PHYS 260 Lecture Notes - Lecture 3: Spin Magnetic Moment, Azimuthal Quantum Number, Gyromagnetic Ratio
25 May 2018
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MAGNETIC MATRIALS
PERMEABILITY
Consider an unmagnetized bar of a magnetic material in a uniform magnetic field. It has been
observed that the bar gets magnetized by induction and gets a polarity. After magnetization,
the magnetic lines in the bar emanate from N-pole pass through the outer region and then re
enter the S-pole. These lines form a closed loop within the magnet by passing from S-pole to N-
pole. It will be interesting to know that the lines of the magnetized bar oppose the lines of the
original field outside the magnet and favour inside the magnet. As a result of this the magnetic
field strength (H) is increased inside the bar and decreased outside it. Similarly the magnetic
flux density (B) becomes high inside the bar and low outside. Thus we find that flux density (B)
is directly proportional to the magnetic field strength (H). Mathematically Bα H, B=µH where µ
is a constant of proportionality and is known as absolute permeability of the medium. If the flux
density is established in air or vacuum or in a non-magnetic material then the above equation
may be written as B0 =µ0H where B0 is the flux density in air or vacuum and µ0 is the absolute
permeability of air or vacuum. The ratio µ/ µ0 is known as the relative permeability of the
medium. It is designed by the symbol µr . Mathematically µr =( µ/ µ0) =(B/B0) it may be noted
that the relative permeability of air or a non-magnetic material is unity.
Magnetization
The term magnetization may be defined as the process of converting non –magnetic bar into a
magnetic bar. The term is almost analogous to the polarization in dielectric materials. The flux
density B=µH=µ0µrH, B= µ0µrH+ µ0H- µ0H = µ0H+ µ0H(µr-1) = µ0 H + µ0M , B= µ0 (H+M) where
the magnetization, M= H(µr-1). It is expressed in ampere/meter. From the equation, we find
that if a magnetic field is applied to a material, the magnetic flux density is equal to the sum of
the effect on vacuum and that on the material. The magnetization may thus be defined as the
magnetic dipole movement per unit volume.
Origin of magnetic moment –Bohr Magnetron
Electric current in atoms:
For simplicity, let us consider the simplest atom of hydrogen in which one electron revolves
round the proton say on a circular path of radius r. At any instant the electron at appoint P and
the proton at the center form an electric dipole, the direction of the dipole goes on changing as
the electron moves further. Obviously the time average of the electric dipole moment is zero
and there will be no electro static field at a distance point due to this dipole.
Again the moving electron is equivalent to a current ring. If v is the velocity of the electron it
will make
2/2/ wrV
revolutions per second and it is equivalent to a current given by
2/ewI
The circular path is equivalent to a magnetic dipole and the magnetic moment due to this
orbital motion of the election is.
2
2/ 2
2ewr
rewIA
el
The angular momentum associated with the electronic motion
is given by mr2w. Hence we can relate the magnetic dipole moment and angular momentum as
me
el 2/
x angular momentum. The minus sign indicates that the dipole moment points in
a direction opposite to the vector representing the angular momentum. A substance therefore
possesses permanent magnetic dipoles if the electrons of its constituent atom have a net non
vanishing angular momentum.
The ratio of the magnetic dipole moment of the electron due to its orbital motion and the
angular moment of the orbital motion is called orbital gyromagnetic ratio, or orbital magneto
mechanical ratio of an electron, represented by
. In our case it is given by
= magnetic
moment / Angular momentum = e/2m. According to modern atomic theory the angular
momentum of an electron in the orbit is determined by the orbital quantum number l which is
restricted to set of values l=0,1,2…..(n-1) where n is the principal quantum number which
determines the energy of the orbit. It can accept only the integer values n=1,2,3,4………… The
corresponding electronic shells are called K,L,M,N……. shells. The angular momentum of
electrons associated with a particular value of l is given by
2/hl . The electrons associated
with the states l = 0,1,2,3…… are called S,P,d,f…. electrons respectively. The strength at the
permanent magnetic dipole is given by lmehllhme elel .4/)2/)(2/(
The
µel
µea
-e
quantity meh
b
4/ is an atomic unit called Bohr magnetron and has a value 24
1027.9
ampere metre2 and this represents the magnetic moment of an elementary permanent
magnetic dipole. From the above equation it is clear that electron in an atom can take only
certain specified values of magnetic moment depending on the value of l. Hence for s electron
l=0 and magnetic moment is zero.
ELECTRON SPIN AND MAGNETIC MOMENT:
Besides rotating on a circular orbit around the positive nucleus, an electron also rotates around
an axis of its own. The magnetic moment associated with spinning of the electron is called spin
magnetic moment
es. Although, strictly speaking it has a quantum-mechanical origin. We can
think of its magnetic moment as being due to the rotation of the electronic charge about one of
the diameters of the electron in a manner similar to that of the earth’s spinning motion around
its north south axis. If we consider the simple case of an electronic charge being spread over a
spherical volume the electron spin would cause different charge elements of this sphere to
form closed currents resulting thereby in a net spin magnetic moment. Clearly this net magnetic
moment would depend upon the detailed structure of the electron and its charge distribution.
It turns out that an equation connects
es with spin angular momentum S by the relation
sme
es )2/(
where the value of the coefficient
called the spin gyro magnetic ratio
depends on the structure of the spinning particle and its charge distribution. The experimental
value of
for an electron is (-2.0024) the negative sign indicating the
es in a direction
opposite to that of s=h/4
for an electron.
24
1027.9
4
m
eh
es
ampere – metre2
Thus the magnitude of magnetic moments due to spin and the orbital motions of an electron
are of the same order of magnitude. It may be mentioned that spin and
es are intrinsic
properties of an electron and exist event for a stationary electron (L=0).
Document Summary
Consider an unmagnetized bar of a magnetic material in a uniform magnetic field. It has been observed that the bar gets magnetized by induction and gets a polarity. After magnetization, the magnetic lines in the bar emanate from n-pole pass through the outer region and then re enter the s-pole. These lines form a closed loop within the magnet by passing from s-pole to n- pole. It will be interesting to know that the lines of the magnetized bar oppose the lines of the original field outside the magnet and favour inside the magnet. As a result of this the magnetic field strength (h) is increased inside the bar and decreased outside it. Similarly the magnetic flux density (b) becomes high inside the bar and low outside. Thus we find that flux density (b) is directly proportional to the magnetic field strength (h).
