3.G.2. First Order Transitions

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,

        ϕ( T,Y,η ) ϕ 0 ( T,Y )+ α 2 η 2 + α 3 η 3 + α 4 η 4                     (3.79)

where α 4 >0  for global stability.  The equilibrium conditions are

        ( ϕ η ) TY =2 α 2 η+3 α 3 η 2 +4 α 4 η 3 =0                        (a)

        ( 2 ϕ η 2 ) TY =2 α 2 +6 α 3 η+12 α 4 η 2 >0                        (b)

Solutions to (a) are

        η=0

and

        2 α 2 +3 α 3 η+4 α 4 η 2 =0                    (c)

Þ    η= 1 8 α 4 ( 3 α 3 ± D )                     (d)

where the discriminant is D=9 α 3 2 32 α 2 α 4 .

As before, the disordered case η=0  requires α 2 >0  or T> T C  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 α 2 <0  or T< T C .  Obviously, this would be a continuous phase transition.

On the other hand, as shown in Fig.3.24, we can adjust α 3  to produce another minimum at η= η D  lower than that at η=0  for the case α 2 >0 .  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 T> T C , this transition preceeds that for the continuous one at T= T C  so that the latter is never observed.  Again with reference to Fig.3.24, the onset of this 1st order transition (curve D) happens when both minima have the same value, i.e.,

ϕ( T,Y, η D )= ϕ 0 ( T,Y )

Þ            α 2 η 2 + α 3 η 3 + α 4 η 4 =0             Þ    α 2 + α 3 η+ α 4 η 2 =0          (e)

Thus, the new phase is determined by conditions (b,c,e).

Now, 4(e) - (c) gives

        2 α 2 + α 3 η=0            Þ    η= 2 α 2 α 3

(c) - 2(e) Þ    α 3 η+2 α 4 η 2 =0       Þ  η= α 3 2 α 4

Combining these 2 expressions, we also have  α 2 = α 3 2 4 α 4 .

Combining (e) with condition (b) gives

        5 α 2 3 α 3 η>0

or, simply, α 2 >0 .  Since α 4 >0 , the sign of h is determined by that of α 3 .