Physics 512 (Spring 2018)
HOMEWORK 4: solutionsdue 8 February 2018
(1) Exact solution of the Weiss modelWe can set external applied field H = 0 and use Weiss molecular field to
find exact solutions. Written dimensionless magnetic moment, m = MMs, and
introducing
y =gµBJλM
kBT
x = exp (y)
the exact solution for the paramagnetic state is:
m = BJ (y)
(Tc = nµ2Bλ/kB and Ms = ngµBJ)For J = 1/2, express T/Tc as a function of m ONLY.Specifically, write down an expression containing only m on one side and
only T/Tc on the other and no other parameters involved.
Solution:For J = 1/2, g = 2 and
BJ (y) = tanh y =ey − e−yey + e−y
=x− 1
x
x+ 1x
=x2 − 1x2 + 1
= m
sox2 =
1 +m
1−mTaking ln of both sides, we have:
2 lnx = 2y = ln1 +m
1−m
with our definitions (and gJ = 1),
y =nµ2Bλ
kB
(gJ)2
T
M
ngµBJ=TcTm
therefore
2TcTm = ln
1 +m
1−mor
T
Tc=
2m
ln (1 +m)− ln (1−m)
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(2) Helical magnetic orderSome materials have distinct layered magnetic structure. For example Dy.
Consider a situation when magnetic moments are ordered ferromagnetically ineach atomic plane, but the planes may have angle θ between directions of themagnetic moments in them. See the illustration below.
The energy can be described by nearest and next nearest neighbour exchangeconstants, J1 and J2 , respectively.
E = −2NS2 (J1 cos θ + J2 cos 2θ)
where N is the number of atoms in each plane, each atom having spin S.(a) Find equilibrium angle(s) θ corresponding to possible ground states in
terms of J1 and J2.(b) Which of the following states possible? ferromagnetic (FM), antiferro-
magnetic (AFM) or helical?(c) Draw the "phase diagram" of possible magnetic states on a (J1, J2) plane.
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Solutions(a) To find possible solution, we need to minimize E with respect to θ, so
dE
dθ= 2NS2 (J1 sin θ + 2J2 sin 2θ) = 0
withsin 2θ = 2 sin θ cos θ
the possible solutions are found from:
sin θ (J1 + 4J2 cos θ) = 0
The solutions are:
sin θ = 0
sin θ = π
cos θ = − J14J2
(b) Therefore, all three states are possible: θ = 0 - FM, θ = π - AFM and
θ = arccos(− J14J2
)- helimagnetic order.
(c) Clearly, the following phase diagram shows possible magnetic states inthis model:
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