Metastable behaviour of the spin-S Ising and Blume- Capel ferromagnets
Muktish Acharyya Department of Physics
Presidency University, Kolkata
8th ISPC meeting , ICTS Bangalore , February 1−3, 2023
Collaborators
M . Naskar , M . Acharyya, E . Vatansever , N . G . Fytas , Phys Rev E , 104 (2021) 014107
1. Moumita Naskar , IMSc Chennai, India 2. Erol Vatansever, Dokuz Eylul University, Turkey 3. Nikolaos G Fytas, Coventry University , UK
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Metastability
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Classical theory of nucleation
E l = −2 hl + σ l
( d −1)
d Energy of formation of droplet (l )
R. Becker and W. Doring, Ann. Phys. (Leipzig) 24 (1935) 719 3
E l
Growth of supercritical droplets
E c = K d σ d
h d −1 n c ∼ exp(− E c
k B T ) ∼ exp (− K d σ d k B Th d−1 )
I ∼ n c ; τ n ∼ 1
I Growth of Single droplet
Coalescence of multiple droplets
τ n ∼ exp ( K d σ d k B T h d−1 )
τ c ∼ exp ( K d σ d
k B T ( d +1 ) h d −1 )
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Model system H =− J
s 2 ∑ ij S i z S z j + D s 2 ∑ i ( S i z ) 2 − h s ∑ i S i z
S z =(− S,− S+ 1 ,−S + 2 ,... + S )
D =0 ( Spin− S Ising model ) D ≠ 0 ( Spin− S Blume −Capel model )
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Monte Carlo simulation
Metropolis single spin flip algorithm
P ( S old z → S new z ) = Min [ 1 , exp (− δ E k B T )]
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Metastable lifetime (MC results)
M . Naskar , M . Acharyya, E . Vatansever , N . G . Fytas , Phys Rev E , 104 (2021) 014107
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Metastable lifetime (Theory)
The multidroplet / coalescence regime
The single droplet / nucleation regime
d dimension
h applied magnetic field
J. D. Gunton and M. Droz, Introduction to the theory of metastable and stable states (Springer-Verlag, 1983)
log ( τ n ) ∼ 1 h d −1 log ( τ c ) ∼ 1
(d +1 ) h d −1
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Metastable lifetime as a function of applied field
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Metastable lifetime as a function of applied field
M . Naskar , M . Acharyya, E . Vatansever , N . G . Fytas , Phys Rev E , 104 (2021) 014107
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How does the metastable volume fraction decay ?
ϕ ∼ exp (−c ( t / τ ) d +1 )
Avrami's law
ϕ = N N
Metastable volume fraction
M. Avrami, J. Chem. Phys. 7 (1939) 1103
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Decay of metastable volume fraction (Ising)
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Decay of metastable volume fraction (Blume-Capel)
M . Naskar , M . Acharyya, E . Vatansever , N . G . Fytas , Phys Rev E , 104 (2021) 014107
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Concluding remarks
1. The MC results of metastable lifetime of Spin-S Ising and Blume-Capel ferromagnet are in good agreement with the classical theory of nucleation.
2. The MC results of the decay of metastable volume fraction are in good agreement with the Avrami’s law.
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