arXiv:cond-mat/0503658 · Physical Review B

Antiferromagnetic Heisenberg model on clusters with icosahedral symmetry

Nikolaos P. Konstantinidis

Published Aug 31, 2005• 91 citations

Abstract

The nearest-neighbor antiferromagnetic Heisenberg model is considered for spins ${s}_{i}=\frac{1}{2}$ and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group ${I}_{h}$. Taking into account the permutational and spin-inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low-energy spectrum for ${s}_{i}=\frac{1}{2}$, where the first excited states are singlets. Frustration also results in a doubly peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. For ${s}_{i}=1$ both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron, a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.

DOI: 10.1103/physrevb.72.064453

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The antiferromagnetic $S=1/2$ Heisenberg model on the C$_{60}$ fullerene geometry

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arXiv:cond-mat/0503658 · Physical Review B

Antiferromagnetic Heisenberg model on clusters with icosahedral symmetry

Nikolaos P. Konstantinidis

Published Aug 31, 2005• 91 citations

arXiv HTML PDF

Abstract

The nearest-neighbor antiferromagnetic Heisenberg model is considered for spins

{s}_{i}=\frac{1}{2}

and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group

{I}_{h}

. Taking into account the permutational and spin-inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low-energy spectrum for

{s}_{i}=\frac{1}{2}

, where the first excited states are singlets. Frustration also results in a doubly peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. For

{s}_{i}=1

both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron, a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.

DOI: 10.1103/physrevb.72.064453

Related papers

The antiferromagnetic $S=1/2$ Heisenberg model on the C $_{60}$ fullerene geometryThe antiferromagnetic $S=1/2$ Heisenberg model on the C$_{60}$ fullerene geometry

Roman Rausch, Cassian Plorin, Matthias Peschke

We solve the quantum-mechanical antiferromagnetic Heisenberg model with spins positioned on vertices of the truncated icosahedron using the density-matrix renormalization group (DMRG). This describes magnetic properties of the undoped C _{60} 60 fullerene at half filling in the limit of strong on-site interaction U U . We calculate the ground state and correlation functions for all possible distances, the lowest singlet and triplet excited states, as well as thermodynamic properties, namely the specific heat and spin susceptibility. We find that unlike smaller C _{20} 20 or C _{32} 32 that are solvable by exact diagonalization, the lowest excited state is a triplet rather than a singlet, indicating a reduced frustration due to the presence of many hexagon faces and the separation of the pentagonal faces, similar to what is found for the truncated tetrahedron. This implies that frustration may be tuneable within the fullerenes by changing their size. The spin-spin correlations are much stronger along the hexagon bonds and exponentially decrease with distance, so that the molecule is large enough not to be correlated across its whole extent. The specific heat shows a high-temperature peak and a low-temperature shoulder reminiscent of the kagomé lattice, while the spin susceptibility shows a single broad peak and is very close to the one of C _{20} 20 .We solve the quantum-mechanical antiferromagnetic Heisenberg model with spins positioned on vertices of the truncated icosahedron using the density-matrix renormalization group (DMRG). This describes magnetic properties of the undoped C _{60} 60 fullerene at half filling in the limit of strong on-site interaction U U . We calculate the ground state and correlation functions for all possible distances, the lowest singlet and triplet excited states, as well as thermodynamic properties, namely the specific heat and spin susceptibility. We find that unlike smaller C _{20} 20 or C _{32} 32 that are solvable by exact diagonalization, the lowest excited state is a triplet rather than a singlet, indicating a reduced frustration due to the presence of many hexagon faces and the separation of the pentagonal faces, similar to what is found for the truncated tetrahedron. This implies that frustration may be tuneable within the fullerenes by changing their size. The spin-spin correlations are much stronger along the hexagon bonds and exponentially decrease with distance, so that the molecule is large enough not to be correlated across its whole extent. The specific heat shows a high-temperature peak and a low-temperature shoulder reminiscent of the kagomé lattice, while the spin susceptibility shows a single broad peak and is very close to the one of C _{20} 20 .

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The magnetic and thermodynamic properties of 13-atom clusters with decahedral and layered geometries have been studied with spin-1/2 using the quantum Heisenberg model. The numerical exact diagonalization technique is applied to solve the quantumspin Hamiltonian for both ferro (FM) and anti-ferromagnetic (AFM) interactions of the clusters. The ground state energies for the FM case are found to be all degenerate while for the AFM case, they are non-degenerate. In the presence of external magnetic field, we found discontinuities in the field-dependent magnetization accompanied by plateaus for the AFM interaction. The thermodynamic entities such as entropy and specific heat as a function of external magnetic field are smeared out with increasing temperature. They also show a strong dependence on the cluster geometry.The magnetic and thermodynamic properties of 13-atom clusters with decahedral and layered geometries have been studied with spin-1/2 using the quantum Heisenberg model. The numerical exact diagonalization technique is applied to solve the quantumspin Hamiltonian for both ferro (FM) and anti-ferromagnetic (AFM) interactions of the clusters. The ground state energies for the FM case are found to be all degenerate while for the AFM case, they are non-degenerate. In the presence of external magnetic field, we found discontinuities in the field-dependent magnetization accompanied by plateaus for the AFM interaction. The thermodynamic entities such as entropy and specific heat as a function of external magnetic field are smeared out with increasing temperature. They also show a strong dependence on the cluster geometry.

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Pyrochlore S=12 Heisenberg antiferromagnet at finite temperature

Robin Schäfer, Imre Hagymási, Roderich Moessner +1 more

We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore $S=\frac{1}{2}$ quantum antiferromagnet, at finite temperature $T$ : canonical typicality for a finite cluster of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ unit cells (i.e., 32 sites), a finite- $T$ matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T=0.57J$ , which is stable across finite size clusters and converged in the series expansion. At $T\ensuremath{\approx}0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47{k}_{B}ln2$ , which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on $T$ of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material ${\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$ .We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore $S=\frac{1}{2}$ quantum antiferromagnet, at finite temperature $T$: canonical typicality for a finite cluster of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ unit cells (i.e., 32 sites), a finite-$T$ matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at $T=0.57J$, which is stable across finite size clusters and converged in the series expansion. At $T\ensuremath{\approx}0.25J$ (the limit of convergence of our method), the residual entropy per spin is $0.47{k}_{B}ln2$, which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on $T$ of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the $S=1$ material ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$ yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material ${\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$.

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Magnetism and thermodynamics of spin- 1∕2 Heisenberg diamond chains in a magnetic field

Bo Gu, Gang Su

The magnetic and thermodynamic properties of spin- $1∕2$ Heisenberg diamond chains are investigated in three different cases: (a) ${J}_{1}$ , ${J}_{2}$ , ${J}_{3}>0$ (frustrated), (b) ${J}_{1}$ , ${J}_{3}<0$ , ${J}_{2}>0$ (frustrated), and (c) ${J}_{1}$ , ${J}_{2}>0$ , ${J}_{3}<0$ (nonfrustrated), where the bond coupling ${J}_{i}$ $(i=1,2,3)>0$ stands for an antiferromagnetic (AF) interaction, and $<0$ for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor $S(q)$ shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ $(a=0,1,2,3,4,5)$ for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with $m=1∕6$ pertains, exhibits the peaks only at $q=0$ , $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the $XXZ$ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound ${\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}$ are fairly compared with our TMRG results.The magnetic and thermodynamic properties of spin-$1∕2$ Heisenberg diamond chains are investigated in three different cases: (a) ${J}_{1}$, ${J}_{2}$, ${J}_{3}>0$ (frustrated), (b) ${J}_{1}$, ${J}_{3}<0$, ${J}_{2}>0$ (frustrated), and (c) ${J}_{1}$, ${J}_{2}>0$, ${J}_{3}<0$ (nonfrustrated), where the bond coupling ${J}_{i}$ $(i=1,2,3)>0$ stands for an antiferromagnetic (AF) interaction, and $<0$ for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor $S(q)$ shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ $(a=0,1,2,3,4,5)$ for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with $m=1∕6$ pertains, exhibits the peaks only at $q=0$, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the $XXZ$ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound ${\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}$ are fairly compared with our TMRG results.

arXiv:0704.0150Physical Review BMay 24, 200774 citations

Study of the excited states of the quantum antiferromagnets

Mohammad G. Merdan

We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$ , the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone $\Gamma(\mathbf q=0)$ and at the magnetic ordering wavevectors $\pm[\mathbf Q= (4\pi/3,0)]$, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.

Nov 10, 20130 citations

The nearest-neighbor antiferromagnetic Heisenberg model is considered for spins

{s}_{i}=\frac{1}{2}

and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group

{I}_{h}

{s}_{i}=\frac{1}{2}

{s}_{i}=1

The antiferromagnetic

S=1/2

Heisenberg model on the C

_{60}

fullerene geometry

The frustrated ferromagnetic spin-

1∕2

Heisenberg chain is studied by means of a low-energy field theory as well as the density-matrix renormalization group and exact diagonalization methods. First, we study the ground-state phase diagram in a magnetic field and find an ``even-odd'' (EO) phase characterized by bound pairs of magnons in the region of two weakly coupled antiferromagnetic chains. A jump in the magnetization curves signals a first-order transition at the boundary of the EO phase, but otherwise the curves are smooth. Second, we discuss thermodynamic properties at zero field, where we confirm a double-peak structure in the specific heat for moderate frustrating next-nearest-neighbor interactions.

We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore

S=\frac{1}{2}

quantum antiferromagnet, at finite temperature

T

: canonical typicality for a finite cluster of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ unit cells (i.e., 32 sites), a finite-

T

matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at

T=0.57J

, which is stable across finite size clusters and converged in the series expansion. At $T\ensuremath{\approx}0.25J$ (the limit of convergence of our method), the residual entropy per spin is

0.47{k}_{B}ln2

, which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on

T

of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the

S=1

material

{\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}

yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material

{\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}

The magnetic and thermodynamic properties of spin-

1∕2

Heisenberg diamond chains are investigated in three different cases: (a)

{J}_{1}

{J}_{2}

{J}_{3}>0

(frustrated), (b)

{J}_{1}

{J}_{3}<0

{J}_{2}>0

(frustrated), and (c)

{J}_{1}

{J}_{2}>0

{J}_{3}<0

(nonfrustrated), where the bond coupling

{J}_{i}

(i=1,2,3)>0

stands for an antiferromagnetic (AF) interaction, and

<0

for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor

S(q)

shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$

(a=0,1,2,3,4,5)

for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with

m=1∕6

pertains, exhibits the peaks only at

q=0

, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the

XXZ

anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound

{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}

are fairly compared with our TMRG results.

\Gamma(\mathbf q=0)

and at the magnetic ordering wavevectors

\pm[\mathbf Q= (4\pi/3,0)]

, the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.