Units¶
Like many PIC codes, Smilei handles only dimension-less variables, normalized to reference quantities.
Basic reference quantities¶
The speed of light, the elementary charge and the electron mass provide the basis of the normalizations in Smilei:
Reference electric charge \(Q_r = e\) (the elementary charge)
Reference mass \(M_r = m_e\) (the electron mass)
Reference velocity \(V_r = c\) (the speed of light)
We can derive from these:
a reference energy \(K_r = m_e c^2\)
a reference momentum \(P_r = m_e c\)
Even with these normalizations, Smilei does not know the scale of the problem: it lacks a reference distance, or equivalently, a reference time.
Arbitrary reference quantities¶
Instead of choosing a physical constant (for example, the electron radius) as a reference, the scale of the problem is not decided a priori, and the user is free to scale the result of the simulation to any value. In fact, quantities are proportional an unknown reference frequency \(\omega_r\), which can be scaled by the user a posteriori.
Usually, \(\omega_r\) will be an important frequency of the problem. For example, if there is a laser, it could be the laser frequency. Or it could be the electron plasma frequency.
From this reference frequency \(\omega_r\), we define:
a reference time \(T_r = 1/\omega_r\)
a reference length \(L_r = c/\omega_r\)
a reference electric field \(E_r = m_e c \omega_r / e\)
a reference magnetic field \(B_r = m_e \omega_r / e\)
a reference particle density \(N_r = \varepsilon_0 m_e \omega_r^2 /e^2\)
a reference current \(J_r = c\, e\, N_r\)
Warning
\(1/N_r\) is a volume, but counter-intuitively, it is not equal to \(L_r^{3}\).
Normalizing all quantities to these references is convenient for resolving Maxwell’s equations, and the charges equation of motion, as it converts them into a dimension-less set of equations:
where \(\mathbf{E}\), \(\mathbf{B}\), \(\mathbf{j}\) and \(\mathbf{\rho}\) are the electric field, magnetic field, current density and charge density, normalized to \(E_r\), \(B_r\), \(J_r\) and \(Q_r N_r\), respectively. \(Z\) and \(\mathbf p\) are a particle’s charge and momentum, normalized to \(Q_r\) and \(P_r\), respectively. Note that the temporal and spatial derivatives are also normalized to \(T_r\) and \(L_r\), respectively.
Tips for the namelist¶
In the namelist, the user must provide all parameters in units of \(Q_r\), \(M_r\), \(V_r\), \(K_r\), \(P_r\), \(T_r\), \(L_r\), \(E_r\), \(B_r\), \(N_r\) or \(J_r\).
This may be cumbersome if you know your input data in other units. However, the namelist is actually a python code that can compute conversions easily.
For example, let us assume that you know your problem size in units of the wavelength. Knowing that the reference wavelength is \(2\pi L_r\), you can multiply all your lengths by \(2\pi\):
from math import pi
wavelength = 2. * pi
cell_length = [0.05 * wavelength]
grid_length = [100. * wavelength]
Problems requiring explicit units¶
Sometimes, Smilei may be requested to compute other things than Maxwell’s equations. That is the case, for example, for computing collisions or ionization. In these situations, equations cannot be normalized to dimension-less terms, and the code must know the value of \(\omega_r\) in physical units. This requires defining an extra parameter in the namelist.
For instance, reference_angular_frequency_SI = 2.*pi*3e8/1e-6 means that
\(L_r = 1\,\mathrm{\mu m} /(2\pi)\).
This information will be used only in some specific parts of the code (collisions, ionization, …)
but not in the main PIC algorithms.
Warning
The outputs of the code are not converted to SI. They are all kept in the reference units listed above.
Quantities integrated over the grid¶
Special care must be taken when considering local quantities that are spatially integrated.
1. The spatially-integrated kinetic energy density
The particle kinetic energy density is naturally in units of \(K_r N_r\). Integrating over space give different results depending on the simulation dimension. In 1D, this space is a length, with units \(L_r\); in 2D, it is a surface, with units \(L_r^2\); and in 3D, it is a volume, with units \(L_r^3\). Overall, the integrated energy has the units \(K_r N_r L_r^D\) where \(D\) is the simulation dimension. Note that we could expect to obtain, in 3D, an energy with units \(K_r\), but counter-intuitively it has the units \(K_r N_r L_r^3\).
These kinetic energies appear, for instance, in the Scalar diagnostics as
Ukin (and associated quantities).
2. The spatially-integrated electromagnetic energy density
The electromagnetic energy density has the units \(E_r^2/\varepsilon_0 = K_r N_r\). Consequently, the spatially-integrated electromagnetic energy density has the units \(K_r N_r L_r^D\); the same as the integrated kinetic energy density above.
These electromagnetic energies appear, for instance, in the Scalar diagnostics as
Uelm (and associated quantities).
3. The space- & time-integrated Poynting flux
The Poynting flux has the units \(E_r B_r / \mu_0 = V_r K_r N_r\). Consequently, the flux integrated over a boundary, and over time, has the units \(V_r K_r N_r L_r^{D-1} T_r = K_r N_r L_r^D\), which is the same as the integrated energy densities above.
This integrated Poynting flux appears, for instance, in the Scalar diagnostics as
Uelm_bnd, PoyXmin, PoyXminInst (and associated quantities).
Macro-particle weights¶
Macro-particles are assigned a statistical weight which measures their contribution to the plasma distribution function. In Smilei, this weight is defined for each particle at the moment of its creation (usually at the beginning of the simulation), and is never modified afterwards. Its definition reads:
As the density is in units of \(N_r\) and the cell hypervolume in units of \(L_r^D\) (where \(D\) is the simulation dimension), then the units of weights is \(N_r L_r^D\).
This definition of weights ensures that they do not depend on the cell hypervolume, i.e. they can be reused in another simulation, as long as \(D\), \(L_r\) and \(N_r\) are unchanged.