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Phase diagrams of a transverse cubic nanowire with diluted surface shell

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Abstract

The effective-field theory with correlations based on the probability distribution technique has been used to investigate the phase diagrams (critical and compensation temperatures) of a transverse antiferromagnetic spin-\(\frac{1}{2}\) Ising cubic nanowire with diluted surface shell. It is found that the phase diagrams of the system are strongly affected by the surface shell parameters. Indeed, two compensation points appear for certain values of Hamiltonian parameters, and the range of appearance of these latter points depends strongly on the surface shell transverse field.

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Acknowledgments

This work has been initiated with the support of URAC: 08, the project RS:02(CNRST) and the Swedish Research Links programme dnr-348-2011-7264 and completed during a visit of A. A. at the Max Planck Institut für Physik Komplexer Systeme Dresden, Germany. The authors would like to thank all the organizations.

Author information

Authors and Affiliations

  1. Laboratoire de Physique des Matériaux et Modélisation des Systèmes, (LP2MS), Unité Associée au CNRST-URAC 08, Physics Department, Faculty of Sciences, University of Moulay Ismail, B.P. 11201, Meknes, Morocco

    M. El Hamri, S. Bouhou, I. Essaoudi & A. Ainane

  2. Max-Planck-Institut für Physik Complexer Systeme, Nöthnitzer Str. 38, 01187, Dresden, Germany

    A. Ainane

  3. Condensed Matter Theory Group, Department of Physics and Astronomy, Uppsala University, 75120, Uppsala, Sweden

    A. Ainane & R. Ahuja

  4. Laboratoire de Chimie et Physique (LCP-A2MC), Institut de Chimie, Physique et Matériaux (ICPM), 1 Bd. Arago, 57070, Metz, France

    F. Dujardin

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  1. M. El Hamri
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  2. S. Bouhou
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  3. I. Essaoudi
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Corresponding author

Correspondence to A. Ainane.

Appendices

Appendix 1: Equations of the core and surface shell magnetizations

Within the framework of the effective field theory with correlations, the longitudinal magnetizations of the core, namely \(m_{{c}_{1}}^{z},\,m_{{c}_{2}}^{z}\) and \(m_{{c}_{3}}^{z}\), and the longitudinal magnetizations of the surface shell, namely \(m_{{s}_{1}}^{z},\,m_{{s}_{2}}^{z}\) and \(m_{{s}_{3}}^{z} \), can be obtained as:

Magnetization of the central spin \(c_{1}\):

$$\begin{aligned} m_{{c}_{1}}^{z} & = \frac{1}{2^{(3N_{2})}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}2^{\mu_{3}}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}} \\ &\quad\times \left( 1-c\right) ^{\mu_{3}}\left( 1-m_{{c}_{1}}^{z}\right) ^{\mu_{1}}\left( 1+m_{{c}_{1}}^{z}\right) ^{N_{2}-\mu_{1}} \\ &\quad\times \left( 1-m_{{c}_{2}}^{z}\right) ^{\mu_{2}}\left( 1+m_{{c}_{2}}^{z}\right) ^{N_{2}-\mu_{2}}\left( c-m_{{s}_{2}}^{z}\right) ^{\nu_{3}} \\ &\quad\times \left( c+m_{{s}_{2}}^{z}\right) ^{N_{2}-\mu_{3}-\nu_{3}}f_{z}(J_{\rm c}(2N_{2}-2(\mu_{1}+\mu_{2})) \\ &\quad+J_{\rm cs}(N_{2}-(\mu_{3}+2\nu_{3})),\Omega_{\rm c}) \end{aligned}$$
(21)

Magnetization of the central spin \(c_{2}\):

$$\begin{aligned} m_{{c}_{2}}^{z}&= \frac{1}{2^{(2N_{1}+2N_{2})}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}2^{\mu_{4}}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{1}} \\& \quad\times C_{\mu_{4}}^{N_{1}}C_{\nu_{4}}^{N_{1}-\mu_{4}}\left( 1-c\right) ^{\mu_{4}}\left( 1-m_{{c}_{1}}^{z}\right) ^{\mu_{1}}\left( 1+m_{{c}_{1}}^{z}\right) ^{N_{2}-\mu_{1}} \\ &\quad\times \left( 1-m_{{c}_{2}}^{z}\right) ^{\mu_{2}}\left( 1+m_{{c}_{2}}^{z}\right) ^{N_{2}-\mu_{2}}\left( 1-m_{{c}_{3}}^{z}\right) ^{\mu_{3}} \\ &\quad\times \left( 1+m_{{c}_{3}}^{z}\right) ^{N_{1}-\mu_{3}}\left( c-m_{{s}_{3}}^{z}\right) ^{\nu_{4}}\left( c+m_{{s}_{3}}^{z}\right) ^{N_{1}-\mu_{4}-\nu_{4}} \\ &\quad\times f_{z}(J_{\rm c}(N_{1}+2N_{2}-2(\mu_{1}+\mu_{2}+\mu_{3})) \\ &\quad+J_{\rm cs}(N_{1}-(\mu_{4}+2\nu_{4})),\Omega_{\rm c}) \end{aligned}$$
(22)

Magnetization of the central spin \(c_{3}\):

$$\begin{aligned} m_{{c}_{3}}^{z}&= \frac{1}{2^{(N_{2}+N_{4})}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{4}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}C_{\mu_{1}}^{N_{4}}C_{\mu_{2}}^{N_{1}}\left( 1-m_{{c}_{2}}^{z}\right) ^{\mu_{1}}\left( 1+m_{{c}_{2}}^{z}\right) ^{N_{4}-\mu_{1}} \\ &\quad\times \left( 1-m_{{c}_{3}}^{z}\right) ^{\mu_{2}}\left( 1+m_{{c}_{3}}^{z}\right) ^{N_{2}-\mu_{2}}f_{z}(J_{\rm c}(N_{2}+N_{4} \\ &\quad-2(\mu_{1}+\mu_{2})),\Omega_{\rm c}) \end{aligned}$$
(23)

Magnetization of the central spin \(s_{1}\):

$$\begin{aligned} m_{{s}_{1}}^{z}&= c\times \frac{1}{2^{2N_{2}}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{1}=0}^{N_{2}-\mu_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}2^{\mu_{1}+\mu_{2}}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\nu_{1}}^{N_{2}-\mu_{1}} \\ &\quad\times C_{\nu_{2}}^{N_{2}-\mu_{2}}\left( 1-c\right) ^{\mu_{1}+\mu_{2}}\left( c-m_{{s}_{1}}^{z}\right) ^{\nu_{1}}\left( c+m_{{s}_{1}}^{z}\right) ^{N_{2}-\mu_{1}-\nu_{1}} \\ &\quad\times \left( c-m_{{s}_{2}}^{z}\right) ^{\nu_{2}}\left( c+m_{{s}_{2}}^{z}\right) ^{N_{2}-\mu_{2}-\nu_{2}} \\ &\quad\times f_{z}(J_{\rm s}(2N_{2}-(\mu_{1}+\mu_{2}+2(\nu_{1}+\nu_{2}))),\Omega_{\rm s}) \end{aligned}$$
(24)

Magnetization of the surface spin \(s_{2}\):

$$\begin{aligned} m_{{s}_{2}}^{z}&= c\times \frac{1}{2^{3N_{1}+N_{2}}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{1}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}2^{\mu_{2}+\mu_{3}+\mu_{4}} \\ &\quad\times C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}}C_{\nu_{2}}^{N_{1}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{\nu_{4}}^{N_{1}-\mu_{4}}\left( 1-c\right) ^{\mu_{2}+\mu_{3}+\mu_{4}} \\ &\quad\times \left( 1-m_{{c}_{1}}^{z}\right) ^{\mu_{1}}\left( 1+m_{{c}_{1}}^{z}\right) ^{N_{1}-\mu_{1}}\left( c-m_{{s}_{1}}^{z}\right) ^{\nu_{2}} \\ &\quad\times \left( c+m_{{s}_{1}}^{z}\right) ^{N_{1}-\mu_{2}-\nu_{2}}\left( c-m_{{s}_{2}}^{z}\right) ^{\nu_{3}}\left( c+m_{{s}_{2}}^{z}\right) ^{N_{2}-\mu_{3}-\nu_{3}} \\ &\quad\times \left( c-m_{{s}_{3}}^{z}\right) ^{\nu_{4}}\left( c+m_{{s}_{3}}^{z}\right) ^{N_{1}-\mu_{4}-\nu_{4}}f_{z}(J_{\rm s}(2N_{1}+N_{2} \\ &\quad-(\mu_{2}+\mu_{3}+\mu_{4}+2(\nu_{2}+\nu_{3}+\nu_{4})))+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm s}) \end{aligned}$$
(25)

Magnetization of the surface spin \(s_{3}\):

$$\begin{aligned} m_{{s}_{3}}^{z}&=c\times \frac{1}{2^{(N_{1}+2N_{2})}}\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}2^{(\mu_{2}+\mu_{3})}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{2}} \\ &\quad\times C_{\mu_{3}}^{N_{2}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}\left( 1-c\right) ^{\mu_{2}+\mu_{3}}\left( 1-m_{{c}_{2}}^{z}\right) ^{\mu_{1}} \\ &\quad\times \left( 1+m_{{c}_{2}}^{z}\right) ^{N_{1}-\mu_{1}}\left( c-m_{{s}_{2}}^{z}\right) ^{\nu_{2}}\left( c+m_{{s}_{2}}^{z}\right) ^{N_{2}-\mu_{2}-\nu_{2}} \\ &\quad\times \left( c-m_{{s}_{3}}^{z}\right) ^{\nu_{3}}\left( c+m_{{s}_{3}}^{z}\right) ^{N_{2}-\mu_{3}-\nu_{3}}f_{z}(J_{\rm s}(2N_{2} \\ &\quad-(\mu_{2}+\mu_{3}+2(\nu_{2}+\nu_{3})))+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm s}) \end{aligned}$$
(26)

Appendix 2

The coefficients A(i,j) of the matrix M are given by

$$\begin{aligned} A(1,1)&= 2^{-\left( 3N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\mu_{1}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{1}}2^{(\mu_{3})}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}} \\ &\quad\times C_{i}^{\mu_{1}}C_{j}^{N_{2}-\mu_{1}}(1-c)^{\mu_{3}}c^{N_{2}-\mu_{3}}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad f_{z}(J_{\rm c}(2N_{2}-2(\mu_{1}+\mu_{2}))+J_{\rm cs}\left( N_{2}-(\mu_{3}+2\nu_{3}\right) ),\Omega_{\rm c}) \end{aligned}$$
(27)
$$\begin{aligned} A(1,2)&= 2^{-\left( 3N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\mu_{2}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{2}}2^{(\mu_{3})}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}} \\ &\quad\times C_{i}^{\mu_{2}}C_{j}^{N_{2}-\mu_{2}}(1-c)^{\mu_{3}}c^{N_{2}-\mu_{3}}\left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm c}(2N_{2} \\ &\quad-2(\mu_{1}+\mu_{2}))+J_{\rm cs}\left( N_{2}-(\mu_{3}+2\nu_{3}\right) ),\Omega_{\rm c}) \end{aligned}$$
(28)
$$\begin{aligned} A(1,5)&= 2^{-\left( 3N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\nu_{3}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{3}-\nu_{3}}2^{(\mu_{3})}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}} \\ &\quad\times C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{i}^{\nu_{3}}C_{j}^{N_{2}-\mu_{3}-\nu_{3}}(1-c)^{\mu_{3}}c^{N_{2}-\mu_{3}-i-j}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm c}(2N_{2}-2(\mu_{1}+\mu_{2}))+J_{\rm cs}\left( N_{2}-(\mu_{3}+2\nu_{3}\right) ),\Omega_{\rm c}) \end{aligned}$$
(29)
$$\begin{aligned} A(2,1)&= 2^{-\left( 2N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\mu_{1}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{1}}2^{(\mu_{4})}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}} \\ &\quad\times C_{\mu_{3}}^{N_{1}}C_{\mu_{4}}^{N_{1}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\mu_{1}}C_{j}^{N_{2}-\mu_{1}}(1-c)^{\mu_{4}}c^{N_{1}-\mu_{4}}\left( -1\right) ^{i} \\ &\quad\times \delta_{i+j,1}f_{z}(J_{\rm c}(N_{1}+2N_{2}-2\left( \mu_{1}+\mu_{2}+\mu_{3}\right) ) \\ &\quad+J_{\rm cs}\left( N_{1}-(\mu_{4}+2\nu_{4})\right) , \Omega_{\rm c}) \end{aligned}$$
(30)
$$\begin{aligned} A(2,2)&= 2^{-\left( 2N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\mu_{2}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{2}}2^{(\mu_{4})}C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}} \\ &\quad\times C_{\mu_{3}}^{N_{1}}C_{\mu_{4}}^{N_{1}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\mu_{2}}C_{j}^{N_{2}-\mu_{2}}(1-c)^{\mu_{4}}c^{N_{1}-\mu_{4}}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm c}(N_{1}+2N_{2}-2\left( \mu_{1}+\mu_{2}+\mu_{3}\right) ) \\ &\quad+J_{\rm cs}\left( N_{1}-(\mu_{4}+2\nu_{4})\right) , \Omega_{\rm c}) \end{aligned}$$
(31)
$$\begin{aligned} A(2,3)&= 2^{-\left( 2N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\mu_{3}}\mathop {\sum }\limits_{j=0}^{N_{1}-\mu_{3}}2^{(\mu_{4})}C_{\mu_{1}}^{N_{2}} \\ &\quad\times C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{1}}C_{\mu_{4}}^{N_{1}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\mu_{3}}C_{j}^{N_{1}-\mu_{3}}(1-c)^{\mu_{4}}c^{N_{1}-\mu_{4}} \\ &\quad\times \left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm c}(N_{1}+2N_{2}-2\left( \mu_{1}+\mu_{2}+\mu_{3}\right) ) \\ &\quad+J_{\rm cs}\left( N_{1}-(\mu_{4}+2\nu_{4})\right) , \Omega_{\rm c}) \end{aligned}$$
(32)
$$\begin{aligned} A(2,6)&= 2^{-\left( 2N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\nu_{4}}\mathop {\sum }\limits_{j=0}^{N_{1}-\mu_{4}-\nu_{4}}2^{(\mu_{4})}C_{\mu_{1}}^{N_{2}} \\ &\quad\times C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{1}}C_{\mu_{4}}^{N_{1}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\nu_{4}}C_{j}^{N_{1}-\mu_{4}-\nu_{4}}(1-c)^{\mu_{4}}c^{N_{1}-\mu_{4}-i-j} \\ &\quad\times \left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm c}(N_{1}+2N_{2}-2\left( \mu_{1}+\mu_{2}+\mu_{3}\right) ) \\ &\quad+J_{\rm cs}\left( N_{1}-(\mu_{4}+2\nu_{4})\right) , \Omega_{\rm c}) \end{aligned}$$
(33)
$$\begin{aligned} A(3,2)&= 2^{-\left( N_{2}+N_{4}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{4}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{i=0}^{\mu_{1}}\mathop {\sum }\limits_{j=0}^{N_{4}-\mu_{1}}C_{\mu_{1}}^{N_{4}}C_{\mu_{2}}^{N_{2}}C_{i}^{\mu_{1}}C_{j}^{N_{4}-\mu_{1}} \\ &\quad\times \left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm c}(N_{2}+N_{4}-2\left( \mu_{1}+\mu_{2}\right) ),\Omega_{\rm c}) \end{aligned}$$
(34)
$$\begin{aligned} A(3,3)&= 2^{-\left( N_{2}+N_{4}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{4}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{i=0}^{\mu_{2}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{2}}C_{\mu_{1}}^{N_{4}}C_{\mu_{2}}^{N_{2}}C_{i}^{\mu_{2}}C_{j}^{N_{2}-\mu_{2}} \\ &\quad\times \left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm c}(N_{2}+N_{4}-2\left( \mu_{1}+\mu_{2}\right) ),\Omega_{\rm c}) \end{aligned}$$
(35)
$$\begin{aligned} A(4,4)&= 2^{-\left( 2N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{1}=0}^{N_{2}-\mu_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{i=0}^{\nu_{1}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{1}-\nu_{1}}2^{(\mu_{1}+\mu_{2})} \\ &\quad\times C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\nu_{1}}^{N_{2}-\mu_{1}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{i}^{\nu_{1}}C_{j}^{N_{2}-\mu_{1}-\nu_{1}}(1-c)^{\mu_{1}+\mu_{2}} \\ &\quad\times c^{2N_{2}-\mu_{1}-\mu_{2}-i-j}\left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm s}(2N_{2} \\ &\quad-\left( \mu_{1}+\mu_{2}+2(\nu_{1}+\nu_{2})\right) ),\Omega_{\rm c}) \end{aligned}$$
(36)
$$\begin{aligned} A(4,5)&= 2^{-\left( 2N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{1}=0}^{N_{2}-\mu_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{i=0}^{\nu_{2}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{2}-\nu_{2}}2^{(\mu_{1}+\mu_{2})} \\ &\quad\times C_{\mu_{1}}^{N_{2}}C_{\mu_{2}}^{N_{2}}C_{\nu_{1}}^{N_{2}-\mu_{1}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{i}^{\nu_{2}}C_{j}^{N_{2}-\mu_{2}-\nu_{2}}(1-c)^{\mu_{1}+\mu_{2}} \\ &\quad\times c^{2N_{2}-\mu_{1}-\mu_{2}-i-j}\left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm s}(2N_{2} \\ &\quad-\left( \mu_{1}+\mu_{2}+2(\nu_{1}+\nu_{2})\right) ),\Omega_{\rm c}) \end{aligned}$$
(37)
$$\begin{aligned} A(5,1)&= 2^{-\left( 3N_{1}+N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{1}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\mu_{1}}\mathop {\sum }\limits_{j=0}^{N_{1}-\mu_{1}} \\ &\quad\times 2^{(\mu_{2}+\mu_{3}+\mu_{4})}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}}C_{\nu_{2}}^{N_{1}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\mu_{1}} \\ &\quad\times C_{j}^{N_{1}-\mu_{1}}(1-c)^{\mu_{2}+\mu_{3}+\mu_{4}}c^{2N_{1}+N_{2}-\mu_{2}-\mu_{3}-\mu_{4}}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm s}(2N_{1}+N_{2}-(\mu_{2}+\mu_{3}+\mu_{4} \\ &\quad+2(\nu_{2}+\nu_{3}+\nu_{4})))+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm c}) \end{aligned}$$
(38)
$$\begin{aligned} A(5,4) &= 2^{-\left( 3N_{1}+N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{1}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\nu_{2}}\mathop {\sum }\limits_{j=0}^{N_{1}-\mu_{2}-\nu_{2}} \\ &\quad\times 2^{(\mu_{2}+\mu_{3}+\mu_{4})}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}}C_{\nu_{2}}^{N_{1}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\nu_{2}} \\ &\quad\times C_{j}^{N_{1}-\mu_{2}-\nu_{2}}(1-c)^{\mu_{2}+\mu_{3}+\mu_{4}}c^{2N_{1}+N_{2}-\mu_{2}-\mu_{3}-\mu_{4}-i-j}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm s}(2N_{1}+N_{2}-(\mu_{2}+\mu_{3}+\mu_{4}+2(\nu_{2}+\nu_{3}+\nu_{4}))) \\ &\quad+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm c}) \end{aligned}$$
(39)
$$\begin{aligned} A(5,5)&= 2^{-\left( 3N_{1}+N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{1}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\nu_{3}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{3}-\nu_{3}} \\ &\quad\times 2^{(\mu_{2}+\mu_{3}+\mu_{4})}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}}C_{\nu_{2}}^{N_{1}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\nu_{3}} \\ &\quad\times C_{j}^{N_{2}-\mu_{3}-\nu_{3}}(1-c)^{\mu_{2}+\mu_{3}+\mu_{4}}c^{2N_{1}+N_{2}-\mu_{2}-\mu_{3}-\mu_{4}-i-j}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm s}(2N_{1}+N_{2}-(\mu_{2}+\mu_{3}+\mu_{4}+2(\nu_{2}+\nu_{3}+\nu_{4}))) \\ &\quad+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm c}) \end{aligned}$$
(40)
$$\begin{aligned} A(5,6)&= 2^{-\left( 3N_{1}+N_{2}\right) }\times c\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{4}=0}^{N_{1}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{1}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{\nu_{4}=0}^{N_{1}-\mu_{4}}\mathop {\sum }\limits_{i=0}^{\nu_{4}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{4}-\nu_{4}} \\ &\quad\times 2^{(\mu_{2}+\mu_{3}+\mu_{4})}C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{1}}C_{\mu_{3}}^{N_{2}}C_{\mu_{4}}^{N_{1}}C_{\nu_{2}}^{N_{1}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{\nu_{4}}^{N_{1}-\mu_{4}}C_{i}^{\nu_{4}} \\ &\quad\times C_{j}^{N_{2}-\mu_{4}-\nu_{4}}(1-c)^{\mu_{2}+\mu_{3}+\mu_{4}}c^{2N_{1}+N_{2}-\mu_{2}-\mu_{3}-\mu_{4}-i-j}\left( -1\right) ^{i}\delta_{i+j,1} \\ &\quad\times f_{z}(J_{\rm s}(2N_{1}+N_{2}-(\mu_{2}+\mu_{3}+\mu_{4}+2(\nu_{2}+\nu_{3}+\nu_{4}))) \\ &\quad+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm c}) \end{aligned}$$
(41)
$$\begin{aligned} A(6,2)&= 2^{-\left( N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\mu_{1}}\mathop {\sum }\limits_{j=0}^{N_{1}-\mu_{1}}2^{(\mu_{2}+\mu_{3})}C_{\mu_{1}}^{N_{1}} \\ &\quad\times C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{i}^{\mu_{1}}C_{j}^{N_{1}-\mu_{1}}(1-c)^{\mu_{2}+\mu_{3}}c^{2N_{2}-\mu_{2}-\mu_{3}} \\ &\quad\times \left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm s}(2N_{2}-(\mu_{2}+\mu_{3}+2(\nu_{2}+\nu_{3}))) \\ &\quad+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm s}) \end{aligned}$$
(42)
$$\begin{aligned} A(6,5)&= 2^{-\left( N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\nu_{2}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{2}-\nu_{2}}2^{(\mu_{2}+\mu_{3})} \\ &\quad\times C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{i}^{\nu_{2}}C_{j}^{N_{2}-\mu_{2}-\nu_{2}}(1-c)^{\mu_{2}+\mu_{3}} \\ &\quad\times c^{2N_{2}-\mu_{2}-\mu_{3}-i-j}\left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm s}(2N_{2} \\ &\quad-(\mu_{2}+\mu_{3}+2(\nu_{2}+\nu_{3})))+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm s}) \end{aligned}$$
(43)
$$\begin{aligned} A(6,6)&= 2^{-\left( N_{1}+2N_{2}\right) }\mathop {\sum }\limits_{\mu_{1}=0}^{N_{1}}\mathop {\sum }\limits_{\mu_{2}=0}^{N_{2}}\mathop {\sum }\limits_{\mu_{3}=0}^{N_{2}}\mathop {\sum }\limits_{\nu_{2}=0}^{N_{2}-\mu_{2}}\mathop {\sum }\limits_{\nu_{3}=0}^{N_{2}-\mu_{3}}\mathop {\sum }\limits_{i=0}^{\nu_{3}}\mathop {\sum }\limits_{j=0}^{N_{2}-\mu_{3}-\nu_{3}}2^{(\mu_{2}+\mu_{3})} \\ &\quad\times C_{\mu_{1}}^{N_{1}}C_{\mu_{2}}^{N_{2}}C_{\mu_{3}}^{N_{2}}C_{\nu_{2}}^{N_{2}-\mu_{2}}C_{\nu_{3}}^{N_{2}-\mu_{3}}C_{i}^{\nu_{3}}C_{j}^{N_{2}-\mu_{3}-\nu_{3}}(1-c)^{\mu_{2}+\mu_{3}} \\ &\quad\times c^{2N_{2}-\mu_{2}-\mu_{3}-i-j}\left( -1\right) ^{i}\delta_{i+j,1}f_{z}(J_{\rm s}(2N_{2} \\ &\quad-(\mu_{2}+\mu_{3}+2(\nu_{2}+\nu_{3})))+J_{\rm cs}(N_{1}-2\mu_{1}),\Omega_{\rm s}) \end{aligned}$$
(44)

where \(N_{1}=1,\,N_{2}=2,\,N_{3}=3,\,N_{4}=4\) and \(N_{6}=6\) denote, respectively, the coordination number, and \(C_{k}^{l}\) are the binomial coefficients \(C_{k}^{l}=\frac{l!}{k!(l-k)!}\).

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El Hamri, M., Bouhou, S., Essaoudi, I. et al. Phase diagrams of a transverse cubic nanowire with diluted surface shell. Appl. Phys. A 122, 202 (2016). https://doi.org/10.1007/s00339-016-9680-z

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