MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 80

a right- or a left-handed screw, depending on the choice of sign. From
the condition 1 /χ ξξ ( Q , 0) T N ), suciently close to T N ,weget

the usual MF result that the order parameter σ Q ∝

∝ ( T −

T ) 1 / 2 .Al-
though 1 /χ ξξ ( Q , 0) becomes negative below T N , the inverse of the actual
susceptibility, 1 /χ ξξ ( Q )=1 /χ ξξ ( Q ,σ Q ), does not. Analogously to the
derivation of A α in (2 . 1 . 21 a ), it may be seen that 1 /χ ξξ ( Q ) is a second
derivative of the free energy, i.e.

( T N −

1 /χ ξξ ( Q )= ∂ 2 f/∂ ( Jσ Q ) 2

1 /χ ξξ ( Q ,σ =0)+12 J 2 B ξξ σ 2 Q = − 2 /χ ξξ ( Q ,σ =0) .

Hence, 1 /χ ξξ ( Q ) is non-negative, as it must be to ensure that the system
is stable, as is also the case for any other component of the susceptibility.
Because is constant, the umklapp contributions to the free

energy in (2.1.24), for which 4 Q is a multiple of the reciprocal-lattice
parameter 4 π/c , cancel. The free energy of the helix is therefore inde-
pendent of the lattice, at least to the fourth power in the magnetization.
If the anisotropy terms in

| J i |

H cf can be neglected, the helix is the most
stable configuration satisfying the condition that = Jσ is constant.

With this constraint, only the two-ion interaction term in the free en-
ergy (2.1.22) may vary, and this may be minimized by the method of
Lagrange multipliers (Nagamiya 1967). We will begin with the weaker

| J i |

constraint; i J i

2 = N ( Jσ ) 2 is constant, which means that we have
to minimize the energy expression

2

+ λ
i

J i

( Jσ ) 2

1

2

U = −

ij J

( ij ) J i · J j −

(2 . 1 . 27 a )

= N
q

− 2 J

( q )+ λ J ( q ) Nλ ( Jσ ) 2 ,

· J ( − q ) −

= q J ( q ) where the introduction of J i exp( i q · R i ), as in (2 . 1 . 10 c ),

yields the second form. Minimizing this expression with respect to

J ( − q ) , we obtain the following equation:

= N −J ( q )+2 λ J ( q ) ∂U/∂ J ( − q ) =0 ,

assuming J ( − q )= J ( q ). For a given value of λ , this condition is only

satisfied if either J ( q ) = 0 ,orif q = q λ ,where J ( q λ )=2 λ ,which

implies that only J ( q λ ) may be non-zero. Introducing this condition

in U , we find

− 2

Nλ ( Jσ ) 2 = ( q λ )( Jσ ) 2 , U = − N J (2 . 1 . 27 b )