def test_get_wavelength(self):
wavelength = ks.get_wavelength(200000)
self.assertEqual(np.round(wavelength * 1000, 3), 2.508)
wavelength = ks.get_wavelength(60000)
self.assertEqual(np.round(wavelength * 1000, 3), 4.866)
with self.assertRaises(TypeError):
ks.get_wavelength('lattice_parameter')
def deficient_kikuchi_line(s_g=0., color_B='black'):
k_len = 1 / ks.get_wavelength(200)
d = .2 # lattice parameter in nm
g = np.linspace(-2, 2, 5) * 1 / d
g_d = np.array([1 / d, 0])
# recirocal lattice
plt.scatter(g, [0] * 5, color='blue')
alpha = -np.arctan(s_g / g_d[0])
theta = -np.arcsin(g_d[0] / 2 / k_len)
k_0 = np.array(
[-np.sin(theta - alpha) * k_len,
np.cos(theta - alpha) * k_len])
k_d = np.array(
[-np.sin(-theta - alpha) * k_len,
np.cos(-theta - alpha) * k_len])
k_i = np.array([-np.sin(theta - alpha) * 3, np.cos(theta - alpha) * 3])
k_i_t = np.array([-np.sin(-alpha) * 3, np.cos(-alpha) * 3])
kk_e = np.array([-np.sin(-theta) * k_len, np.cos(-theta) * k_len])
kk_d = np.array([-np.sin(theta) * k_len, np.cos(theta) * k_len])
# Ewald Sphere
ewald_sphere = patches.Circle((k_0[0], k_0[1]),
radius=np.linalg.norm(k_0),
clip_on=False,
zorder=10,
linewidth=1,
edgecolor=color_B,
fill=False)
plt.gca().add_artist(ewald_sphere)
# K_0
plt.plot([k_0[0], k_0[0]], [k_0[1], k_0[1] + 4],
color='gray',
linestyle='-',
alpha=0.3)
plt.gca().arrow(k_0[0] + k_i[0],
k_0[1] + k_i[1],
-k_i[0],
-k_i[1],
head_width=0.3,
head_length=0.4,
fc=color_B,
ec=color_B,
length_includes_head=True)
plt.plot([k_0[0] + k_i_t[0], k_0[0] - k_i_t[0]],
[k_0[1] + k_i_t[1], k_0[1] - k_i_t[1]],
color='black',
linestyle='--',
alpha=0.5)
plt.scatter(k_0[0], k_0[1], color='black')
plt.gca().arrow(k_0[0],
k_0[1],
-k_0[0],
-k_0[1],
head_width=0.3,
head_length=0.4,
fc=color_B,
ec=color_B,
length_includes_head=True)
plt.gca().annotate("K$_0$", xytext=(-k_0[0] / 2, 0), xy=(k_0[0] / 2, 0))
plt.gca().arrow(k_0[0],
k_0[1],
-k_d[0],
-k_d[1],
head_width=0.3,
head_length=0.4,
fc=color_B,
ec=color_B,
length_includes_head=True)
# K_e exces line
plt.gca().arrow(k_0[0],
k_0[1],
-kk_e[0],
-kk_e[1],
head_width=0.3,
head_length=0.4,
fc='red',
ec='red',
length_includes_head=True)
plt.gca().annotate("excess",
xytext=(k_0[0] - kk_e[0], -1),
xy=(-kk_e[0] + k_0[0], 0))
plt.plot([k_0[0] - kk_e[0], k_0[0] - kk_e[0]], [-1, 1], color='red')
# k_d deficient line
plt.gca().arrow(k_0[0],
k_0[1],
-kk_d[0],
-kk_d[1],
head_width=0.3,
head_length=0.4,
fc='blue',
ec='blue',
length_includes_head=True)
plt.plot([k_0[0] - kk_d[0], k_0[0] - kk_d[0]], [-1, 1], color='blue')
plt.gca().annotate("deficient",
xytext=(k_0[0] - kk_d[0], -1),
xy=(k_0[0] - kk_d[0], 0))
# s_g excitation Error of HOLZ reflection
plt.gca().arrow(g_d[0],
g_d[1],
0,
s_g,
head_width=0.3,
head_length=0.4,
fc='k',
ec='k',
length_includes_head=True)
plt.gca().annotate("s$_g$",
xytext=(g_d[0] * 1.01, g_d[1] + s_g / 3),
xy=(g_d[0] * 1.01, g_d[1] + s_g / 3))
theta = np.degrees(theta)
alpha = np.degrees(alpha)
bragg_angle = patches.Arc((k_0[0], k_0[1]),
width=5.5,
height=5.5,
theta1=90 + theta - alpha,
theta2=90 - alpha,
fc='black',
ec='black')
if alpha > 0:
deviation_angle = patches.Arc((k_0[0], k_0[1]),
width=6,
height=6,
theta1=90 - alpha,
theta2=90,
fc='black',
ec='red')
else:
deviation_angle = patches.Arc((k_0[0], k_0[1]),
width=6,
height=6,
theta1=90,
theta2=90 - alpha,
fc='black',
ec='red')
plt.gca().annotate(r"$\theta$",
xytext=(k_0[0] + k_i_t[0] / 1.3, k_0[1] + 2),
xy=(k_0[0] + k_i_t[0], k_0[1] + 2))
plt.gca().annotate(r"$\alpha$",
xytext=(k_0[0] + k_i_t[0] / 1.3, k_0[1] + 3),
xy=(k_0[0] + k_i_t[0], k_0[1] + 3),
color='red')
plt.gca().add_patch(bragg_angle)
plt.gca().add_patch(deviation_angle)
plt.xlim(-12, 12)
plt.ylim(-2, k_0[1] * 1.4)
plt.gca().set_aspect('equal')
def deficient_holz_line(exact_bragg=False,
shift=False,
laue_zone=1,
color='black'):
"""
Ewald sphere construction to explain Laue Circle and deficient HOLZ lines
Parameters:
exact_bragg: boolean
whether to tilt into exact Bragg condition or along zone axis
shift: boolean
whether to shift exact Bragg condition onto zone axis origin
laue_zone: int
first or second Laue zone only
color: string
color of wave vectors and Ewald sphere
"""
k_0 = [0, 1 / ks.get_wavelength(600)]
d = .5 # lattice parameter in nm
if laue_zone == 0:
s_g = 1 / d + 0.6
else:
s_g = 1
g = np.linspace(-5, 6, 12) * 1 / d
g_d = np.array([5. / d + laue_zone * 1 / d / 2, laue_zone * 1 / d])
g_sg = g_d.copy()
g_sg[1] = g_d[1] + s_g # point on Ewald sphere
# reciprocal lattice
plt.scatter(g[:-1], [0] * 11, color='red')
plt.scatter(g - 1 / d / 2, [1 / d] * 12, color='blue')
shift_x = shift_y = 0.
d_theta = d_theta1 = d_theta2 = 0
if exact_bragg:
d_theta1 = np.arctan((1 / d * laue_zone + s_g) / g_d[0])
d_theta2 = np.arctan((1 / d * laue_zone) / g_d[0])
d_theta = -(d_theta1 - d_theta2)
s_g = 0
s = np.sin(d_theta)
c = np.cos(d_theta)
k_0 = [-s * k_0[1], c * k_0[1]]
if shift:
shift_x = -k_0[0]
shift_y = np.linalg.norm(k_0) - k_0[1]
d_theta = np.degrees(d_theta)
k_0[0] += shift_x
k_0[1] += shift_y
# Ewald Sphere
ewald_sphere = patches.Circle((k_0[0], k_0[1]),
radius=np.linalg.norm(k_0),
clip_on=False,
zorder=10,
linewidth=1,
edgecolor=color,
fill=False)
plt.gca().add_artist(ewald_sphere)
plt.gca().arrow(g[-1] + 1 / d / 4,
1 / d / 2,
0,
1 / d / 2,
head_width=0.3,
head_length=0.4,
fc='k',
ec='k',
length_includes_head=True)
plt.gca().arrow(g[-1] + 1 / d / 4,
1 / d / 2,
0,
-1 / d / 2,
head_width=0.3,
head_length=0.4,
fc='k',
ec='k',
length_includes_head=True)
plt.gca().annotate("$|g_{HOLZ}|$",
xytext=(g[-1] + 1 / d / 3, 1 / d / 3),
xy=(g[-1] + 1 / d / 3, 1 / d / 3))
# k_0
plt.scatter(k_0[0], k_0[1])
plt.gca().arrow(k_0[0],
k_0[1],
-k_0[0] + shift_x,
-k_0[1] + shift_y,
head_width=0.3,
head_length=0.4,
fc=color,
ec=color,
length_includes_head=True)
plt.gca().annotate("K$_0$",
xytext=(k_0[0] / 2, k_0[1] / 3),
xy=(k_0[0] / 2, k_0[1] / 2))
# K_d Bragg of HOLZ reflection
plt.gca().arrow(k_0[0],
k_0[1],
-k_0[0] + g_d[0] + shift_x,
-k_0[1] + g_d[1] + s_g + shift_y,
head_width=0.3,
head_length=0.4,
fc=color,
ec=color,
length_includes_head=True)
plt.gca().annotate("K$_d$",
xytext=(k_0[0] + (g_d[0] - k_0[0]) / 2, k_0[1] / 2),
xy=(6.5 / d / 2, k_0[1] / 2))
# s_g excitation Error of HOLZ reflection
if s_g > 0:
plt.gca().arrow(g_d[0],
g_d[1],
0,
s_g,
head_width=0.3,
head_length=0.4,
fc='k',
ec='k',
length_includes_head=True)
plt.gca().annotate("s$_g$",
xytext=(g_d[0] * 1.01, g_d[1] + s_g / 3),
xy=(g_d[0] * 1.01, g_d[1] + s_g / 3))
# Bragg angle
g_sg = g_d
g_sg[1] = g_d[1] + s_g
plt.plot([0 + shift_x, g_sg[0] + shift_x], [0 + shift_y, g_d[1] + shift_y],
color=color,
linewidth=1,
alpha=0.5,
linestyle='--')
plt.plot([k_0[0], g_sg[0] / 2 + shift_x], [k_0[1], g_sg[1] / 2 + shift_y],
color=color,
linewidth=1,
alpha=0.5,
linestyle='--')
# d_theta = np.degrees(np.arctan(k_0[0]/k_0[1]))
bragg_angle = patches.Arc(
(k_0[0], k_0[1]),
width=k_0[1],
height=k_0[1],
theta1=-90 + d_theta,
theta2=-90 + d_theta +
np.degrees(np.arcsin(np.linalg.norm(g_sg / 2) / k_0[1])),
fc=color,
ec=color)
plt.gca().annotate(r"$\theta $",
xytext=(k_0[0] / 1.3, k_0[1] / 1.5),
xy=(k_0[0] / 2 + g_d[0] / 4, k_0[1] / 2))
plt.gca().add_patch(bragg_angle)
# deviation/tilt angle
if np.abs(d_theta) > 0:
if shift:
deviation_angle = patches.Arc((k_0[0], k_0[1]),
width=k_0[1] * 1.5,
height=k_0[1] * 1.5,
theta1=-90 + d_theta,
theta2=-90,
fc=color,
ec=color,
linewidth=3)
plt.gca().annotate(r"$d \theta $",
xytext=(k_0[0] - 1.3, k_0[1] / 3.7),
xy=(k_0[0] + g_d[0] / 4, k_0[1] / 2))
plt.gca().arrow(shift_x,
-2,
0,
2,
head_width=0.5,
head_length=0.6,
fc=color,
ec='black',
length_includes_head=True,
linewidth=3)
plt.gca().annotate("deficient line",
xytext=(shift_x * 2, -2),
xy=(shift_x, 0))
else:
deviation_angle = patches.Arc((0, 0),
width=k_0[1],
height=k_0[1],
theta1=np.degrees(d_theta2),
theta2=np.degrees(d_theta1),
fc=color,
ec=color,
linewidth=3)
plt.gca().annotate(r"$d \theta $",
xytext=(g_d[0] * .8, 1 / d / 3),
xy=(g_d[0], 1 / d))
plt.gca().add_patch(deviation_angle)
plt.gca().set_aspect('equal')
plt.xlim(-14, 14)
plt.ylim(-4, k_0[1] * 1.1)