
The first thing to note about diamagnetism is that all
atoms and therefore all materials show diamagnetic behavior. 


Diamagnetism thus is always superimposed on all other forms of magnetism. Since
it is a small effect, it is hardly noticed, however. 


Diamagnetism results because all matter contains electrons  either "orbiting"
the nuclei as in insulators or in the valence band (and lower bands) of semiconductors, or being "free", e.g.
in metals or in the conduction band of semiconductors. All these electrons can respond to a (changing) magnetic field. Here
we will only look at the (much simplified) case of a bound electron orbiting a nucleus in a circular
orbit. 

The basic response of an orbiting electron
to a changing magnetic field is a precession of the orbit, i.e. the
polar vector describing the orbit now moves in a circle around the magnetic field vector H: 
 


The angular vector w characterizing the blue orbit of the
electron will experience a force from the (changing) magnetic field that forces it into a circular movement on the green
cone. 


Why do we emphasize "changing" magnetic fields? Because there is no
way to bring matter into a magnetic field without changing it  either be switching it on or by moving the material into
the field. 

What exactly happens to the orbiting electron?
The reasoning given below follows the semiclassical approach contained within Bohr's atomic model. It gives essentially
the right results (in cgs units!). 


The changing magnetic field, dH/dt, generates a force
F on the orbiting electron via inducing a voltage and thus an electrical field E. We can always express
this as 
 
F = m*_{e} · a 
=  m*_{e} · 
dv dt  := e · E 




With a = acceleration = dv/dt
= e · E/m*_{e}. 

Since dH/dt primarily induces a voltage V,
we have to express the field strength E in terms of the induced voltage V. Since the electron
is orbiting and experiences the voltage during one orbit, we can write: 
 



With L = length of orbit = 2p ·
r, and r = radius of orbit. 


V is given by the basic equations of induction, it is

 



With F = magnetic flux = H · A;
and A = area of orbit = p · r^{2}. The minus
sign is important, it says that the effect of a changing magnetic fields
will be opposing the cause in accordance with Lenz's law. 

Putting everything together we obtain 
 
dv dt 
= 
e · E_{ }
m*_{e}  = 
V · e_{ }
L · m*_{e}  = – 
e · r_{ }
2 m*_{e} 
·  dH
dt 



The total change in v will be given by integrating: 
 
Dv = 
v_{2}
ó õ v_{1} 
dv = – 
e · r_{ }
2m*_{e}  ·
 H
ó õ 0 
dH  = –
 e · r · H_{ }
2 m*_{e} 



The magnetic moment
m_{orb} of the undisturbed electron was m_{orb} = ½ · e · v ·
r 


By changing v by Dv, we change m_{orb}
by Dm_{orb}, and obtain 
 
Dm_{orb} = 
e · r · Dv 2
 = – 
e^{2} · r ^{2} · H
4m*_{e} 



That is more or less the equation for diamagnetism
in the primitive electron orbit model. 


What comes next is to take into account that the magnetic field does not have
to be perpendicular to the orbit plane and that there are many electrons. We have to add up the single electrons and average
the various effects. 


Averaging over all possible directions of H (taking into account
that a field in the plane of the orbit produces zero effect) yields for the average
induced magnetic moment almost the same formula: 
 
Dm_{orb} = <Dm_{orb}>  = – 
e^{2} · <r>^{2} ·
H 6m*_{e} 




<r> denotes that we average over the orbit radii at the
same time 

Considering that not just one, but all z
electrons of an atom participate, we get the final formula: 
 
Dm = <Dm_{orb}> = – 
e^{2} · z · r
^{2} · H 6 m*_{e} 



The additional magnetization M caused by Dm
is all the magnetization there is for diamagnets; we thus we can drop the D and get 




With the definition
for the magnetic susceptibility
c = M/H we finally obtain for the relevant material parameter for
diamagnetism 
 
c_{dia} = –

e^{2} · z · <r>
^{2} 6 m*_{e} · V 
= – 
e^{2} · z · <r>
^{2} 6 m*_{e} 
· r_{atom} 




With r_{atom} = number of atoms per
unit volume 

Plugging in numbers will yield c values around
– (10^{–5}  10^{–7}) in good agreement with experimental values. 