Sample synthesis and basic characterizations

Our powder sample of CdEr₂Se₄(batch number ATF450 and ATF451) and CdEr₂S₄(batch number ATF503) were synthesized through the solid-state reaction by Dr. Vladimir Tsurkan at University of Augsburg. To reduce the neutron absorption of the natural Cd element,¹¹⁴Cd isotope enriched starting material with an absorption cross section of 41.6 barn was used for the synthesis (Trace Sci. Int. Corp.). Sample quality of CdEr₂Se₄was checked with synchrotron X-ray diffraction at the Materials Science beamline (MS) at the Swiss Light Source (SLS) of PSI.

The quality of CdEr₂S₄was checked with laboratory X-ray powder diffraction at University of Augsburg.

Fig. 3.1 presents the Rietveld refinement result for the batch ATF450 of CdEr₂Se₄using the FullProf program [115]. The space groupF d¯3m is used. During the refinement, the Cd atoms at the 8asite of [1/8 1/8 1/8] and the Er atoms at the 16dsite of [1/2 1/2 1/2] are allowed to exchange their positions but no improvement can be observed, which suggests negligible anti-site disorder in our sample. The best fit has a unit cell length of 11.5959 Å and the Se atoms at the 32esite withx=0.2561. The corresponding agreement factors areR_p=11.0,R_{w p}=15.9, andχ²=65.63. The relatively large agreement factors are due to secondary impurity phases that can be identified as erbium selenides. However, the maximum intensity of the impurity phase is only 0.5% of that for CdEr₂Se₄, suggesting negligible impurity concentration for most of our studies. A similar good refinement is achieved for the batch ATF451 of CdEr₂Se₄, which allows us to combine these two batches for the neutron scattering experiments.

3.2. Crystal electric field transitions in CdEr2Se4and CdEr2S4

2theta

0 10 20 30 40 50 60

Intensity (a. u.)

-2000 0 2000 4000 6000 8000

Figure 3.1 – Refinement result for the synchrotron X-ray diffraction data of CdEr2S4. Data points are shown as red circles. The calculated pattern is shown as black solid line. The vertical bars show the positions of the Bragg peaks. And the blue line at the bottom shows the difference of data and calculated intensities.

3.2.2 Crystal electric field transitions in CdEr2Se4

The crystal electric field transitions in CdEr₂Se₄were measured with the time-of-flight spec-trometers IN4 and IN6 at various temperatures between 1.5 and 200 K. For the measurements, about 9 g of¹¹⁴CdEr₂Se₄powder was filled into an annular-shaped aluminum can with out-er/inner diameters of 12/8 mm. Depending on the required energy range and resolution, incoming neutron wavelengths of 1.21 and 2.41 (5.12 and 5.92) Å were chosen for the measure-ments on IN4 (IN6).

Fig. 3.2 presents typical Q-E maps measured on IN4 and IN6 at different temperatures. Differ-ent Q-dependences of the intensities suggest their differDiffer-ent origins. For example, in Fig. 3.2a, intensities at∼17 meV increase with Q, which indicates phonon excitations as their origin;

while intensities at 10 meV decrease with Q in accordance with the magnetic form factor of Er³⁺, suggesting crystal electric field transitions as their origin. The temperature dependence of the measured spectra in Fig. 3.2 arises from the population of the CEF states and detailed balance. At low temperatures, only the ground state is populated, which allows upwards transitions to the CEF excited states observed in Fig. 3.2a. With increasing temperature, the CEF excited states get populated, and downwards transitions from the excited states become observable in Fig. 3.2d.

Under theD_3dsymmetry of the crystal electric field, the ground state manifold⁴I¹⁵

2 of Er³⁺

splits into 5Γ⁺₄ doublets and 3Γ⁺₅⊕Γ⁺₆ doublets [116]. Therefore, up to 7 CEF transitions shall be observable at the base temperature of 1.5 K. Treating the coefficientsB^m_l in eqn.

(3.2) as fitting parameters, we can extract the ground state information from the positions and intensities of the measured CEF transitions. Our fits are performed with the McPhase program [117], and the standardχ²is evaluated as the controlling parameter. A combined fit

IN4T = 2 K IN4 T = 50 K

IN6T = 1.6 K IN6

T = 50 K

a b

c d

Figure 3.2 –(a,b)Q-E map for the crystal field transitions measured on IN4 with incoming neutron wavelength of 1.21 Å at 2 and 50 K. The measurements with an empty can have been subtracted as the background.(c,d)Q-E map for the crystal field transitions measured on IN6 with incoming wavelength of 5.1 Å at 1.6 and 50 K. Intensities at 1.6 meV in panel c and d are due to a known malfunction of the detectors.

was performed for the data measured on IN4 with incoming wavelengths of 1.21 and 2.41 Å at 2 K, and the CEF transitions at other temperatures are calculated using the fitted parameters at 2 K. For the IN6 experiment, we used the data measured with an incoming wavelength of 5.12 Å at 50 K. The overall fitting results are shown in Fig. 3.3. The fitted parameters in the Wybourne format are listed in Tab. 3.1. The dominance of a negativeB₂⁰(see the text in Tab.

3.1) is quite obvious, indicating that the ground state should have an Ising character due to a large component of|15/2,±15/2〉. The exact expression for the ground state doublet extracted from the IN4 data is:

|±〉 = ±0.906|7.5,±7.5〉 +0.386|7.5,±4.5〉 ±0.159|7.5,±1.5〉 (3.11)

−0.073|7.5,∓1.5〉 ±0.004|7.5,∓4.5〉,

which is dominated by the|7.5,±7.5〉component and hasg-factors ofg_⊥=0 andg_∥=16.5.

Noticing that thezaxis of theD_3dsymmetry is along the local [111] directions, our inelastic neutron scattering study provides microscopic evidence for the local Ising character of the Er³⁺spins in CdEr₂Se₄.

The Ising character of Er³⁺in CdEr2Se4is very different from that in Er2Ti2O7[118, 119]. For the latter system, Er³⁺exhibits an easy-plane character with the local [111] direction as the hard axis. This difference originates from the different local crystal environments of the Er³⁺

3.2. Crystal electric field transitions in CdEr2Se4and CdEr2S4

0

1

2

3 012 012 E (meV)-6-4-20

Intensity (a. u.) 2 1 0

00.10.20.30.40.5 00.10.20.30.4 00.10.20.30.4 E (meV)024681012

Intensity (a. u.)

00.10.20.30.4

00.010.020.030.040.05 00.010.020.030.04 00.010.020.030.04 E (meV)010203000.010.020.030.04Intensity (a. u.)

5.12 Å, 1.6 K 5.12 Å, 25 K 5.12 Å, 50 K 5.12 Å, 100 K

2.41 Å, 2 K 2.41 Å, 25 K 2.41 Å, 50 K 2.41 Å, 100 K

1.21 Å, 2 K 1.21 Å, 25 K 1.21 Å, 50 K 1.21 Å 100 K

a b c d

e f g h

i j k l Figure3.3–FitsofthecrystalfieldtransitionsinCdEr2Se4.Experimentaldataareshownasbluecirclesandthefittedresultisshownasthered linewiththeparametersinTab.3.1.(a-d)MeasurementsareperformedonIN6withanincomingwavelengthof5.12Å.Theexperimental datapointsareobtainedthroughintegrationintheregion1.05<|Q|<1.15Å.(e-h)MeasurementsareperformedonIN4withanincoming wavelengthof2.41Å.Theexperimentaldatapointsareobtainedthroughintegrationintheregion1.0<|Q|<2.0Å.(i-l)Measurementsare performedonIN4withanincomingwavelengthof1.21Å.Theexperimentaldatapointsareobtainedthroughintegrationintheregion 1.7<|Q|<3.7Å.

Table 3.1 – The Wybourne crystal field parameters (meV) for CdEr₂Se₄. The correspond-ing Stevens parameters (meV) areB₂⁰= −0.032636,B₄⁰= −0.000598,B₄³= −0.012837,B₆⁰= 0.000003,B₆³= −0.000051,B₆⁶=0.000019 for IN4.

L⁰₂ L⁰₄ L³₄ L⁰₆ L³₆ L⁶₆ IN4 (2 K) -25.70 -107.73 -97.74 25.31 -19.06 9.51 IN6 (50 K) -26.00 -107.80 -96.73 25.17 -18.12 9.30

ions [113]. Along the [111] direction, there is a pair of O²⁻anions that are closer to the Er³⁺ ions compared to the O²⁻neighbours along other directions. However, no such shorter Er-Se bonds exist in CdEr2Se4along the [111] direction, and instead, the nearest Er³⁺cations are close to the [111] direction. Such a charge reversal leads to a change in the sign ofB₂⁰, thus explaining the Ising character for Er³⁺in CdEr₂Se₄.

Compared to the pyrochlore spin ice compounds of dysprosium and holmium titanites, the CEF excitation gap in CdEr2Se4is relatively small (see Fig. 3.5a). In Dy2Ti2O7, the excitation gap is as high as 20 meV, while in CdEr₂Se₄, it is only∼4 meV. As is pointed out by Molavianet al., perturbations through the excited states might renormalize the exchange interactions and influence the ground state properties. Another difference from the pyrochlore spin ices is the relatively large|J,J_z≤7/2〉components for the ground state doublet (3.11) of CdEr₂Se₄. As is discussed in the beginning of this section,|J,J_z〉components with|J_z| ≤7/2 allow relatively stronger transverse couplings between the pseudospins and enable the tunnelling within the ground state doublet.

It is worth mentioning that the symmetry of the Er³⁺ground state doublet in CdEr₂Se₄is distinctive from the usual magnetic dipoles [120–122]. Under theD_3d three-fold rotation operator ˆR(C3)=exp(−i2πJˆz/3), the ground states in eqn. (3.11) only pick up a negative sign, which indicates that each state shall transform as a one-dimensional representation (Γ⁺₅ or Γ⁺₆) and the doublet should haveΓ⁺₅ ⊕Γ⁺₆ symmetry . Components of thisΓ⁺₅⊕Γ⁺₆ doublet transform differently under theD_3dmirror plane operation: while thexandzcomponents change their signs as the magnetic dipole, they component however stays the same as an octupole component [120]. As in the usual quantum spin ice where theJ_xandJ_y couplings perturb the ice state dominated byJz, an octupole quantum spin ice state might be realized where the J_x andJ_z couplings perturb the ice states dominated by J_y. For this octupole quantum spin ice, the magnetic field (or the electric field following the notation of the field theory) defined by theycomponent of the pseudospin will follow a symmetry different from the magnetic dipoles and lead to different correlations for the pseudospins [120, 122].

3.2.3 Crystal electric field transitions in CdEr2S4

Similar investigations of the crystal electric field transitions in CdEr₂S₄were performed on IN4 and IN6 at various temperatures between 1.5 and 100 K. For the measurements, about

3.2. Crystal electric field transitions in CdEr2Se4and CdEr2S4

0

2

4

6

8 0246 0246 E (meV)-8-6-4-20

Intensity (a. u.) 6 4 2 0

0

0.005

0.010.015

0.02 0

0.005

0.010.015 0

0.005

0.010.015 E (meV)01020300

0.005

0.010.015

Intensity (a. u.)

5.12 Å, 1.5 K 5.12 Å, 25 K 5.12 Å, 50 K 5.12 Å, 100 K

1.11 Å, 1.5 K 1.11 Å, 25 K 1.11 Å, 50 K 1.11 Å 100 K

a b c d

i j k l

00.5

11.5 0

0.51 0

0.51 E (meV)024681012

Intensity (a. u.)

00.5

1

2.21 Å, 1.5 K 2.21 Å, 25 K 2.21 Å, 50 K 2.21 Å, 100 K

e f g h Figure3.4–FitsofthecrystalfieldtransitionsinCdEr2S4.Experimentaldataareshownasbluecirclesandthefittedresultisshownasthered linewiththeparametersinTab.3.2.(a-d)MeasurementsareperformedonIN6withanincomingwavelengthof5.12Å.Theexperimental datapointsareobtainedthroughintegrationintheregion1.05<|Q|<1.15Å.(e-h)MeasurementsareperformedonIN4withanincoming wavelengthof2.22Å.Theexperimentaldatapointsareobtainedthroughintegrationintheregion1.0<|Q|<2.0Å.(i-l)Measurementsare performedonIN4withanincomingwavelengthof1.12Å.Theexperimentaldatapointsareobtainedthroughintegrationintheregion 1.7<|Q|<3.7Å.

Table 3.2 – The Wybourne crystal field parameters (meV) for CdEr₂S₄. The correspond-ing Stevens parameters (meV) areB₂⁰= −0.037055,B₄⁰= −0.000681,B₄³= −0.014928,B₆⁰= 0.000003,B₆³= −0.000058,B₆⁶=0.000028 for IN4.

L⁰₂ L⁰₄ L³₄ L⁰₆ L³₆ L⁶₆ IN4 (1.5 K) -29.18 -122.72 -113.66 25.97 -21.89 14.41

IN6 (50 K) -26.84 -122.10 -100.94 25.72 -23.84 13.27

4.5 g of¹¹⁴CdEr2S4powder was filled into an annular-shaped aluminum can with outer/inner diameters of 10/8 mm. Setups with incoming neutron wavelengths of 1.12, 2.22, and 3.06 Å (5.12 Å) were employed for the measurements on IN4 (IN6).

Fig. 3.4 presents the fitting results for the integrated Q-dependence of the IN4 data with the 1.12 and 2.22 Å setups and the IN6 data with the 5.12 Å setup using the McPhase program.

As for CdEr₂Se₄, fits are performed at 1.5 K (50 K) for IN4 (IN6) and the results for the other temperatures are calculated with the fitted parameters listed in Tab. 3.2. As is compared in Fig. 3.5, the CEF levels in CdEr2S4are a little bit higher in energy than those in CdEr2Se4, which is due to the relatively smaller size of the CdEr₂S₄unit cell and explains the higher CEF parameters in Tab. 3.2. As will be discussed in section (3.4.2), the difference in CEF levels directly influences the high-temperature monopole dynamics in this two compounds. Despite of the difference in the CEF level energies, the ground state doublet in CdEr₂S₄fitted from the IN4 data is very similar to that in CdEr₂Se₄:

|±〉 = ±0.904|7.5,±7.5〉 +0.391|7.5,±4.5〉 ±0.145|7.5,±1.5〉 (3.12)

−0.094|7.5,∓1.5〉 ±0.006|7.5,∓4.5〉,

which suggests similar Ising character for the Er³⁺spins in these two compounds.

Similarity of the anisotropy for the Er³⁺ spins in CdEr₂Se₄ and CdEr₂S₄ is also observed

0 meV 3.955.65 8.939.76 26.28 29.11, 29.21

0 meV 5.086.91 10.47 11.35 30.46 33.78, 34.00

CdEr₂Se₄ CdEr₂S₄

a b

Figure 3.5 – Fitted crystal electric levels in CdEr₂Se₄(a)and CdEr₂S₄(b).

3.3. Quasi-static spin-spin correlations in CdEr2Se4and CdEr2S4

0 1 2 3 4 5

0 1 2 3 4 5

CdEr₂S₄ CdEr₂Se₄ H (T)

M (μ B/Er)

2 K 25 K 50 K 100 K

Figure 3.6 – Magnetization measurements for CdEr2Se4and CdEr2S4at 2, 25, 50, and 100 K. CdEr₂Se₄data are shown as triangles, CdEr₂S₄data are shown as circles. The results of calculations using the CEF parameters for CdEr₂Se₄listed in Tab. 3.1 are shown as the red lines.

indirectly from the magnetization measurements. Fig. 3.6 presents theM-H relation at 2, 25, 50, and 100 K for powder samples of CdEr₂Se₄and CdEr₂S₄without isotope-enriched element that were measured with the Quantum Design PPMS in the Laboratory for Scientific Developments and Novel Materials (LDM) at PSI. TheM-Hresults for CdEr₂Se₄and CdEr₂S₄ almost overlap with each other. Using the CEF parameters listed in Tab. 3.1, we calculate the magnetization using the SAFiCF code from Duc Le [123]. The calculation results agree very well with the measurements, which confirms the correctness of our inelastic neutron study of the CEF transitions.

3.3 Quasi-static spin-spin correlations in CdEr

₂

Se

₄

and CdEr

₂

S

₄

In this section, we present our neutron scattering study of the quasi-static spin-spin correla-tions in CdEr₂Se₄and CdEr₂S₄. As is discussed in section 2.1.3, the momentum-dependence of the neutron diffuse scattering atE∼0 meV is a direct probe of the quasi-static spin-spin cor-relations, and representative examples related to spin ice can be found for Ho₂Ti₂O₇[96, 124], Dy₂Ti₂O₇[125, 126], Tb₂Ti₂O₇[104], and Yb₂Ti₂O₇[127].

3.3.1 Non-polarized neutron diffuse scattering study of CdEr₂Se₄

Non-polarized neutron diffuse scattering experiments were performed on the diffractometer D20 with a³He cryostat at ILL for temperatures between 0.5 and 20 K. For all the measurements, about 9 g of¹¹⁴CdEr₂Se₄powder was filled into an annular-shaped copper can with outer/inner diameters of 6/10 mm, and the setup with 2.41 Å incoming neutron wavelength was employed.

0.5 1 1.5 2 2.5

Figure 3.7 –(a)Non-polarized neutron diffuse scattering results for CdEr₂Se₄measured on D20 at 0.5, 1.5, 4, and 8 K. The data at 20 K has been subtracted as the background. The inset compares the temperature dependence of the Bragg peak intensity at 1.22 Å⁻¹and the integrated diffuse scattering intensity in 1.325<Q<1.375 Å⁻¹. (b)Detailed temperature dependence of the intensities in the region 1-1.7 Å⁻¹in between 0.5 and 1.2 K.

Fig. 3.7a presents typical diffraction data measured at 0.5, 1.5, 4.0, and 8.0 K. Data at 20 K has been subtracted as the background. As can be seen from the figure, broad peaks around 0.6 and 1.4 Å⁻¹start to form below 8 K, with their intensities growing with decreasing temperature.

Such a signal is compatible with what is normally observed for classical spin ices [79] and is a strong indication for the ice correlations in CdEr₂Se₄.

Near the base temperature of 0.5 K, a series of sharp peaks are also observed. As is detailed in the inset of Fig. 3.7a and Fig. 3.7b, the temperature evolution of these sharp peaks is different from that of the broad diffuse intensities: the sharp peaks saturate at temperatures below 0.8 K while the broad peaks from the diffuse scattering continue to grow. Thus the origin of the sharp peaks should be different from the main feature. Combined with the synchrotron X-ray diffraction data shown in section 3.2.1, we attribute the sharp peaks observed in Fig. 3.7b to the long-range magnetic order of erbium selenide Er_xSe_y.

3.3. Quasi-static spin-spin correlations in CdEr2Se4and CdEr2S4

Figure 3.8 –x yzpolarization analysis for the diffuse scattering of CdEr₂Se₄measured at 70 mK on D7, showing the separation of the magnetic scattering component from the nuclear coherent/isotope incoherent scattering and the nuclear spin incoherent scattering.

3.3.2 Polarized neutron diffuse scattering study of CdEr₂Se₄

The non-polarized neutron diffuse scattering results shown in Fig. 3.7a are not purely mag-netic; they contain four contributions: nuclear coherent, nuclear isotope incoherent, nuclear spin incoherent, and magnetic scatterings. For a paramagnetic sample without long-range magnetic order, the magnetic component can be separated out using thex yzpolarization analysis technique [128]. With thez(x y) axis perpendicular to (inside of) the scattering plane, thex yz polarization analysis technique exploits the different polarization dependence of the scattering processes: nuclear coherent and isotope incoherent scattering do not flip the spin; nuclear spin incoherent scattering has a 2/3 probability to flip the spin; and diffuse magnetic scattering has flipping probabilities depending on the relative angle between the polarization direction and the scattering vector. Thus the magnetic scattering cross section can be separated out using the 6-points method expressed as:

µdσ

where Tsf (Tnsf) refers to the total spin-flip (non-spin-flip) intensities and the (dσ/dΩ)_nuc term contains both the nuclear coherent and isotope incoherent scattering.

Ourx yzpolarized neutron diffuse scattering experiment was performed on the diffractometer D7 with a dilution cryostat at ILL. For this measurement, 5.26 g of¹¹⁴CdEr2Se4powder sample was filled into an annular-shaped copper can with outer/inner diameters of 15/14 mm and the setup with 4.8 Å incoming neutron wavelength was used. The measured transmission of

the sample was∼60 %, which enables a reliable polarization analysis. The efficiencies of the detectors were calibrated using a vanadium standard with a transmission comparable to the sample. And the efficiencies of the flipper and polarizer/analyzer system were calibrated with a quartz standard. To compensate for the out-of-plane scattering due to the finite height of the detectors, the 10-points method was used, which adds the polarization analysis along the x±ydirections to the normalx yzdirections and the details can be found in reference [129].

Fig. 3.8 presents thex yzpolarization analysis results for CdEr₂Se₄measured at 70 mK. The magnetic scattering has been separated from the nuclear coherent/isotope incoherent scat-tering and the nuclear spin incoherent scatscat-tering. Similar to the results with un-polarized neutrons shown in Fig. 3.7a, broad diffuse signal can be observed around 0.6 and 1.4 Å, which suggests short-range ice-correlations in this system.

Mean-field analysis

The mean-field treatment provides a first approximation in understanding the spin correla-tions in CdEr₂Se₄[79]. Denoting theαcomponent (α=x,y,z) of theν-th unit-length spin (ν=1,2,3,4) in then-th primitive unit cell asS_n,ν,α, the Hamiltonian on the pyrochlore lattice can be explicitly expressed as:

where theDaterm represents the easy-axis anisotropy and ˆn_νis the unit vector along the easy axis of theν-th spin. Fourier transform of the real-space couplingJ_n,ν,α;n⁰_,ν⁰_,βleads to the 12×12 coupling matricesJ_k;ν,α;ν0,βin reciprocal space, which is then diagonalized:

X

ν⁰,β

J_k;ν,α;ν⁰_,βu_k;ν^(ρ)_0,β=λ⁽_k^ρ)u⁽_k;ν,α^ρ) , (3.15)

whereλ^(ρ)_k withρ=1,2,...,12 denotes the eigenvalues andu^(ρ)_k denotes the corresponding eigenvectors. The global maximum ofλ^(ρ)_k determines the long-range order transition under the mean-field approximation, with the transition temperatureT_c and ordered structure

〈Sn,ν,α〉as:

k_BT_c=2

3[λ^(ρ)_k ]_max,

〈S_n,ν,α〉 = 〈S^(ρ)_k 〉u^(ρ)_k;ν,αexp(ik·r_n,ν)+c.c. . (3.16)

3.3. Quasi-static spin-spin correlations in CdEr2Se4and CdEr2S4

The paramagnetic susceptibility atT_MF>T_c can be approximated by the eigenvalues and eigenfunctions:

whereNis the total number of the unit cell,V is the volume of the system, andµis the size of the magnetic moment. Using Eq. (2.4), the cross section of the magnetic scattering can be expressed as: whereCis a constant,f(Q) is the magnetic form factor,τis the reciprocal lattice vector, and r_νdenotes the position of theν-th atom in the first primitive cell.

We implemented this mean-field calculation using Matlab on the MCC cluster at LDM of PSI. To account for the Ising character of the Er³⁺spin, a high anisotropy ofD_a=810 K was used. The dipolar interaction withD_dp=0.616 K was cut beyond the length of 20 unit cells.

500 randomQ-vectors were generated for each of the|Q|values for the powder averaged calculation. SinceT_cis an effective temperature that can be different from the real ordering temperature, the mean-field temperatureTMFwas used as a fitting parameter. Fig. 3.9 presents the fitted results withJ₁=0 for the magnetic scattering measured on D7. In this case,T_cis calculated to be 1.600 K, and the fittedT_MFis 1.622, 1.635, 1.710, 3.6, and 21.6 K for the data measured at 0.07, 0.5, 1.5, 5, and 20 K, respectively. The good quality of the fit confirms the correctness of the dipolar spin ice model.

However, in the mean-field calculation, a finiteJ₁does not affect the quality of the fit: as long as the nearest-neighbour couplingJ_eff=J₁/3+5D_dp/3 is ferromagnetic, a good fit can always be achieved through the adjustment of the fitting parameterTMF. In order to fix the coupling strengths, we resort to Monte Carlo simulations.

Monte Carlo simulation

In the first chapter, a brief introduction to Monte Carlo simulations has been presented.

Differently from that in the mean-field calculations, temperature in Monte Carlo simulations can be equalized to the real value, which enables a reliable determination of the coupling strengths. To this purpose, a 6×6×6 supercell with periodic boundary conditions has been implemented using the ALPS package. Following the model of eqn. (3.3), couplings between the Ising spins include the nearest-neighbour exchange interactionJ₁, the second-neighbour exchange interactionJ₂and the dipolar interactions with a cut beyond the length of 3 unit cells.

Previous investigations of the dipolar spin ice model have revealed equivalence between the effective nearest-neighbour coupling strength and the temperature position of the maximum

Q (Å^-1)

0.5 1 1.5 2 2.5

Intensity (a. u.)

0 5 10 15 20 25 30 35 40

0.07 K 0.5 K 1.5 K 5 K 20 K

Figure 3.9 – Mean-field calculation results for the magnetic scattering data measured on D7.

Data atT =0.07, 0.5, 1.5, 5, and 20 K are shifted along the y axis by 9, 18, 27, 36, and 45, respectively. Blue circles are the experimental data. Red lines are the mean-field calculation results.

of the specific heat peak. Thus the reportedC_p(T) peak centered at∼1 K enables us to fixJ_eff also to 1 K, meaning thatJ₁should be ferromagnetic−0.026 K (antiferromagnetic 0.078 K) for Ising spins with local (global) axes. Similar to the mean-field calculations, 500 randomQ -vectors were generated at each|Q|value for the powder average. 5k sweeps of thermalization and 10k sweeps of measurement were used for the simulations at 1.5, 5, and 20 K. And for the simulations at 0.07 and 0.5 K, an increased number of 10k sweeps of thermalization and 40k sweeps of measurement were used in order to compensate for the longer auto-correlation times.

Fig. 3.10 presents the Monte Carlo simulation results for the D7 magnetic scattering data with

Fig. 3.10 presents the Monte Carlo simulation results for the D7 magnetic scattering data with

Dans le document Neutron scattering investigation of the spin correlations in frustrated spinels (Page 49-74)