Nuclei

Nuclear ForcesShort Ranged Force ( ~ femtometers 10-15)

In short range , they are very strong(even stronger than Coulomb and Gravitation) . Other than this they are very weak.

Nuclear force does not depend on charge , Hence it is same for neutron-neutron , proton-proton and neutron-proton

From Graph: r > r0 attractive

r<ro repulsive

R0

Strong force region

Basic Facts A = Mass number = No of protons + Number of electrons

Z = Atomic Number = No of protons

Radius of Nucleus A∝ 1/3

Volume = (4/3) π R3 A∝

Density = Mass / Volume = Mass number /Volume = Constant

So Density of nucleus is independent of Mass number .

Density of nucleus is very high . As it was concluded in Rutherford’s Experiment

Mass Energy EquivalenceMass is condensed form of Energy

Energy Equivalent for Mass : E = mc2

Energy Equivalent of 1 amu = 931 MeV. ~930

So , If mass is known in amu just multiply it by 931 to get energy equivalent .

Energy ConceptsTo break things we need energy . Example to break a chemical bond energy is required , To break an ice cube ,energy is required .

When things are formed on their own , energy is released .

Binding Energy

Since we are breaking , Energy should be added . Energy Equal to Binding Energy is required to break nucleus . So Binding Energy Stability

Applying Energy Conservation

Energy of Nucleus + Binding Energy = Energy of nucleons

Mnucleusc2 + Binding Energy = Mnucleons c2

Mnucleons > Mnucleus .. This is called mass defect (खु�ल्ले� का� mass ज्या�दा� हो ता� हो�) So Binding Energy = (Mass Defect) c2 = (Mnucleons-Mnucleus) c2

Consider a nucleus having Atomic Number Z and Mass number A and Atomic mass M

Number of Proton = Z

Number of neutrons = A- Z

Mass of Nucleons = Z mproton + (A-Z ) mneutron

Energy Equivalent = (Z mproton + (A-Z ) mneutron) c2

Mass of Nucleus = M

Energy Equivalent = Mc2

Energy of Nucleus + Binding Energy = Energy of Nucleons

Mc2 + Binding Energy =(Z mproton + (A-Z ) mneutron) c2

But Binding Energy is not a good measure , Rather (Binding Energy / nucleon) is better

Example Protons = 8 Neutrons = 16-8=8

But Actual Mass = Atomic Mass = ( Mass Defect !!)

Energy of Nucleus + Binding Energy = Energy of nucleons

(15.99493)x931 + Binding = (16.12744)X931

Binding Energy = 127.5 MeV No of nucleon =16

Binding energy/ nucleons = 127.5 / 16 =7.96 MeV

Binding Energy CurveCurve is on Binding Energy /Nucleon

Most stable : Iron Group

30< A<70 almost constant , because nuclear force is

short ranged .

Binding Energy / nucleon is low for light as well as heavy elements

Interpretations of curve Light elements can combine to form a bigger element So that

BE / nucleon increase . Hence they are suitable for Fusion

Similarly ,Heavy Elements can break into two to form elements

in mid . So they are suitable for fission

Middle Elements are stable , so they are not suitable for any nuclear reactions

Alpha Decay Alpha Particle ( ) is emitted

Energy Calculations :

Initial Mass = Mx Initial Energy = Mx c2

Final Mass = (My + MHe ) c2

If Initial Mass is higher Initial energy is more . So , Energy will be released(spontaneous)

If Final Mass is higher Final energy is more . So Energy will be absorbed .

Energy released is carried by alpha particle in form of kinetic energy

Qvalue = (initial mass – final mass ) c2 = (Mx – My-MHe ) c2

Qvalue> 0 for spontaneous (Reactions with negative Q will not occur)

Example Uranium

Initial Mass = 238.05079 u

Final Mass = 234.04363 + 4.00260 = 238.04623 u

Initial Mass > Final Mass

Initial Energy > Final Energy Energy will be Released Spontaneous

Energy released = (238.05079 -238.04623 ) X 931

= 0.00456 X931 MeV = 4.35 MeV

1 u = 931 MeV

Beta - decay Electron is emitted . Ve is antineutrino ( zero mass)

Neutron breaks into electron and proton

Explanation

LHS : Protons = Z Neutrons = A-Z Electrons = Z

RHS : Since One Neutron has converted into electron + proton

Protons = Z+1 Neutrons = (A-Z)-1 Electrons = Z+1

So New Atomic number = Z+1 New Mass Number = (Z+1) +(A-Z)-1 = A !

Z Z +1 A unchanged

Energy Calculations

Initial Mass = Mx

Initial Energy = Mxc2

Final Mass = (My + Me)

Final Energy = (My + Me)c2

Qvalue = ( Initial Mass – Final Mass ) c2 = (Mx- My -Me)c2

Qvalue > 0 Spontaneous

This energy ( Q) is carried away as kinetic energy by the electron and antineutrino.

Beta plus decay Proton breaks into Neutron and positive electron

Explanation :

LHS : Protons : Z Neutrons : A-Z

Now one proton converts into positive electron and neutron

RHS : Proton : Z-1 Neutron : (A-Z+1)

New Atomic number : Z-1 New Mass Number = (Z-1) + (A-Z+1 ) = A !

Z Z-1 A Unchanged

Energy Calculations

Initial Mass = Mx

Initial Energy = Mxc2

Final Mass = (My + Me)

Final Energy = (My + Me)c2

Qvalue = ( Initial Mass – Final Mass ) c2 = (Mx- My -Me)c2

Qvalue > 0 Spontaneous

This energy(Q) is carried away as kinetic energy by the electron and neutrino.

Gamma Decay Like electrons , nuclei also have energy states .

But energy gap is very high in case of nucleus . In case of electrons it is relatively small ( 13.6 eV, 10.2eV etc)

But in case of nucleus it is of the order of MeV ( Mega = 10 6 , as in megabytes)

When nucleus , jumps down from excited state to lower state energy is released in the form of photons . Efinal- Einitial = h v

Example Fact : Beta / alpha decay are generally accompanied by gamma decay

In this example , Cobalt first goes beta minus decay , In beta

Minus decay Z increases by 1 ( 2728)

Then it undergoes two gamma decays

First gamma : 1.17 Mev 1.17X106 = 1240/ λ λ =10-3 nm

Second gamma : 1.33 MeV

Radioactivity First Order Reaction

N = Number of nuclei present

N0 initial number of nuclei

So number of nuclei will decrease with time .

Decay rate :

So ,

Putting value of N in above equation

R is also called activity of sample

Half life : Time at which number of nuclei present become half of initial nuclei

T1/2 depends only on decay constant .not on N

Average life : Life a nuclei is expected to live .

Like some humans die at early age , some at high age . The average comes in middle which is called life expectancy .

Moderator – For slowing down Neutrons

Safety rods – to absorb neutrons