
With the results of the elasticity theory we can get approximate formulas for
the line energy of a dislocation and the elastic interaction with other defects, i.e. the
forces acting on dislocations. 


The energy of a dislocation comes from the elastic part that is contained in
the elastically strained bonds outside the radius r_{0} and
from the energy stored in the core, which is of course energy sitting in the distorted bonds, too, but is not amenable to elasticity theory.



The total energy per unit length E_{ul}
is the sum of the energy contained in the elastic field, E_{el}, and the energy in the core, E_{core} .

 
E_{ul}  = 
E_{el} + E_{core} 




Do not confuse energies E with Youngs modulus
Y which is often (possibly here) written as E, too! From the context it is always clear what is meant. 


Using
the formula for the strain energy for a volume element given before,
integration over the total volume will give the total elastic energy E_{el} of the dislocation. The
integration is easily done for the screw dislocation; in what follows the equations are always
normalized to a unit of length. 


dE_{ el}(screw)  = 
p · r · dr · (s_{Q z} · e_{Q z} + s_{zQ} · e_{zQ}) 
= 
4p · r · dr · G · (e_{Q z})^{2} 
E_{el}(screw)  = 
G · b^{2} 4p 
·  R
ó õ r_{o}  
dr r  = 
G · b^{2} 4p 
· ln  R
r_{o} 




The integration runs from r_{0}, the core radius of the dislocation
to R, which is some as yet undetermined external radius of the elastic
cylinder containing the dislocation. In principle, R should go to infinity, but this is not sensible as we
are going to see. 

The integration for the edge dislocation is much more difficult to do, but the
result is rather simple, too: 
 
E_{el}(edge)  = 
G · b^{2} 4p(1 – n)  · ln 
R r_{o} 




So, apart from the factor (1 – n), this is the same
result as for the screw dislocation. 

Let us examine these equations. There are a number of interesting properties;
moreover, we will see that there are very simple approximations to be gained: 

1.
The total energy U of a dislocation is proportional to its length L. 
 
U _{ }  = 
E_{ ul} · L = L · (E_{el} +
E _{ core}) 




Since we always have the principle of minimal energy (entropy does not play a role in this
case), we can draw a important conclusion: 


A dislocation tends to be straight between its two "end
points" (usually dislocation knots). That is a first rule about
the direction a dislocation likes to assume. 

2.
The line energy of an edge dislocation is always larger than that of a screw dislocation
since (1 – n) < 1. With n
» 1/3, we have E_{screw}
» 0,66 ·E_{edge} . 


This means that a dislocation tends to have as large a screw component as possible. This is
a second rule about the direction a dislocation likes to assume which may be in contradiction to the first one. It is quite possible that a dislocations needs
to zigzag to have as much screw character as geometrially allowed  it then cannot be straight at the same time. 

3.
The elastic part of the energy depends (logarithmically) on the crystal size
(expressed in R),
for an infinite crystal it is infinite (¥)!
Does this make any sense? 


Of course this doesn't make sense. Infinite crystals, however, do not make sense either. And
in finite crystals, even in big finite crystals, the energy is finite! 


Moreover, in most real crystals it is not
the outer dimension that counts, but the size of the grains which are usually quite small. In addition, if there are
many dislocations with different signs of the Burgers vector, their strain fields will (on average) tend to cancel each
other. So for practical cases we have a finite energy. 

4.
The elastic part of the energy also depends (logarithmically) on the core radius r_{0}. 

5.
The energy is a weak function of the crystal (or grain size) R. 


Taking an extremely small value for r_{0}, e.g. 0,1 nm, we obtain
for ln (R/r_{0}) an extreme range of 20,7  2,3 if we pick extreme values for R
of 100 mm or 1 nm, respectively. A more realistic range for R
would be 100 µm  10 nm, giving ln (R/r_{0}) = 13,8  4,6. Grain size
variations, within reasonable limits, thus only provide a factor of 2  3 for energy variations. 

We will now deduce an approximation for the line energy that is sufficiently good
for most purposes. 


We equate r_{0} with the magnitude of the Burgers vector, b
. This makes sense because the Burgers vector is a direct measure of the "strength"
of a dislocation, i.e. the strength of the displacement in the core region. 


We need a value for the energy in the core of the dislocations, which so far we have not dealt
with. Since there is no easy way of calculating that energy, we could equate it in a first approximation with the energy
of melting. That would make sense because the dislocation core is comparable in its degree of distortion to the liquid state.
More sophisticated approaches end up with the best simple value: 
 



There are, however, other approaches, too. 


The total energy for r_{0} = b then becomes 
 
E_{tot}  = 
G · b^{2} 4p 
· 
æ ç è  ln 
R b  + 2 
ö ÷ ø 



More generally, the following formula is often used 
 
E_{tot}  = 
G · b^{2} 4p(1 – n)
 · 
æ ç è  ln 
R b  + B 
ö ÷ ø 




with B = pure number best approximating the core energy of the particular case.
Often B = 1 is chosen, leading to 


E_{tot}  = 
G · b^{2} 4p (1 – n)
 · 
æ ç è  ln 
R b  + 1 
ö ÷ ø 
  
  
  
 = 
G · b^{2} 4p(1 –n )
 · 
æ ç è  ln 
e ·R b 
ö ÷ ø 




For the last equation bear in mind that ln(e) = 1. 

The ln term is not very important. To give an example; it is exactly
4p for e · R = 3,88 x 10^{4} b, i.e. for R
» 5 µm ; the total energy in this case would be E_{tot} = 2G
· b^{2} . 


In a very general way we can write 
 
E_{tot}  = 
a · G · b^{2}_{ } 


 



And a (from measurements) is found to be a
» 1,5 ....0,5. If we do not care for factors in the order of unity,
we get the final very simple formula for the line energy
of a dislocation 
 
