### 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 1^{st}
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 1^{st} 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 1^{st} 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
.