## Dante Gatteschi, Roberta Sessoli, and Jacques Villain

Print publication date: 2006

Print ISBN-13: 9780198567530

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567530.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 25 February 2017

# (p.349) APPENDIX J SPECIFIC HEAT

Source:
Molecular Nanomagnets
Publisher:
Oxford University Press

# J.1 Specific heat at equilibrium and at high frequency

The equilibrium specific heat C eq of a spin is given, as seen in Chapter 3, by

(J.1)
$Display mathematics$
The quantity of interest is the magnetic specific heat, but the subscript ‘mag’ has been omitted because there is no ambiguity. The magnetic specific heat is assumed to be decoupled from the lattice, and the lattice specific heat is never considered in this appendix.

If the spin has (2s +1) energy levels E m, the mean energy<ℋ> is

(J.2)
$Display mathematics$
which is a generalization of the formulae seen in Chapter 3. The calculation yields
(J.3)
$Display mathematics$
where z = ∑m exp(−β Em).

As in Section 3.2.3 we now wish to consider a spin in a double potential well, and a high frequency ω≫1/τ, when the spin has almost no chance to go jump to the other part of the double well. Then one has to consider the specific heat C + (resp. C ) of a spin confined in the right-hand (resp. left-hand) well. The eigenstates|m> will be assumed to be localized. In analogy with (J.3)

(J.4)
$Display mathematics$
where $∑ m +$ and $∑ m −$ respectively, designate states localized in the right and left hand well, and $z = ∑ m ± exp ( − βE m )$.

At equilibrium, the spin has probabilities z +/z and z /z to be in the right or left-hand well, respectively. Therefore the high-frequency specific heat C(ω) = C uni as defined in section 10.8 is

$Display mathematics$
(p.350) or according to (J.4)
(J.5)
$Display mathematics$
Subtraction of (J.5) from (J.3) yields
(J.6)
$Display mathematics$
where
(J.7)
$Display mathematics$

Since z = z + + z , (J.6) reads

(J.8)
$Display mathematics$
which is positive.

# J.2 Frequency-dependent specific heat

The next problem is to interpolate between the low-frequency specific heat C(0) = Ceq and the high-frequency specific heat C uni. It is appropriate to introduce a frequency-dependent specific heat C(ω) which should first be defined. The definition should be an extension of the equilibrium definition C = ∂U/∂T (at constant magnetic field, for instance). In the presence of a sinusoidal thermal excitation of frequency ω, the energy U(t) and the temperature T(t) are functions of time,

(J.9)
$Display mathematics$
and
(J.10)
$Display mathematics$
An appropriate definition of C(ω) is
(J.11)
$Display mathematics$
which generalizes the equilibrium property C = ∂U/∂ T.

(p.351) The next task is to calculate δU(t) when δT (t) is known. This requires knowledge of the probability pm(t) to be in state m at time t. Indeed

(J.12)
$Display mathematics$
where pm(t) is the probability to be in state m at time t. This probability depends on a single quantity p +(t), the probability to be in the right-hand well at time t. Indeed, inside each part of the double well, there is thermal equilibrium. Therefore, for all states |m) of the right-hand well,
(J.13)
$Display mathematics$
while for all states of the left-hand well,
(J.14)
$Display mathematics$
with p (t) = 1 − p +(t).

The only additional thing needed is an equation which determines the evolution of p+(t). This equation is

(J.15)
$Display mathematics$
where τ is the relaxation time already introduced before, and $p T +$ is the probability to be in the right-hand well when the system is at equilibrium at temperature T. It depends on t because T is given by (J.9). Of course, $p T +$ is readily written as the sum of the Boltzmann probabilities z −1 exp[− E m/(k B T)] on all states m of the right-hand well.

The above equations are easy to solve. Straightforward algebra (Fominaya et al. 1999) leads to (10.15).