1. Nuclear Radius (r):r - Nuclear Physics Lecture 2

1. Nuclear Radius (r):

1 /

3 R = r A

r = Nuclear radius parameter (constant) = 1.2 – 1.5 fm r A = Mass Number

Nuclear radius of Li – 6 : 2.217 fm Nuclear radius of Rn – 216 ? 10.897 fm

To estimate the density of nuclear matter

= ≈ Nuclear mass n p n p

M Zm Nm Am

3 Nuclear volume V = π R = π r A

3 Am m

M p p n

ρ = = = Matter density

Independent of A

3 V π r A π r

3 − 27 −

15 m ≈ 10 kg , r ≈ 10 m p m p

3 = ≈ 10 kg / m

3 Density of water 10 kg/m !

π r

2. Nuclear spin (I):

Nucleons have intrinsic spin angular momentum S = 1/2 (in unit of ħ)

In addition, nucleons posses orbital angular momenta about the CM of nucleus – quantum number L Total angular momentum of the nucleus (commonly termed as nuclear spin) I = L + S Quantum Mechanically h p = S ( + S 1 )

Spin angular momentum S h p = L L

( + 1 ) Spin angular momentum L h p = I ( +

I 1 )

I Spin angular momentum

7 Ground state spin of He is 0 & Li is 3/2

3. Statistics of Nuclei:

Nuclear spin can be 0 or some integer or half – integer Accordingly nuclei follow Bose – Einstein or Fermi – Dirac statistics Nuclei having I = n (n = 0, 1, 2, 3, ….) follow BE statistics Nuclei having I = (n + 1/2) [n = 0, 1, 2, 3, ….] follow FD statistics

7 He (I = 0) follows BE statistics & Li (I = 3/2) follows FD statistics

4. Parity of Nuclei:

Quantum mechanically the nucleus is described by a wave function

ψ (r ) r r ( r → − r )

The space inversion is described by the parity operator which

Pˆ r r ˆ

operates as

P r r ψ ( ) = ψ ( − )

If the Hamiltonian of nuclei remains invariant under space inversion, the

r r r ˆ

change in wave function under parity operation is

P r r r ψ ( ) = ψ ( − ) = ± ψ ( ) r r

The nucleus is said to have even parity for & odd parity

r r ψ ( − ) = ψ + ( ) r r

for

r r ψ ( − ) = − ψ ( )

7 Ground state parity of He is even & Li is odd

5. Magnetic dipole moment of Nuclei:

The nucleons, like electron, carry intrinsic magnetic moment. The intrinsic magnetic moment for proton is μ = 2.7927 μ & for neutron μ p N n

= – 1.9131 μ N

µ = N

is the nuclear magneton = 5.0571 × 10 J/T

2 m

µ = [Bohr magneton = 9.2849 × 10 J/T ]

2 m

e Neutron, though electrically neutral, has intrinsic magnetic moment!

5. Magnetic dipole moment of Nuclei:

In addition to intrinsic magnetic moment, the contribution comes from orbital motion as well, but for proton only. No contribution to the

nuclear magnetic moment comes from orbital motion of neutrons

Total nuclear magnetic moment is the vector sum of the intrinsic magnetic moments of protons and neutrons, and magnetic moment due to orbital motion of the proton

6. Electric moments of Nuclei:

Nucleus is positively charged with azimuthally symmetric charge distribution Electrostatic potential due to this charge distribution has multipole components Most dominating component is due to monopole – equal to total charge (+Ze)

The electric dipole moment of a nucleus in its ground state vanishes

Next higher order term comes from quadrupole moment defined as

2 Q z r r d = 3 ' − ' ( ' ) '

ρ τ ( )

∫

6. Electric moments of Nuclei:

Q = 0 for spherical charge distribution, Q < 0 for oblate and Q > 0 for prolate

charge distribution Measurement of Q yields an idea on the shape of nucleus Nuclear force binds the protons & neutrons inside a tiny volume

(1) Nuclear force is the strongest force in nature

The nuclear force is stronger than the electromagnetic & far stronger than the gravitational force The attractive (negative) force has a maximum at a distance of about 1 fm with a force of about 25,000 N Particles much closer than a distance of 0.8 fm experience a large repulsive (positive) force Particles separated by a distance greater than 1 fm are still attracted (Yukawa potential), but the force falls as an exponential function of distance

(2) Nuclear force is short – ranged

Acts in fm range Powerfully attractive between nucleons at distances of about 1 fm Rapidly decreases to insignificance at distances beyond about 2.5 fm Becomes repulsive at distances less than 0.7 fm

Nuclear potential Independent of the charge of the interacting particles The force between two protons is same as the force between two neutrons or between a proton and a neutron within the nuclear distances. Symbolically

(3) Nuclear force is charge independent

Coulomb repulsion between protons becomes important for r > 3 fm

( ) ( ) ( ) n p p p n n nuc

− = − = −

(4) Nuclear force is charge symmetric

The strength of the nuclear force is same for the protons and neutrons, i.e. if all the neutrons in a nucleus were replaced by protons (or the vice-versa), the strength of the nuclear force remains unchanged. Symbolically,

n − n = p − p = p − n ( ) ( ) ( ) nuc

(5) Nuclear force is spin dependent

Experimental evidences show that nuclear force acting between the nucleons depends on mutual orientation of the spin of the nucleons

2 In deuteron ( H ) the spins of the proton and neutron are parallel

(6) Nuclear force shows saturation property

One nucleon in the nucleus interacts with limited number of nucleons nearest to it (since the force is short – ranged) In heavy nuclei, nuclear size is larger than the range of nuclear force A nucleon senses approximately a constant number of neighbourhood nucleons It results in a constant binding fraction (binding energy per nucleon)

Nuclear force is a fundamental interaction – strong interaction. It acts

between quarks and mediated by gluons (detailed discussion to be followed –

Elementary Particles)

A device for measurement of isotopic mass of nuclei Atoms with one or two electrons removed, become positive ions A beam of positive ions produced in a discharge tube is collimated into a fine beam by two narrow slits (S )

1 The fine beam enters into a velocity

selector region The velocity selector consists of two plane parallel plates (A, B) which produces a uniform electric field (E), and an electromagnet which produces a uniform magnetic field (B) These two fields (E & B) are mutually perpendicular and perpendicular to the beam direction

The ions with their velocity v = E/B do not experience any force within the velocity selector and pass through the slit (S )

2 Only those ions with their velocity v = E/B enter the mass spectrograph from the velocity selector through the slit (S )

2 The positive ions with same velocity are acted upon by a magnetic field B’

perpendicular to v Ions are deflected in a circular path of radius r & strike the photographic plate

2 mv B ' qv = r B ' qr BB ' qr m = = v E Ions with different masses trace different semicircular paths of different radii and produce dark spot on the photographic plate The distance between the opening of the chamber and the dark spot on the plate yields the diameter 2r from which r can be measured.

Since B, B’, E, q are known, m can be precisely determined

https://www.youtube.com/watch?v=CxNnOf3POoA

Problems

1. In a mass spectrometer, a singly charged positive ion is accelerated through a potential difference of 1000 volt. It then travels through a uniform magnetic field of 1000 Gauss and deflected through a circular path of radius 18.2 cm. Calculate the (i) speed of the ion, (ii) mass of the ion and (iii) mass number . [CU – 2015]

Problems 2. Singly ionized Argon ions are mass analyzed by a Bainbridge mass spectrograph.

4 The electric and magnetic fields in the velocity filter are 1.5 × 10 V/m and 0.4 T

respectively. After coming out of the velocity filter, the ions enter a magnetic field of 0.9 T. Find the distances between the ion focus lines on the photographic plate for

three isotopes: Ar , Ar and Ar . [CU – 2016]

1. Nuclear Radius (r):r - Nuclear Physics Lecture 2