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23 """Module for the handling of Frame Order."""
24
25
26 from math import cos, pi, sqrt
27 from numpy import sinc
28 try:
29 from scipy.integrate import tplquad
30 except ImportError:
31 pass
32
33
34 from lib.frame_order.matrix_ops import pcs_pivot_motion_full, pcs_pivot_motion_full_qrint, rotate_daeg
35
36
38 """Generate the rotated 2nd degree Frame Order matrix for the isotropic cone.
39
40 The cone axis is assumed to be parallel to the z-axis in the eigenframe.
41
42 @param matrix: The Frame Order matrix, 2nd degree to be populated.
43 @type matrix: numpy 9D, rank-2 array
44 @param Rx2_eigen: The Kronecker product of the eigenframe rotation matrix with itself.
45 @type Rx2_eigen: numpy 9D, rank-2 array
46 @param cone_theta: The cone opening angle.
47 @type cone_theta: float
48 @param sigma_max: The maximum torsion angle.
49 @type sigma_max: float
50 """
51
52
53 populate_2nd_eigenframe_iso_cone(matrix, cone_theta, sigma_max)
54
55
56 return rotate_daeg(matrix, Rx2_eigen)
57
58
59 -def pcs_numeric_int_iso_cone(theta_max=None, sigma_max=None, c=None, r_pivot_atom=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None):
60 """Determine the averaged PCS value via numerical integration.
61
62 @keyword theta_max: The half cone angle.
63 @type theta_max: float
64 @keyword sigma_max: The maximum torsion angle.
65 @type sigma_max: float
66 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units).
67 @type c: float
68 @keyword r_pivot_atom: The pivot point to atom vector.
69 @type r_pivot_atom: numpy rank-1, 3D array
70 @keyword r_ln_pivot: The lanthanide position to pivot point vector.
71 @type r_ln_pivot: numpy rank-1, 3D array
72 @keyword A: The full alignment tensor of the non-moving domain.
73 @type A: numpy rank-2, 3D array
74 @keyword R_eigen: The eigenframe rotation matrix.
75 @type R_eigen: numpy rank-2, 3D array
76 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations).
77 @type RT_eigen: numpy rank-2, 3D array
78 @keyword Ri_prime: The empty rotation matrix for the in-frame isotropic cone motion, used to calculate the PCS for each state i in the numerical integration.
79 @type Ri_prime: numpy rank-2, 3D array
80 @return: The averaged PCS value.
81 @rtype: float
82 """
83
84
85 result = tplquad(pcs_pivot_motion_full, -sigma_max, sigma_max, lambda phi: -pi, lambda phi: pi, lambda theta, phi: 0.0, lambda theta, phi: theta_max, args=(r_pivot_atom, r_ln_pivot, A, R_eigen, RT_eigen, Ri_prime))
86
87
88 SA = 4.0 * pi * sigma_max * (1.0 - cos(theta_max))
89
90
91 return c * result[0] / SA
92
93
94 -def pcs_numeric_int_iso_cone_qrint(points=None, theta_max=None, sigma_max=None, c=None, full_in_ref_frame=None, r_pivot_atom=None, r_pivot_atom_rev=None, r_ln_pivot=None, A=None, R_eigen=None, RT_eigen=None, Ri_prime=None, pcs_theta=None, pcs_theta_err=None, missing_pcs=None, error_flag=False):
95 """Determine the averaged PCS value via numerical integration.
96
97 @keyword points: The Sobol points in the torsion-tilt angle space.
98 @type points: numpy rank-2, 3D array
99 @keyword theta_max: The half cone angle.
100 @type theta_max: float
101 @keyword sigma_max: The maximum torsion angle.
102 @type sigma_max: float
103 @keyword c: The PCS constant (without the interatomic distance and in Angstrom units).
104 @type c: numpy rank-1 array
105 @keyword full_in_ref_frame: An array of flags specifying if the tensor in the reference frame is the full or reduced tensor.
106 @type full_in_ref_frame: numpy rank-1 array
107 @keyword r_pivot_atom: The pivot point to atom vector.
108 @type r_pivot_atom: numpy rank-2, 3D array
109 @keyword r_pivot_atom_rev: The reversed pivot point to atom vector.
110 @type r_pivot_atom_rev: numpy rank-2, 3D array
111 @keyword r_ln_pivot: The lanthanide position to pivot point vector.
112 @type r_ln_pivot: numpy rank-2, 3D array
113 @keyword A: The full alignment tensor of the non-moving domain.
114 @type A: numpy rank-2, 3D array
115 @keyword R_eigen: The eigenframe rotation matrix.
116 @type R_eigen: numpy rank-2, 3D array
117 @keyword RT_eigen: The transpose of the eigenframe rotation matrix (for faster calculations).
118 @type RT_eigen: numpy rank-2, 3D array
119 @keyword Ri_prime: The empty rotation matrix for the in-frame isotropic cone motion, used to calculate the PCS for each state i in the numerical integration.
120 @type Ri_prime: numpy rank-2, 3D array
121 @keyword pcs_theta: The storage structure for the back-calculated PCS values.
122 @type pcs_theta: numpy rank-2 array
123 @keyword pcs_theta_err: The storage structure for the back-calculated PCS errors.
124 @type pcs_theta_err: numpy rank-2 array
125 @keyword missing_pcs: A structure used to indicate which PCS values are missing.
126 @type missing_pcs: numpy rank-2 array
127 @keyword error_flag: A flag which if True will cause the PCS errors to be estimated and stored in pcs_theta_err.
128 @type error_flag: bool
129 """
130
131
132 for i in range(len(pcs_theta)):
133 for j in range(len(pcs_theta[i])):
134 pcs_theta[i, j] = 0.0
135 pcs_theta_err[i, j] = 0.0
136
137
138 num = 0
139 for i in range(len(points)):
140
141 theta, phi, sigma = points[i]
142
143
144 if theta > theta_max:
145 continue
146 if sigma > sigma_max or sigma < -sigma_max:
147 continue
148
149
150 pcs_pivot_motion_full_qrint(theta_i=theta, phi_i=phi, sigma_i=sigma, full_in_ref_frame=full_in_ref_frame, r_pivot_atom=r_pivot_atom, r_pivot_atom_rev=r_pivot_atom_rev, r_ln_pivot=r_ln_pivot, A=A, R_eigen=R_eigen, RT_eigen=RT_eigen, Ri_prime=Ri_prime, pcs_theta=pcs_theta, pcs_theta_err=pcs_theta_err, missing_pcs=missing_pcs)
151
152
153 num += 1
154
155
156 for i in range(len(pcs_theta)):
157 for j in range(len(pcs_theta[i])):
158
159 pcs_theta[i, j] = c[i] * pcs_theta[i, j] / float(num)
160
161
162 if error_flag:
163 pcs_theta_err[i, j] = abs(pcs_theta_err[i, j] / float(num) - pcs_theta[i, j]**2) / float(num)
164 pcs_theta_err[i, j] = c[i] * sqrt(pcs_theta_err[i, j])
165 print("%8.3f +/- %-8.3f" % (pcs_theta[i, j]*1e6, pcs_theta_err[i, j]*1e6))
166
167
169 """Populate the 1st degree Frame Order matrix in the eigenframe for an isotropic cone.
170
171 The cone axis is assumed to be parallel to the z-axis in the eigenframe.
172
173 @param matrix: The Frame Order matrix, 1st degree.
174 @type matrix: numpy 3D, rank-2 array
175 @param angle: The cone angle.
176 @type angle: float
177 """
178
179
180 for i in range(3):
181 for j in range(3):
182 matrix[i, j] = 0.0
183
184
185 matrix[2, 2] = (cos(angle) + 1.0) / 2.0
186
187
189 """Populate the 2nd degree Frame Order matrix in the eigenframe for the isotropic cone.
190
191 The cone axis is assumed to be parallel to the z-axis in the eigenframe.
192
193
194 @param matrix: The Frame Order matrix, 2nd degree.
195 @type matrix: numpy 9D, rank-2 array
196 @param tmax: The cone opening angle.
197 @type tmax: float
198 @param smax: The maximum torsion angle.
199 @type smax: float
200 """
201
202
203 for i in range(9):
204 for j in range(9):
205 matrix[i, j] = 0.0
206
207
208 sinc_smax = sinc(smax/pi)
209 sinc_2smax = sinc(2.0*smax/pi)
210 cos_tmax = cos(tmax)
211 cos_tmax2 = cos_tmax**2
212
213
214 fact_sinc_2smax = sinc_2smax*(cos_tmax2 + 4.0*cos_tmax + 7.0) / 24.0
215 fact_cos_tmax2 = (cos_tmax2 + cos_tmax + 4.0) / 12.0
216 fact_cos_tmax = (cos_tmax + 1.0) / 4.0
217
218
219 matrix[0, 0] = fact_sinc_2smax + fact_cos_tmax2
220 matrix[1, 1] = fact_sinc_2smax + fact_cos_tmax
221 matrix[2, 2] = sinc_smax * (2.0*cos_tmax2 + 5.0*cos_tmax + 5.0) / 12.0
222 matrix[3, 3] = matrix[1, 1]
223 matrix[4, 4] = matrix[0, 0]
224 matrix[5, 5] = matrix[2, 2]
225 matrix[6, 6] = matrix[2, 2]
226 matrix[7, 7] = matrix[2, 2]
227 matrix[8, 8] = (cos_tmax2 + cos_tmax + 1.0) / 3.0
228
229
230 matrix[0, 4] = matrix[4, 0] = -fact_sinc_2smax + fact_cos_tmax2
231 matrix[0, 8] = matrix[8, 0] = -(cos_tmax2 + cos_tmax - 2.0) / 6.0
232 matrix[4, 8] = matrix[8, 4] = matrix[0, 8]
233
234
235 matrix[1, 3] = matrix[3, 1] = fact_sinc_2smax - fact_cos_tmax
236 matrix[2, 6] = matrix[6, 2] = sinc_smax * (cos_tmax2 + cos_tmax - 2.0) / 6.0
237 matrix[5, 7] = matrix[7, 5] = matrix[2, 6]
238