Calculating the intensities of magnetic scattering
In this program for the
calculation of intensity of magnetic scattering the following expression is
used: I=kL|F|2exp(-2W(q))
where L is the Lorentz factor, F is the magnetic structural amplitude, exp(-2W(q))-the
Debye-Waller factor. The current version of the program supports only single
type of the Lorentz factor: L=1/Sin(Θ)Sin(2Θ),
assuming the experiment with the "bathing" sample.
F is given as:
F(q)=F'(q)-(eF'(q))e,
where e is normalized vector directed along the scattering vector. As
the program is intended for search of only approximate coincidence between
experimental and calculated diffraction patterns, other factors influencing on
the intensity are neglected. As the integral intensities is given in "counts",
all the other factors, identical for both of nuclear and magnetic intensity, is
taken into account by appropriate normalization of the experimental
intensities.
The case of the one - arm star.
For the one-arm star, in conformity
with the requirement of reality of vector of magnetic moment
Snj
for any translation vector tn,
only ferromagnetic and antiferromagnetic structures is possible.
The ferromagnetic structure.
The structural amplitude of the
magnetic scattering normalized on the number of unit cells in the crystal is
given as (for q = b):
F'(q)|q=b=∑jexp(-iqrj)S0jfj(q),
F(q)|q=b=∑jexp(-iqrj)S0jfj(q)(mj-(emj)e) (*),
where mj is the normalized vector directed along the magnetic moment of j-th atom.
The antiferromagnetic structure.
For this type of
magnetic structure the structural amplitude is defined by the expression
similar to (*). The difference is the maximums of coherent scattering are
observed for the scattering vectors q = k + b.
The case of the two-arms star.
The case of the two-arms star.
The ferromagnetic spiral (FS).
For this type of
magnetic ordering the structural amplitudes normalized on the number of unit
cells in the crystal are defined by:
F'(q)|q=b=m∑jexp(-iqrj)S0jmfj(q),
where m is the normalized vector directed along the spiral axis.
F(q)|q=b=(m-(em)e)∑jexp(-iqrj)S0jmfj(q),
F(q)|q=b+k=∑jexp(-iqrj)1/2S0jm1m2fj(q)((m1j-(em1j)e)-i(m2j-(em2j)e)),
F(q)|q=b-k=∑jexp(-iqrj)1/2S0jm1m2fj(q)((m1j-(em1j)e)+i(m2j-(em2j)e)),
F(q)Re|q=b+k=∑j1/2S0jm1m2fj(q)(Cos(qrj)(m1j-(em1j)e)-Sin(qrj)(m2j-(em2j)e)),
F(q)Im|q=b+k=-∑j1/2S0jm1m2fj(q)(Sin(qrj)(m1j-(em1j)e)+Cos(qrj)(m2j-(em2j)e)),
F(q)Re|q=b-k=∑j1/2S0jm1m2fj(q)(Cos(qrj)(m1j-(em1j)e)+Sin(qrj)(m2j-(em2j)e)),
F(q)Im|q=b-k=∑j1/2S0jm1m2fj(q)(-Sin(qrj)(m1j-(em1j)e)+Cos(qrj)(m2j-(em2j)e)),
The elliptic spiral (ES).
By analogy with the
FS-structure the expressions for the real and imaginary parts of structural
amplitude defined by:
F(q)Re|q=b+k=∑j1/2S0jfj(q)(Cos(qrj)(m1j-(em1j)e)-pjSin(qrj)(m2j-(em2j)e)),
F(q)Im|q=b+k=-∑j1/2S0jfj(q)(Sin(qrj)(m1j-(em1j)e)+pjCos(qrj)(m2j-(em2j)e)),
F(q)Re|q=b-k=∑j1/2S0jfj(q)(Cos(qrj)(m1j-(em1j)e)+pjSin(qrj)(m2j-(em2j)e)),
F(q)Im|q=b-k=∑j1/2S0jfj(q)(-Sin(qrj)(m1j-(em1j)e)+pjCos(qrj)(m2j-(em2j)e)),
The longitude spin wave combined with the transverse spin wave (LSW/TSW).
F(q)|q=b+k=F(q)|q=b-k=∑jexp(-iqrj)1/2S0jfj(q)(mj-(emj)e)
