Uniaxial and fourfold basal anisotropy in GdRh₂Si₂

The growth of the single crystals of GdRh₂Si₂ and their characterisation are explained in reference [12]. We used a platelet shaped sample with an area of about 1 × 1.3 mm² and a thickness of about 0.5 mm.

Continuous-wave ESR was performed both at a frequency of ν_rf = 34 GHz (Q-band) and at 9.4 GHz (X-band). A Bruker Elexsys II spectrometer was used. In case of the Q-band experiments the resonator was a cylindrical Bruker ER5106QTW cavity and for X-band measurements the resonator used was a rectangular Bruker ER4122SHQ. In both cases a helium gas-flow cryostat allowed a temperature stability of less than 1 K. To measure the angular dependence in the naturally grown (aa)-plane, the samples were mounted with their growth plane perpendicular to the rotation axis of a goniometer. In order to perform rotations in the (ac)-plane, an a-axis was aligned parallel with the rotation axis, the sample being glued on a vertical facet of a glass holder. The external field H₀ was swept up to 1.8 T. The spectra shown in this work are the first derivative of the absorbed microwave power, owing to the use of lock-in technique with field modulation.

Looking for the eigenoscillations of two antiferromagnetically coupled magnetisation sublattices, these two magnetisations are written as ${\boldsymbol{M}}^{\left(i\right)}={\boldsymbol{M}}_{0}^{\left(i\right)}+{\boldsymbol{m}}^{\left(i\right)}$ with i = 1, 2, meaning a decomposition into their static parts (index 0) and dynamic parts (small letters), where $\left\vert {\boldsymbol{m}}^{\left(i\right)}\right\vert \ll \left\vert {\boldsymbol{M}}_{0}^{\left(i\right)}\right\vert$. Their equations of motion read

$$\begin{equation}\mathrm{i}\omega {\boldsymbol{m}}^{\left(1\right)}+\gamma {\boldsymbol{m}}^{\left(1\right)}{\times}{\boldsymbol{H}}_{\text{eff0}}^{\left(1\right)}+\gamma {\boldsymbol{M}}_{0}^{\left(1\right)}{\times}{\boldsymbol{h}}_{\text{eff}}^{\left(1\right)}=0,\end{equation} \tag{ 1 }$$

$$\begin{equation}\mathrm{i}\omega {\boldsymbol{m}}^{\left(2\right)}+\gamma {\boldsymbol{m}}^{\left(2\right)}{\times}{\boldsymbol{H}}_{\text{eff0}}^{\left(2\right)}+\gamma {\boldsymbol{M}}_{0}^{\left(2\right)}{\times}{\boldsymbol{h}}_{\text{eff}}^{\left(2\right)}=0.\end{equation} \tag{ 2 }$$

Here, ${\boldsymbol{H}}_{\text{eff}\enspace 0}^{\left(i\right)}$ and ${\boldsymbol{h}}_{\text{eff}}^{\left(i\right)}$ are the static and dynamic effective fields acting on the respective sublattice i with $\left\vert {\boldsymbol{h}}_{\text{eff}}^{\left(i\right)}\right\vert \ll \left\vert {\boldsymbol{H}}_{\text{eff}\enspace 0}^{\left(i\right)}\right\vert$, and γ is the gyromagnetic ratio. Equations (1) and (2) are derived from the respective Landau–Lifshitz equations by linearising in the small quantities m⁽ⁱ⁾ and ${\boldsymbol{h}}_{\text{eff}}^{\left(i\right)}$ and by setting the damping to zero [13]. Effective fields are expressed via the free energy density F:

$$\begin{equation}\begin{gathered}{c}{H}_{\text{eff}\enspace \alpha }^{\left(i\right)}=-\frac{\partial F}{\partial {M}_{\alpha }^{\left(i\right)}},\enspace {\boldsymbol{H}}_{\text{eff0}}^{\left(i\right)}={\left.{\boldsymbol{H}}_{\text{eff}}^{\left(i\right)}\right\vert }_{\boldsymbol{m}=0},\enspace {h}_{\text{eff}\enspace \alpha }^{\left(i\right)}=\sum _{\beta }{\left.\frac{\partial {H}_{\text{eff}\enspace \alpha }^{\left(i\right)}}{\partial {m}_{\beta }^{\left(i\right)}}\right\vert }_{\boldsymbol{m}=0}{m}_{\beta }^{\left(i\right)},\end{gathered}\end{equation} \tag{ 3 }$$

where α and β denote Cartesian components x, y, z. The internal energy in the system to be investigated by ESR consists of exchange energy F_e, Zeeman energy F_Z, uniaxial anisotropy energy F_u and basal anisotropy energy F_b:

$$\begin{align}\hfill U& ={F}_{\text{e}}+{F}_{\text{Z}}+{F}_{\text{u}}+{F}_{\text{b}}\hfill \\ \hfill & ={\Lambda}{\boldsymbol{M}}_{1}\cdot {\boldsymbol{M}}_{2}-{\boldsymbol{H}}_{0}\cdot \left({\boldsymbol{M}}_{1}+{\boldsymbol{M}}_{2}\right)\hfill \\ \hfill & \quad +\enspace {K}_{\text{u}}\left({\mathrm{sin}}^{2}\enspace {\theta }_{1}+{\mathrm{sin}}^{2}\enspace {\theta }_{2}\right)+{K}_{\text{b}}\left[{\mathrm{sin}}^{2}\left(2{\varphi }_{1}\right)+{\mathrm{sin}}^{2}\left(2{\varphi }_{2}\right)\right].\hfill \end{align} \tag{ 4 }$$

Here, Λ is the dimensionless antiferromagnetic exchange constant, H₀ the external magnetic field, K_u the first order uniaxial anisotropy constant and K_b the first order basal anisotropy constant. The definitions of the polar angles θ_i correspond, in figure 2(a), to the angles between the M⁽ⁱ⁾ and the axis of uniaxial anisotropy (z-axis), and the azimuthal angles φ_i are, as is illustrated in that same sketch, defined within the (aa)-plane. Note that the above definition of basal anisotropy energy matches the one in reference [1].

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With ${M}_{0}=\left\vert {\boldsymbol{M}}_{0}^{\left(1\right)}\right\vert =\left\vert {\boldsymbol{M}}_{0}^{\left(2\right)}\right\vert$ and z₀ as the unit vector along z, the effective fields are, again after linearising in the m⁽ⁱ⁾,

$$\begin{equation}\begin{aligned}\hfill {\boldsymbol{H}}_{\text{eff0}}^{\left(1\right)}& =-{\Lambda}{\boldsymbol{M}}_{0}^{\left(2\right)}+{\boldsymbol{H}}_{0}+\frac{2{K}_{\mathrm{u}}}{{M}_{0}^{2}}{\boldsymbol{z}}_{0}\left({\boldsymbol{M}}_{0}^{\left(1\right)}\cdot {\boldsymbol{z}}_{0}\right)+{\boldsymbol{H}}_{\text{b}\enspace 0}^{\left(1\right)},\hfill \\ \hfill {\boldsymbol{h}}_{\text{eff}}^{\left(1\right)}& =-{\Lambda}{\boldsymbol{m}}^{\left(2\right)}+\frac{2{K}_{\mathrm{u}}}{{M}_{0}^{2}}{\boldsymbol{z}}_{0}\left({\boldsymbol{m}}^{\left(1\right)}\cdot {\boldsymbol{z}}_{0}\right)+{\boldsymbol{h}}_{\text{b}}^{\left(1\right)},\hfill \\ \hfill {\boldsymbol{H}}_{\text{eff0}}^{\left(2\right)}& =-{\Lambda}{\boldsymbol{M}}_{0}^{\left(1\right)}+{\boldsymbol{H}}_{0}+\frac{2{K}_{\mathrm{u}}}{{M}_{0}^{2}}{\boldsymbol{z}}_{0}\left({\boldsymbol{M}}_{0}^{\left(2\right)}\cdot {\boldsymbol{z}}_{0}\right)+{\boldsymbol{H}}_{\text{b}\enspace 0}^{\left(2\right)},\hfill \\ \hfill {\boldsymbol{h}}_{\text{eff}}^{\left(2\right)}& =-{\Lambda}{\boldsymbol{m}}^{\left(1\right)}+\frac{2{K}_{\mathrm{u}}}{{M}_{0}^{2}}{\boldsymbol{z}}_{0}\left({\boldsymbol{m}}^{\left(2\right)}\cdot {\boldsymbol{z}}_{0}\right)+{\boldsymbol{h}}_{\text{b}}^{\left(2\right)}.\hfill \end{aligned}\end{equation} \tag{ 5 }$$

The effective fields of basal anisotropy, ${\boldsymbol{H}}_{\text{b}\enspace 0}^{\left(i\right)}$ and ${\boldsymbol{h}}_{\text{b}}^{\left(i\right)}$, will be explained below for the two cases of field rotation separately. Equations (1) and (2) define in this way a coupled linear equation system, with six independent variables, namely the three Cartesian components of the two dynamic magnetisations m⁽ⁱ⁾. Defining a 6 × 6-matrix A, this is summarised as

$$\begin{equation}A\left(\begin{matrix}\hfill {\boldsymbol{m}}_{1}\hfill \\ \hfill {\boldsymbol{m}}_{2}\hfill \end{matrix}\right)=0.\end{equation} \tag{ 6 }$$

In order to get nontrivial solutions, we set det A = 0, providing either resonance frequencies ν_res for a given field or resonance fields H_res for a given microwave frequency. Of particular interest are two special cases, for which (6) is solved analytically [13] for K_u < 0 as is the case in GdRh₂Si₂: firstly, when K_b = 0 and H₀ is in the basal plane, one of the two resonance frequencies is

$$\begin{equation}{\omega }_{\text{res}}=\gamma {H}_{0}\sqrt{1-{K}_{\text{u}}/\left({\Lambda}{M}_{0}^{2}\right)},\end{equation} \tag{ 7 }$$

where H₀ = |H₀|. This value is very close to the paramagnetic value ω_res = γH₀ as long as the exchange field ΛM₀ is large as compared to the anisotropy field H_u = 2K_u/M₀. Secondly, for K_b = 0 and with H₀ along the c-axis, one of the resonances occurs at ω_res = 0, corresponding to a Goldstone mode, which is provided from the spontaneous symmetry breaking by the antiferromagnetic order parameter.

2.3.1. Out-of-plane rotations

When rotating the field within the (ac)-plane, and as long as we deal with the static field and magnetisation parts only, we may benefit from the mirror symmetry with respect to the (ac)-plane (see figure 2(a)) and write θ = θ₁ = θ₂ as well as φ = φ₁ = φ₂. For further calculations, it is convenient to introduce a primed coordinate system x', y', z' with x' = x, but tilted by the angle 90° − θ' towards H₀, having y' within the purple plane of figure 2(a) and z' perpendicular to it. Now we introduce the angles ${\varphi }^{\prime }=\sphericalangle \left({\boldsymbol{M}}^{\left(1\right)},{x}^{\prime }\right)=\sphericalangle \left({\boldsymbol{M}}^{\left(2\right)},-{x}^{\prime }\right)$, θ' = ∢(y', z) and $\delta =\sphericalangle \left({\boldsymbol{M}}^{\left(1\right)},{\boldsymbol{H}}_{0}\right)=\sphericalangle \left({\boldsymbol{M}}^{\left(2\right)},{\boldsymbol{H}}_{0}\right)$. The relationships between convenient angles θ', φ' and the angles θ, φ and δ used in (4) read

$$\begin{equation}\begin{gathered}{c}\mathrm{cos}\enspace \theta =\mathrm{cos}\enspace {\theta }^{\prime }\enspace \mathrm{sin}\enspace {\varphi }^{\prime },\\ \mathrm{sin}\enspace \varphi =\mathrm{tan}\enspace {\theta }^{\prime }\enspace \mathrm{cot}\enspace \theta ,\\ \mathrm{cos}\enspace {\varphi }^{\prime }=\mathrm{sin}\enspace \theta \enspace \mathrm{cos}\enspace \varphi ,\\ \mathrm{cos}\enspace \delta =\mathrm{sin}\enspace {\varphi }^{\prime }\enspace \mathrm{cos}\enspace \left({\theta }^{\prime }-{\theta }_{\mathrm{H}}\right).\end{gathered}\end{equation} \tag{ 8 }$$

In this way, the free energy is expressed in terms of θ' and φ' as

$$\begin{align}\hfill F& =-{\Lambda}{M}_{0}^{2}\enspace \mathrm{cos}\enspace 2{\varphi }^{\prime }-2{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace {\varphi }^{\prime }\enspace \mathrm{cos}\left({\theta }^{\prime }-{\theta }_{\mathrm{H}}\right)\hfill \\ \hfill & \quad -2{K}_{\text{u}}\enspace {\mathrm{cos}}^{2}\enspace {\theta }^{\prime }\enspace {\mathrm{sin}}^{2}\enspace {\varphi }^{\prime }+2{K}_{\text{b}}\frac{{\mathrm{sin}}^{2}\enspace 2{\varphi }^{\prime }\enspace {\mathrm{sin}}^{2}\enspace {\theta }^{\prime }}{{\left(1-{\mathrm{cos}}^{2}\enspace {\theta }^{\prime }\enspace {\mathrm{sin}}^{2}\enspace {\varphi }^{\prime }\right)}^{2}}.\hfill \end{align} \tag{ 9 }$$

The two equilibrium angles θ₀' and φ₀' are found via the minimum of F.

When setting up the matrix A as defined by (6), however, the two sublattices with the respective effective fields acting on them need to be treated separately and the contracted form (9) for F is not suitable. Moreover, F shall be represented in a Cartesian form. In the primed coordinate system, ${\boldsymbol{H}}_{0}=\left(0,\mathrm{cos}\left({\theta }_{0}^{\prime }-{\theta }_{\mathrm{H}}\right),\mathrm{sin}\left({\theta }_{0}^{\prime }-{\theta }_{\mathrm{H}}\right)\right)$, ${\boldsymbol{M}}_{0}^{\left(1\right)}=\left(\mathrm{cos}\enspace {\varphi }_{0}^{\prime },\mathrm{sin}\enspace {\varphi }_{0}^{\prime },0\right)$, ${\boldsymbol{M}}_{0}^{\left(2\right)}=\left(-\mathrm{cos}\enspace {\varphi }_{0}^{\prime },\mathrm{sin}\enspace {\varphi }_{0}^{\prime },0\right)$, z₀ = (0, cos θ₀', sin θ₀') and the calculation of the matrix A using the equations of motion (1), (2) and the effective fields (5) is straightforward, except for the basal anisotropy fields.

With ${\boldsymbol{M}}^{\left(i\right)}=\left({M}_{x}^{\left(i\right)},{M}_{y}^{\left(i\right)},{M}_{z}^{\left(i\right)}\right)$ in the unprimed system and ${\boldsymbol{M}}^{\left(i\right)}=\left({M}_{{x}^{\prime }}^{\left(i\right)},{M}_{{y}^{\prime }}^{\left(i\right)},{M}_{{z}^{\prime }}^{\left(i\right)}\right)$ in the primed system we have ${M}_{x}^{\left(i\right)}={M}_{{x}^{\prime }}^{\left(i\right)}$, ${M}_{y}^{\left(i\right)}={M}_{{y}^{\prime }}^{\left(i\right)}\enspace \mathrm{sin}\enspace {\theta }_{0}^{\prime }-{M}_{{z}^{\prime }}^{\left(i\right)}\enspace \mathrm{cos}\enspace {\theta }_{0}^{\prime }$. From figure 2(a) follows

$$\begin{equation}\mathrm{sin}\enspace {\varphi }_{i}=\frac{{M}_{y}^{\left(i\right)}}{{\left[{\left({M}_{x}^{\left(i\right)}\right)}^{2}+{\left({M}_{y}^{\left(i\right)}\right)}^{2}\right]}^{1/2}}.\end{equation} \tag{ 10 }$$

Note that now φ_i stands for the azimuthal angle due to both static and dynamic magnetisation parts. The two parts i of the basal anisotropy energy U_b in (4) are then expressed in a Cartesian way as

$$\begin{equation}{U}_{\text{b}i}={K}_{\text{b}}\enspace {\mathrm{sin}}^{2}\enspace 2{\varphi }_{i}=4{K}_{\text{b}}\frac{{\left({M}_{{x}^{\prime }}^{\left(i\right)}\right)}^{2}{\left({M}_{{y}^{\prime }}^{\left(i\right)}\enspace \mathrm{sin}\enspace {\theta }_{0}^{\prime }-{M}_{{z}^{\prime }}^{\left(i\right)}\enspace \mathrm{cos}\enspace {\theta }_{0}^{\prime }\right)}^{2}}{{\left[{\left({M}_{{x}^{\prime }}^{\left(i\right)}\right)}^{2}+{\left({M}_{{y}^{\prime }}^{\left(i\right)}\enspace \mathrm{sin}\enspace {\theta }_{0}^{\prime }-{M}_{{z}^{\prime }}^{\left(i\right)}\enspace \mathrm{cos}\enspace {\theta }_{0}^{\prime }\right)}^{2}\right]}^{2}},\end{equation} \tag{ 11 }$$

where static and dynamic magnetisation parts are still unseparated. Effective field components ${H}_{\text{b}\enspace {\alpha }^{\prime }}^{\left(i\right)}$ in direction α' are obtained as ${H}_{\text{b}\enspace {\alpha }^{\prime }}^{\left(i\right)}=-\partial F/\partial {M}_{{\alpha }^{\prime }}^{\left(i\right)}$. Magnetisation components are subsequently separated into static and dynamic parts. Following (3), the static parts of the field components ${H}_{\text{b}\enspace 0\enspace {\alpha }^{\prime }}^{\left(i\right)}$ are given by setting m = 0 and the dynamic field components ${h}_{\text{b}\enspace {\alpha }^{\prime }}^{\left(i\right)}$ are obtained as linearisations in the ${m}_{{\alpha }^{\prime }}^{\left(i\right)}$. The assumption ${M}_{{y}^{\prime }\enspace 0}^{\left(i\right)}\ll {M}_{{x}^{\prime }\enspace 0}^{\left(i\right)}$, reflecting that the exchange field is the dominating interaction gives important simplifications. Concerning the static parts of the two sublattices, the only difference between them is the reversed static x-component: ${M}_{0\enspace {x}^{\prime }}^{\left(2\right)}=-{M}_{0\enspace {x}^{\prime }}^{\left(1\right)}$.

The resulting terms needed for the calculation of θ₀', φ₀' and A are documented in appendix A.

2.3.2. In-plane rotations

Following figure 2(b), the primed coordinate system is now defined by z' = z, but with y' ∥ H₀, being thus rotated by the angle φ_H around z with respect to the unprimed framework. The static part of the total magnetic energy may be written

$$\begin{align}\hfill F& =-{\Lambda}{M}_{0}^{2}\enspace \mathrm{cos}\left({\varphi }_{1}^{\prime }+{\varphi }_{2}^{\prime }\right)-{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace {\varphi }_{1}^{\prime }-{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace {\varphi }_{2}^{\prime }\hfill \\ \hfill & \quad +{K}_{\text{u}}\left({\mathrm{sin}}^{2}\enspace 9{0}^{{\circ}}+{\mathrm{sin}}^{2}\enspace 9{0}^{{\circ}}\right)\hfill \\ \hfill & \quad +{K}_{\text{b}}\left\{{\mathrm{sin}}^{2}\left[2\left({\varphi }_{1}^{\prime }-{\varphi }_{\mathrm{H}}\right)\right]+{\mathrm{sin}}^{2}\left[2\left({\varphi }_{2}^{\prime }+{\varphi }_{\mathrm{H}}\right)\right]\right\}.\hfill \end{align} \tag{ 12 }$$

Now the equilibrium angles ${\left({\varphi }_{1}^{\prime }\right)}_{0}$ and ${\left({\varphi }_{2}^{\prime }\right)}_{0}$ are found as the ones minimising F.

In the primed coordinate system H₀ = (0, H₀, 0), ${\boldsymbol{M}}_{0}^{\left(1\right)}=\left({M}_{0}\enspace \mathrm{cos}{\left({\varphi }_{1}^{\prime }\right)}_{0},{M}_{0}\enspace \mathrm{sin}{\left({\varphi }_{1}^{\prime }\right)}_{0},0\right)$ and ${\boldsymbol{M}}_{0}^{\left(2\right)}=\left(-{M}_{0}\enspace \mathrm{cos}{\left({\varphi }_{2}^{\prime }\right)}_{0},{M}_{0}\enspace \mathrm{sin}{\left({\varphi }_{2}^{\prime }\right)}_{0},0\right)$. As before, exchange, Zeeman and uniaxial terms in (1), (2) and (5) can be easily calculated. The basal anisotropy energy shall again be expressed in terms of Cartesian magnetisation components, so that basal terms can be introduced into matrix A, too. The angles φ₁' and φ₂' are regarded as due to both static and dynamic magnetisation. The K_b-term in (12) is expanded into sines and cosines of only φ₁' or φ₂', and using the just mentioned representations of the ${\boldsymbol{M}}_{0}^{\left(i\right)}$, the basal energy parts become

$$\begin{align}\hfill {F}_{\text{b}\enspace i}& ={K}_{\text{b}}/{M}_{0}^{4}\left\{2{M}_{{x}^{\prime }}^{\left(i\right)}{M}_{{y}^{\prime }}^{\left(i\right)}\enspace \mathrm{cos}\enspace 2{\varphi }_{\mathrm{H}}\right.\hfill \\ \hfill & \quad -{\left.\left[{\left({M}_{{x}^{\prime }}^{\left(i\right)}\right)}^{2}-{\left({M}_{{y}^{\prime }}^{\left(i\right)}\right)}^{2}\right]\mathrm{sin}\enspace 2{\varphi }_{\mathrm{H}}\right\}}^{2}.\hfill \end{align} \tag{ 13 }$$

Here again, H_bα' = −∂U/∂M_α'. The procedure is then completely analogous to the out-of-plane treatment: separation into static and dynamic magnetisations, determination of effective fields as described by (3) using approximations ${M}_{{y}^{\prime }\enspace 0}^{\left(i\right)}\ll {M}_{{x}^{\prime }\enspace 0}^{\left(i\right)}$ and construction of A's matrix entries following (1), (2), (5) and (6). Note that, unlike the out-of-plane case, in general ${M}_{{x}^{\prime }}^{\left(2\right)}\ne -{M}_{{x}^{\prime }}^{\left(1\right)}$ and ${M}_{{y}^{\prime }}^{\left(2\right)}\ne {M}_{{y}^{\prime }}^{\left(1\right)}$.

Appendix B lists the detailed terms used in the calculation of ${\left({\varphi }_{1}^{\prime }\right)}_{0}$, ${\left({\varphi }_{2}^{\prime }\right)}_{0}$ and A.

2.3.3. Numerical procedure

A few representative ESR spectra from the antiferromagnetically ordered state in GdRh₂Si₂ are plotted in figure 3, showing the field derivative of the absorbed microwave power, dP_abs/dH₀, versus the external field H₀. The fits use an asymmetric Lorentzian derivative, where the asymmetry is due to the conductivity of the sample [2]. The asymmetric lineshape indeed physically makes sense, because the penetration depth amounts to about 3 μm in the paramagnetic regime [14], much smaller than the thickness of the sample of about 0.5 mm. A dropping resistivity below T_N towards about 3 μΩcm at T = 5 K [12] results in even smaller penetration depths in the ordered state. It is seen that these fits provide an excellent description of the measured spectra, confirming the idea of well-localised and strong Gd³⁺ magnetic moments. Obtained parameters are the resonance field H_res, the linewidth ΔH, the asymmetry ratio D/A and the line intensity I, which corresponds to the area under the non-derived spectrum. While the focus shall here be on the resonance fields, it is noteworthy that the linewidth, in the range of 400 Oe, typically increases with increasing resonance field, as is also observed in the ordered states of many other localised-moment magnets [15–17]. D/A mostly assumes values between 0.5 and 1.0, in accordance with the expectation for a conductive system. Concerning H_res, a clear anisotropy is distinguished when rotating the external field out of plane (left frames in figure 3), but also within the (aa)-plane (right frames). The figure demonstrates that decreasing temperature enhances the magnetocrystalline anisotropy, which is because the sublattice magnetisations approach their saturated values.

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The resonance fields for out-of-plane rotations at different temperatures are shown in figure 4. At the temperature of 80 K and at θ_H = 90°, meaning that the field is along the a-axis, the resonance field amounts to 12.2 kOe, close to the value of 12.1 kOe given for a paramagnetic moment with g = 2 at the frequency ν_rf = 34 GHz. Following (7), which holds for this field direction as long as basal anisotropy is negligible and when uniaxial anisotropy is much smaller than the exchange field, the paramagnetic value is indeed expected here. Resonance fields diverge when approaching H₀ ∥ c and the observation of the resonance gets limited by the largest possible external field. As the resonance fields increase with decreasing temperature, only an angular range of ±30° around the a-axis is covered at T = 20 K. Diverging resonance fields for H₀ ∥ c indicate that for a fixed external field the resonance frequency gets suppressed down to zero, thus reaching the Goldstone mode which cannot be observed experimentally. We can conclude that the quasi-paramagnetic mode at θ_H = 90° in easy-plane antiferromagnets with resonance condition (7) transforms continuously into the Goldstone mode also mentioned in section 2.3 when tilting towards the hard axis. Resonance fields at T = 5 K were analyzed, too, but are not displayed here because of a vanishing ESR signal over a wide angular range.

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Fit curves in figure 4 demonstrate that the theoretical model of coupled oscillating magnetisation sublattices satisfactorily describes the measured data. Free fit parameters are only K_u and K_b and the g-factor is fixed to g = 2. Other parameters, which are supplied by external measurements, are the microwave frequency ν_rf, the sublattice magnetisations M₀ and the exchange constant Λ. We calculate the temperature dependence of M₀ as ${M}_{0}\left(T\right)={M}_{\text{s}}\sqrt{1-T/{T}_{\text{N}}}$ with the saturation value M_s = 400 G which one gets for the Gd³⁺ moment μ = 7 μ_B using lattice constants given in reference [12]. Λ can be calculated within a mean-field theory from the Néel-temperature T_N, the Curie–Weiss-temperature Θ_CW and the Curie constant C as Λ = (T_N + Θ_CW)/C = 710 [18]. The fitted anisotropy constants are discussed in section 3.2.

In-plane resonance fields are shown in figure 5. Here, a fourfold anisotropy is clearly visible. From the point of view of the uniaxial anisotropy, the (aa)-plane is an easy plane, but the additional basal anisotropy changes the a-axes (φ_H = 0°, 90°) into hard axes within this plane, whereas the [110]-axes are the easy axes. Thus, according to the definition in (4), K_b is negative in GdRh₂Si₂. As in the out-of-plane case, the anisotropy simulations fit the experimental data very well over a broad temperature range between 20 and 80 K. This is the case for T = 5 K, too, but the data are not shown here due to spectra of bad quality over a wider angular range. As before, the free fitting parameters are K_u and K_b only.

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ESR data were additionally recorded at X-band microwave frequency. The resonance fields generally reflect the same behaviour as described above for the Q-band. The spectra can be well evaluated for temperatures T > 60 K. At lower temperatures, the signal weakens and gets disturbed by nonresonant low-field features in the spectra [2] due to a spin–flip transition aligning the magnetic sublattices from their ground state configuration to nearly perpendicular to the external field [1]. For 60 K and 80 K, we applied the parameters from the Q-band fits to the frequency of 9.4 GHz. Whereas the agreement to experimental data at 80 K is still acceptable, the simulation does not match the X-band data at 60 K. While the experimental data has, for rotations in the (aa)-plane, its minima at 1800 Oe, the simulation drops down to less than 600 Oe, where it becomes discontinuous. Section 3.3 takes up this issue in more detail.

Figure 6 summarises the anisotropy constants determined from the fits in section 3.1. Both kinds of field rotations provide an uniaxial constant K_u in the same order of magnitude, see figure 6(a). Rotations in the (aa)-plane seem more trustworthy, because they reproduce, except for the lowest temperature, a monotonous increase with decreasing temperature. Surprisingly, out-of-plane rotations, i.e. in the (ac)-plane, only suggest a rather constant K_u over the given temperature range and are, thus, considered less suitable for providing the uniaxial anisotropy constant. Indeed, the divergence of H_res towards the c-axis (see figure 4) is an intrinsic feature of easy-plane antiferromagnets and in first respect is not dependent on the magnitude of the negative K_u. Error bars in figure 6 are roughly estimated by variation of the respective anisotropy constant until the square deviations χ² of the fit get doubled. Thus, they indicate the borders where deviations in K would result in a similar error in the fit as compared to the remaining χ² at its minimum. Obviously, for K_u there is no satisfactory overlap between (aa)- and (ac)-measurements at 60 and 80 K. On the low-temperature side, the result for 5 K must even more be considered with caution. Experimentally it was difficult to obtain an ESR signal of reasonable strength at this temperature and it even was weakened further during both kinds of rotations, so that we could only access limited angular ranges. The reason therefore could not be clarified. Thus, we consider at least the value obtained for K_u at the lowest temperature unlikely and the seeming maximum at 20 K shall not be taken too serious. The neglection of the point at 5 K would allow an extrapolation of the more reliable (aa)-data to an order-of-magnitude estimation of the lowest-temperature value K_u ≈ −10 × 10⁶ erg cm⁻³, corresponding to an anisotropy field H_u = |2K_u/M| = 50 kOe. As shown by figure 6(a), a precise determination of K_u in this system is generally hard to be realized by antiferromagnetic resonance. The reason why especially the (ac)-fits do not allow a satisfactory determination of K_u is that, contrary to the expectation, large variations of this constant only lead to a minor vertical shift of calculated resonance fields in figure 4. The main contribution to the shifts stems from the basal anisotropy constant K_b. The divergence of H_res towards H ∥ c is characteristic for the considered type of antiferromagnet, irrespective of the detailed size of K_u, as long as K_u < 0. The fact of minor vertical shifts of calculated H_res upon large modification of K_u has been proven in the in-plane plots H_res(φ_H) (figure 5), too. This is the reason, why also the (aa)-points in figure 6(a) partially have large errors.

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Concerning the physical origin of anisotropy in GdRh₂Si₂, several mechanisms may be mentioned. Because of the spin-only magnetic moments of Gd³⁺ (ground state ⁸S_7/2), admixtures of excited atomic states are needed to account for a possible single-ion anisotropy, which however would only explain an anisotropy field less than 0.1 kOe and a similar estimate is obtained for the anisotropy due to dipolar interaction [19]. Instead exchange anisotropy as a microscopic origin of anisotropy has been discussed in the literature [19, 20].

Contrary to K_u, it must be noted that the basal anisotropy constant K_b (figure 6(b)) is generally more consistent between both kinds of measurements, out-of-plane and in-plane rotations. Error bars are small in the latter case. At 80 K the drop of K_b as determined by the (aa)-plane experiments seems more realistic and makes these measurements again preferable over the (ac)-plane experiments. And indeed it is considered natural that in-plane-rotations provide the basal anisotropy constant much more accurately. Nevertheless it is surprising that also the out-of-plane measurements give rather accurate results for the strength of the basal anisotropy. At low temperatures, K_b amounts to approximately −7000 erg cm⁻³. The smallness of the basal anisotropy as compared to the dominant uniaxial term is worth to note and the reason why it is usually neglected [13].

Pure simulations of resonance frequencies ν_res as a function of external field H₀ are plotted in figure 7. M₀ = 400 G and Λ = 710 are chosen as the low-temperature parameters for GdRh₂Si₂, K_u = −1 × 10⁶ erg cm⁻³ is taken in an order of magnitude realistic for GdRh₂Si₂ as well as g = 2. In figure 7(a) the field angles θ_H = 70° and φ_H = 0° are selected and K_b, the anisotropy parameter which is mainly discussed in this work, is varied in steps of 1000 erg cm⁻³. Starting with K_b = 0 a proportionality between H₀ and ν_res is observed with a slightly smaller slope than one would expect for a g = 2 paramagnet. That slope, however, is almost restored when setting θ_H = 90° (not shown in the figure), as can be seen from (7). When turning K_b positive, zero field resonances occur at finite frequencies, meaning that an excitation gap opens. This is because for positive K_b the easy axes of the basal anisotropy are along the a-axes, which also correspond to the approximate magnetisation directions in our calculation. Thus even at zero external field the sublattice magnetisations underlie the existing internal anisotropy field, yielding finite resonance frequencies at H₀ = 0. For K_b < 0 the curves in figure 7(a) bend in the opposite way and for low external fields the simulations fail. A spin reorientation transition takes place. At zero and low external fields the sublattice magnetisations are, in reality, along the diagonals between the a-axes. With increasing H₀, the M⁽ⁱ⁾ turn continuously towards directions close to the a-axes and ν_res decreases, because internal and external fields counteract. When the magnetisation has become perpendicular to the external field, the resonance frequency can again increase with increasing field which is indeed seen in the simulations. As our model is based on the latter magnetisation configuration, the low field features are not captured in figure 7(a). Empirically, the excitation gaps for K_b > 0 observed in this figure match the well-known expression ${\omega }_{\text{res}}=\gamma \sqrt{2{\Lambda}{M}_{0}{H}_{\text{an}}}$ [13], when defining the basal anisotropy field as ${H}_{\text{an}}={H}_{\text{b}}{:=}\left\vert 8{K}_{\text{b}}/{M}_{0}\right\vert$.

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The situation is similar in figure 7(b), where K_b = −5000 erg cm⁻³ is fixed and the angle θ_H of the external field within the (ac)-plane is varied. The spin-reorientation transition is always observed, but its field value increases when tilting H₀ away from the (aa)-plane. Moreover, the resonance frequency above this transition gets more and more suppressed. At θ_H = 0° one ends up with the aforementioned Goldstone mode ν_res = 0. In figure 7(c) the same anisotropy constants are used, but the rotation now takes place within the (aa)-plane. For φ_H = 0° the same situation as in figure 7(b) at θ_H = 90° is encountered. Increasing to φ_H = 1° already lifts the minimal resonance frequency from 0 to 10 GHz, rendering the simulations possible even below the transition. The significant difference to out-of-plane simulations is here, that the azimuthal angles of the two sublattice magnetisations may differ from each other when employing in-plane simulations. As anticipated before, ν_res-curves in figure 7(c) approach a constant and finite value at zero field. The upwards trend is continued up to φ_H = 45°, where the situation now corresponds exactly to the case of φ_H = 0° and K_b = +5000 erg cm⁻³ in figure 7(a).

These field-frequency plots help to understand why the X-band data on GdRh₂Si₂ are not suitably described by our model. With too large anisotropy constants, the excitation gap is not overcome by the microwave energy. At higher temperatures, anisotropies are small and the excitation gap may still be overcome. But even then, the steep slopes in ν_res(H₀) when the external field is in the (ac)-plane or close to it as well as strong shifts of H_res at ν_rf = 9.4 GHz during field rotations destabilize the simulation.

$$\begin{align*}\hfill \frac{\partial F}{\partial \theta }& =2{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace \varphi \enspace \mathrm{sin}\left(\theta -{\theta }_{\mathrm{H}}\right)+2{K}_{\text{u}}\enspace \mathrm{sin}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace \varphi \hfill \\ \hfill & \quad +\frac{2{K}_{\text{b}}}{{D}^{3}}\left\{{\mathrm{sin}}^{2}\enspace 2\varphi \enspace \mathrm{sin}\enspace 2\theta \left[1-{\mathrm{sin}}^{2}\enspace \varphi \left(1+{\mathrm{sin}}^{2}\enspace \theta \right)\right]\right\},\hfill \\ \hfill \frac{{\partial }^{2}F}{\partial {\theta }^{2}}& =2{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace \varphi \enspace \mathrm{cos}\left(\theta -{\theta }_{\mathrm{H}}\right)+4{K}_{\text{u}}\enspace \mathrm{cos}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace \varphi \hfill \\ \hfill & \quad +\frac{2{K}_{\text{b}}}{{D}^{4}}{\mathrm{sin}}^{2}\enspace 2\varphi \left\{\left(2D\enspace \mathrm{cos}\enspace 2\theta -3\enspace {\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace \varphi \right)\right.\hfill \\ \hfill & \enspace \quad {\times}\left.\left[1-{\mathrm{sin}}^{2}\enspace \varphi \left(1+{\mathrm{sin}}^{2}\enspace \theta \right)\right]-D\enspace {\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace \varphi \right\},\hfill \\ \hfill \frac{\partial F}{\partial \varphi }& =2{\Lambda}{M}_{0}^{2}\enspace \mathrm{sin}\enspace 2\varphi -2{H}_{0}{M}_{0}\enspace \mathrm{cos}\enspace \varphi \enspace \hfill \\ \hfill & \quad {\times}\mathrm{cos}\left(\theta -{\theta }_{\mathrm{H}}\right)-2{K}_{\text{u}}\enspace {\mathrm{cos}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace 2\varphi \hfill \\ \hfill & \quad +\frac{{K}_{\text{b}}}{{D}^{3}}\left(4D\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace 4\varphi +{\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{3}\enspace 2\varphi \right),\hfill \\ \hfill \frac{{\partial }^{2}F}{\partial {\varphi }^{2}}& =4{\Lambda}{M}_{0}^{2}\enspace \mathrm{cos}\enspace 2\varphi +2{H}_{0}{M}_{0}\enspace \mathrm{sin}\enspace \varphi \enspace \mathrm{cos}\enspace \left(\theta -{\theta }_{\mathrm{H}}\right)\hfill \\ \hfill & \quad -4{K}_{\text{u}}\enspace {\mathrm{cos}}^{2}\enspace \theta \enspace \mathrm{cos}\enspace 2\varphi +\frac{{K}_{\text{b}}}{{D}^{4}}\left\{\left[16D\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{cos}\enspace 4\varphi \right.\right.\hfill \\ \hfill & \quad -{\mathrm{sin}}^{2}\enspace 2\theta \enspace \mathrm{sin}\enspace 2\varphi \enspace \mathrm{sin}\enspace 4\varphi +6\enspace {\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace 2\varphi \hfill \\ \hfill & \quad \left.{\times}\enspace \mathrm{cos}\enspace 2\varphi \right]D+3\enspace {\mathrm{cos}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace 2\varphi \hfill \\ \hfill & \quad \left.{\times}\left(4D\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace 4\varphi +{\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{3}\enspace 2\varphi \right)\right\},\hfill \\ \hfill \frac{{\partial }^{2}F}{\partial \theta \partial \varphi }& =2{H}_{0}{M}_{0}\enspace \mathrm{cos}\enspace \varphi \enspace \mathrm{sin}\enspace \left(\theta -{\theta }_{\mathrm{H}}\right)+2{K}_{\text{u}}\enspace \mathrm{sin}\enspace 2\theta \enspace \mathrm{sin}\enspace 2\varphi \hfill \\ \hfill & \quad +\frac{{K}_{\text{b}}}{{D}^{4}}\left\{\left[4\enspace \mathrm{sin}\enspace 2\theta \enspace \mathrm{sin}\enspace 4\varphi \left({\mathrm{sin}}^{2}\enspace \theta \enspace {\mathrm{sin}}^{2}\enspace \varphi +D\right)\right.\hfill \right.\\ \hfill & \left.\quad +\mathrm{sin}\enspace 4\theta \enspace {\mathrm{sin}}^{3}\enspace 2\varphi \right]D-3\enspace \mathrm{sin}\enspace 2\theta \enspace {\mathrm{sin}}^{2}\enspace \varphi \hfill \\ \hfill & \quad {\times}\left.\left(4D\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace 4\varphi +{\mathrm{sin}}^{2}\enspace 2\theta \enspace {\mathrm{sin}}^{3}\enspace 2\varphi \right)\right\},\hfill \end{align*}$$

where D = 1 − cos² θ sin² φ.

Using ${M}_{x\enspace 0}{:=}{M}_{x\enspace 0}^{\left(1\right)}={M}_{0}\enspace \mathrm{cos}\varphi$, ${M}_{y\enspace 0}{:=}{M}_{y\enspace 0}^{\left(1\right)}={M}_{0}\enspace \mathrm{sin}\varphi$, η:=M_y0 sin θ/M_x0, ω_e := γΛM₀, ω_u := 2K_uγ/M₀ and $\kappa {:=}8{K}_{\text{b}}\gamma {M}_{y\enspace 0}\enspace \mathrm{sin}\enspace \theta /{\left({M}_{x\enspace 0}\right)}^{2}$, the matrix elements are

$$\begin{align*}\hfill {A}_{11}& =\gamma {M}_{0}{k}_{zx}\enspace \mathrm{sin}\enspace \varphi +\mathrm{i}\omega ,\hfill \\ \hfill {A}_{12}& =\gamma {H}_{0}\enspace \mathrm{sin}\left(\theta -{\theta }_{\mathrm{H}}\right)+{\omega }_{\text{u}}\enspace \mathrm{sin}\enspace 2\theta \enspace \mathrm{sin}\enspace \varphi \hfill \\ \hfill & \quad +\gamma {M}_{0}{k}_{zy}\enspace \mathrm{sin}\enspace \varphi +\kappa \enspace \mathrm{cos}\enspace \theta ,\hfill \\ \hfill {A}_{13}& ={\omega }_{\text{e}}\enspace \mathrm{sin}\enspace \varphi -\gamma {H}_{0}\enspace \mathrm{cos}\left(\theta -{\theta }_{\mathrm{H}}\right)+{\omega }_{\text{u}}\enspace \mathrm{sin}\enspace \varphi \hfill \\ \hfill & \quad {\times}\left({\mathrm{sin}}^{2}\enspace \theta -{\mathrm{cos}}^{2}\enspace \theta \right)+\gamma {M}_{0}{k}_{zz}\enspace \mathrm{sin}\enspace \varphi +\kappa \enspace \mathrm{sin}\enspace \theta ,\hfill \\ \hfill {A}_{16}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace \varphi ,\hfill \\ \hfill {A}_{21}& =-\gamma {H}_{0}\enspace \mathrm{sin}\left(\theta -{\theta }_{\mathrm{H}}\right)-\frac{{\omega }_{\text{u}}}{2}\mathrm{sin}2\enspace \theta \enspace \mathrm{sin}\enspace \varphi \hfill \\ \hfill & \quad -\gamma {M}_{0}{k}_{zx}\enspace \mathrm{cos}\enspace \varphi -\kappa \enspace \mathrm{cos}\enspace \theta ,\hfill \\ \hfill {A}_{22}& =-\frac{{\omega }_{\text{u}}}{2}\mathrm{sin}\enspace 2\theta \enspace \mathrm{cos}\enspace \varphi -\gamma {M}_{0}{k}_{zy}\enspace \mathrm{cos}\enspace \varphi +\mathrm{i}\omega ,\hfill \\ \hfill {A}_{23}& ={\omega }_{\text{e}}\enspace \mathrm{cos}\enspace \varphi -{\omega }_{\text{u}}\enspace {\mathrm{sin}}^{2}\enspace \theta \enspace \mathrm{cos}\enspace \varphi -\gamma {M}_{0}{k}_{zz}\enspace \mathrm{cos}\enspace \varphi +\kappa \eta ,\hfill \\ \hfill {A}_{26}& ={\omega }_{\text{e}}\enspace \mathrm{cos}\enspace \varphi ,\hfill \\ \hfill {A}_{31}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace \varphi +\gamma {H}_{0}\enspace \mathrm{cos}\left(\theta -{\theta }_{\mathrm{H}}\right)+{\omega }_{\text{u}}\enspace {\mathrm{cos}}^{2}\enspace \theta \enspace \mathrm{sin}\enspace \varphi \hfill \\ \hfill & \quad +\gamma {M}_{0}\left({k}_{yx}\enspace \mathrm{cos}\enspace \varphi -{k}_{xx}\enspace \mathrm{sin}\enspace \varphi \right)-\kappa \enspace \mathrm{sin}\enspace \theta ,\hfill \\ \hfill {A}_{32}& =-{\omega }_{\text{e}}\enspace \mathrm{cos}\enspace \varphi +{\omega }_{\text{u}}\enspace {\mathrm{cos}}^{2}\enspace \theta \enspace \mathrm{cos}\enspace \varphi \hfill \\ \hfill & \quad +\gamma {M}_{0}\left({k}_{yy}\enspace \mathrm{cos}\enspace \varphi -{k}_{xy}\enspace \mathrm{sin}\enspace \varphi \right)-\kappa \eta ,\hfill \\ \hfill {A}_{33}& =\frac{{\omega }_{\text{u}}}{2}\mathrm{sin}\enspace 2\theta \enspace \mathrm{cos}\enspace \varphi +\gamma {M}_{0}\left({k}_{yz}\enspace \mathrm{cos}\enspace \varphi -{k}_{xz}\enspace \mathrm{sin}\enspace \varphi \right)+\mathrm{i}\omega ,\hfill \\ \hfill {A}_{34}& ={\omega }_{\text{e}}\enspace \mathrm{sin}\enspace \varphi ,\hfill \\ \hfill {A}_{35}& =-{\omega }_{\text{e}}\enspace \mathrm{cos}\enspace \varphi ,\hfill \\ \hfill {A}_{44}& =-\gamma {M}_{0}{k}_{zx}\enspace \mathrm{sin}\enspace \varphi +\mathrm{i}\omega ,\hfill \\ \hfill {A}_{55}& =\frac{{\omega }_{\text{u}}}{2}\mathrm{sin}\enspace 2\theta \enspace \mathrm{cos}\enspace \varphi +\gamma {M}_{0}{k}_{zy}\enspace \mathrm{cos}\enspace \varphi +\mathrm{i}\omega ,\hfill \\ \hfill {A}_{66}& =-\frac{{\omega }_{\text{u}}}{2}\mathrm{sin}\enspace 2\theta \enspace \mathrm{cos}\enspace \varphi \hfill \\ \hfill & \quad -\gamma {M}_{0}\left({k}_{yz}\enspace \mathrm{cos}\enspace \varphi -{k}_{xz}\enspace \mathrm{sin}\enspace \varphi \right)+\mathrm{i}\omega ,\hfill \end{align*}$$

and further A₄₃ = A₁₆, A₄₅ = A₁₂, A₄₆ = A₁₃, A₅₃ = A₃₅, A₅₄ = A₂₁, A₅₆ = −A₂₃, A₆₁ = A₃₄, A₆₂ = A₂₆, A₆₄ = A₃₁, A₆₅ = −A₃₂ as well as A₁₄ = A₁₅ = A₂₄ = A₂₅ = A₃₆ = A₄₁ = A₄₂ = A₅₁ = A₅₂ = A₆₃ = 0. The coefficients k_αβ, connecting dynamic magnetisation component β to dynamic basal field component α are

$$\begin{align*}\hfill {k}_{xx}& =-24{K}_{\text{b}}{\eta }^{2}/{M}_{x\enspace 0}^{2},\hfill \\ \hfill {k}_{xy}& =16{K}_{\text{b}}\eta \enspace \mathrm{sin}\enspace \theta /{M}_{x\enspace 0}^{2},\hfill \\ \hfill {k}_{xz}& =-8{K}_{\text{b}}{M}_{y\enspace 0}\enspace \mathrm{sin}\enspace 2\theta /{M}_{x\enspace 0}^{3},\hfill \\ \hfill {k}_{yy}& =-8{K}_{\text{b}}\enspace {\mathrm{sin}}^{2}\enspace \theta /{M}_{x\enspace 0}^{2},\hfill \\ \hfill {k}_{yz}& =4{K}_{\text{b}}\enspace \mathrm{sin}\enspace 2\theta /{M}_{x\enspace 0}^{2},\hfill \\ \hfill {k}_{zz}& =-8{K}_{\text{b}}\enspace {\mathrm{cos}}^{2}\enspace \theta /{M}_{x\enspace 0}^{2}\hfill \end{align*}$$

and k_yx = k_xy, k_zx = k_xz, k_zy = k_yz.

Using ${M}_{x\enspace 0}^{\left(1\right)}={M}_{0}\enspace \mathrm{cos}{\varphi }_{1}$, ${M}_{y\enspace 0}^{\left(1\right)}={M}_{0}\enspace \mathrm{sin}{\varphi }_{1}$, ${M}_{x\enspace 0}^{\left(2\right)}=-{M}_{0}\enspace \mathrm{cos}{\varphi }_{2}$, ${M}_{y\enspace 0}^{\left(2\right)}={M}_{0}\enspace \mathrm{sin}{\varphi }_{2}$, ω_e := γΛM₀, ω_u := 2K_uγ/M₀ and $\kappa {:=}2{K}_{\text{b}}/{M}_{0}^{4}$, the matrix elements are

$$\begin{align*}\hfill {A}_{13}& ={\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{2}-\gamma {H}_{0}+{\omega }_{\text{u}}\enspace \mathrm{sin}\enspace {\varphi }_{1}-\gamma {H}_{\text{b}\enspace 0\enspace y}^{\left(1\right)},\hfill \\ \hfill {A}_{16}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{1},\hfill \\ \hfill {A}_{23}& ={\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{2}-{\omega }_{\text{u}}\enspace \mathrm{cos}\enspace {\varphi }_{1}+\gamma {H}_{\text{b}\enspace 0\enspace x}^{\left(1\right)},\hfill \\ \hfill {A}_{26}& ={\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{1},\hfill \\ \hfill {A}_{31}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{2}+\gamma {H}_{0}+\gamma {H}_{\text{b}\enspace 0\enspace y}^{\left(1\right)}\hfill \\ \hfill & \quad +\gamma {M}_{0}\left( {k}_{yx}^{\left(1\right)}\enspace \mathrm{cos}\enspace {\varphi }_{1}-{k}_{xx}^{\left(1\right)}\enspace \mathrm{sin}\enspace {\varphi }_{1} \right),\hfill \\ \hfill {A}_{32}& =-{\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{2}-\gamma {H}_{\text{b}\enspace 0\enspace x}^{\left(1\right)}\hfill \\ \hfill & \quad +\gamma {M}_{0}\left( {k}_{yy}^{\left(1\right)}\enspace \mathrm{cos}\enspace {\varphi }_{1}-{k}_{xy}^{\left(1\right)}\enspace \mathrm{sin}\enspace {\varphi }_{1} \right),\hfill \\ \hfill {A}_{43}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{2},\hfill \\ \hfill {A}_{46}& ={\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{1}-\gamma {H}_{0}+{\omega }_{\text{u}}\enspace \mathrm{sin}\enspace {\varphi }_{2}-\gamma {H}_{\text{b}\enspace 0\enspace y}^{\left(2\right)},\hfill \\ \hfill {A}_{53}& =-{\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{2},\hfill \\ \hfill {A}_{56}& =-{\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{1}+{\omega }_{\text{u}}\enspace \mathrm{cos}\enspace {\varphi }_{2}+\gamma {H}_{\text{b}\enspace 0\enspace x}^{\left(2\right)},\hfill \\ \hfill {A}_{64}& =-{\omega }_{\text{e}}\enspace \mathrm{sin}\enspace {\varphi }_{1}+\gamma {H}_{0}+\gamma {H}_{\text{b}\enspace 0\enspace y}^{\left(2\right)}\hfill \\ \hfill & \quad -\gamma {M}_{0}\left( {k}_{yx}^{\left(2\right)}\enspace \mathrm{cos}\enspace {\varphi }_{2}+{k}_{xx}^{\left(2\right)}\enspace \mathrm{sin}\enspace {\varphi }_{2} \right),\hfill \\ \hfill {A}_{65}& ={\omega }_{\text{e}}\enspace \mathrm{cos}\enspace {\varphi }_{1}-\gamma {H}_{\text{b}\enspace 0\enspace x}^{\left(2\right)}\hfill \\ \hfill & \quad -\gamma {M}_{0}\left({k}_{yy}^{\left(2\right)}\enspace \mathrm{cos}\enspace {\varphi }_{2}+{k}_{xy}^{\left(2\right)}\enspace \mathrm{sin}\enspace {\varphi }_{2}\right),\hfill \end{align*}$$

and further A₁₁ = A₂₂ = A₃₃ = A₄₄ = A₅₅ = A₆₆ = iω, A₁₂ = A₁₄ = A₁₅ = A₂₁ = A₂₄ = A₂₅ = A₃₆ = A₄₁ = A₄₂ = A₄₅ = A₅₁ = A₅₂ = A₅₄ = A₆₃ = 0 as well as A₃₄ = −A₁₆, A₃₅ = −A₂₆, A₆₁ = −A₄₃, A₆₂ = −A₅₃. Static basal field components and dynamic basal coefficients for the two sublattices i = 1, 2 are

$$\begin{align*}\hfill {H}_{\text{b}\enspace 0\enspace x}^{\left(i\right)}& =2\kappa \left[\frac{3}{2}{\left({M}_{x\enspace 0}^{\left(i\right)}\right)}^{2}{M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}-{\left({M}_{x\enspace 0}^{\left(i\right)}\right)}^{3}\enspace {\mathrm{sin}}^{2}\enspace 2{\varphi }_{\mathrm{H}}\right],\hfill \\ \hfill {H}_{\text{b}\enspace 0\enspace y}^{\left(i\right)}& =\kappa {\left( {M}_{x\enspace 0}^{\left(i\right)} \right)}^{2}\left[ {M}_{x\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}\right.\hfill \\ \hfill & \left.\quad +2{M}_{y\enspace 0}^{\left(i\right)}\left( {\mathrm{sin}}^{2}\enspace 2{\varphi }_{\mathrm{H}}-2\enspace {\mathrm{cos}}^{2}\enspace 2{\varphi }_{\mathrm{H}} \right) \right],\hfill \\ \hfill {k}_{xx}^{\left(i\right)}& =2\kappa \left[{M}_{x\enspace 0}^{\left(i\right)}{M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}-{\left({M}_{x\enspace 0}^{\left(i\right)}\right)}^{2}\enspace {\mathrm{sin}}^{2}\enspace 2{\varphi }_{\mathrm{H}}\right.\hfill \\ \hfill & \quad -\left.2{\left({M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{cos}\enspace 2{\varphi }_{\mathrm{H}}-{M}_{x\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 2{\varphi }_{\mathrm{H}}\right)}^{2}\right],\hfill \\ \hfill {k}_{xy}^{\left(i\right)}& =\kappa \left[4\left({M}_{x\enspace 0}^{\left(i\right)}\enspace \mathrm{cos}\enspace 2{\varphi }_{\mathrm{H}}+{M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 2{\varphi }_{\mathrm{H}}\right)\right.\hfill \\ \hfill & \quad {\times}\left.\left({M}_{x\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 2{\varphi }_{\mathrm{H}}-{M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{cos}\enspace 2{\varphi }_{\mathrm{H}}\right)\right.\hfill \\ \hfill & \quad -\left.4{M}_{x\enspace 0}^{\left(i\right)}{M}_{y\enspace 0}^{\left(i\right)}\enspace {\mathrm{cos}}^{2}\enspace 2{\varphi }_{\mathrm{H}}+{\left({M}_{x\enspace 0}^{\left(i\right)}\right)}^{2}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}\right],\hfill \\ \hfill {k}_{yx}^{\left(i\right)}& =\kappa {M}_{x\enspace 0}^{\left(i\right)}\left[3{M}_{x\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}\right.\hfill \\ \hfill & \left.\quad -4{M}_{y\enspace 0}^{\left(i\right)}\left(2\enspace {\mathrm{cos}}^{2}\enspace 2{\varphi }_{\mathrm{H}}-{\mathrm{sin}}^{2}\enspace 2{\varphi }_{\mathrm{H}}\right)\right],\hfill \\ \hfill {k}_{yy}^{\left(i\right)}& =2\kappa {M}_{x\enspace 0}^{\left(i\right)}\left[{M}_{x\enspace 0}^{\left(i\right)}\left({\mathrm{sin}}^{2}\enspace 2{\varphi }_{\mathrm{H}}-2\enspace {\mathrm{cos}}^{2}\enspace 2{\varphi }_{\mathrm{H}}\right)\right.\hfill \\ \hfill & \left.\quad -3{M}_{y\enspace 0}^{\left(i\right)}\enspace \mathrm{sin}\enspace 4{\varphi }_{\mathrm{H}}\right].\hfill \end{align*}$$