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Publication Figures

Publication Date:
2019-05-13

First Author:
Jacob B. J. Chapman

Title:
The Dynamics of Magnetism in Fe-Cr Alloys with Cr Precipitation

Paper Identifier:
CP/18/276

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Figure Reference Title Description Number of Figure Data Items Identifier Download Figure Details
Fig. 1 Fig 1: SD_MC_comp2 Comparison between time averaged spin dynamics against Monte Carlo (MC) samplingcite{Lavrentiev_JNucMat_2009}, both implementing the magnetic cluster expansion (MCE) Hamiltonian (Eq. ref{eqn:mce}) for (a) the magnetic moment of pure Fe and (b) the averaged magnitude of the Fe magnetic moments against temperature. One can see excellent agreement between the two methods. A small discrepancy can be observed around the $T_C$. It can be accounted for by the larger system size being employed for spin dynamics simulations, which permits longer wavelength magnons. We used a box size of 25$times$25$times$25 unit cells box for spin dynamics simulation, where 16$times$16$times$16 unit cells box were used for MC MCE simulations. 3 CF/18/277 Download
Figure 2 Fig 2: Mvx5 Figure 2. Magnetic properties of Fe$_{1-x}$Cr$_{x}$~alloys simulated using spin dynamics. (a) The average magnetisation of Fe, of Cr and of the whole system of disordered Fe$_{1-x}$Cr$_{x}$ solid solutions at $T=25$K. For each composition, three randomly generated configurations are modelled using SD (circles) and compared with Ref.~onlinecite{Lavrentiev_SSP_2011} (connected triangles). (b) to (e) Snapshots of magnetic moments in systems with different Cr concentrations (b) Pure Fe at 100~K, in a ferromagnetic state. (c) Fe$_{0.9}$Cr$_{0.1}$ at 100~K. Cr moments are anti-parallel to Fe moments in the dilute limit. (d) Pure Cr at 100~K, with moments ordered antiferromagnetically. (e) Fe$_{0.75}$Cr$_{0.25}$ at 25~K. Higher Cr concentrations leads to frustration and complex non-collinear arrangement of magnetic moments. 2 CF/18/280 Download
Figure 3 Fig 3: Tc_ss_comp_4 Magnetic properties of disordered Fe$_{1-x}$Cr$_{x}$ solid solutions with different Cr concentrations. (a) Magnetisation with temperature. (b) Magnetic susceptibility (Eq. ref{eqn:sus}). Inset: Susceptibility around the Curie temperature $T_C$. The points of inflection for $x=0$, 5 and 25 at.%Cr are indicated by arrows . (c) Curie temperature $T_C$ variation with Cr concentration as determined from the susceptibility. Data from Monte Carlo calculations and experimentcite{Lavrentiev_JPhysCM_2012} are given for comparison. The determination of $T_C$ presented in Ref. onlinecite{Lavrentiev_JPhysCM_2012} is via observation of peak in the specific heat. Error bars are set as the temperature interval in sampling, which is 20K. 1 CF/18/282 Download
Figure 4 Fig 4: MvT_clusters3 Averaged magnetisation of Fe-Cr alloys as a function of temperature, concentration and cluster size. All precipitates are spherical clusters in bcc $alpha'$ phase with a local concentration of 100% Cr. Remaining Cr required to construct supercells with overall concentration $x$ are dispersed randomly in the Fe-matrix. System Cr concentrations $x$ of (a) 9% (b) 12% (c) 15% (d) 18% (e) 22% (f) 25% for radii of 6 to 26AA. A radius of 0AA $;$ corresponds to a disordered solid solution. 1 CF/18/285 Download
Figure 5 Fig 5: cluster_9pc2 Magnetic moments of a spherical $alpha'$ precipitate with radius = 18AA~ at $T=100$K. The overall chemical composition contains 9 at.% Cr. An instantaneous snapshot is taken at the final time step of the production run. (a) Magnetic moments of Fe (orange) and Cr (white). (b) Magnetic moments colourised using double conical space. (c) Focused view near the interface. (d) Magnetic moment-space representation of all moments in the system at the final timestep. (e) Magnetic moment space map colourised according to the sum of the occupational site variables of the nearest neighbours (8=all Fe, 0=50%Fe-50%Cr, -8=all Cr). 1 CF/18/287 Download
Figure 6 Fig 6: 600-800_dM Plot illustrating the effect of microstructure on the magnetisation for Fe 9 at.%Cr alloys at 600K and 800K. The lower limits correspond to the disordered solid solution (SS). The upper limits corresponds to complete phase separation (FS). 1 CF/18/289 Download
Figure 7 Fig 7: Tcvx4 Variation of the Curie temperature with respect to $alpha'$ cluster size for different nominal Cr concentrations. The Curie temperature is maximised for any nominal concentration $x$ when the effective concentration of the $alpha$ is $x'approx 5-6$ at.%. 1 CF/18/291 Download
Figure 8 Fig 8: xc3 Magnetic properties of spherical Cr clusters of radius $r_c=4$~nm in Fe. (a) Averaged magnetisation and (b) susceptibility. Evaluated as a function of distance $|mathbf{r}-mathbf{r_0}|$ from the centre of the precipitate (inset). 1 CF/18/293 Download
Figure 9 Fig 9: ideal_corr6 Radially resolved autocorrelation and first nearest neighbour correlation functions at $T=$100K, 500K, 800K and 1200K. They are taken as averaged values for magnetic moments located at 13.1, 35.1, 37.0, 40.1, 45.0 and 70.0AA~ away from the centre of a Cr cluster in Fe with radius equals to 40AA. 1 CF/18/295 Download
Figure 10 Fig 10: c2_vt4 (a) 1st nearest neighbour radially resolved correlation function (1nRRCF) in the long time limit as a function of radius $|mathbf{r}-mathbf{r}_0|$ from the centre of the Cr precipitate. The 1nRRCF is converged and plotted for time t=200ps at 100K, 500K, 800K and 1200K. (b) Averaged magnetic configurations of representative unit cells for $|mathbf{r}-mathbf{r}_0|$ = (i) 10.1AA, (ii) 37.15AA, (iii) 40.0AA~ and (iv) 69.97AA~ at $T=$100K. 1 CF/18/297 Download
Figure 11 Fig 11: xc_mxvt4 (i) Radially averaged magnetisation and (ii) magnetic susceptibility as a function of radius from the centre of a spherical $alpha'$ cluster with radius of 40AA. The Cr precipitate has a concentration of (a) 80 at.%Cr, (b) 60 at.%Cr, (c) 40 at.%Cr and (d) 20 at.%Cr. Three samples are simulated for each case. The averaged value is shown as the solid line. The region between the smallest and largest values measured for each simulation is shaded. 1 CF/18/299 Download
Figure 12 Fig 12: momspace_clusters2 Time averaged magnetic moment space diagrams for the Cr clusters of different local concentrations considered in Fig. ref{fig:xc_mxvt} at 100K, 500K, 900K and 1500K. Dots correspond to origin centred Fe moments and blue for Cr. Each map is shown with respect to a scale of 1$mu_B$. Magnetic moments were averaged over every time step during the 5ps production run. 2 CF/18/301 Download
Figure 13 Fig 13: interface_benchmark3 Magnetic properties across a (001) Fe-Cr interface. (a) Magnetisation with respect to bulk Fe and with bulk Fe scaled proportional to the nominal concentration (50 at.%Cr). (b) Snap-shot of the magnetic moments at the interface at 100K. Fe magnetic moments are in orange. Cr are in grey. (c) Magnetic moment space diagram for the interface system at 100K (i) Coloured with respect to atom type. Blue corresponds to Cr moments. Red corresponds to Fe, and (ii) the sum of the nearest neighbours occupational site variables. (d) Layer averaged magnetisation for each (001) layer as a function of temperature. 1 CF/18/304 Download
Figure 14 Fig 14: interface_x3 Magnetic susceptibility as a function of temperature for each (001) layer in a Fe/Cr superlattice. 1 CF/18/306 Download
Figure 15 Fig 15: interface_corr4 Time-displaced spin-spin autocorrelation and nearest neighbour correlation functions of each (001) layer in a Fe/Cr superlattice at 100K, 800K and 1200K. Fe/Cr interface between layers 80 & 1, and 40 & 41. Curves of interest are labelled according to the atoms in the layer and their proximity to the interface. The Fe interfacial layers (1,40) are labelled as Fe(1). The next layers (2,39) are labelled Fe(2), Cr interface layers (41,80) are labelled Cr(1), and so on. 1 CF/18/308 Download