X-Ray N-Beam Takagi-Taupin Dynamical Theory and N-Beam Pinhole Topographs Experimentally Obtained and Computer-Simulated

In the field of X-ray crystal structure analysis, while the absolute values of structure factors are directly observed, phase information is lost in general. However, this problem (phase problem) has been overcome mainly by the direct method developed by Hauptmann and Karle except for protein crystals. In the case of protein crystal structure analysis, the isomorphous replacement method and/or anomalous dispersion method are mainly used to solve the phase problem. Phasing the structure factors is sometimes the most difficult process in protein crystallography.

On the other hand, Okitsu and his coauthors [13,14,16,17] extended the Takagi-Taupin (T-T) dynamical diffraction theory [18][19][20] to n-beam cases (n ∈ {3, 4, 5, 6, 8, 12}) and presented a numerical method to solve the theory. They showed six-beam pinhole topographs experimentally obtained and computer-simulated based on the n-beam T-T equation, between which excellent agreements were found. In reference [16], it was shown that the n-beam T-T equation can deal with X-ray wave field in an arbitrary-shaped crystal.
In the n-beam method to solve the phase problem in the protein crystal structure analysis, one of the difficulty is the shape of the crystal which is complex in general. Then, the above advantage of n-beam T-T equation over the E-L dynamical theory is important. The present authors have derived an n-beam T-T equation applicable for arbitrary number of n, which will be published elsewhere.
The n-beam T-T equation was derived in references [13,17] from Takagi's fundamental equation of the dynamical theory [19]. In section 2 of the present chapter, however, the n-beam E-L theory is described at first. The n-beam T-T equation is derived by Fourier-transforming the n-beam E-L dynamical theory. Then, it is also described that the E-L theory can be derived from the T-T equation. This reveals a simple relation between the E-L and T-T formulations of X-ray dynamical diffraction theory. This equivalence between the E-L and T-T formulations has been implicitly recognized for many years but is explicitly described for the first time. In section 5, experimentally obtained and computer-simulated pinhole topographs are shown for n ∈ {3, 4, 5, 6, 8, 12}, which verifies the theory and the computer algorithm to solve it.

Description of the n-beam Ewald-Laue dynamical diffraction theory
The fundamental equation with the E-L formulation is given by [1,2] Here, k i is wavenumber of the ith numbered Bloch wave whose wave vector is k 0 + h i where k 0 is the wave vector of the forward-diffracted wave in the crystal, K(= 1/λ) is the wavenumber of X-rays in vacuum, D i and D j are complex amplitude vectors of the ith and jth numbered Bloch waves, ∑ is an infinite summation for all combinations of i and j, By applying an approximation that k i + K ≈ 2k i to (1), the following equation is obtained, where Let the electric displacement vector D i be represented by a linear combination of scalar amplitudes as follows: i e (1) i .
Here, e i and e (1) i are unit vectors perpendicular to s i , where s i is a unit vector parallel to k i . s i , e (0) i and e (1) i construct a right-handed orthogonal system in this order. (2) can be described as follows: where, i, j ∈ {0, 1, · · · , n − 1}, n ∈ {3, 4, 5, 6, 8, 12}, l, m Here, S and C are polarization factors defined by where i and j are ordinal numbers of waves (i, j ∈ {0, 1, 2, · · · , n − 1}) and l and m are ordinal numbers of polarization state (l, m ∈ {0, 1}). When deriving (3) from (2), all reciprocal lattice nodes lying on the surface of Ewald sphere are assumed to be on a circle in reciprocal space. Then number of waves n are limited to be n ∈ {3, 4, 5, 6, 8, 12} even in the case of cubic crystals with the highest symmetry. Θ B is the angle spanned by − → PQ and k i which is an identical value for every i (i ∈ {0, 1, 2, · · · , n − 1}), where P and Q are centers of the Ewald sphere and the circle on which the reciprocal lattice nodes lie, respectively. ξ is such a value that where P 1 is the common initial point of k i [whose terminal points are H i (i ∈ {0, 1, · · · , n − 1})] and P 1 is a point on the sphere whose distance from the origin O of reciprocal space is K.
Hereafter, this surface of sphere is approximated as a plane whose distance from O is K in the vicinity of the Laue point La whose distance from H i (i ∈ {0, 1, · · · , n − 1}) is the identical value K. For description in the next section, it is described here that − − → P 1 P 1 is represented by a linear combination of s i , e i and e (1) i as follows: (2) and (3) can also be represented using matrices and vector as follows: ξ cos Θ B ED = AD.
Here, E is a unit matrix of size 2n, D is a amplitude column vector of size 2n whose qth element is D (m) j (q = 2j + m + 1) and A is a square matrix of size 2n whose element a p,q is given by i,0 β (1) .
Here, p = 2i + l + 1 and δ p,q is Kronecker delta. 2n couples of ξ and D can be obtained by solving eigenvalue-eigenvector problem of (7). This problem was solved by Colella [3] for the first time. Dispersion surfaces on which the initial point P 1 of wave vectors of Bloch waves should be, is given as 2n sets of eigenvalues for (7).

Derivation of the n-beam Takagi-Taupin equation from the Ewald-Laue theory
In this section, the n-beam theory of T-T formulation is derived by Fourier-transforming the n-beam E-L theory described by (3).
A general solution of dynamical diffraction theory is considered to be coherent superposition of Bloch plane-wave system when X-ray wave fieldD(r) is given as follows: where r is the location vector. For the following description, r is described by a linear combination of s i , e i and e (1) i as follows, i e (1) i .
The amplitude of the ith component wave whose polarization state is l is described as, In this section, the amplitude of plane wave whose wave vector is k i and polarization state is Substituting (4) with j = 0, (5), (6) and (9) into (10), i ) dS k . (11) Here,

D.S.
Δk dS k means an integration over the dispersion surfaces in reciprocal space and i ) is a term that does not depend on s i . ∂D (l) i (r)/∂s i can be calculated as follows: Substituting (3) into (12), Incidentally, when the crystal is perfect, the electric susceptibility χ(r) is represented by Fourier series as However, when the crystal has a lattice displacement field of u(r), the electric susceptibility is approximately given by χ[r − u(r)] and represented by Fourier series as follows, Then, in the case of crystal with a lattice displacement field of u(r), χ h i −h j can be replaced by . Therefore, the following equation is obtained from (13), where, The above equation is nothing but the n-beam T-T equation that appeared as eq. (4) in reference [17].

Derivation of the n-beam E-L dynamical theory from the T-T equation
In this section, it is described that the n-beam E-L theory given by (3) can be derived from the n-beam T-T equation (14).
When plane-wave X-rays are incident on the crystal to excite 2n tie points on the dispersion surfaces, each Bloch plane-wave system is described bỹ Even when D (14) with u(r) = 0, . (15) Applying the same procedure as used when deriving (11), Comparing (15) and (16), the same equation as (3) is obtained. The equivalence between the n-beam E-L and T-T X-ray dynamical diffraction theories (n ∈ {3, 4, 5, 6, 8, 12}) described by a Fourier transform as defined by (10) is verified. As far as the present authors know, description on this equivalence between the E-L and T-T dynamical diffraction theories for two-beam case is found just in section 11.3 of Authier's book [1]. Figure 1(a) and 1(b) are schematic drawings for explanation of the algorithm to solve the n-beam T-T equation (14) for a six-beam case whose computer-simulated and experimentally obtained results are shown in Figure 9 of the present chapter. Vectors (1) in Figure  1(a) are parallel to s i . When the length of (1) is sufficiently small compared with the extinction length −1/(χ 0 K) of the forward diffraction, The T-T equation (14) can be approximated by  (14) in a six-beam case whose results are shown in Figure 9.

Algorithm to solve the theory
The above equation (17) can be described using matrix and vectors as follows: where, a p = −i 1 2 πK Here, A and D are column vectors of size 2n whose pth and qth elements are a p and d q , respectively, and B is a square matrix of size 2n whose element of the pth column and the qth raw is b p,q .   Figure 1(b) are parallel to the wave vectors of 0 0 0-forward-diffracted and 4 0 4-, 4 2 6-, 0 6 6-, 2 6 4and 2 2 0-reflected X-rays. As a boundary condition on the crystal surface, amplitude array D even (i,l,n x ,n y ) has nonzero value (unity) when (i,l,n x ,n y )=(0,0,0,0) or (i,l,n x ,n y )=(0,1,0,0). On the first layer, nonzero X-ray amplitudes D odd (j, m, n x , n y ) are calculated when (n x , n y ) = [n x (i), n y (i)] (i ∈ {0, 1, · · · , n − 1}). Here, [n x (i), n y (i)] = (0, 0),(0, 2), (1,3), (3,3), (3,2) and (1, 0) for i = 0, 1, 2, 3, 4, 5, respectively. In general, D even (i,l,n x ,n y ) [or D odd (i,l,n x ,n y )] is calculated as D (l) i (R (1) ) by substituting D odd (j,m,n x − n x (i),n y − n y (i)) [or D even (j,m,n x − n x (i),n y − n y (i))] into D (17). The calculation was performed layer by layer scanning n x and n y in a range of NMin[n x (i)] ≤ n x ≤ N Max[n x (i)] and NMin[n y (i)] ≤ n y ≤ N Max[n y (i)], where N is the ordinal number of layer. The values of χ h i −h j were calculated by using XOP version 2.3 [6].

Phase-retarder system
When taking four-, five-, six-and eight-beam pinhole topographs shown in section 5, the horizontally polarized synchrotron X-rays monochromated to be 18.245 keV with a water-cooled diamond monochromator system at BL09XU of SPring-8 were incident on the 'rotating four-quadrant phase retarder system' [15,17].  2.198, 1.565 and 2.633 mm were mounted on tangential-bar type goniometers such that the deviation angles Δθ PR n from the exact Bragg condition of 1 1 1 reflection in an asymmetric Laue geometry can be controlled. See Figure 4 in reference [17] for more detail. The four tangential-bar type goniometers were mounted in a χ-circle goniometer [see Figure 3 (b)] such that the whole system of the phase retarders can be rotated around the beam axis of transmitted X-rays. The rotation angle of the χ-circle χ PR and Δθ PR n were controlled as summarized in Table 3 in reference [17] such that horizontal-linearly (LH), vertical-linearly (LV), right-screwed circularly (CR), left-screwed circularly (CL), −45 • -inclined-linearly (L−45) and +45 • -inclined-linearly (L+45) polarized X-rays were generated to be incident on the sample crystal.
In the cases of three-and twelve-beam pinhole topographs, horizontally polarized synchrotron X-rays monochromated to be 18.245 keV and to be 22.0 keV, respectively, but not transmitted through the phase retarder system were incident on the sample crystals.  [17] showing the experimental setup when the six-beam pinhole topographs shown in reference [17] were taken. Also in the case of n-beam pinhole topographs (n ∈ {3, 4, 5, 6, 8, 12}) the [1 1 1]-oriented floating-zone silicon crystal with thickness 9.6 mm (for three-, four-, five-, six-and eight-beam topographs) and 10.0 mm (for twelve-beam topographs) were also mounted on the four-axis goniometer whose χ-, φ-, ωand θ-axes can be rotated. Transmitted X-rays through the sample and two reflected X-rays were searched by three PIN photo diodes as shown in Figure 4. The positions of the two PIN photo diodes for detecting the reflected X-rays were determined using a laser beam guide reflected by a mirror. The mirror was mounted at the sample position on the goniometer whose angular positions were calculated such that the mirror reflects the laser beam to the direction of X-rays to be reflected by the sample crystal.  Figure 7 in reference [17]).

Sample crystal
After adjusting the angular position of the goniometer such that the n-beam simultaneous reflection condition was satisfied, the size of slit S in Figure 3 (a) was limited to be 25×25 μm.
N images of n-beam pinhole topographs were simultaneously recorded on an imaging plate placed behind the sample crystal.

Three-beam case
Three-beam case is the most primitive case of X-ray multiple reflection.   were obtained with an incidence of +45 • -inclined-linearly, −45 • -inclined-linearly and right-screwed-circularly polarized X-rays, respectively, generated with the phase retarder system or assumed in the simulation.   ∈ {a, b, c}) are experimentally obtained and computer-simulated four-beam X-ray pinhole topographs with an incidence of +45 • -inclined-linearly, −45 • -inclined-linearly and right-screwed-circularly polarized X-rays whose photon energy was 18.245 keV. The exposure time for [E(x)] was 1800 s. Between the horizontal and vertical components of incident X-rays, there is difference not in amplitude but in phase among Figures [Y(a) S}), which reveals that the wave fields excited by horizontal-and vertical-linearly polarized components of the incident X-rays interfere with each other. In the case of cubic crystals, five reciprocal lattice nodes (including the origin of reciprocal space) can ride on a circle in reciprocal space. For understanding such a situation, refer to Figure 1 of reference [17]. Remarking on the directions of KEL (1) and KEL (2), these knife-edge patterns are directed to 0 0 0-forward-diffracted and 3 3 3-reflected X-ray images, respectively. Then, KEL (1) and KEL (2) are considered to suggest the strong energy exchange mechanism between 0 0 0-forward-diffracted and 5 5 5-reflected X-ray wave fields and between 3 3 3-and 5 5 5-reflected X-ray wave fields. Such knife-edge patterns are found also in three-, four-, six-and eight-beam pinhole topograph images shown in the present chapter. While experimental and computer-simulated six-beam pinhole topograph images whose shapes are regular hexagons have been reported in reference [14,16,17], shown in this section are six-beam pinhole topographs whose Borrmann pyramid is not a regular hexagonal pyramid.

Six-beam case
Such six-beam pinhole topographs experimentally obtained and computer-simulated are shown in Figure 9.  (2)] indicated by arrows in Figure 9[S(b)] are found also in Figure 9[E(b)]. Circular patterns that were found in the central part of the six-beam pinhole topographs [14,16,17] cannot be found in the present case. A 'heart-shaped' pattern (HSP) is found also in Figure 9[E(b)].    Twelve is the largest number of n for the n-beam T-T equation (14) that restricts a condition that n reciprocal lattice nodes should ride on a circle in reciprocal space. A very bright region (VBR), 'V-shaped' pattern (VSP), central circle (CC) and 'U-shaped' pattern indicated by arrows in Figure 12[S(b)] are found also in Figure 12[E(b)].
The equivalence between the E-L and T-T formulations of the n-beam X-ray dynamical diffraction theory, which has been implicitly recognized for two-beam case theory, was explicitly described in the present chapter. Whereas the former theory can be calculated by solving an eigenvalue-eigenvector problem, the latter can be calculated by solving a partial differential equation. This equivalence is very similar to that between the Heisenberg and Schrödinger pictures of quantum mechanics and is very important.
Whereas this chapter has been described with focusing on the n-beam case that n ∈ {3, 4, 5, 6, 8, 12}, the n-beam X-ray dynamical diffraction theory applicable to the case of arbitrary number of n, which is effective and important for solving the phase problem in protein crystal structure analysis, will be described elsewhere. In the case of protein crystallography, the situation that arbitrary number of reciprocal lattice nodes are very close to the surface of the Ewald sphere, cannot be avoided. In protein crystallography, the n-beam X-ray dynamical diffraction theory for arbitrary number of n is necessary.
Yasuhiko Imai and Yoshitaka Yoda Japan Synchrotron Radiation Research Institute, Japan