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Molecular Nanomagnets$

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

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(p.349) APPENDIX J SPECIFIC HEAT

(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)
C eq = T < > .
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)
< > = Σ m = s s E m exp [ β E m ] Σ m = s s exp [ β E m ]
which is a generalization of the formulae seen in Chapter 3. The calculation yields
(J.3)
C eq = 1 k B T 2 1 z m E m 2 exp ( βE m ) 1 k B T 2 [ 1 z m E m exp ( βE m ) ] 2
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)
C ± = 1 k B T 2 1 z ± ± m E m 2 exp ( βE m ) 1 k B T 2 [ 1 z ± ± m E m exp ( βE m ) ] 2
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

C uni = z + z C + + z z C
(p.350) or according to (J.4)
(J.5)
C uni = 1 k B T 2 1 z m E m 2 exp ( β E m ) 1 z k B T 2 { 1 z + [ m + E m exp ( β E m ) ] 2 + 1 z [ m E m exp ( β E m ) ] 2 } .
Subtraction of (J.5) from (J.3) yields
(J.6)
C eq C uni = 1 z 2 k B T 2 { z z + + 2 + z z 2 [ + + ] 2 }
where
(J.7)
± = m ± E m exp ( βE m ) .

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

(J.8)
C eq C uni = 1 z 2 k B T 2 { z z + + 2 + z + z 2 2 + } = 1 z 2 k B T 2 { z z + + z + z } 2
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)
T ( t ) = T 0 + δT ( t ) = T 0 + δT 0 cos ( ωt ) = T 0 + Re [ δ T 0 exp ( iωt ) ]
and
(J.10)
U ( t ) = U 0 + δU ( t ) = U 0 + δU 0 cos ( ωt ϕ ) = U 0 + δ U 0 exp [ ( i ϕ ) exp ( iωt ) ] .
An appropriate definition of C(ω) is
(J.11)
C ( ω ) = δ U 0 exp ( i ϕ ) δ T 0
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)
U ( t ) s s p m ( t ) E m
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)
p m ( t ) = p + ( t ) z + exp ( β E m )
while for all states of the left-hand well,
(J.14)
p m ( t ) = p ( t ) z exp ( βE m )
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)
t p + ( t ) = 1 τ [ p + ( t ) p T + ( t ) ]
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).