Binding energy in atoms and nuclei

[Sec. 4.1 Dunlap]

The binding energy of an atom is the energy released as all the constituent particles (n, p and e) come together FROM INFINITY under both the STRONG force and the EM force.

The binding energy is something that is LOST from the atomic system. Thus it is not something that the system possesses.

CONCEPT OF BINDING ENERGY

CALCULATION OF BINDING ENERGY

Total Energy Total Energy

EBcXcZmNmZm Nenp . M 2AZ

2

2

2

22

atom mass - tsconstituen mass

Z .

c

cXMNmZm

cXMcZmNmmEBAZnH

NAZenp

ANOTHER WAY OF VIEWING BINDING ENERGY

+

ATOM Constituents at infinity

The opposite way of seeing binding energy - is that if B.E. (MeV) is put into the atom then there is just enough energy available to split all the constituents of the atoms apart and get them to rest at infinity.

SINGLE NEUTRON SEPARATION ENERGY

The same method can be used to easily compute the “Single Neutron Separation Energy” – which is the energy required to “pull” a neutron out of the nucleus.

21

1

221

12

MM

M M

cXmXS

cmcXcXS

NAZnN

AZn

nNAZN

AZn

Note we don’t have to measure Sn directly.

SINGLE PROTON SEPARATION ENERGY

The same clever strategy applies to finding the “Single Proton Separation Energy” Sp. But note here there is a difference – we must be careful in counting electron mass.

22211

2 M M cmcmcYcXS epNAZN

AZp

21

1

211

MM

MM

cXmY

cXmmYS

NAZHN

AZ

NAZepN

AZp

pS [Mass of Final Products – Mass of Initial atom] c2

ALPHA PARTICLE DECAY ENERGYIn a nuclear decay energy is given out in the separation of particles. This energy is often referred to as the “Q” of the reaction. Clearly the Q is the negative of the particle separation energy.

Q M M 222

42

2 cmcYcX NAZN

AZ

He

MM4

242

22

42

BYBXB

cmYXQ

NAZN

AZ

NAZN

AZ

Eq 8.2

Eq. 8.3

Eq. 8.4

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CALCULATION OF BINDING ENERGY

Total Energy Total Energy

EBcXcZmNmZm Nenp . M 2AZ

2

2

2

22

atom mass - tsconstituen mass

Z .

c

cXMNmZm

cXMcZmNmmEBAZnH

NAZenp

Mass Defect• Mass defect (M.D) is another way of saying nuclear

B.E. It is simply the nuclear B.E. expressed not as MeV but in mass units (MeV/c2)

M

M Z ..

XNmZm

XZmNmmDMAZnH

NAZenp

2

2

22

atom mass - tsconstituen mass

M

M Z .

c

cXNmZm

cXcZmNmmEBAZnH

NAZenp

= Mass constituents of atom – mass of atom

Mass Excess• Do not confuse Mass Excess with Mass Defect

(or Binding Energy). Mass Excess is just a CONVENIENT WAY to write down the mass of a nucleus in amu (u). 1u = 931.5MeV

AuX NAZM

This is just a common sense thing. The mass of a nucleus can get very large if expressed in MeV and will always be approximately equal to Au because it is made up of A nucleons. It is thus convenient to tabulate rather than the whole nuclear mass.

2 uM cAXAZ MeV

Can either be expressed in u or MeV

Mass Excess – Example on 238U

(238U)= .0507826 u

M(238U)=238+ .0507826 u =238.0507826 u

= 238.0507826 x 931.494 MeV/c2

= 221,742 . 875 MeV/c2

Armed with this information we can work out the B.E. of 238U

Mass Deficit + Binding Energy of

92 proton mass = 86,319 . 736 MeV /c2

146 neutron mass = 137,174 . 446 MeV /c2

92 electron mass= 47 . 012 MeV /c2

Mass constituents = 223,541 . 194 MeV /c2

M(238U) observed = 221,742 . 875 MeV/c2

Mass Defect = 1,798 . 319 MeV/c2

Binding Energy = 1,798 . 319 MeV

Electronic B.E = . 795 MeV

Nuclear B.E. = 1,797 . 52 MeV

B.E/nucleon = 1,797.52/238= 7.55MeV

14623892 U

How much is electronic binding energy?

There are two types of binding energy in the atom – Strong Nuclear B.E. and the Electromagnetic B.E. of the electrons to the nucleus.

Nuclear EM

5 7/3Nuclear 2.08 10

A A AZ Z Z

AZ

B X B X B X

B X Z

7/3238 592 U 2.08 10 92

0.795MeV

EMB

THE FAMOUS B/A (binding energy per nucleon) CURVE