Appendix E: Solutions
241
expression is first rewritten to E =−NS 2 J 1 cos θ −NS 2 J 2 2 cos 2 θ −NS 2 J 2 . Then the
energy is minimized with respect to θ : ∂E/∂θ = NS 2 J 1 sin θ + 4NS 2 J 2 cos θ sin θ =
0 ⇒ sin θ(J 1 + 4J 2 cos θ) = 0. The solutions are θ = 0 (ferromagnetic), 180
(antiferromagnetic) and cos θ =− J 1 /4J 2 (helical). (d) J 1 = 1, J 2 =− 0.3; θ =
arccos(1/1.2) = 0.586 rad = 33.56 ◦ ; J 1 =−1, J 2 =−0.3; θ = arccos(−1/1.2) =
2.556 rad = 146.44 ◦ .
241
expression is first rewritten to E =−NS 2 J 1 cos θ −NS 2 J 2 2 cos 2 θ −NS 2 J 2 . Then the
energy is minimized with respect to θ : ∂E/∂θ = NS 2 J 1 sin θ + 4NS 2 J 2 cos θ sin θ =
0 ⇒ sin θ(J 1 + 4J 2 cos θ) = 0. The solutions are θ = 0 (ferromagnetic), 180
(antiferromagnetic) and cos θ =− J 1 /4J 2 (helical). (d) J 1 = 1, J 2 =− 0.3; θ =
arccos(1/1.2) = 0.586 rad = 33.56 ◦ ; J 1 =−1, J 2 =−0.3; θ = arccos(−1/1.2) =
2.556 rad = 146.44 ◦ .