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Scattering intensity

The dynamical structure factor could be obtained from the inelastic scattering of the X-ray under the following conditions [7]: i) the scattering process is dominated by the Thompson term and there is no electronic excitation in the energy transfers region, ii) adiabatic approximation is valid, so that the center of mass of the electronic clouds could be identified with the nuclear one. The first assumption is simultaneously satisfied with X-ray with energy in the $\approx 10\div 25~keV$ energy range used for IXS experiments, when we look at energy transfers $\leq 0.4 ~eV$ ($\leq 100 ~THz$). The second one is a very general assumption in condensed matter theory. Under these assumptions we can write the inelastic cross section for one-phonon process as follows. First, we decompose the moment as $\bf {Q}=\bf {G}+\bf {q}$ (in the reciprocal lattice vector $\bf {G}$ plus the phonon pseudo-momentum $\bf {q}$; hence we obtain for a given angle $\Omega$ and frequency $\omega$:


$\displaystyle (\frac{\partial^2\sigma}{\partial\Omega\partial\omega}) \propto$ $\textstyle \frac{k_i}{k_f} (\bf {\hat{e}_i} \cdot \bf {\hat{e}_f})^2
\sum_{\bf ...
...bf {R}_{\kappa}}
~\bf {Q} \cdot \bf {e} (\kappa \vert^{\bf {q}}_\jmath)~\vert^2$    
  $\textstyle \times \langle n_\jmath+ \frac{1}{2}\pm \frac{1}{2}\rangle
~\delta(\omega\mp\omega_\jmath (\bf {q})) ~\Delta(\bf {Q}\mp(\bf {G}+\bf {q}))$   (17)

where $m_{\kappa}$ is the mass for the atom $\kappa$ in the unit cell, $e^{W_{\kappa}}$ is the Debye-Waller factor (which is not calculated in the present version of OpenPhonon), and $n_\jmath$ is the Bose factor for the mode $\jmath$. The phonon polarization term is $\bf {e} (\kappa \vert^{\bf {q}}_\jmath)=u_{\kappa}\sqrt{m_{\kappa}}$ of Eq. 14.

Then the the inelastic cross section can be written as:


$\displaystyle (\frac{\partial^2\sigma}{\partial\Omega\partial\omega}) \propto$ $\textstyle \frac{k_i}{k_f} (\bf {\hat{e}_i} \cdot \bf {\hat{e}_f})^2
\sum_{\bf ...
...\kappa}}
~\bf {Q} \cdot \bf {e} (\kappa \vert^{\bf {G}+\bf {q}}_\jmath)~\vert^2$    
  $\textstyle \times \langle n_\jmath+ \frac{1}{2}\pm \frac{1}{2}\rangle
~\delta(\omega\mp\omega_\jmath (\bf {q})) ~\Delta(\bf {Q}\mp(\bf {G}+\bf {q}))$   (18)

This corresponds to the neutron scattering cross section for 1-phonon scattering provided one replaces the nuclear average scattering length $\bar{b}_{\kappa}$ with the atomic form factor $f_{\kappa}(\bf {Q})$ for the atom $\kappa$ in the unit cell, and the polarisation factor of the Thompson scattering $(\bf {\hat{e}_i} \cdot \bf {\hat{e}_f})^2$. The latest term is fixed $=1$ in this version, because most of the experiments considered where in the scattering confguration with $(\bf {\hat{e}_i} \cdot \bf {\hat{e}_f})^2=1$ [4] or $(\bf {\hat{e}_i} \cdot \bf {\hat{e}_f})^2\approx 1$ (low $\bf {Q}$). Moreover for high energy resolution inelastic X-ray scattering we have $\frac{k_i}{k_f} \cong 1$. For a more detailed description of the IXS cross section, see also the review of E. Burkel [8] and references therein.

In OpenPhonon the calculation is made calling the program scattering_shifted.py in the following way :

python scattering_shifted.py storeall paramfile
where paramfile is the name of a file like this :
 
BrillShift_List=[(6,0,0)]
scatt_dictio_f0={ 'Cu_G0':'Cu1+', 'O_G0':'O','O_G1':'0',\
     'Nd_G0':'Nd','Nd_G1':'Nd'   }
scatt_dictio_f12={ 'Cu_G0':'Cu', 'O_G0':'O','O_G1':'0',\
     'Nd_G0':'Nd','Nd_G1':'Nd'   }
Lambda=0.783867
Temp = 15.0
A sample input file is delivered in the distribution archive under the name scatt_shinput. The intensities are calculated over the set of points defined by the variable $Q$ defined in the previous input files, at which points the eigenvectors and frequencies have been calculated and stored in the file store all. The variable $BrillShift\_List$ specifies a set of shift. So one can calculate once the dispersion from $\Gamma$ to $X$ (just to to give an example ) and use the resulting eigenvectors to calculate the scattering intensities at several reciprocal vector zones.

The atomic scattering factors are read using the Dabax database. The $f0$ factors are read from WaasKirf compilation and f1,f2 from Windt datafile.

To specify the scattering factors of the atoms one has to specify the wavelength of X-rays and the two dictionaries, one for $f0$ and the other for (f1,f2). The dictionary keywords are the equivalence class names and their values are Dabax record names like "Cu" for neutral copper or "Cu1+" for copper ion. These names must exist in the database otherwise an error occurs. Finally the temperature ( in Kelvin) has to be entered to properly take into account the Bose-Einstein statistics.

The results are stored in the file scatt_results in the form:
Qx,Qy,Qz,frequency(1),intensity_stoke(1),intensity_antistoke(1),..,
frequency(j),intensity_stoke(j),intensity_antistoke(j),..,
frequency(N),intensity_stoke(N),intensity_antistoke(N)
where j indicates the mode in increasing $\Gamma$ frequency order.


next up previous contents
Next: Calculation of the partial Up: Files structure and use Previous: Use of an arbitrary   Contents
Alessandro Mirone 2003-11-17