If η is a scalar, or tensor of even rank, odd power terms of η can also be scalars if properly constructed. Transitions in such systems are 1st order. For simplicity, we shall assume η to be a scalar and write,
where for global stability. The equilibrium conditions are
Solutions to (a) are
where the discriminant is .
As before, the disordered case requires or to satisfy (b).
If we follow the procedure of the last section, we can obtain from (d) an expression for h for the ordered phase valid for or . Obviously, this would be a continuous phase transition.
On the other hand, as shown in Fig.3.24, we
to produce another minimum at
lower than that at
for the case
. Obviously, transition to this new minimum will
result in a discontinuous change in the slope of j and hence signifies a 1st order phase transition. Furthermore, since we are still in the region
this transition preceeds that for the continuous one at
so that the latter is
Þ Þ (e)
Thus, the new phase is determined by conditions (b,c,e).
Now, 4(e) - (c) gives
(c) - 2(e) Þ Þ
Combining these 2 expressions, we also have .
Combining (e) with condition (b) gives
or, simply, . Since , the sign of h is determined by that of .