Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0512005 · Physical Review B

Low-temperature magnetization and the excitation spectrum of antiferromagnetic Heisenberg spin rings

Larry Engelhardt, Marshall Luban

Published Feb 23, 2006• 38 citations
arXivHTMLPDF

Abstract

Accurate results are obtained for the low-temperature magnetization vs magnetic field of Heisenberg spin rings consisting of an even number $N$ of intrinsic spins $s=1∕2,1,3∕2,2,5∕2,3,7∕2$ with nearest-neighbor antiferromagnetic exchange by employing a numerically exact quantum Monte Carlo method. A straightforward analysis of this data, in particular, the values of the level-crossing fields, provides accurate results for the lowest-energy eigenvalue ${E}_{N}(S,s)$ for each value of the total spin quantum number $S$. In particular, the results are substantially more accurate than those provided by the rotational band approximation. For $s\ensuremath{\leqslant}5∕2$, data are presented for all even $N\ensuremath{\leqslant}20$, which are particularly relevant for experiments on finite magnetic rings. Furthermore, we find that for $s\ensuremath{\geqslant}3∕2$, the dependence of ${E}_{N}(S,s)$ on $s$ can be described by a scaling relation, and this relation is shown to hold well for ring sizes up to $N=80$ for all intrinsic spins in the range $3∕2\ensuremath{\leqslant}s\ensuremath{\leqslant}7∕2$. Considering ring sizes in the interval $8\ensuremath{\leqslant}N\ensuremath{\leqslant}50$, we find that the energy gap between the ground state and the first excited state approaches zero proportional to $1∕{N}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}\ensuremath{\approx}0.76$ for $s=3∕2$ and $\ensuremath{\alpha}\ensuremath{\approx}0.84$ for $s=5∕2$. Finally, we demonstrate the usefulness of our present results for ${E}_{N}(S,s)$ by examining the ${\mathrm{Fe}}_{12}$ ring-type magnetic molecule, leading to a more accurate estimate of the exchange constant for this system than has been obtained heretofore.

DOI
10.1103/physrevb.73.054430

Related papers

Spin dynamics of finite antiferromagnetic Heisenberg spin rings

O. Waldmann

The spin pair-correlation function of finite bipartite antiferromagnetic Heisenberg quantum spin rings was studied numerically by means of exact diagonalization techniques. For large spins, the spectrum of the spin pair-correlation function is of a remarkably simple structure: It consists of few characteristic peaks at low, and a broad featureless signal at high temperatures. This arises as the energy spectrum exhibits a set of parallel rotational bands at low energies emerging into a quasi-continuum of states, with transitions from the lowest rotational band to the quasi-continuum being highly suppressed. The energies of the rotational bands can be accurately described by a generalized dispersion relation that depends on the total spin quantum number and, as usual, on the shift quantum number. These regularities are better fulfilled the larger the spin length but the smaller the ring size. It will be shown that all these features are associated with the underlying sublattice structure of the spin rings, and it is argued that they are valid for a more general class of finite Heisenberg systems than rings.

Physical review. B, Condensed matter•Dec 19, 2001•92 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.

•Nov 10, 2013•0 citations

Low-energy excitations of two-dimensional diluted Heisenberg quantum antiferromagnets

Ling Wang, Anders W. Sandvik

We study the low-energy dynamics of $S=1/2$ antiferromagnetic Heisenberg clusters constructed by diluting a square lattice at vacancy concentration $p$ at and below the percolation threshold ${p}^{\ensuremath{\ast}}\ensuremath{\approx}0.407$. The finite-size scaling behavior of the average excitation gap, $⟨\ensuremath{\Delta}⟩\ensuremath{\sim}{L}^{\ensuremath{-}z}$, where $L$ is the cluster length, is obtained using quantum Monte Carlo results for an upper bound ${\ensuremath{\Delta}}^{\ensuremath{\ast}}$ to $\ensuremath{\Delta}$, derived from sum rules. At the percolation threshold, we obtain a dynamic exponent $z=3.6\ifmmode\pm\else\textpm\fi{}0.1\ensuremath{\approx}2{D}_{f}$ for clusters with singlet $(S=0)$ ground state. Here ${D}_{f}=91/48$ is the fractal dimensionality of the percolating cluster. We argue that this large dynamic exponent---roughly twice that expected for quantum-rotor excitations---is a consequence of weakly interacting localized effective magnetic moments, which form due to local sublattice imbalance. This picture is supported by an extremal-value analysis of local spectral gaps, which delivers an exponent relation (between $z$ and two exponents characterizing the local-gap distribution) reproduced by our simulation data. However, the average $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ over all clusters, which have mostly ground-state spin $S>0$, scales with a smaller exponent than for the $S=0$ clusters alone; $z\ensuremath{\approx}1.5{D}_{f}$. Lanczos exact diagonalization for small clusters show that typically, $S\ensuremath{\rightarrow}S\ensuremath{-}1$ in the lowest-energy excitations while the dominant spectral weight originates from $S\ensuremath{\rightarrow}S+1$ excitations. Thus, the scaling of $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ for clusters with ground state $S>0$ does not reflect the lowest-energy excitations but the higher $S\ensuremath{\rightarrow}S+1$ excitations. This result can be understood within a valence bond picture. To further explore the scenario of localized moments, we introduce a classical dimer-monomer aggregation model to study the distribution of nearest-neighbor sites forming dimers (which are the objects used in mapping to the quantum-rotor model) and unpaired spins (monomers). The monomers are localized and, thus, effective magnetic moments should form in the spin system. We also study the lowest triplet excitation of $S=0$ clusters using quantum Monte Carlo calculations in the valence bond basis. The triplet is concentrated at some of the classical monomer regions, confirming the mechanism of moment formation. The number of spins (and moment regions) affected by the excitation scales as a nontrivial power of the cluster size. For a dimer-diluted bilayer Heisenberg model with weak interlayer coupling (where the system remains N\'eel ordered), there is no sublattice imbalance. In this case we find $z\ensuremath{\approx}{D}_{f}$, consistent with quantum-rotor excitations. For a single layer at $p<{p}^{\ensuremath{\ast}}$ we find $z\ensuremath{\approx}2=D$, which indicates that the weakly interacting localized moment mechanism is valid only exactly at the percolation point. There is a crossover behavior close to the percolation point.

arXiv:0909.5211•Physical Review B•Feb 16, 2010•32 citations

Evolution of thermodynamic properties and inelastic neutron scattering intensities for spin- 12 antiferromagnetic quantum rings

Jason T. Haraldsen

This study examines the increasing complexity in the magnetic properties of small $n=3,\phantom{\rule{0.16em}{0ex}}4,\phantom{\rule{0.16em}{0ex}}5,\phantom{\rule{0.16em}{0ex}}\mathrm{and}\phantom{\rule{0.16em}{0ex}}6$ spin-$\frac{1}{2}$ quantum rings. Using an exact diagonalization of the isotropic Heisenberg Hamiltonian with nearest and next-nearest neighbor interactions, the energy eigenstates, magnetic specific heat capacity, magnetic susceptibility, and inelastic neutron scattering structure factors are determined for variable next-nearest neighbor interactions. Here, it is shown that the presence of spin exchange symmetry breaking, multiple ground states, and nonzero total spin ground states can greatly complicate the energy eigenstates and excitations for these systems. Overall, the energy eigenstates and structure factor intensities are presented in closed form, while the thermodynamic properties detail the effect of a crossing interaction in the rings. The goal of this work is to provide insight into the evolution of the magnetic properties and spin excitations within these systems.

arXiv:1605.05167•Physical review. B./Physical review. B•Aug 31, 2016•10 citations

Magnetization Curve at Zero Temperature for the Antiferromagnetic Heisenberg Linear Chain

Robert B. Griffiths

The highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton, $g$ the electron $g$ factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.

Physical Review•Feb 3, 1964•402 citations

Antiferromagnetic Heisenberg spin-1 chain: Magnetic susceptibility of the Haldane chain described using scaling

Jean Souletie, Marc Drillon, Pierre Rabu +1 more

The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins $S=1$, whose size $n$ is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For $n$ large, the coefficients of the expression converge towards ${C}_{1}=0.125$, ${W}_{1}=0.451J$, ${C}_{2}=0.564$, ${W}_{2}=1.793J$ ($J$ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at $T=0$ and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of ${\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}$ and of $\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}$ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.

Physical Review B•Aug 18, 2004•16 citations
Accurate results are obtained for the low-temperature magnetization vs magnetic field of Heisenberg spin rings consisting of an even number NNN of intrinsic spins s=1∕2,1,3∕2,2,5∕2,3,7∕2s=1∕2,1,3∕2,2,5∕2,3,7∕2s=1∕2,1,3∕2,2,5∕2,3,7∕2 with nearest-neighbor antiferromagnetic exchange by employing a numerically exact quantum Monte Carlo method. A straightforward analysis of this data, in particular, the values of the level-crossing fields, provides accurate results for the lowest-energy eigenvalue EN(S,s){E}_{N}(S,s)EN​(S,s) for each value of the total spin quantum number SSS. In particular, the results are substantially more accurate than those provided by the rotational band approximation. For s\ensuremath⩽5∕2s\ensuremath{\leqslant}5∕2s\ensuremath⩽5∕2, data are presented for all even N\ensuremath⩽20N\ensuremath{\leqslant}20N\ensuremath⩽20, which are particularly relevant for experiments on finite magnetic rings. Furthermore, we find that for s\ensuremath⩾3∕2s\ensuremath{\geqslant}3∕2s\ensuremath⩾3∕2, the dependence of EN(S,s){E}_{N}(S,s)EN​(S,s) on sss can be described by a scaling relation, and this relation is shown to hold well for ring sizes up to N=80N=80N=80 for all intrinsic spins in the range 3∕2\ensuremath⩽s\ensuremath⩽7∕23∕2\ensuremath{\leqslant}s\ensuremath{\leqslant}7∕23∕2\ensuremath⩽s\ensuremath⩽7∕2. Considering ring sizes in the interval 8\ensuremath⩽N\ensuremath⩽508\ensuremath{\leqslant}N\ensuremath{\leqslant}508\ensuremath⩽N\ensuremath⩽50, we find that the energy gap between the ground state and the first excited state approaches zero proportional to 1∕N\ensuremathα1∕{N}^{\ensuremath{\alpha}}1∕N\ensuremathα, where \ensuremathα\ensuremath≈0.76\ensuremath{\alpha}\ensuremath{\approx}0.76\ensuremathα\ensuremath≈0.76 for s=3∕2s=3∕2s=3∕2 and \ensuremathα\ensuremath≈0.84\ensuremath{\alpha}\ensuremath{\approx}0.84\ensuremathα\ensuremath≈0.84 for s=5∕2s=5∕2s=5∕2. Finally, we demonstrate the usefulness of our present results for EN(S,s){E}_{N}(S,s)EN​(S,s) by examining the Fe12{\mathrm{Fe}}_{12}Fe12​ ring-type magnetic molecule, leading to a more accurate estimate of the exchange constant for this system than has been obtained heretofore.
The spin pair-correlation function of finite bipartite antiferromagnetic Heisenberg quantum spin rings was studied numerically by means of exact diagonalization techniques. For large spins, the spectrum of the spin pair-correlation function is of a remarkably simple structure: It consists of few characteristic peaks at low, and a broad featureless signal at high temperatures. This arises as the energy spectrum exhibits a set of parallel rotational bands at low energies emerging into a quasi-continuum of states, with transitions from the lowest rotational band to the quasi-continuum being highly suppressed. The energies of the rotational bands can be accurately described by a generalized dispersion relation that depends on the total spin quantum number and, as usual, on the shift quantum number. These regularities are better fulfilled the larger the spin length but the smaller the ring size. It will be shown that all these features are associated with the underlying sublattice structure of the spin rings, and it is argued that they are valid for a more general class of finite Heisenberg systems than rings.
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 Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/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 study the low-energy dynamics of S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg clusters constructed by diluting a square lattice at vacancy concentration ppp at and below the percolation threshold ${p}^{\ensuremath{\ast}}\ensuremath{\approx}0.407$. The finite-size scaling behavior of the average excitation gap, $⟨\ensuremath{\Delta}⟩\ensuremath{\sim}{L}^{\ensuremath{-}z}$, where LLL is the cluster length, is obtained using quantum Monte Carlo results for an upper bound ${\ensuremath{\Delta}}^{\ensuremath{\ast}}$ to $\ensuremath{\Delta}$, derived from sum rules. At the percolation threshold, we obtain a dynamic exponent $z=3.6\ifmmode\pm\else\textpm\fi{}0.1\ensuremath{\approx}2{D}_{f}$ for clusters with singlet (S=0)(S=0)(S=0) ground state. Here Df=91/48{D}_{f}=91/48Df​=91/48 is the fractal dimensionality of the percolating cluster. We argue that this large dynamic exponent---roughly twice that expected for quantum-rotor excitations---is a consequence of weakly interacting localized effective magnetic moments, which form due to local sublattice imbalance. This picture is supported by an extremal-value analysis of local spectral gaps, which delivers an exponent relation (between zzz and two exponents characterizing the local-gap distribution) reproduced by our simulation data. However, the average $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ over all clusters, which have mostly ground-state spin S>0S>0S>0, scales with a smaller exponent than for the S=0S=0S=0 clusters alone; $z\ensuremath{\approx}1.5{D}_{f}$. Lanczos exact diagonalization for small clusters show that typically, $S\ensuremath{\rightarrow}S\ensuremath{-}1$ in the lowest-energy excitations while the dominant spectral weight originates from $S\ensuremath{\rightarrow}S+1$ excitations. Thus, the scaling of $⟨{\ensuremath{\Delta}}^{\ensuremath{\ast}}⟩$ for clusters with ground state S>0S>0S>0 does not reflect the lowest-energy excitations but the higher $S\ensuremath{\rightarrow}S+1$ excitations. This result can be understood within a valence bond picture. To further explore the scenario of localized moments, we introduce a classical dimer-monomer aggregation model to study the distribution of nearest-neighbor sites forming dimers (which are the objects used in mapping to the quantum-rotor model) and unpaired spins (monomers). The monomers are localized and, thus, effective magnetic moments should form in the spin system. We also study the lowest triplet excitation of S=0S=0S=0 clusters using quantum Monte Carlo calculations in the valence bond basis. The triplet is concentrated at some of the classical monomer regions, confirming the mechanism of moment formation. The number of spins (and moment regions) affected by the excitation scales as a nontrivial power of the cluster size. For a dimer-diluted bilayer Heisenberg model with weak interlayer coupling (where the system remains N\'eel ordered), there is no sublattice imbalance. In this case we find $z\ensuremath{\approx}{D}_{f}$, consistent with quantum-rotor excitations. For a single layer at $p<{p}^{\ensuremath{\ast}}$ we find $z\ensuremath{\approx}2=D$, which indicates that the weakly interacting localized moment mechanism is valid only exactly at the percolation point. There is a crossover behavior close to the percolation point.
This study examines the increasing complexity in the magnetic properties of small n=3,4,5,and6n=3,\phantom{\rule{0.16em}{0ex}}4,\phantom{\rule{0.16em}{0ex}}5,\phantom{\rule{0.16em}{0ex}}\mathrm{and}\phantom{\rule{0.16em}{0ex}}6n=3,4,5,and6 spin-12\frac{1}{2}21​ quantum rings. Using an exact diagonalization of the isotropic Heisenberg Hamiltonian with nearest and next-nearest neighbor interactions, the energy eigenstates, magnetic specific heat capacity, magnetic susceptibility, and inelastic neutron scattering structure factors are determined for variable next-nearest neighbor interactions. Here, it is shown that the presence of spin exchange symmetry breaking, multiple ground states, and nonzero total spin ground states can greatly complicate the energy eigenstates and excitations for these systems. Overall, the energy eigenstates and structure factor intensities are presented in closed form, while the thermodynamic properties detail the effect of a crossing interaction in the rings. The goal of this work is to provide insight into the evolution of the magnetic properties and spin excitations within these systems.
The highest and lowest energies as a function of the total spin are computed for the class of "unbound" states in the Bethe formalism for the linear chain of spin-\textonehalf{} atoms with a Heisenberg exchange interaction between nearest neighbors. The lowest energies are used to compute the magnetization curve for the infinite antiferromagnetic chain in the limit of zero temperature. At zero temperature and in zero field, the magnetic susceptibility has the value 0.050661 $\frac{{g}^{2}{\ensuremath{\mu}}^{2}}{J}$, where $\ensuremath{\mu}$ is the Bohr magneton, ggg the electron ggg factor, and the interaction between neighboring spins is of the form $2J{\mathbf{S}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{2}$.
The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1S=1S=1, whose size nnn is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For nnn large, the coefficients of the expression converge towards C1=0.125{C}_{1}=0.125C1​=0.125, W1=0.451J{W}_{1}=0.451JW1​=0.451J, C2=0.564{C}_{2}=0.564C2​=0.564, W2=1.793J{W}_{2}=1.793JW2​=1.793J (JJJ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at T=0T=0T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}Y2​BaNiO5​ and of Ni(C2H8N2)2NO2ClO4\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}Ni(C2​H8​N2​)2​NO2​ClO4​ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.