T al. [10] is taken into account so that you can receive right final results: Hsp- ph = 1 1 z z F(i, j, k)Qi Sz Sk – 4 R(i, j, r, s)Qi Q j Srz Ss h.c. j 2 i,j,k i,j,r,s (four)where F and R would be the spin-phonon coupling constants in the very first and second order. The anharmonic phonon-phonon interactions are provided by: H ph= 1 2! 1 four!0i ai ai three! B(i, j, r)Qi Q j Qri i,j,r i,j,r,sA(i, j, r, s) Qi Q j Qr Qs ,(five)where Qi and 0i are the regular coordinate and frequency of the lattice mode. From the phonon Green’s function, defined by way of the phonon creation a and annihilation a Ziritaxestat Protocol operators Gij (t) = ai (t); a (6) j is observed the phonon power and phonon damping = sp- ph ph- ph (7)applying the complete Hamiltonian and also the approach of Tserkovnikov [31]. The Ising model inside a transverse field describes the ferroelectric properties. It can be applied to order-disorder (KH2 PO4 ) and displacive (BaTiO3 ) variety ferroelectrics [32,33]. The Hamiltonian reads: 1 He = Bix – (1 – x ) Jij Biz Bz , (8) j two ij i where Bix , Biz are the spin-1/2 operators on the pseudo-spins, Jij denotes the pseudo-spin interaction, could be the tunneling frequency, and x would be the concentration from the doped ions at Y states. The Y ion displacement and the FeO6 octahedral distortion cause the spontaneous polarization [34,35], which is calculated to be: Ps = 1 NiBix ; 0;1 NiBiz .(9)Hme defines the magnetoelectric interaction involving the two -Irofulven supplier subsystems: Hme = – (Ps eij ) (Si S j ).ij(ten)exactly where may be the coupling constant and eij could be the unit vector along the direction between the nearest-neighbours Fe3 -ions.Nanomaterials 2021, 11, 2731 Nanomaterials 2021, 11,4 of 11 4 ofThe band gap energy Eg of YFO is defined by the difference amongst the valence and also the band gap energy Eg of YFO is defined by the difference among the valence and conduction bands: conduction bands: Eg = ( k = 0) – – ( k = k ). (11) Eg = ( k = 0) – – ( k = k ). (11) The electronic energies The electronic energies (k ) = k – I Szz (12) (k) = k – 2 I S (12) 2 are observed from the Green’s function g(k, ) = ck, ; ck , = , ci and ci are are observed in the Green’s function g(k, ) = ck, ; c , = , ci and ci are k Fermi operators, and I will be the s-d interaction constant [36]. Fermi operators, and I is definitely the s-d interaction continual [36]. 3. Final results and Discussion three. Results and Discussion z A specific Fe-spin is fixed in the center of your nanoparticle with an icosahedral symmeA specific Fe-spin is fixed within the center from the nanoparticle with an icosahedral symmetry. All spins are included into shells numbered by n = 1, …, N. n = 1 denotes the central try. All spins are integrated into shells numbered by n = 1, …, N. n = 1 denotes the central spin and n = N represents the surface shell [37]. spin and n = N represents the surface shell [37]. The numerical calculations are made working with the following model parameters: J = -13.8 cm-11 , The numerical calculations are produced using1the following model parameters:1 J = -13.eight cm- , -1 , J = 575 cm-1 , = 21.four cm- , D = four.25 cm-1 , K = 0.09 cm- , = 1.four cm-1 , J = -3.45 cm -1 J = -3.45 cm , J = 575 cm-1 , = 21.4 cm-1 , D = four.25 cm-1 , K = 0.09 cm-1 , = 1.four cm-1 , TN = 640 K, TC = 420 K [2,38], F = 21 cm-11 R = -18 cm-11 B = – three cm-11 and a = six.six cm-11 , , , . TN = 640 K, TC = 420 K [2,38], F = 21 cm- , R = -18 cm- , B = – three cm- , along with a = six.six cm- .three.1. Size and Shape Dependence on the Magnetization 3.1. Size and Shape Dependence in the Magnetization We’ll very first demonstrate the siz.