Blume-Capel model

The Blume-Capel model according to Martin Blume and Hans Willem Capel is a generalization of the Ising model in solid state physics . It describes a situation in which the spins of several interacting particles can align parallel, anti-parallel or orthogonal to an external magnetic field , which in addition to the Ising model also covers the orthogonal case. In other words, the Blume-Capel model describes spins with a magnitude , while the Ising model describes spins with a magnitude . ${\displaystyle s=1}$ ${\displaystyle s={\tfrac {1}{2}}}$

definition

The Hamilton operator on which the Blume-Capel model is based , whose eigenvalues ​​are the possible energies of the system, is: ${\displaystyle {\mathcal {H}}}$

${\displaystyle {\mathcal {H}}=-D\sum _{i}(1-s_{i}^{z2})-J\sum _{\langle i,j\rangle }s_{i}^{z}s_{j}^{z}-\mu H\sum _{i}s_{i}^{z}}$

It is

• ${\displaystyle D}$the zero-field splitting , which indicates the energy difference between the singlet and the doublet ${\displaystyle s^{z}=0}$ ${\displaystyle s^{z}=\pm 1}$
• ${\displaystyle J}$ the strength of the interaction of neighboring spins
• ${\displaystyle \mu }$the magnetic moment of the spins
• ${\displaystyle H}$the strength of the external magnetic field
• ${\displaystyle s_{i}^{z}}$the component of the -th spin. ${\displaystyle z}$ ${\displaystyle i}$

The notation under the sum is intended to express that only the next neighbors are added.

The biggest difference to the Ising model is the additional, parameter- dependent term in the Hamilton operator. For only two possible orientations of the spin, as in the Ising model, this term would be a constant and, like a constant term in a potential , physically meaningless. ${\displaystyle D}$

properties

Depending on the value of , the system assumes different basic states and shows different behavior during the phase transition . ${\displaystyle D}$

Denote the number of nearest neighbors and the number of spins in the system, then: ${\displaystyle n}$ ${\displaystyle N}$

• For the ground state is magnetically disordered and all the spins are orthogonal to the magnetic field . The ground state energy is included . ${\displaystyle D>{\tfrac {1}{2}}nJ}$ ${\displaystyle s_{i}=0}$ ${\displaystyle E_{0}=-ND}$
• For the ground state is degenerate . ${\displaystyle D={\tfrac {1}{2}}nJ}$
• For the ground state is completely magnetically ordered . That means, all spins take on the value and the ground state energy is attached . In this area there is a Curie temperature at which the system changes from a magnetically disordered to a magnetically ordered state. This phase transition is ${\displaystyle D<{\tfrac {1}{2}}nJ}$ ${\displaystyle s_{i}=1}$ ${\displaystyle E_{0}=-{\tfrac {1}{2}}NnJ}$ ${\displaystyle T_{\mathrm {C} }}$
  • for a second order phase transition . The Curie temperature decreases from at to at (therein is the Boltzmann constant ) ${\displaystyle -\infty <D<{\tfrac {1}{3}}nJ\ln 4}$ ${\displaystyle {\frac {nJ}{k_{\mathrm {B} }}}}$ ${\displaystyle D=-\infty }$ ${\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}$ ${\displaystyle D={\tfrac {1}{3}}nJ\ln 4}$ ${\displaystyle k_{\mathrm {B} }}$
  • for a first order phase transition in which the magnetization jumps abruptly from zero to a finite value. The Curie temperature continues to decrease from at to K at . ${\displaystyle {\tfrac {1}{3}}nJ\ln 4<D<{\tfrac {1}{2}}nJ}$ ${\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}$ ${\displaystyle D={\tfrac {1}{3}}nJ\ln 4}$ ${\displaystyle 0}$  ${\displaystyle D={\tfrac {1}{2}}nJ}$

Individual evidence

1. ↑ Martin Blume: Theory of the First-Order Magnetic Phase Change in UO ₂ . In: Physical Review . tape 141 , no. 2 , 1965, p. 517-524 , doi : 10.1103 / PhysRev.141.517 (English).
2. ^ Hans Willem Capel: On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting . In: Physica . tape 32 , no. 5 , 1966, pp. 966-988 , doi : 10.1016 / 0031-8914 (66) 90027-9 (English).