9 Three-Dimensional Magnetically-Oriented Microcrystal Array : A Large Sample for Neutron Diffraction Analysis

Structure determination of biomolecules is of great importance because the structure is closely related to biological functions. Some proteins are activated when binding with ligands; a drug molecule functions by binding to a specific protein site (Sousa et al., 2000; Graves et al., 1994). The structure determination is important not only for biomolecules but also for many inorganic, organic, and polymeric crystals that are key materials in materials and pharmaceutical sciences.


Introduction
Structure determination of biomolecules is of great importance because the structure is closely related to biological functions. Some proteins are activated when binding with ligands; a drug molecule functions by binding to a specific protein site (Sousa et al., 2000;Graves et al., 1994). The structure determination is important not only for biomolecules but also for many inorganic, organic, and polymeric crystals that are key materials in materials and pharmaceutical sciences.
Currently, three major methods to solve the structure of molecules are known: solution nuclear magnetic resonance (NMR) (Herrmann et al., 2002), X-ray and neutron single crystal analysis (Blundell et al., 1976;Kendrew et al., 1958;Blake et al., 1965), and X-ray powder diffraction (Margiolaki et al, 2008;Hariss et al., 2004) methods. The solution NMR method has an advantage over diffraction methods because it does not require crystals. However, it can be applied only to proteins with lower molecular weights. X-ray and neutron singlecrystal analysis is the most powerful method, but it is sometimes difficult to grow a crystal (Ataka et al., 1986) sufficiently large for conventional or synchrotron single-crystal X-ray measurement. The size requirement is much more severe for neutron diffraction measurements (Niimura et al., 1999). The X-ray powder method can be performed if microcrystalline powders are available, but an appropriate refinement of many parameters is needed to obtain a reliable result. Preferential orientation of a powder sample is utilized to produce single-crystal-like diffraction data (Wessels et al., 1999). However, the data quality strongly depends on the quality of the orientation.
It is well known that feeble magnetic materials such as most biological, organic, polymeric and inorganic materials respond to applied magnetic fields, although the response is weak. A number of studies on the magnetic alignment of such materials have been reported (Maret et al., 1985;Asai et al., 2006). The study of the feeble interaction of diamagnetic materials with an applied magnetic field and its application are now recognized as an immerging area of science and technology, and named "Magneto-Science".
We have recently proposed a method that enables to convert a diamagnetic or paramagnetic microcrystalline powder to a "pseudo single crystal (PSC)" (T. Kimura et al. 2005; T. Kimura, www.intechopen.com Neutron Diffraction 180 2009a) (Fig. 1). A PSC is a composite in which microcrystals are oriented three-dimensionally in a resin matrix. A PSC is also referred to as a "three-dimensional magnetically-oriented microcrystal array" (3D-MOMA) (F. Kimura et al., 2011). This method enables us to perform a single-crystal analysis of a material through alignment of its powder sample. The composite is fabricated using magnetic alignment of a microcrystal suspension under an elliptically rotating magnetic field, followed by consolidation of the suspending matrix.
The degree of alignment strongly depends on the diamagnetic anisotropy of the crystal, applied field strength, and size of microcrystals to be aligned. If these conditions are appropriately satisfied, the obtained 3D-MOMA can produce well-separated diffraction spots that allow single-crystal analysis. This method can be applied to biaxial crystals including the triclinic, monoclinic, and orthorhombic systems which have three different magnetic susceptibility values. We only obtain fiber diffraction patterns for uniaxial crystals even if we apply the 3D-MOMA method. This method does not work for the cubic system. However, the 3D-MOMA method has some drawbacks including the limitation of the suspending medium, the difficulty of recovering a precious sample, and the broadening of diffraction spots by consolidation. A solution to these problems may be measurement of the X-ray diffraction patterns directly from a three-dimensional magnetically-oriented microcrystal suspension (3D-MOMS) without consolidation of the suspending medium. Xray diffraction from magnetically oriented solutions of macromolecular assemblies was reported (Glucksman et al., 1986;Samulski et al., 1986). Also, small-angle X-ray scattering of colloidal platelets (van der Beek et al., 2006) and molecular aggregates (Gielen et al., 2009) under magnetic fields were reported. In-situ X-ray (Kohama et al., 2007) and neutron (Terada et al., 2008) diffraction measurements under applied magnetic fields were reported. We showed a preliminary result (Matsumoto et al., 2011) of the X-ray diffraction from a MOMS achieved with and without sample rotation in a static magnetic field. We believe that the magnetic method we introduce here is of considerable use for the crystal structure determination of materials that do not grow large sizes necessary for the diffraction measurement. Especially, our method has advantages when applied to single crystal neutron diffraction analyses of proteins where a single crystal sample much larger than that needed for the X-ray diffraction measurement is required.

Theoretical background of preparation of a 3D-MOMA
A crystal has a diamagnetic susceptibility tensor , which is expressed in terms of the principal diamagnetic susceptibility axes ( 3 < 2 < 1 <0). When the crystal is exposed to a magnetic field B, it has a magnetic energy, expressed with respect to the laboratory coordinate system as follows where  0 is the magnetic permeability of the vacuum, V is the volume of the crystallite, A is the transformation matrix defined by the Eulerian angles (, , , and the superscript t indicates the transpose. We consider an applied elliptical magnetic field that periodically changes in direction and intensity on the xy plane as follows: Here,  is the angular velocity of the field rotation, B is the intensity of the field, and b x and b y are the longer and shorter axes of the ellipse (b y < b x ), respectively. If  is far larger than a certain rate   (the intrinsic rate of magnetic response of a particle exposed to the static field of the intensity B) then the easy magnetization axis,  1 , cannot follow the rotation of the field. This condition is called "Rapid Rotation Regime (RRR)", defined as ||>>1/2, where  is expressed as  = 6 0 / a B 2 in the case that the particle shape is spherical (T. Kimura et al., 2000) where  a and  are magnetic anisotropy and the viscosity of the surrounding medium.
In the RRR, a crystal could be assumed to be placed in an time-averaged magnetic potential expressed by the following equation: Inserting eq (2) into eq (3) and performing the integration, we obtain E av as an analytical function of the Eulerian angles. Expanding eq (3) around ===0 and truncating the higher terms, we obtain (T. Kimura et al., 2005)   where the constant terms are not shown.
where k B is the Boltzmann constant and T is the temperature. To obtain a good 3D-MOMA, we should minimize and equalize the values of  ( 2  and  (( +  )) 2  by appropriately choosing the values of b x and b y . The crystallite volume V is also important. The smaller the V, the larger the fluctuations. Typically, the size necessary is about the order of micrometers, depending on the field strength used and the magnetic anisotropy of the crystal. To obtain a high quality diffraction pattern, the rotation speed  is also important. It should satisfy ||>>1/2 (RRR) (T. Kimura et al., 2005).
In general, the susceptibility axes of a biaxial crystal do not necessarily coincide with the crystallographic a, b, c axes, except the orthorhombic system ( Fig. 2a) (F. Kimura et al., 2010a). The orthorhombic system includes three point groups, that is, 222, mm2, and mmm. Both for 222 and mmm, three crystallographic axes are 2-fold axes that coincide with the susceptibility axes. As a result, the rotation about the susceptibility axes through an angle of π does not create new crystal orientations. On the other hand, mm2 has just one 2-fold axis. Therefore, the rotation about the susceptibility axes can produce a new orientation. For the monoclinic system (point group 2, space group P2 1 ), its twofold axis (b axis) coincides with one of the magnetic susceptibility axes (Nye, 1985). The crystallographic and magnetic axes are shown schematically in Fig. 2b, where the b axis and axis are placed perpendicular to the plane of the diagram (F. Kimura et al., 2010b). Since the three magnetic susceptibility axes are mutually perpendicular, the other two magnetic susceptibility axes are placed on the plane of the diagram. The a and c axes are on the same plane. However, there is no rule to relate the axes with the a and c axes. If the crystal is rotated by an angle ofπaround each of the magnetic susceptibility axes, four different crystal orientations are obtained, as shown in Figs. 2b (a)-(d). These four orientations have the same magnetic energy when they are placed in a given magnetic field. Because of the twofold symmetry (point group 2) along the b axis, the crystal orientations of Figs. 2b(a) and 2b(c) are identical; those of Figs. 2b(b) and 2b(d) are also identical. Thus, there are two different orientations with the same magnetic energy. In Table 1, possible crystal orientations with an equal magnetic energy are summarized for crystals belonging to the orthorhombic, monoclinic, and triclinic systems.

Generation of modulated magnetic field
A 3D-MOMA of a biaxial crystal was prepared under modulated magnetic fields. In the previous section, the calculation was performed using the elliptical field. In the actual experiment, there are various versions for the elliptical field. We used two different dynamic magnetic fields (T. Kimura et al., 2009b). The first one is an amplitude modulated dynamic field (T. Kimura et al., 2005). This is generated by a four-pole electromagnet, in which one pair of poles generates a x B field strength sinusoidally oscillating at  , and the other generates y B field strength oscillating at the same frequency as the previous one but with a phase shift of / 2  . This combination created a magnetic field expressed as (c o s ,s i n , 0 ) xy Bt Bt   , shown in Fig. 3. The second one is a frequency modulated dynamic magnetic field (T. Kimura et al., 2006). In this setup, a sample is non-uniformly rotated in a uniform static magnetic field. A sample-rotating unit was placed in a uniform horizontal magnetic field, shown in Fig. 4. The rotation axis was vertical. The rotation was not uniform. In the figure, the x-, y-, and z-axes are laboratory coordinates and the x'-, y'-, and z'-axes are imbedded in the rotating plate. The x-axis is parallel to the magnetic field. The z'-and z-axes are parallel to vertical direction. The angle between the x'-axis before rotating and x-axis is /2 The three parameters,  s ,  q , and , must be selected appropriately in order to obtain sharp diffraction spots. The half-width of a spot is a result of the fluctuations of the  1 and  3 axes about the x' and z' axes, respectively. In terms of diffraction analysis, it is advantageous that the magnitudes of the two fluctuations are equal. www.intechopen.com

Sample preparation 4.1 Preparation of 3D -MOMA
Micro-crystallites were mixed with a UV (Ultra Violet) curable monomer. The concentration of the crystallites in the monomer is 10-30 v/v%. The suspension is poured into any size of plastic container, exposed to a dynamic magnetic field, followed by UV light irradiation to polymerize the resin precursor to fix the alignment. See experimental details in Table 2.

Preparation of MOMS
Typical experimental setting (KU model 10-1) is shown in Fig. 5. Sample suspension is poured into a capillary, and then it is placed on a rotating unit equipped with a pair of magnets.

X-ray results
We fabricated 3D-MOMAs using frequency modulated magnetic field except LiCoPO 4 . We fabricated 3D-MOMAs of LiCoPO 4 using both amplitude and frequency modulated magnetic fields. Alignment of these 3D-MOMAs are almost the same. Here, we only show the results obtained from the MOMA using frequency modulated magnetic field.

Alanine (T. Kimura et al., 2006)
The top figure in Fig. 6 shows the XRD powder pattern obtained from the experiment in which magnetic field was not applied. A sparse ring pattern is observed in this case. The diffraction spots are broad compared to those from a real single crystal, but they are sufficiently distinguishable to be assigned by the software program. Since the lattice parameter is the largest for the b axis, the spots in Fig. 6(b) are relatively less resolved. The half widths of the spots obtained by using an azimuthal plot are ca. 3for most of the spots in all of the patterns in Figs. 6 (a), (b), and (c). Incidentally, the half widths for an actual original single crystal are ca. 0.5. The fluctuations in  and + are complex functions in the case of the frequency-modulated magnetic field used in the present study. However, they can be roughly estimated by using eqs (5) and (6). Using the values of B=5 T, b x =1, b y =0.5, T=300 K, and a rough estimation of  1- 2 and  2- 3 =10 -7 , we can obtain the estimation for the fluctuation of  and + ca. 1for V =1000 m 3 . This value should be compared with the half widths of the diffraction spots. The experimental values are larger than those obtained by the theoretical estimation. Using the values of the viscosity ( =1.2 Pa s) and the shape factor (F = 0.064, corresponding to the assumed aspect ratio of 11 for crystallites), we can obtain the estimation, =1 min. The time required for the completion of the alignment might be estimated as five times of , i.e., ca. 5 min. This value is far shorter than 2 h actually applied in the experiment. These discrepancies have several possible explanations, including the existence of smaller crystallites, imperfect dispersion, shrinkage of the resin during the UV cure, the flow caused by sample rotation, insufficient rotation speed, imperfect switching from one rotation to the other, etc.

LiCoPO 4 (T. Kimura et al., 2009b)
A typical diffraction of the 3D-MOMA image is shown in Fig. 7. The estimated mosaicity is 3.9°, which is larger than that observed for normal single crystals on the same instrument (around 0.8°). The cell constants and an orientation matrix for data collection corresponded to a primitive orthorhombic cell with dimensions a = 10.202 (6) (1994), who used a 0.090  0.108  0.158 mm single crystal. The structure was solved by direct methods and expanded using Fourier techniques. The present result was compared with that from the literature, showing that the atomic coordinates determined in this study are in excellent agreement with those determined using a traditional single crystal. The R1 and wR2 values were 6.59 and 16.8%, respectively.

Sucrose (F. Kimura et al., 2010b)
The sucrose crystal belongs to the monoclinic system (point group 2, space group P2 1 ). Assuming an initial random orientation of crystallites in a suspension, the probability of finding two different orientations with the same magnetic energy is equal. These two orientations produce a diffraction pattern similar to that produced by a twin crystal. The diffraction image was analyzed using software designed for twin structures. The cell constants correspond to a primitive monoclinic cell with dimensions a = 7.7735 (12)   In Fig. 8, the structure solved from the MOMA is compared with that reported previously (Hanson et al., 1973). They show a good agreement. The twin structure is shown in Fig. 9. The magnetic susceptibility axes, as discussed previously (Fig. 9), are also shown. In principle, these magnetic susceptibility axes are attributed to each of the easy (largest)  1 , hard (smallest) 3 , and intermediate (medium)  2 magnetic susceptibility axes. The angle between the magnetic axis,  1 , and crystallographic c axis is determined to be ca 16°. This value is different from a previously reported value of -1° 5' (Finke, 1909). The reason for the difference is unclear at present.

Lyzozyme (F. Kimura et al., 2011)
The fluctuations of the  1 and  3 axes were estimated from the half-widths determined from the rocking curve and the azimuthal β scan, respectively. The rotation axis of the scan coincided with the  3 axis. The data was collected using  scans at 1.0° steps. The three parameters,  s ,  q , and α, must be selected appropriately in order to obtain sharp diffraction spots. The half-width of a spot is a result of the fluctuations of the  1 and  3 axes about the x' and z'-axes, respectively. In terms of diffraction analysis, it is advantageous that the magnitudes of the two fluctuations are equal. Sample rotation speed  s > 1/(2τ)should be met (Rapid Rotation Regime (RRR)). Next, an appropriate choice of α and q is necessary in order to minimize and equalize the fluctuations of the  1 and  3 axes.
In alanine, LiCoPO 4 , and sucrose studies describing the single-crystal analysis of MOMAs, =90° was appropriate. However, a theoretical study (T. Kimura, 2009a) shows that the equalization cannot be achieved for some sets of ( 1 ,  2 ,  3 ) if α is fixed to 90°. Since the three values of the magnetic susceptibility of lysozyme are unknown, we need to find out an appropriate value of  by trial and error.  Table 3. Fluctuation of  1 and  3 for three sets of parameters( s ,  q , α), determined using the (400) diffraction spot.
We tested several combinations of the parameters, and  =10°,  s =10 rpm, and  q =60 rpm was found to be the most suitable regarding minimization and equalization of the halfwidths. This condition was therefore chosen for the preparation of a MOMA for X-ray analysis. In Fig. 10, photograph and microphotograph of fabricated lysozyme MOMA are shown. In Fig. 11, diffraction patterns taken from three directions are shown. Well-separated diffraction spots were obtained. The directions of the magnetic axes as deduced from the preparation procedure are indicated in the figure. From the results of the indexing described in the next paragraph, the crystal belongs to the orthorhombic system. For this crystal www.intechopen.com system, the magnetic axes correspond to the crystallographic axes. From the figure, we find that the cell dimensions increase in the order  1 ,  2 , and  3 . Combining the indexing results, we conclude that  1 = c,  2 = a, and  3 = b. The alignment of the c axis parallel to the magnetic field has been reported for orthorhombic lysozyme crystals. ( Sato et al., 2000). Figure 12 shows a detailed diffraction pattern with resolution rings. The X-ray results are summarized in Table 4. The indexing was determined as follows: space group, P2 1 2 1 2 1 ; lattice constants, a=51.26 Å, b=59.79 Å , c=29.95 Å , and V=91803 Å 3 . This data was compared with that reported on the lysozyme single crystal (PDB code 2ZQ4) in Table 5. The cell dimensions obtained in this study are shorter than those reported in the literature. The shrinkage was attributed to the dehydration of the crystal. X-ray diffraction of dehydrated lysozyme crystals in triclinic (Kachalova et al., 1991) and monoclinic (Nagendra et al., 1998) forms has been reported. The graphical display is shown in Fig. 13 Kimura et al., 2011): Cryst. Growth Des. Vol.11(2011  In measuring the pole figure, a 3D-MOMA sample was set on a goniometer approximately in the direction shown in Fig. 14(a). The angles and correspond to the operation angles of the goniometer. The angle runs from 0 to 180° at a given value of the angle that runs from 0 to -180°. At this setting, the diffractions corresponding to the (040), (002), (120), etc. are expected to appear at locations shown in the figure. In Fig. 14(b), the measured pole figure is displayed. Here, all the diffraction spots are displayed in the same figure. A half sphere was scanned so that spots for {120} were all observed. These spots appear around =0, -180° and =45, 135° because a/b = 1/2. On the other hand, a spot for (040) was only scanned on one side. Instead, a close examination of (040) was made by scanning in the vicinity of =0° and =90°. A contour plot of the intensity of (040) is shown in Fig. 15 as a function of and . The average fwhm over two angles is ca. 4°. A 2θ profile for the (040) diffraction is shown in Fig. 16 as a typical example. The peak is clearly distinguished from the background. This high signal-to-noise ratio was unexpected because no care was taken to reduce the background incoherent scattering when choosing the resin precursor. This might be attributed to the fact that the coherent diffraction is extremely increased because the microcrystals are aligned three-dimensionally.  After subtraction of the baseline, the intensity of the peak is calculated by integration and corrected for Lorentz factor; no absorption correction was performed. The structure refinement was performed with Shelxl-97 using the crystallographic data of neutron diffraction of L-alanine reported in the literature (Wilson et al., 2005), where the atom coordinates, the temperature factors, and the anisotropic temperature factors were fixed. The results are summarized in Table 6. The values of R1 and wR2 are 0.184 and 0.368, respectively. In the present preliminary study, the integration was performed only along the 2θ direction, and the integration with respect to and directions was not performed. In addition, absorption correction was not made. Hence, the intensity data could include some error. However, the experimental and calculated results appear to agree satisfactorily, indicating that the MOMA can provide diffraction data that will lead to a successful structure analysis. Figure 17 shows the diffraction patterns obtained for  = 0 rpm (static) and 20 rpm. At  = 0 rpm, the {002} diffractions did not appear, and the {120} diffractions appeared on the equator. This suggests that the reciprocal vectors G{002} are aligned in the direction of the applied magnetic field (||x ) and the reciprocal vectors G{120} are distributed randomly on the yz plane (see Fig. 5). Because the crystal has three mutually orthogonal 2-fold axes (a, b, and c axes), the diffractions belonging to {120} are not distinguishable in a MOMA, neither are the diffractions {002}. With an increase in rotation speed, the {120} diffractions on the equator disappeared and moved to locations around 45° and 135° when  = 20 rpm. At the same time, the {002} diffractions appeared on the meridian at this rotation speed. This indicates that the b* axis was directed to the z-axis, and there was rotational symmetry about the b* axis. The alignment of the  3 axis (||b*) in the direction of the rotation axis (||z) is based on the behavior of a magnetically uniaxial particle (T. Kimura, 2009;Yamaguchi et al., 2010). We successfully obtained two fiber X-ray diffraction patterns from a microcrystal suspension (crystal size of ca. 2-20 μm) oriented in a magnetic field of 1 T generated by permanent magnets. Diffraction spots were sharp (ca. 2-3° in half width) and well separated. The results suggest a potential use of MOMS for crystal structure analysis of solid materials that do not grow into large crystals but are obtained in the form of microcrystal suspensions.

Possibility of MOMS for application of structure analysis (Matsumoto et al., 2011)
To summarize, we have introduced a new method, 3D-MOMA (three-dimensional magnetically oriented microcrystal array) method. We use a 3D-MOMA that is a composite in which microcrystals are three-dimensionally aligned. With a 3D-MOMA, we can obtain diffraction equivalent to that from a real single crystal, successfully determining the molecular structure. With this method we can convert a suspension of biaxial microcrystals with sizes of 10 m or larger to a 3D-MOMA of centimeter sizes by using magnetic fields of 2 T or more. Successful examples of single crystal X-ray analyses of several crystals, including inorganic, organic, and protein crystals, by using 3D-MOMA are presented. A preliminary result of neutron diffraction of 3D-MOMA is also reported. Though the application to the neutron diffraction is preliminary, the MOMA method has high potential for the neutron diffraction method because a large size 3D-MOMA can easily be prepared. The problems to be solved for the future development of this method in the area of neutron single crystal diffraction analyses are discussed.