expected to be stable though clearly we’ve seen a

narrow band of stability in the N vs A plane

for light nuclides: N=Z (A = 2Z)

for heavy nuclides: A>2Z

There’s a repulsive Coulomb energy building up with all those protons!

Surface effect

Coulomb Energy The work done in collecting and compacting

total charge Q into a sphere of volume . 3

3

4R

potential at the surface of a spherical concentration of charge

with charge density: 3)3/4( r

Q

3/1)1( AZZaE cc

3/13/2)1(

AZZaAaAaB

csv

volume term

grows with

addition of

each interacting

nuclei

surface term

corrects for

nuclei at the

surface (not

completely

surrounded by

nearest neighbors)

Coulomb term

accounts for

(repulsive) energy

built up in the

accumulation

of protons

A = 127

isobars

-88

-86

-84

-80

-88

-86

-84

-80

52 54 50 56

Atomic number, Z

Ma

ss

de

fec

t D

(M

eV

)

Like the “closed shell” electronic configurations of the noble gases

we see “magic numbers” recurring…marking tightly bound,

extraordinarily stable nuclear configurations.

including to a lesser extent, marked stability for nuclei with

paired protons and/or paired neutrons

(like the closed energy levels of atomic orbitals

where electrons with opposite spins cancel)

Since nuclei with paired protons or paired neutrons

tend to be more tightly bound, we introduce the

pairing energy term:

4/3

4/3

0)(

A

a

A

a

p

p

A

=4He (with Z=N=2) and 16O (Z=N=8) are doubly magic!

even Z, N

odd Z, N

Z = N = 0

)()2/(

)1(

2

3/13/2

AA

ZAa

AZZaAaAaB

sym

csv

volume term

grows as

nuclei added

surface term

corrects for

surface nuclei

Coulomb term

repulsion due

to protons

symmetry term

drives number of

protons ≈ neutrons

pairing energy term

increased stability

for paired nuclei

2),()(),( cZABmZAZmZAM np

),(/)2/(

)1(),(

2

3/13/2

ZAAZAa

AZZaAaAaZAB

sym

csv

av=15.5 MeV

as=16.8 MeV

ac=0.72 MeV

asym=23.0 MeV

ap=34.0 MeV

Thank You