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Theoretical Investigation of Phase Diagrams and Compensation Behaviors of a Ferrimagnetic Mixed-Spin (3/2,2) Ising Nanowire with Cylindrical Core-Shell Structure

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Abstract

In this work, we use effective field theory (EFT) based on the probability distribution technique to study the magnetic properties (phase diagrams and magnetization curves) of a cylindrical mixed-spin (3/2, 2) nanowire. The system has a core-shell composition, where the core consists of \(3/2-\)spins coupled to the \(2-\)spins of the shell in a ferrimagnetic manner. The effect of reduced coupling constants, uniaxial anisotropy on the system, as well as different types of magnetization such as P-type, Q-type, and the N-type characteristic of compensation behavior are investigated.

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Acknowledgements

The authors would like to acknowledge nancial support from the National Centre for Scientific and Technical Research (CNRST) -Morocco.

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Authors and Affiliations

  1. LPHE, Modeling & Simulations, Faculty of Science, Mohammed V University in Rabat, Rabat, Morocco

    A. Arbaoui, K. Htoutou, L. B. Drissi & R. Ahl Laamara

  2. CRMEF, Centre Régional des Métiers de l’Education et de la Formation, Fes-Meknes, Morocco

    K. Htoutou

  3. CPM, Centre of Physics and Mathematics, Faculty of Science, Mohammed V University in Rabat, Rabat, Morocco

    L. B. Drissi & R. Ahl Laamara

  4. Hassan II Academy of Science and Technology, Rabat, Morocco

    L. B. Drissi

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  1. A. Arbaoui
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  2. K. Htoutou
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  3. L. B. Drissi
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  4. R. Ahl Laamara
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Appendix

Appendix

The probability distributions of the spin states \((S=3/2)\) and \((\mu =2)\) are as follows:

$$\begin{aligned} p_{i}\left( S_{iz}\right)= & {} \frac{1}{48}[\left( -3-2(m_{iz}^{c})_{\alpha }+12(q_{iz}^{c})_{\alpha }+8(r_{iz}^{c})_{\alpha }\right) \delta \left( S_{iz}+\frac{3}{2}\right) \\ \nonumber&+3\left( 9+18(m_{iz}^{c})_{\alpha }-4(q_{iz}^{c})_{\alpha }-8(r_{iz}^{c})_{\alpha }\right) \delta \left( S_{iz}+\frac{1}{2}\right) \nonumber \\ \nonumber&+3\left( 9-18(m_{iz}^{c})_{\alpha }-4(q_{iz}^{c})_{\alpha }+8(r_{iz}^{c})_{\alpha }\right) \delta \left( S_{iz}-\frac{1}{2}\right) \nonumber \\ \nonumber&+\left( -3+2(m_{iz}^{c})_{\alpha }+12(q_{iz}^{c})_{\alpha }-8(r_{iz}^{c})_{\alpha }\right) \delta \left( S_{iz}-\frac{3}{2}\right)] \end{aligned}$$
(11)

and :

$$\begin{aligned} p_{k}(\mu _{kz})= & {} \frac{1}{24}[(24-30(q_{iz}^{s})_{\eta }+6(t_{z}^{s})_{\eta })\delta \left( \mu _{kz}^{s}\right) \\ \nonumber&+(2(m_{z}^{s})_{\eta }-(q_{z}^{s})_{\eta }-2(r_{z}^{s})_{\eta }+(t_{z}^{s})_{\eta })\delta \left( \mu _{kz}+2\right) \nonumber \\ \nonumber&+(-16(m_{z}^{s})_{\eta }+16(q_{z}^{s})_{\eta }+4(r_{z}^{s})_{\eta }-4(t_{z}^{s})_{\eta })\delta \left( \mu _{kz}+1\right) \nonumber \\ \nonumber&+(16(m_{z}^{s})_{\eta }+16(q_{z}^{s})_{\eta }-4(r_{z}^{s})_{\eta }-4(t_{z}^{s})_{\eta })\delta (\mu _{kz}-1) \nonumber \\ \nonumber&+(-2(m_{z}^{s})_{\eta }-(q_{z}^{s})_{\eta }+2(r_{z}^{s})_{\eta }+(t_{z}^{s})_{\eta })\delta \left( \mu _{kz}-2\right)].\end{aligned}$$
(12)

For the core sites, we have :

$$\begin{aligned} \langle \langle (S_{z})_{1}^{p}\rangle \rangle = &\; \frac{1}{48^{N_{1}^{c}}}\underset{\nu _{1}=0}{\overset{N_{1}^{c}}{\sum }}\underset{\nu _{2}=0}{\overset{N_{1}^{c}-\nu _{1}}{\sum }}\underset{\nu _{3}=0}{\overset{N_{1}^{c}-\nu _{1}-\nu _{2}}{\sum }C_{\nu _{1}}^{N_{1}^{c}}}{\small C}_{\nu _{2}}^{N_{1}^{c}-\nu _{1}}{\small C}_{\nu _{3}}^{N_{1}^{c}-\nu _{1}-\nu _{2}}\\&\times \left( -3-2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}+8(r_{z}^{c})_{2}\right) ^{\nu _{1}}\\ &\times \left( 27-54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}+24(r_{z}^{c})_{2}\right) ^{\nu _{2}}\\ &\times \left( 27+54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}-24(r_{z}^{c})_{2}\right) ^{\nu _{3}}\\ &\times \left( -3-2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}+8(r_{z}^{c})_{2}\right) ^{N_{0}-\nu _{1}-\nu _{2}-\nu _{3}}\\ &\times G_{pz}(\frac{1}{2}\left[ 3N_{1}^{c}-6\nu _{1}-4\nu _{2}-2\nu _{3}\right] ,D_{z}^{(c)})\end{aligned}$$
(13)
$$\begin{aligned} \langle \langle (S_{z})_{2}^{p}\rangle \rangle= & {} \frac{1}{24^{N_{2}^{s}}}\frac{1}{48^{N_{2}^{c}}}\underset{\nu _{1}=0}{\overset{N_{2}^{c,1}}{\sum }}\underset{\nu _{2}=0}{\overset{N_{2}^{c,1}-\nu _{1}}{\sum }}\underset{\nu _{3}=0}{\overset{N_{2}^{c,1}-\nu _{1}-\nu _{2}}{\sum }}\underset{\gamma _{1}=0}{\overset{N_{2}^{c,2}}{\sum }}\underset{\gamma _{2}=0}{\overset{N_{2}^{c,2}-\gamma _{1}}{\sum }}\underset{\gamma _{3}=0}{\overset{N_{2}^{c,2}-\gamma _{1}-\gamma _{2}}{\sum }}\underset{k_{1}=0}{\overset{N_{2}^{s,1}}{\sum }}\\ &\underset{k_{2}=0}{\overset{N_{2}^{s,1}-k_{1}}{\sum }}\underset{k_{3}=0}{\overset{N_{2}^{s,1}-k_{1}-k_{2}}{\sum }}\underset{k_{4}=0}{\overset{N_{2}^{s,1}-k_{1}-k_{2}-k_{3}}{\sum }}\underset{j_{1}=0}{\overset{N_{2}^{s,2}}{\sum }}\underset{j_{2}=0}{\overset{N_{2}^{s,2}-j_{1}}{\sum }}\underset{j_{3}=0}{\overset{N_{2}^{s,2}-j_{1}-j_{2}}{\sum }}\underset{j_{4}=0}{\overset{N_{2}^{s,2}-j_{1}-j_{2}-j_{3}}{\sum }}\\ &{\small C}_{\nu _{1}}^{N_{2}^{c,1}}{\small C}_{\nu _{2}}^{N_{2}^{c,1}-\nu _{1}}{\small C}_{\nu _{3}}^{N_{2}^{c,1}-\nu _{1}-\nu _{2}}{\small C}_{\gamma _{1}}^{N_{2}^{c,2}}{\small C}_{\gamma _{2}}^{N_{2}^{c,2}-\gamma _{1}}{\small C}_{\gamma _{3}}^{N_{2}^{c,2}-\gamma _{1}-\gamma _{2}}{\small C}_{k_{1}}^{N_{2}^{s,1}}{\small C}_{k_{2}}^{N_{2}^{s,1}-k_{1}} \\&{\small C}_{k_{3}}^{N_{2}^{s,1}-k_{1}-k_{2}}{\small C}_{k_{4}}^{N_{2}^{s,1}-k_{1}-k_{2}-k_{3}}{\small C}_{j_{1}}^{N_{2}^{s,2}}{\small C}_{j_{2}}^{N_{2}^{s,2}-j_{1}}{\small C}_{j_{3}}^{N_{2}^{s,2}-j_{1}-j_{2}}{\small C}_{j_{4}}^{N_{2}^{s,2}-j_{1}-j_{2}-j_{3}}\\ &\times \left( -3+2(m_{z}^{c})_{1}+12(q_{z}^{c})_{1}-8(r_{z}^{c})_{1}\right) ^{\nu _{1}}\times( 27-54(m_{z}^{c})_{1}\\ &-12(q_{z}^{c})_{1}+24(r_{z}^{c})_{1}) ^{\nu _{2}}\times \left.( 27+54(m_{z}^{c})_{1}-12(q_{z}^{c})_{1}-24(r_{z}^{c})_{1}\right.) ^{\nu _{3}}\\ &\times \left( -3-2(m_{z}^{c})_{1}+12(q_{z}^{c})_{1}+8(r_{z}^{c})_{1}\right) ^{N_{2}^{c,1}-\nu _{1}-\nu _{2}-\nu _{3}}\times ( -3\\ &+2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}-8(r_{z}^{c})_{2}) ^{\gamma _{1}}\times ( 27-54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}\\&+24(r_{z}^{c})_{2}) ^{\gamma _{2}}\times\left( 27+54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}-24(r_{z}^{c})_{2}\right) ^{\gamma _{3}}\\ &\times \left( -3-2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}+8(r_{z}^{c})_{2}\right) ^{N_{2}^{c,2}-\gamma _{1}-\gamma _{2}-\gamma _{3}}\times (24-30(q_{z}^{s})_{1}\\&+6(t_{z}^{s})_{1})^{k_{1}}\times (2(m_{z}^{s})_{1}-(q_{z}^{s})_{1}-2(r_{z}^{s})_{1}+(t_{z}^{s})_{1})^{k_{2}}\times (-16(m_{z}^{s})_{1}\\ &+16(q_{z}^{s})_{1}+4(r_{z}^{s})_{1}-4(t_{z}^{s})_{1})^{k_{3}}\times (16(m_{z}^{s})_{1}+16(q_{z}^{s})_{1}-4(r_{z}^{s})_{1}\\ &-4(t_{z}^{s})_{1})^{k_{4}}\times (-2(m_{z}^{s})_{1}-(q_{z}^{s})_{1}+2(r_{z}^{s})_{1}+(t_{z}^{s})_{1})^{N_{2}^{s,1}-k_{1}-k_{2}-k_{3}-k_{4}}\\ &\times (24-30(q_{z}^{s})_{2}+6(t_{z}^{s})_{2})^{j_{1}}\times (2(m_{z}^{s})_{2}-(q_{z}^{s})_{2}-2(r_{z}^{s})_{2}+(t_{z}^{s})_{2})^{j_{2}}\\ &\times (-16(m_{z}^{s})_{2}+16(q_{z}^{s})_{2}+4(r_{z}^{s})_{2}-4(t_{z}^{s})_{2})^{j_{3}}\times(16(m_{z}^{s})_{2}\\&+16(q_{z}^{s})_{2}-4(r_{z}^{s})_{2}-4(t_{z}^{s})_{2})^{j_{4}}\times (-2(m_{z}^{s})_{2}-(q_{z}^{s})_{2}+2(r_{z}^{s})_{2}\\&+(t_{z}^{s})_{2})^{N_{2}^{s,2}-j_{1}-j_{2}-j_{3}-j_{4}}\times G_{pz}^{(c)}(\frac{1}{2}[3(N_{2}^{c,1}+N_{2}^{c,2})\\ &-6(\nu _{1}+\gamma _{1})-4(\nu _{2}+\gamma _{2})-2(\nu _{3}+\gamma _{3})] +R_{cs}[2(N_{2}^{s,1}+N_{2}^{s,2})\\&-2(k_{1}+j_{1})-3(k_{3}+j_{3})-4(k_{2}+j_{2})-(k_{4}+j_{4})],D_{z}^{(c)}), \end{aligned}$$
(14)

and for the shell sites, we have :

$$\begin{aligned} \langle \langle (\mu _{z})_{1}^{p}\rangle \rangle= & {} \frac{1}{24^{N_{1}^{s}}}\frac{1}{48^{N_{1}^{c}}}\underset{\nu _{1}=0}{\overset{N_{1}^{c}}{\sum }}\underset{\nu _{2}=0}{\overset{N_{1}^{c}-\nu _{1}}{\sum }}\underset{\nu _{3}=0}{\overset{N_{1}^{c}-\nu _{1}-\nu _{2}}{\sum }}\underset{k_{1}=0}{\overset{N_{1}^{s}}{\sum }}\underset{k_{2}=0}{\overset{N_{1}^{s}-k_{1}}{\sum }}\underset{k_{3}=0}{\overset{N_{1}^{s}-k_{1}-k_{2}}{\sum }}\underset{k_{4}=0}{\overset{N_{1}^{s}-k_{1}-k_{2}-k_{3}}{\sum }}\\ &{\small C}_{\nu _{1}}^{N_{1}^{c}}{\small C}_{\nu _{2}}^{N_{1}^{c}-\nu _{1}}{\small C}_{\nu _{3}}^{N_{1}^{c}-\nu _{1}-\nu _{2}}{\small C}_{k_{1}}^{N_{1}^{s}}{\small C}_{k_{2}}^{N_{1}^{s}-k_{1}}{\small C}_{k_{3}}^{N_{1}^{s}-k_{1}-k_{2}}{\small C}_{k_{4}}^{N_{1}^{s}-k_{1}-k_{2}-k_{3}}\\&{\small C}_{k_{4}}^{N_{1}^{s}-k_{1}-k_{2}-k_{3}}\times \left( -3+2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}-8(r_{z}^{c})_{2}\right) ^{\nu _{1}}\\ &\times \left( 27-54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}+24(r_{z}^{c})_{2}\right) ^{\nu _{2}}\times ( 27+54(m_{z}^{c})_{2}\\&-12(q_{z}^{c})_{2}-24(r_{z}^{c})_{2}) ^{\nu _{3}}\times( -3-2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}\\ &+8(r_{z}^{c})_{2}) ^{N_{1}^{c}-\nu _{1}-\nu _{2}-\nu _{3}}\times (24-30(q_{z}^{s})_{2}+6(t_{z}^{s})_{2})^{k_{1}}\\&\times (2(m_{z}^{s})_{2}-(q_{z}^{s})_{2}-2(r_{z}^{s})_{2}+(t_{z}^{s})_{2})^{k_{2}}\times (-16(m_{z}^{s})_{2}\\&+16(q_{z}^{s})_{2}+4(r_{z}^{s})_{2}-4(t_{z}^{s})_{2})^{k_{3}}\times (16(m_{z}^{s})_{2}+16(q_{z}^{s})_{2}\\ &-4(r_{z}^{s})_{2}-4(t_{z}^{s})_{2})^{k_{4}}\times (-2(m_{z}^{s})_{2}-(q_{z}^{s})_{2}+2(r_{z}^{s})_{2}\\&+(t_{z}^{s})_{2})^{N_{1}^{s}-k_{1}-k_{2}-k_{3}-k_{4}}\times F_{pz}^{(s)}(\frac{1}{2}[ 3N_{1}^{c}-6\nu _{1}\\&-4\nu _{2}-2\nu _{3}]+R_{cs}[2N_{1}^{s}-2k_{1}-3k_{3}-4k_{2}-k_{4}] ,D_{z}^{(s)}), \end{aligned}$$
(15)
$$\begin{aligned} \langle \langle (\mu _{z})_{2}^{p}\rangle \rangle= & {} \frac{1}{24^{N_{2}^{s}}}\frac{1}{48^{N_{2}^{c}}}\underset{\nu _{1}=0}{\overset{N_{2}^{c}}{\sum }}\underset{\nu _{2}=0}{\overset{N_{2}^{c}-\nu _{1}}{\sum }}\underset{\nu _{3}=0}{\overset{N_{2}^{c}-\nu _{1}-\nu _{2}}{\sum }}\underset{k_{1}=0}{\overset{N_{2}^{s}}{\sum }}\underset{k_{2}=0}{\overset{N_{2}^{s}-k_{1}}{\sum }}\underset{k_{3}=0}{\overset{N_{2}^{s}-k_{1}-k_{2}}{\sum }}\underset{k_{4}=0}{\overset{N_{2}^{s}-k_{1}-k_{2}-k_{3}}{\sum }}\\&{\small C}_{\nu _{1}}^{N_{2}^{c}}{\small C}_{\nu _{2}}^{N_{2}^{c}-\nu _{1}}{\small C}_{\nu _{3}}^{N_{2}^{c}-\nu _{1}-\nu _{2}}{\small C}_{k_{1}}^{N_{2}^{s}} {\small C}_{k_{2}}^{N_{2}^{s}-k_{1}}{\small C}_{k_{3}}^{N_{2}^{s}-k_{1}-k_{2}}{\small C}_{k_{4}}^{N_{2}^{s}-k_{1}-k_{2}-k_{3}}\\&\times \left( -3+2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}-8(r_{z}^{c})_{2}\right) ^{\nu _{1}}\times( 27-54(m_{z}^{c})_{2}\\&-12(q_{z}^{c})_{2}+24(r_{z}^{c})_{2}) ^{\nu _{2}}\times ( 27+54(m_{z}^{c})_{2}-12(q_{z}^{c})_{2}\\&-24(r_{z}^{c})_{2}) ^{\nu _{3}}\times ( -3-2(m_{z}^{c})_{2}+12(q_{z}^{c})_{2}\\&+8(r_{z}^{c})_{2}) ^{N_{2}^{c}-\nu _{1}-\nu _{2}-\nu _{3}}\times (24-30(q_{z}^{s})_{1}+6(t_{z}^{s})_{1})^{k_{1}}\\ &\times (2(m_{z}^{s})_{1}-(q_{z}^{s})_{1}-2(r_{z}^{s})_{1}+(t_{z}^{s})_{1})^{k_{2}} \times (-16(m_{z}^{s})_{1}\\ &+16(q_{z}^{s})_{1}+4(r_{z}^{s})_{1}-4(t_{z}^{s})_{1})^{k_{3}}\times (16(m_{z}^{s})_{1}+16(q_{z}^{s})_{1}\\&-4(r_{z}^{s})_{1}-4(t_{z}^{s})_{1})^{k_{4}}\times (-2(m_{z}^{s})_{1}-(q_{z}^{s})_{1}+2(r_{z}^{s})_{1}\\ &+(t_{z}^{s})_{1})^{N_{2}^{s}-k_{1}-k_{2}-k_{3}-k_{4}}\times F_{pz}^{(s)}(\frac{1}{2}[ 3N_{2}^{c}-6\nu _{1}-4\nu _{2}-2\nu _{3}]\\ &+R_{cs}[2N_{2}^{s}-2k_{1}-3k_{3}-4k_{2}-k_{4}] ,D_{z}^{(s)}). \end{aligned}$$
(16)

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Arbaoui, A., Htoutou, K., Drissi, L.B. et al. Theoretical Investigation of Phase Diagrams and Compensation Behaviors of a Ferrimagnetic Mixed-Spin (3/2,2) Ising Nanowire with Cylindrical Core-Shell Structure. J Supercond Nov Magn 34, 3413–3423 (2021). https://doi.org/10.1007/s10948-021-05985-w

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