Bulk Properties
1. Nuclear Radius (
r
):
r
0= Nuclear radius parameter
(constant)
r
0= 1.2 – 1.5 fm
A
= Mass Number
3 / 1 0
A
r
R
=
Nuclear radius of Li – 6 : 2.217 fm
Nuclear radius of Rn – 216 ?
Bulk Properties
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
Spin angular momentum
Spin angular momentum
Spin angular momentum
Bulk Properties
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
Bulk Properties
4. Parity of Nuclei:
Quantum mechanically the nucleus is described by a wave function
The space inversion is described by the parity operator which
operates as
If the Hamiltonian of nuclei remains invariant under space inversion, the
change in wave function under parity operation is
The nucleus is said to have even parity for & odd parity
for
Bulk Properties
5. Magnetic dipole moment of Nuclei:
The nucleons, like electron, carry intrinsic magnetic moment. The
intrinsic magnetic moment for proton is
μ
p= 2.7927
μ
N& for neutron
μ
n= – 1.9131
μ
Nis the nuclear magneton = 5.0571 × 10
-27J/T
[Bohr magneton
= 9.2849 × 10
-24J/T ]
Neutron, though electrically neutral, has intrinsic magnetic moment!
Bulk Properties
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
Bulk Properties
6. Electric moments of Nuclei:
(
3
z
'
2r
'
2)
ρ
(
r
'
)
d
τ
'
Q
=
∫
−
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
(+
Z
e)
The electric dipole moment of a nucleus in its ground state vanishes
Bulk Properties
6. Electric moments of Nuclei:
Q = 0 for spherical charge distribution, Q < 0 for oblate and Q > 0 for prolate charge distribution
Nuclear Force
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
Nuclear Force
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
(2) Nuclear force is short – ranged
Nuclear potential
Nuclear Force
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
Nuclear Force
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,
(4) Nuclear force is charge symmetric
(
n
−
n
) (
=
p
−
p
)
nuc=
(
p
−
n
)
(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
Nuclear Force
(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 –
Bainbridge Mass Spectrometer
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 (S1)
The fine beam enters into a velocity
Bainbridge Mass Spectrometer
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
Bainbridge Mass Spectrometer
Only those ions with their velocity v = E/B enter the mass spectrograph from the
velocity selector through the slit (S2)
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
Bainbridge Mass Spectrometer
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.
Bainbridge Mass Spectrometer
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]
2. Singly ionized Argon ions are mass analyzed by a Bainbridge mass spectrograph.
The electric and magnetic fields in the velocity filter are 1.5 × 104 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
