Download Zeeman Effect

Zeeman Effect


B
• Let us have a magnetic dipole  in a magnetic field B.
• Torque  on a magnetic dipole in the magnetic field of
density B is
•  =  B sin 
• Where  is the angle between  and B.


B
• The torque is maximum when the dipole is perpendicular to
the field, and torque is minimum when dipole is parallel to
it.
• Potential energy of dipole inside magnetic field depends on
the magnitude of  of magnetic moment and the orientation
of the moment with respect to the field.
• So potential energy at any other orientation  due to applied
field B is




2
2
U m   d B sin  d
  B cos 
•When  points in the same direction as B then Um=-B
Means its minimum value.
•Potential energy Um is zero at = 90° means
when  is perpendicular to B.
• The magnetic moment of the orbital electron in a hydrogen
atom depends on its angular momentum L.
• When electron is moving in a circular orbit, for that current
loop the magnetic moment is  = IA, where I is the current
and A is the area it encloses.
• Again an electron that makes f rev/sec in a circular orbit of
radius r is equivalent to a current of –ef and so its magnetic
moment is  = -efr2 …………….(1)
• Linear speed v of the electron is v =  r = 2fr
• Angular momentum L = mvr = 2fmr2 …………(2)
• Comparing (1) and (2)
•  = -efr2 = - (e/2m)L…………(3)
• - (e/2m) is referred as gyromagnetic ratio. –ve sign means
that  is in opposite direction to that of L.
L
B
-e

• The magnetic potential energy of an atom in a
magnetic field is
• Um = -B cos  = (e/2m)LB cos 
• We know that LZ = L cos 
• Means cos  ml
l l  1
• So
 e 
U m  ml 
 B  ml  B B
 2m 
Here the quantity under bracket is called
Bohr magneton
• So in a magnetic field the energy of a perticular
atomic state depends on the value of ml as well as on
n.
• A state of total quantum number n breaks up into
several substates when the atom is in a magnetic
field and their energies are slightly more or slightly
less than the energy of the state in the absence of the
flield.
• this phenomenon leads to s splitting of individual
spectral lines into separate lines when atoms radiate
ina magnetic field.
• The spacing of the lines depends on the magnitude
of the field.
• The splitting of spectral lines by magnetic field is
called the Zeeman effect.
• Changes in ml are restricted to Δml =0, ±1
• Normal Zeeman effect consists of the splitting of a
spectral line of frequency 0 into three components
whose frequencies are
B
e
1   0   B   0 
B
h
4m
 2 0
B
e
 3   0  B   0 
B
h
4m
Zeeman Effect: Splits m values
• Orbital magnetic moment L interacts with an
external magnetic field B and separates
m=1
degenerate energy levels.
m=0
m = –1
l=1
NO B
Field
l=0
B Field
m=0
With magnetic field
ml = 2
ml = 1
No field
l=2
ml = 0
ml = -1
ml = -2
h0
ml = 1
ml = 0
ml = -1
l=1
Δml = -1
Δml = 0
Δml = +1
With magnetic field
ml = 2
ml = 1
No field
l=2
ml = 0
ml = -1
h0
ml = -2
eB
h 0 
2m
h 0
h 0 
eB
2m
ml = 1
ml = 0
ml = -1
l=1
Δml = -1
Δml = 0
Δml = +1
• A sample of Certain element is placed in a 0.300-T
magnetic field and suitably excited. How far apart
are the Zeeman components of the 450-nm spectral
line of this element?
• Seperation of the Zeeman component is
eB
 
4m

c

, d  
cd

2
2
  eB
 

c
4mc
2