• General info
• Defining reference scales for static calculation
  • Fermi momentum
  • Chemical potentials
  • Examples
    • Fermi momentum is fixed by density in the box center
• Defining reference scales for time-dependent calculations

General info

During the computation process, W-SLDA codes exploit information about typical scales present in the problem. Precisely, reference scales define typical orders of magnitude for computed quantities. The most important reference scale is Fermi momentum. For uniform system it is defined as

• k_F^{(1D)}=\frac{\pi n}{2}
• k_F^{(2D)}=\sqrt{2\pi n}
• k_F^{(3D)}=(3\pi^2 n)^{1/3}

Other reference scales computed automatically from k_F are:

• \varepsilon_F=\frac{1}{2}k_F^2 - Fermi energy,
• E_{\textrm{ffg}}=\frac{3}{5}N\varepsilon_F - energy of free Fermi gas.

Finally, chemical potentials also serve as reference scales for static problems:

• \mu_{\uparrow} - chemical potential is spin-up particles (particles of type a),
• \mu_{\downarrow} - chemical potential is spin-down particles (particles of type b).

Defining reference scales for static calculation

Fermi momentum

There are the following methods of defining the k_F reference scale:

• via input file: k_F is provided by user in input file. To activate this mode you need to uncomment tag referencekF:

referencekF             1.0    # hard set for reference value of kF

• via problem-definition.h file: (VERSION>=2022.02.21) by editing function:

/**
 * This function computes Fermi momentum, which is used as the reference value. 
 * Other reference scales are set automatically to: eF=kF^2/2, Effg=(3/5)*N*eF (N-total number of particles)
 * For more details see: https://gitlab.fizyka.pw.edu.pl/wtools/wslda/-/wikis/Reference%20scales
 * NOTE units are: hbar=m=k_b=1
 * @param it iteration number
 * @param h_densities structure with densities, see (wiki) documentation for list of fields
 * @param params array of input parameters, before call of this routine the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * @return value of Fermi momentum for your problem
 * */
double referencekF(int it, wslda_density h_densities, double *params, size_t extra_data_size, void *extra_data)
{
    if(input->referencekF>0.0) return input->referencekF; // take it from input file
    
    // define here your prescription for computing kF
    // ...
    // default: extract max density and use it for definition of kF
    double max_dens=0.0, kF;
    int ixyz;
    for(ixyz=0; ixyz<h_densities.nx*h_densities.ny*h_densities.nz; ixyz++) 
        if(h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz]>max_dens) max_dens=h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz];
    
    // depending on dimensionality of the problem
    if(NY==1 && NZ==1) kF = 0.5*M_PI*max_dens;                // 1D
    else if(NZ==1)     kF = pow(2.0*M_PI*max_dens,1./2.);     // 2D
    else               kF = pow(3.*M_PI*M_PI*max_dens,1./3.); // 3D
    
    return kF;
}

Chemical potentials

Chemical potentials are adjusted automatically when mode with fixed particle number is executed. For mode with fixed chemical potential see here.

Examples

Fermi momentum is fixed by density in the box center

double referencekF(int it, wslda_density h_densities, double *params, size_t extra_data_size, void *extra_data)
{
    if(input->referencekF>0.0) return input->referencekF; // take it from input file
    
    // Fermi momentum is fixed by density in the box center
    // DETERMINE LOCAL SIZES OF ARRAYS (CODE DIMENSIONALITY DEPENDENT)
    int lNX=h_densities.nx, lNY=h_densities.ny, lNZ=h_densities.nz; // local sizes

    // take value of density in box center and save it to extra_data
    int ixyz = lNZ/2 + lNZ*lNY/2 + lNZ*lNY*lNX/2;
    double dens = h_densities.rho_a[ixyz]+h_densities.rho_b[ixyz];

    
    // depending on dimensionality of the problem
    double kF;
    if(NY==1 && NZ==1) kF = 0.5*M_PI*dens;                // 1D
    else if(NZ==1)     kF = pow(2.0*M_PI*dens,1./2.);     // 2D
    else               kF = pow(3.*M_PI*M_PI*dens,1./3.); // 3D
    
    return kF;
}

Defining reference scales for time-dependent calculations

All reference scales are provided together with an initial state, i.e. binary files produced by static codes contain this information. Presently there is no option of changing values for reference scales.