def test_energy():
energy = Accelerator(energy=300e3)
assert energy.energy == 300e3
assert np.isclose(energy.wavelength, energy2wavelength(300e3))
energy.energy = 200e3
assert np.isclose(energy.wavelength, energy2wavelength(200e3))
def point_resolution(Cs: float, energy: float):
"""
Calculate the point resolution.
Parameters
----------
Cs: float
Spherical aberration [Å].
energy: float
Electron energy [eV].
Returns
-------
float
The point resolution.
"""
return (energy2wavelength(energy)**3 * np.abs(Cs) / 6)**(1 / 4)
def scherzer_defocus(Cs, energy):
"""
Calculate the Scherzer defocus.
Parameters
----------
Cs: float
Spherical aberration [Å].
energy: float
Electron energy [eV].
Returns
-------
float
The Scherzer defocus.
"""
return np.sign(Cs) * np.sqrt(
3 / 2 * np.abs(Cs) * energy2wavelength(energy))
def wavelength(self) -> float:
"""
Relativistic wavelength [Å].
"""
self.check_is_defined()
return energy2wavelength(self.energy)
def _evaluate_potential(self):
from sympy.physics.wigner import wigner_3j
self.grid.check_is_defined()
self.accelerator.check_is_defined()
potential = np.zeros(self.gpts, dtype=np.complex64)
kx, ky = spatial_frequencies(self.gpts, self.sampling)
kz = self.momentum_transfer
kt, phi = polar_coordinates(kx, ky)
k = np.sqrt(kt**2 + kz**2)
theta = np.pi - np.arctan(kt / kz)
radial_grid = np.arange(0, np.max(k) * 1.05, 1 / max(self.extent))
l, ml = self._bound_state
lprime, mlprime = self._continuum_state
for lprimeprime in range(max(l - lprime, 0), np.abs(l + lprime) + 1):
prefactor1 = (np.sqrt(4 * np.pi) * ((-1.j)**lprimeprime) * np.sqrt(
(2 * lprime + 1) * (2 * lprimeprime + 1) * (2 * l + 1)))
jk = None
for mlprimeprime in range(-lprimeprime, lprimeprime + 1):
if ml - mlprime - mlprimeprime != 0: # Wigner3j selection rule
continue
# Evaluate Eq. (14) from Dwyer Ultramicroscopy 104 (2005) 141-151
prefactor2 = (
(-1.0)**(mlprime + mlprimeprime) *
float(wigner_3j(lprime, lprimeprime, l, 0, 0, 0)) * float(
wigner_3j(lprime, lprimeprime, l, -mlprime,
-mlprimeprime, ml)))
if np.abs(prefactor2) < 1e-12:
continue
if jk is None:
jk = interp1d(
radial_grid,
self.overlap_integral(radial_grid, lprimeprime))(k)
Ylm = sph_harm(mlprimeprime, lprimeprime, phi, theta)
potential += prefactor1 * prefactor2 * jk * Ylm
potential *= np.prod(self.gpts) / np.prod(self.extent)
# Multiply by orbital filling
# if orbital_filling_factor:
potential *= np.sqrt(4 * l + 2)
# Apply constants:
# sqrt(Rdyberg) to convert to 1/sqrt(eV) units
# 1 / (2 pi**2 a0 kn) as as per paper
# Relativistic mass correction to go from a0 to relativistically corrected a0*
# divide by q**2
kn = 1 / energy2wavelength(self.energy + self.energy_loss)
potential *= relativistic_mass_correction(self.energy) / (
2 * units.Bohr * np.pi**2 * np.sqrt(units.Rydberg) * kn * k**2)
return potential
def momentum_transfer(self):
k0 = 1 / energy2wavelength(self.energy)
kn = 1 / energy2wavelength(self.energy + self.energy_loss)
return k0 - kn
def point_resolution(Cs, energy):
return (energy2wavelength(energy)**3 * np.abs(Cs) / 6)**(1 / 4)
def scherzer_defocus(Cs, energy):
return np.sign(Cs) * np.sqrt(
3 / 2 * np.abs(Cs) * energy2wavelength(energy))
def test_energy2wavelength():
assert np.isclose(energy2wavelength(300e3), 0.01968748889772767)