def calc_Darwin_width(crystal='Si',
energy=8,
h=1,
k=1,
l=1,
rel_angle=1,
debye_temp_factor=1):
'''
Calculates Darwin width and intrinsic resolution for p and s polarizations. It uses xraylib.
Parameters:
- crystal: crystal material. [str]
- energy: energy in keV. [float]
- h, k, l: Miller indexes. [int]
- rel_angle: relative angle [float]
- debye_temp_factor: Debye Temperature Factor. [float]
Returns:
- dw_s: Angular Darwin width for s polarization in rad. [float]
- dw_p: Angular Darwin width for p polarization in rad. [float]
- resolution_s: Intrinsic resolution for s polarization. [float]
- resolution_p: Intrinsic resolution for p polarization. [float]
References:
A simple formula to calculate the x-ray flux after a double-crystal monochromator - M. S. del Rio; O. MATHON.
'''
cryst = xraylib.Crystal_GetCrystal(crystal)
bragg = xraylib.Bragg_angle(cryst, energy, h, k, l)
FH = xraylib.Crystal_F_H_StructureFactor(cryst, energy, h, k, l,
debye_temp_factor, rel_angle)
FHbar = xraylib.Crystal_F_H_StructureFactor(cryst, energy, -h, -k, -l,
debye_temp_factor, rel_angle)
C = abs(np.cos(2 * bragg))
dw_p = 1e10 * 2 * C * (xraylib.R_E / cryst['volume']) * (
xraylib.KEV2ANGST * xraylib.KEV2ANGST / (energy * energy)) * np.sqrt(
abs(FH * FHbar)) / np.pi / np.sin(2 * bragg)
dw_s = 1e10 * 2 * (xraylib.R_E / cryst['volume']) * (
xraylib.KEV2ANGST * xraylib.KEV2ANGST / (energy * energy)) * np.sqrt(
abs(FH * FHbar)) / np.pi / np.sin(2 * bragg)
resolution_p = dw_p / np.tan(bragg)
resolution_s = dw_s / np.tan(bragg)
return dw_s, dw_p, resolution_s, resolution_p
def F0(self, energy):
"""
Calculate F0 from Zachariasen.
:param energy: photon energy in eV.
:return: F0
"""
energy_in_kev = energy / 1000.0
F_0 = xraylib.Crystal_F_H_StructureFactor(self._crystal, energy_in_kev,
0, 0, 0, self._debyeWaller,
1.0)
return F_0
def FH_bar(self, energy, rel_angle=1.0):
"""
Calculate FH_bar from Zachariasen.
:param energy: photon energy in eV.
:return: FH_bar
"""
energy_in_kev = energy / 1000.0
F_H_bar = xraylib.Crystal_F_H_StructureFactor(
self._crystal, energy_in_kev, -self.millerH(), -self.millerK(),
-self.millerL(), self._debyeWaller, rel_angle)
return F_H_bar
def calc_rocking_curve_shift(crystal='Si',
energy=8,
h=1,
k=1,
l=1,
rel_angle=1,
debye_temp_factor=1):
'''
Calculates the angular shift of the Rocking Curve. It uses xraylib.
Parameters:
- crystal: crystal material. [str]
- energy: energy in keV. [float]
- h, k, l: Miller indexes. [int]
- rel_angle: relative angle [float]
- debye_temp_factor: Debye Temperature Factor. [float]
Returns:
- w0: Angular shift of the Rocking Curve in rad. [float]
References:
Elements of modern X-ray physics / Jens Als-Nielsen, Des McMorrow – 2nd ed. Cap. 6.
'''
# Function:
def calc_g(d, r0, Vc, F):
return abs((2 * d * d * r0 / (Vc)) * F)
# Calculating rocking curve shift:
cryst = xraylib.Crystal_GetCrystal(crystal)
bragg = xraylib.Bragg_angle(cryst, energy, h, k, l)
r0 = physical_constants['classical electron radius'][0]
F0 = xraylib.Crystal_F_H_StructureFactor(cryst, energy, 0, 0, 0,
debye_temp_factor, rel_angle)
d = 1e-10 * xraylib.Crystal_dSpacing(cryst, h, k, l)
V = 1e-30 * cryst['volume']
g0 = calc_g(d, r0, V, F0)
w0 = (g0 / (np.pi)) * np.tan(bragg)
return w0
def FH(self, energy):
"""
Calculate FH from Zachariasen.
:param energy: photon energy in eV.
:return: FH
"""
energy_in_kev = energy / 1000.0
F_H = xraylib.Crystal_F_H_StructureFactor(self._crystal, energy_in_kev,
self.millerH(),
self.millerK(),
self.millerL(),
self._debyeWaller, 1.0)
return F_H
energy = 8
debye_temp_factor = 1.0
rel_angle = 1.0
bragg = xraylib.Bragg_angle(cryst, energy, 1, 1, 1)
print(" Bragg angle: Rad: {} Deg: {}".format(bragg, bragg * 180 / math.pi))
q = xraylib.Q_scattering_amplitude(cryst, energy, 1, 1, 1, rel_angle)
print(" Q Scattering amplitude: {}".format(q))
#notice the 3 return values!!!
f0, fp, fpp = xraylib.Atomic_Factors(14, energy, q, debye_temp_factor)
print(" Atomic factors (Z = 14) f0, fp, fpp: {}, {}, i*{}".format(
f0, fp, fpp))
FH = xraylib.Crystal_F_H_StructureFactor(cryst, energy, 1, 1, 1,
debye_temp_factor, rel_angle)
print(" FH(1,1,1) structure factor: ({}, {})".format(FH.real, FH.imag))
F0 = xraylib.Crystal_F_H_StructureFactor(cryst, energy, 0, 0, 0,
debye_temp_factor, rel_angle)
print(" F0=FH(0,0,0) structure factor: ({}, {})".format(F0.real, F0.imag))
# Diamond diffraction parameters
cryst = xraylib.Crystal_GetCrystal("Diamond")
if cryst is None:
raise KeyError('Diamond crystal not found')
print("")
print("Diamond 111 at 8 KeV. Incidence at the Bragg angle:")
bragg = xraylib.Bragg_angle(cryst, energy, 1, 1, 1)
print(" Bragg angle: Rad: {} Deg: {}".format(bragg, bragg * 180 / math.pi))
def calc_Darwin_curve(delta_theta=np.linspace(-0.00015, 0.00015, 5000),
crystal='Si',
energy=8,
h=1,
k=1,
l=1,
rel_angle=1,
debye_temp_factor=1,
use_correction=True,
save_txt=True,
save_fig=True,
filename_to_save='Darwin_curve'):
'''
Calculates the Darwin curve. It does not considers absortion. Valid for s-polarization only.
Parameters:
- delta_theta: array containing values of (Theta - Theta_Bragg) in rad. [array]
- crystal: crystal material. [str]
- energy: energy in keV. [float]
- h, k, l: Miller indexes. [int]
- rel_angle: relative angle [float]
- debye_temp_factor: Debye Temperature Factor. [float]
- use_correction: if True, considers the corrected Bragg angle due to refraction. [boolean]
- save_txt: if True, saves the Darwin Curve in a .txt file. [boolean]
- save_fig: if True, saves a figure with the Darwin Curve in .png. [boolean]
- filename_to_save: name to save the figure and the .txt file. [string]
Returns:
- delta_theta: Array containing values of (Theta - Theta_Bragg) in rad. [array]
- R: Intensity reflectivity array. [array]
- zeta_total: Total Darwin width (delta_lambda/lambda). [float]
- zeta_FWHM: Darwin width FWHM (delta_lambda/lambda). [float]
- w_total: Total Angular Darwin width (delta_Theta) in rad. [float]
- w_FWHM: Angular Darwin width FWHM (delta_Theta) in rad. [float]
- w0: Angular shift of the Rocking Curve in rad. [float]
References:
Elements of modern X-ray physics / Jens Als-Nielsen, Des McMorrow – 2nd ed. Cap. 6.
'''
# Functions:
def calc_g(d, r0, Vc, F):
return abs((2 * d * d * r0 / (Vc)) * F)
def calc_xc(zeta, g, g0):
return np.pi * zeta / g - g0 / g
def calc_zeta_total(d, r0, Vc, F):
return (4 / np.pi) * (d) * (d) * (r0 * abs(F) / Vc)
def Darwin_curve(xc):
R = []
for x in xc:
if (x >= 1):
r = (x - np.sqrt(x * x - 1)) * (x - np.sqrt(x * x - 1))
if (x <= 1):
r = 1
if (x <= -1):
r = (x + np.sqrt(x * x - 1)) * (x + np.sqrt(x * x - 1))
R.append(r)
return np.array(R)
# Calculating Darwin curve:
cryst = xraylib.Crystal_GetCrystal(crystal)
bragg = xraylib.Bragg_angle(cryst, energy, h, k, l)
zeta = delta_theta / np.tan(bragg)
r0 = physical_constants['classical electron radius'][0]
FH = xraylib.Crystal_F_H_StructureFactor(cryst, energy, h, k, l,
debye_temp_factor, rel_angle)
F0 = xraylib.Crystal_F_H_StructureFactor(cryst, energy, 0, 0, 0,
debye_temp_factor, rel_angle)
d = 1e-10 * xraylib.Crystal_dSpacing(cryst, h, k, l)
V = 1e-30 * cryst['volume']
g = calc_g(d, r0, V, FH)
g0 = calc_g(d, r0, V, F0)
xc = calc_xc(zeta, g, g0)
R = Darwin_curve(xc)
zeta_total = calc_zeta_total(d, r0, V, FH)
zeta_FWHM = (3 / (2 * np.sqrt(2))) * zeta_total
w_total = zeta_total * np.tan(bragg)
w_FWHM = (3 / (2 * np.sqrt(2))) * w_total
w0 = (g0 / (np.pi)) * np.tan(bragg)
# Correcting curve offset (due to refraction):
if (use_correction):
delta_theta = delta_theta - w0
# Saving .txt file:
if (save_txt):
filename = filename_to_save + '.txt'
with open(filename, 'w') as f:
f.write('#Delta_Theta[rad] Intensity_Reflectivity \n')
for i in range(len(R)):
f.write('%.6E\t%.6E \n' % (delta_theta[i], R[i]))
# Plotting Graph:
plt.figure()
plt.plot(delta_theta, R, linewidth=1.8, color='black')
plt.fill_between(delta_theta, R, alpha=0.9, color='C0')
plt.ylabel('Intensity reflectivity', fontsize=13)
plt.xlabel('$\Delta$' + '$\Theta$' + ' [rad]', fontsize=13)
plt.xscale('linear')
plt.yscale('linear')
plt.minorticks_on()
plt.ticklabel_format(style='sci', axis='x', scilimits=(0, 0))
plt.tick_params(which='both',
axis='both',
direction='in',
right=True,
top=True,
labelsize=12)
plt.grid(which='both', alpha=0.2)
plt.tight_layout()
textstr = '\n'.join((r'$\zeta_{total}=$%.4E' % (zeta_total, ),
r'$\zeta_{FWHM}=$%.4E' % (zeta_FWHM, ),
r'$\omega_{total}=$%.4E rad' % (w_total, ),
r'$\omega_{FWHM}=$%.4E rad' % (w_FWHM, )))
props = dict(boxstyle='round', facecolor='wheat',
alpha=0.5) # wheat # gray
plt.text(0.05,
0.95,
textstr,
transform=plt.gca().transAxes,
fontsize=10,
verticalalignment='top',
bbox=props)
plt.show()
if (save_fig):
plt.savefig(filename_to_save + '.png', dpi=600)
return delta_theta, R, zeta_total, zeta_FWHM, w_total, w_FWHM, w0