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Nano-Engineering Research Center, Institute of Engineering Innovation, Graduate School of Engineering, The University of Tokyo, 2-11-16 Yayoi, Bunkyo-ku, Tokyo 113-8656, Japan
Yasuhiko Imai
Japan Synchrotron Radiation Research Institute, SPring-8, 1-1-1 Kouto, Mikazuki-cho, Sayo-gun, Hyogo 679-5198, Japan
Yoshitaka Yoda
Japan Synchrotron Radiation Research Institute, SPring-8, 1-1-1 Kouto, Mikazuki-cho, Sayo-gun, Hyogo 679-5198, Japan
*Address all correspondence to:
1. Introduction
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, It has been recognized for many years since the suggestion by Lipscomb in 1949 [12] that the phase information can be physically extracted, at least in principle, from X-ray diffraction profiles of three-beam cases in which transmitted and two reflected beams are simultaneously strong in the crystal. This suggestion was verified by Colella [3] that stimulated many authors [2, 4, 5, 21, 22] and let them investigate the multiple-beam (n-beam) method to solve the phase problem in protein crystallography.
The most primitive n-beam diffraction is the cases n=3. The shape of three-beam rocking curve simply depends on the triplet phase invariant. In the case of protein crystallography, however, it is extremely difficult to realize such three-beam cases that transmitted and only two reflected beams are strong in the crystal, which is due to the extremely high density of reciprocal lattice nodes owing to the large size of unit cell of the crystal. Therefore, X-ray n-beam dynamical diffraction theory is necessary to solve the phase problem in protein crystallography. The Ewald-Laue (E-L) dynamical diffraction theory [7, 11] was extended to the three-beam cases in the late 1960’s [8, 9, 10]. The numerical method to solve the n-beam (n≤3) E-L theory was given by Colella [3]. Colella’s method [3] to solve the n-beam E-L theory is applicable only to the case of crystals with planar surfaces.
1.03 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.
2. Derivation of the n-beam Takagi-Taupin equation
2.1. Description of the n-beam Ewald-Laue dynamical diffraction theory
The fundamental equation with the E-L formulation is given by [1, 2]
ki2-K2ki2Di=∐ϥhi-hjDj⊤ki.E1
Here, ki is wavenumber of the ith numbered Bloch wave whose wave vector is k0+hi where k0 is the wave vector of the forward-diffracted wave in the crystal, K(=1/ϙ) is the wavenumber of X-rays in vacuum, Di and Dj are complex amplitude vectors of the ith and jth numbered Bloch waves, ∐ is an infinite summation for all combinations of i and j, ϥhi-hj is Fourier coefficient of electric susceptibility and [Dj]⊤ki is component vector of Dj perpendicular to ki, respectively.
By applying an approximation that ki+K≇2ki to (1), the following equation is obtained,
ϜiDi=K2∐ϥhi-hjDj⊤ki,E2
whereϜi=ki-K.
Let the electric displacement vector Di be represented by a linear combination of scalar amplitudes as follows:
Di=Di(0)ei(0)+Di(1)ei(1).
1.07 Here, ei(0) and ei(1) are unit vectors perpendicular to si, where si is a unit vector parallel to ki. si, ei(0) and ei(1) construct a right-handed orthogonal system in this order. (2) can be described as follows:
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. ζB is the angle spanned by PQ⃗ and ki 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. Ϝ is such a value that
P1P1'⃗=-ϜPQ⃗/PQ⃗,E5
where P1' is the common initial point of ki [whose terminal points are Hi(i∇{0,1,⋮,n-1})] and P1 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 Hi(i∇{0,1,⋮,n-1}) is the identical value K. For description in the next section, it is described here that P1P1'⃗ is represented by a linear combination of si, ei(0) and ei(1) as follows:
P1P1'⃗=-ϜcosζBsi+ϕi(0)sinζBei(0)+ϕi(1)sinζBei(1).P1 is such a point that
P1La⃗=Kϐ(0)e0(0)+ϐ(1)e0(1).E6
(2) and (3) can also be represented using matrices and vector as follows:
ϜcosζBED=AD.E7
Here, E is a unit matrix of size 2n, D is a amplitude column vector of size 2n whose qth element is Dj(m)(q=2j+m+1) and A is a square matrix of size 2n whose element ap,q is given by
Here, p=2i+l+1 and ϒp,q is Kronecker delta. 2n couples of Ϝ 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 P1' of wave vectors of Bloch waves should be, is given as 2n sets of eigenvalues for (7).
2.2. 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 field D̡(r) is given as follows:
In this section, the amplitude of plane wave whose wave vector is ki and polarization state is l is denoted by Di(l)(βk) in place of Di(l) in order to clarify this value depends on βk. Di(l)(r) in (8) is represented by superposing coherently Di(l)(βk) as follows:
Di(l)(r)=∪βkD.S.Di(l)(βk)exp-i2Ϟβk⋄rdSk.E10
Substituting (4) with j=0, (5), (6) and (9) into (10),
Here, ∪βkD.S.dSk means an integration over the dispersion surfaces in reciprocal space and Ti(Ϝ,ei(0),ei(1)) is a term that does not depend on si. ∁Di(l)(r)/∁si can be calculated as follows:
Incidentally, when the crystal is perfect, the electric susceptibility ϥ(r) is represented by Fourier series as ϥ(r)=∐hiϥhiexp[-i2Ϟhi⋄r]. However, when the crystal has a lattice displacement field of u(r), the electric susceptibility is approximately given by ϥ[r-u(r)] and represented by Fourier series as follows,
ϥ[r-u(r)]=∐hiϥhiexp[i2Ϟhi⋄u(r)]exp(-i2Ϟhi⋄r).
Then, in the case of crystal with a lattice displacement field of u(r), ϥhi-hj can be replaced by ϥhi-hjexp[i2Ϟ(hi-hj)⋄u(r)]. Therefore, the following equation is obtained from (13),
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 Ri(0)R(1)⃗ in Figure 1(a) are parallel to si. When the length of Ri(0)R(1)⃗ is sufficiently small compared with the extinction length -1/(ϥ0K) of the forward diffraction, The T-T equation (14) can be approximated by
Here, A and D are column vectors of size 2n whose pth and qth elements are ap and dq, respectively, and B is a square matrix of size 2n whose element of the pth column and the qth raw is bp,q.
Figure 2 is a top view of Figure 1(b). The X-ray amplitudes Dj(m)(Ri(1)) were calculated from the X-ray amplitudes at the incidence point D0(l)(Rinc) of the crystal surface. In this case, 000-forward-diffracted and 4¯04-, 4¯26-, 066-, 264- and 220-reflected X-rays are simultaneously strong. The angle spanned by nx- and ny-axes is 120∗. RincRi(1)⃗ in Figure 1(b) are parallel to the wave vectors of 000-forward-diffracted and 4¯04-, 4¯26-, 066-, 264- and 220-reflected X-rays. As a boundary condition on the crystal surface, amplitude array Deven(i,l,nx,ny) has nonzero value (unity) when (i,l,nx,ny)=(0,0,0,0) or (i,l,nx,ny)=(0,1,0,0). On the first layer, nonzero X-ray amplitudes Dodd(j,m,nx,ny) are calculated when (nx,ny) = [nx'(i),ny'(i)](i∇{0,1,⋮,n-1}). Here, [nx'(i),ny'(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, Deven(i,l,nx,ny) [or Dodd(i,l,nx,ny)] is calculated as Di(l)(R(1)) by substituting Dodd(j,m,nx-nx'(i),ny-ny'(i)) [or Deven(j,m,nx-nx'(i),ny-ny'(i))] into Dj(m)(Ri(0)) in (17). The calculation was performed layer by layer scanning nx and ny in a range of NMin[nx'(i)]≣nx≣NMax[nx'(i)] and NMin[ny'(i)]≣ny≣NMax[ny'(i)], where N is the ordinal number of layer. The values of ϥhi-hj were calculated by using XOP version2.3 [6].
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].
Figure 3 shows (a) a schematic drawing of the phase retarder system and (b) a photograph of it. [100]-oriented four diamond crystals PRn(n∇{1,2,3,4}) with thicknesses of 1.545, 2.198, 1.565 and 2.633 mm were mounted on tangential-bar type goniometers such that the deviation angles βϖPRn from the exact Bragg condition of 111 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 ϥ-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 ϥ-circle ϥPR and βϖPRn 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.
4.2. Sample crystal
Figure 4 is a reproduction of Fig 7 in reference [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 [11¯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 ϥ-, ϳ-, ϧ- and ϖ-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.
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 Ϛm.
N images of n-beam pinhole topographs were simultaneously recorded on an imaging plate placed behind the sample crystal.
Three-beam case is the most primitive case of X-ray multiple reflection. Figures 5[E(a)] and 5[S(a)] are 000-forward-diffracted and 4¯04- and 044-reflected X-ray pinhole topograph images. Figures 5[E(b)] and 5[S(b)] are enlargements of 044-reflected images from Figures 5[E(a)] and 5[S(a)], respectively. Fine-fringe regions #1 and #2 ([FFR(1)] and [FFR(2)]) and ‘Y-shaped’ bright region (YBR) indicated by arrows in Figure 5[S(b)], are found also in Figure 5[E(b)].
5.2. Four-beam case
Figures 6[E(x)] and 6[S(x)](x∇{a,b,c}) show experimentally obtained and computer-simulated pinhole topographs of 000-forward-diffracted, and 066-, 6¯28- and 6¯2¯4-reflected X-ray images, respectively. [Y(a)], [Y(b)] and [Y(c)](Y∇{E,S}) 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.
Figures 7[E(x)] and 7[S(x)](x∇{a,b,c}) are enlargements of 6¯28-reflected X-ray images from Figures 6[E(x)] and 6[S(x)]. In Figure 7[S(a)], fine-fringe region #1 [FFR(1)], fine-fringe region #2 [FFR(2)] and knife-edge line (KEL) are indicated by arrows. These characteristic patterns are also observed in Figure 7[E(a)]. In Figures 7[E(b)] and 7[S(b)], while FFR(2) is observed at the same position, a pattern like a fish born (PFB) is observed in place of [FFR(1)]. KEL in Figures 7[E(b)] and 7[S(b)] are fainter. Furthermore, an arched line (AL) and a bright region (BR) not observed in Figures 7[E(a)] and 7[S(a)] are observed in Figure 7[E(b)] and 7[S(b)]. In Figures 7[E(c)] and 7[S(c)], almost all the characteristic patterns above-mentioned are observed.
Between the horizontal and vertical components of incident X-rays, there is difference not in amplitude but in phase among Figures [Y(a)], [Y(b)] and [Y(c)] (Y∇{E,S}), which reveals that the wave fields excited by horizontal- and vertical-linearly polarized components of the incident X-rays interfere with each other.
5.3. Five-beam case
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].
Figures 8[E(a)] and 8[S(a)] are experimentally obtained and computer-simulated five-beam pinhole topographs. Figures 8[E(b)] and 8[S(b)] are enlargements of 5¯55-reflected X-ray images from Figures 8[E(a)] and 8[S(a)]. Knife-edge patterns #1 and #2 [KEL(1) and KEL(2)] and ‘harp-shaped’ pattern (HpSP) indicated by arrows in Figure 8[S(b)] are observed also in Figure 8[E(b)].
Remarking on the directions of KEL(1) and KEL(2), these knife-edge patterns are directed to 000-forward-diffracted and 333-reflected X-ray images, respectively. Then, KEL(1) and KEL(2) are considered to suggest the strong energy exchange mechanism between 000-forward-diffracted and 5¯55-reflected X-ray wave fields and between 333- and 5¯55-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.
5.4. Six-beam case
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.
Such six-beam pinhole topographs experimentally obtained and computer-simulated are shown in Figure 9. Figures 9[E(b)] and 9[S(b)] are enlargements of 264- and 066-reflected X-ray images from Figures 9[E(a)] and 9[S(a)]. Knife-edge patterns [KEL(1) and KEL(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)].
5.5. Eight-beam case
Figure 10[E(a)] and 10[S(a)] are eight-beam X-ray pinhole topographs experimentally obtained and computer-simulated, respectively, with an incidence of horizontal-linearly polarized X-rays. Figure 10[E(b)] and 10[S(b)] were obtained with an incidence of vertical-linearly polarized X-rays. Figure 11[E(x)] and 11[S(x)] (x∇{a,b}) are enlargements of 000-forward-diffracted X-ray images from Figures 10[E(x)] and 10[S(x)], respectively.
In Figure 11[S(a)], A ‘harp-shaped’ pattern (HpSP), knife-edge line (KEL), ‘hook-shaped’ pattern (HkSP), ‘Y-shaped’ pattern (YSP) and ‘nail-shaped’ patterns are indicated by arrows. All these characteristic patterns are observed also in Figure 11[E(a)]. NSP is also observed in Figures 11[E(b)] and 11[S(b)]. However, HpSP in Figures 11[E(b)] and 11[S(b)] are rather fainter compared with Figures 11[E(a)] and 11[S(a)].
5.6. Twelve-beam case
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. Figures 12[E(a)] and 12[S(a)] are experimentally obtained and computer-simulated tweleve-beam pinhole topographs. Figures 12[E(b)] and 12[S(b)] are enlargements of 242-reflected X-ray images from Figures 12[E(a)] and 12[S(a)].
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 n-beam (n∇{3,4,5,6,8,12}) Takagi-Taupin equation and computer algorithm to solve it were verified from excellent agreements between experimentally obtained and computer-simulated three-, four-, five-, six-, eight- and twelve-beam pinhole topographs.
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.
The part of theoretical study and computer simulation of the present work was conducted in Research Hub for Advanced Nano Characterization, The University of Tokyo, supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
HITACHI SR-11000 and SGI Altix ICE 8400EX super computer systems of the Institute for Solid State Physics of The University of Tokyo were used for the computer simulations.
The preliminary experiments were performed at AR-BL3A of the Photon Factory AR under the approval of the Photon Factory Program Advisory Committee (Proposals No. 2003G202 and No. 2003G203). The main experiments were performed at BL09XU of SPring-8 under the approval of Japan Synchrotron Radiation Research Institute (JASRI) (Proposals No. 2005B0714 and No. 2009B1384).
The present work is one of the activities of Active Nano-Characterization and Technology Project financially supported by Special Coordination Fund of the Ministry of Education, Culture, Sports, Science and Technology of the Japan Government.
The authors are indebted to Dr. Y. Ueji Dr. X.-W. Zhang and Dr. G. Ishiwata for their technical support in the present experiments and also to Professor Emeritus S. Kikuta for his encouragements and fruitful discussions for the present work.
References
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2.ChangS.L.2004X-Ray N-Beam Takagi-Taupin Dynamical Theory and N-Beam Pinhole TopographsExperimentally Obtained and Computer-SimulatedSPringer.