Exact Solution of the Decorated Ising Model with Nonmagnetic Impurities

E. A. Shapoval

Published Jul 1, 1968• 0 citations

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Exact Solution of the Decorated Ising Model with Nonmagnetic Impurities

E. A. Shapoval

Published Jul 1, 1968• 0 citations

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Exact solution of the mixed-spin Ising model on a decorated square lattice with two different kinds of decorating spins on horizontal and vertical bonds

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The mixed-spin- $(1∕2,{S}_{\mathrm{B}},{S}_{\mathrm{C}})$ Ising model on a decorated square lattice with two different kinds of decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ $({S}_{\mathrm{B}}\ensuremath{\ne}{S}_{\mathrm{C}})$ placed on its horizontal and vertical bonds, respectively, is exactly solved by establishing a precise mapping relationship with the corresponding spin- $1∕2$ Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy acting on both types of decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ is examined, in particular. If decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ are integer and half-odd-integer, respectively, or if the reverse is the case, the model under investigation displays a very peculiar critical behavior that had bearing on the spontaneously ordered ``quasi-one-dimensional'' spin system, which appears as a result of the single-ion anisotropy strengthening. We have found convincing evidence that this remarkable spontaneous ordering virtually arises even though all integer-valued decorating spins tend toward their ``nonmagnetic'' spin state $S=0$ and the system becomes disordered only upon further increase of the single-ion anisotropy. The single-ion anisotropy parameter is also at an origin of various temperature dependences of the total magnetization when imposing the pure ferrimagnetic or the mixed ferro-ferrimagnetic character of the spin arrangement.The mixed-spin-$(1∕2,{S}_{\mathrm{B}},{S}_{\mathrm{C}})$ Ising model on a decorated square lattice with two different kinds of decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ $({S}_{\mathrm{B}}\ensuremath{\ne}{S}_{\mathrm{C}})$ placed on its horizontal and vertical bonds, respectively, is exactly solved by establishing a precise mapping relationship with the corresponding spin-$1∕2$ Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy acting on both types of decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ is examined, in particular. If decorating spins ${S}_{\mathrm{B}}$ and ${S}_{\mathrm{C}}$ are integer and half-odd-integer, respectively, or if the reverse is the case, the model under investigation displays a very peculiar critical behavior that had bearing on the spontaneously ordered ``quasi-one-dimensional'' spin system, which appears as a result of the single-ion anisotropy strengthening. We have found convincing evidence that this remarkable spontaneous ordering virtually arises even though all integer-valued decorating spins tend toward their ``nonmagnetic'' spin state $S=0$ and the system becomes disordered only upon further increase of the single-ion anisotropy. The single-ion anisotropy parameter is also at an origin of various temperature dependences of the total magnetization when imposing the pure ferrimagnetic or the mixed ferro-ferrimagnetic character of the spin arrangement.

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The mixed-spin-

(1∕2,{S}_{\mathrm{B}},{S}_{\mathrm{C}})

Ising model on a decorated square lattice with two different kinds of decorating spins

{S}_{\mathrm{B}}

and

{S}_{\mathrm{C}}

$({S}_{\mathrm{B}}\ensuremath{\ne}{S}_{\mathrm{C}})$ placed on its horizontal and vertical bonds, respectively, is exactly solved by establishing a precise mapping relationship with the corresponding spin-

1∕2

Ising model on an anisotropic square (rectangular) lattice. The effect of uniaxial single-ion anisotropy acting on both types of decorating spins

{S}_{\mathrm{B}}

and

{S}_{\mathrm{C}}

is examined, in particular. If decorating spins

{S}_{\mathrm{B}}

and

{S}_{\mathrm{C}}

are integer and half-odd-integer, respectively, or if the reverse is the case, the model under investigation displays a very peculiar critical behavior that had bearing on the spontaneously ordered ``quasi-one-dimensional'' spin system, which appears as a result of the single-ion anisotropy strengthening. We have found convincing evidence that this remarkable spontaneous ordering virtually arises even though all integer-valued decorating spins tend toward their ``nonmagnetic'' spin state

S=0

and the system becomes disordered only upon further increase of the single-ion anisotropy. The single-ion anisotropy parameter is also at an origin of various temperature dependences of the total magnetization when imposing the pure ferrimagnetic or the mixed ferro-ferrimagnetic character of the spin arrangement.