Motif MASTER

Possible types of stars and magnetic structures

Presented program now works only with two types of stars: the one-arm star {k} and the two-arms star {k, -k}. The basic expression for magnetic moment of atom with index j in the n-th unit cell is written as: S_nj=Σ_Lexp(-ik_Lt_n)S^L_0j(*), where summation is taken over all the arms of the star.

The source of limitation defining possible types of stars is the following condition: the vectors of magnetic moments S_nj must be real, whereas Fourier - components S^L_0j is generally complex.

Let us consider some consequences of this condition.

The case of the one-arm star.

In this case the expression (*) gets the form: S_nj=S_nj exp(-ikt_n)S_0j. The condition of reality of the moment vector S_nj requires reality of the value exp(-ikt_n). The value exp(-ikt_n) is real when Sin(kt_n)=0. This implies that (kt_n)=πm where m is an integer. As (kt_n)=2π(k_xt_nx+k_yt_ny+k_zt_nz) then 2(k_xt_nx+k_yt_ny+k_zt_nz)=m. This in turn implies that the components of propagation vector k_x, k_y, k_zcan accept only 0 or 1/2 value. This means that for the one - arm star only ferromagnetic or antiferromagnetic structures is possible.

The case of the two-arms star.

In this case the expression (*) gets the form: S_nj=S^k_0jexp(-ikt_n)+S^-k_0jexp(ikt_n). The vector S_nj is real for any translation vector t_n when S^k*_0j=S^-k_0j The two - arms star {k, -k} corresponds to the spiral or half - ordered magnetic structures. This program allows working with the following types of structures:

• The superposition of the longitude spin wave (LSW) and the transverse spin wave (TSW). For this type of magnetic ordering S^k_0j=¹/₂S_0jm, where m is the normalized vector of direction, along which the magnetic moments of all atoms in given sublattice are directed S_nj=¹/₂S_0jm(exp(-ikt_n)+exp(ikt_n)).
• The ferromagnetic spiral (FS) transforming into the simple spiral (SS).
  S^k_0j=¹/₂S_0jm1m2(m_1j+im_2j),
  S_0j=S_0jmm_j+¹/₂S_0jm1m2((m_1j+im_2j)exp(-ikt_n)+(m_1j-im_2j)exp(ikt_n)), where S_0jm1m2 is the value of projection of magnetic moment of atom j onto the plane containing the vectors m₁, m₂, and S_0jm is the projection of vector of magnetic moment on the rotation axis m.
• The elliptic spiral (ES).
  S^k_0j=¹/₂S_0j(m_1j+ip_jm_2j),
  S_nj=¹/₂S_0j>((m_1j+ip_jm_2j)exp(-ikt_n)+(m_1j-ip_jm_2j)exp(ikt_n)), where p_j-spiral ellipse parameter for the magnetic moments of atoms of j-th sublattice, m is the normalized vector directed along rotation axis.
  The vectors m₁, m₂, m form orthonormalized basis. The vector m₁ is defined from the condition t_n=0.
  For the ferromagnetic spiral S_0j=S_0jmm_j+S_0jm1m2m_1j, S_0jm=(S_0jm_j), hence m_1j is the normalized vector directed along the vector S_0j-(S_0jm_j)m_j.
  For the elliptic spiral S_0j=S_0jm_1j, that is the m_1j is simply the normalized vector directed along magnetic moment of the j-th atom.