## 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 (

The most primitive

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

In the

The

## 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]

Here,

By applying an approximation that

Let the electric displacement vector

1.07 Here,

Here,

where

where

(2) and (3) can also be represented using matrices and vector as follows:

Here,

Here,

### 2.2. Derivation of the n -beam Takagi-Taupin equation from the Ewald-Laue theory

In this section, the

A general solution of dynamical diffraction theory is considered to be coherent superposition of Bloch plane-wave system when X-ray wave field

where

The amplitude of the

In this section, the amplitude of plane wave whose wave vector is

Substituting (4) with

Here,

(12) |

(13) |

Incidentally, when the crystal is perfect, the electric susceptibility

Then, in the case of crystal with a lattice displacement field of

The above equation is nothing but the

### 2.3. Derivation of the n -beam E-L dynamical theory from the T-T equation

In this section, it is described that the

When plane-wave X-rays are incident on the crystal to excite

Even when

Applying the same procedure as used when deriving (11),

(16) |

Comparing (15) and (16), the same equation as (3) is obtained. The equivalence between the

## 3. Algorithm to solve the theory

Figure 1

(17) |

The above equation (17) can be described using matrix and vectors as follows:

Here,

Figure 2 is a top view of Figure 1

## 4. Experimental

### 4.1. 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].

Figure 3 shows

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

After adjusting the angular position of the goniometer such that the

## 5. Results and discussion

### 5.1. Three-beam case

Three-beam case is the most primitive case of X-ray multiple reflection. Figures 5[

### 5.2. Four-beam case

Figures 6

Figures 7

Between the horizontal and vertical components of incident X-rays, there is difference not in amplitude but in phase among Figures [

### 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[

Remarking on the directions of

### 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[

### 5.5. Eight-beam case

Figure 10[

In Figure 11[

### 5.6. Twelve-beam case

Twelve is the largest number of

A very bright region (

## 6. Summary

TheThe equivalence between the E-L and T-T formulations of the

Whereas this chapter has been described with focusing on the