Orientation Fluctuation of Microcrystals under Three-Dimensionally

Apr 4, 2013 - and that the principal axes of fluctuation are inclined with respect to the longitude and latitude of the reciprocal lattice sphere. Mag...
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Orientation Fluctuation of Microcrystals under Three-Dimensionally Constraining Dynamic Magnetic Field Tsunehisa Kimura,* Tatsuya Tanaka, Guangjie Song, Kenji Matsumoto, Keiji Fujita, and Fumiko Kimura Division of Forest and Biomaterials Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan ABSTRACT: The orientation fluctuation of a reciprocal lattice vector of a biaxial microcrystal suspended in a liquid medium under a dynamic magnetic field is investigated theoretically to evaluate the broadening of the X-ray diffraction spots from a suspension of the microcrystals. The dynamic magnetic fields considered are those that confine the three magnetic susceptibility axes to the laboratory coordinates. The fluctuation is expressed in terms of the magnetic anisotropy of the biaxial microcrystal and the type of applied dynamic magnetic field. We find that for most reciprocal lattice vectors, the fluctuation is anisotropic, and that the principal axes of fluctuation are inclined with respect to the longitude and latitude of the reciprocal lattice sphere.

M

agnetic alignment of diamagnetic crystals1,2 provides a novel means to prepare a composite in which micro- and nanocrystals are oriented in a polymer matrix. This composite we refer to as MOMA, which denotes a magnetically oriented microcrystal array,3 can exhibit superior physical properties, and greatly assist in the determination of crystal structure by diffraction methods. The X-ray diffraction obtained from a MOMA prepared from a microcrystalline powder is comparable to that obtained from a real single crystal, enabling single crystal analysis using a microcrystalline powder sample. In fact, we have previously used MOMA to demonstrate the X-ray-based determination of the single-crystal structure of inorganic,4 organic,5 and protein3 crystals. The manner in which crystals are aligned and the degree of alignment depend on both the magnetic field applied and the magnetic anisotropy of the crystal to be aligned.6−11 For example, if a dynamic magnetic field is applied to a biaxial crystal (having three different magnetic susceptibilities χ1 > χ2 > χ3), the magnetic axes are oriented in a three-dimensional manner.12−16 Narrowing the half-width of the diffraction spots obtained from a MOMA is important in order to achieve wellresolved diffraction patterns suitable for the crystal structure analyses. For this purpose, the parameters of the applied dynamic magnetic field need be optimized based on the ratio of the magnetic anisotropy, rχ = (χ2 − χ3)/(χ1 − χ2), of the crystal under consideration. However, in many cases, this value is not known. In our previous papers, we derived the intensity of the fluctuation in a reciprocal lattice vector of a microcrystal under a static or a uniformly rotating magnetic field and related it to the half-width of the X-ray diffraction spots in order to determine the value of rχ.17,18 In the present paper, we develop a general formula for expressing the intensity of the anisotropic fluctuation in a reciprocal lattice vector of a biaxial crystal subjected to a dynamic magnetic field that confines each of the three magnetic axes three-dimensionally with respect to the laboratory coordinates. In light of the general description, a revised scheme for the determination of rχ is also proposed. © 2013 American Chemical Society

Let us consider a biaxial microcrystal, whose magnetic susceptibility tensor χ has three different principal values χ1 > χ2 > χ3 and which is suspended in a viscous medium. The anisotropic magnetic energy of this microcrystal, when it is placed in a magnetic field B, is expressed as E(ϕ,θ,ψ) = −(V/2 μ0) tB(tAχA)B, where μ0 is the magnetic permeability of the vacuum, V is the volume of the microcrystal, and A is the transformation matrix defined using the Euler angles (ϕ, θ, and ψ) (in this paper, the Z1Y2X3 convention is employed, as explained in appendix A of ref 19). These Euler angles define the orientation of the χ1χ2χ3 coordinates with respect to the laboratory xyz coordinates. We assume that the applied magnetic field is a dynamic one whose frequency is extremely high, whereby the susceptibility axes of the crystal cannot follow the frequency [this is referred to as the rapid rotation regime (RRR)20]. Under the appropriate choice of dynamic magnetic field,14,21 the χ1, χ2, and χ3 axes are confined to the laboratory x, y, and z coordinates, respectively. The anisotropic magnetic energy can be expanded around its minimum, to obtain E(ω) = tωCω = Cxωx 2 + Cyωy 2 + Czωz 2

(1)

where the isotropic and higher terms are neglected. The rotation vector is defined as ω = t(ωx,ωy,ωz), where ωx, ωy, and ωz correspond to the infinitesimally small values of ψ, θ, and ϕ, respectively. The matrix C is a diagonal matrix with its positive components Cx, Cy, and Cz defined by Cx = (2μ0 )−1B2 KxV (χ2 − χ3 ), Cy = (2μ0 )−1B2 K yV (χ1 − χ3 ), Cz = (2μ0 )−1B2 K zV (χ1 − χ2 )

(2)

Received: February 5, 2013 Revised: March 26, 2013 Published: April 4, 2013 1815

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where Kx, Ky, and Kz are positive constants, characteristic of the type of the applied dynamic magnetic field. In the case of an amplitude modulated elliptical magnetic field, defined by B(t) = B(bx cos Ω t, by sin Ωt, 0) with 0 < by < bx ≤ 1 and Ω as the rotation speed, we have14 Kx = by 2 /2,

K y = bx 2 /2,

K z = (bx 2 − by 2)/2

Let us define the direction of a reciprocal lattice vector G, corresponding to the (hkl) plane, by two angles Θ and Φ with respect to the χ1χ2χ3 coordinates embedded in the crystal lattice, as shown in Figure 2. As will be discussed later, the

(3)

In the case of a frequency-modulated elliptical magnetic field21 [bx = by, and the speed of rotation Ω changes between ωs and ωq (ωs < ωq) at every 90°], we have20,22 Kx = Ky = Kz =

(π + 2)ωs + (π − 2)ωq 2π (ωs + ωq ) (π − 2)ωs + (π + 2)ωq 2π (ωs + ωq )

, Figure 2. Angles Θ and Φ, which characterize the direction of the reciprocal lattice vector G with respect to the χ1χ2χ3 coordinates.

,

2(ωq − ωs) π (ωs + ωq )

definition of G is different when a static field is concerned. The orientation of G fluctuates due to the fluctuation in ω. The laboratory xyz coordinates are not suitable for describing the orientation fluctuation in G because the displacement ΔG due to the fluctuation in ω, is restricted to the surface of the reciprocal lattice sphere (the sphere with a radius G). Therefore, we transform the coordinates xyz to the new coordinates x′y′z′, where the z′ axis coincides with the direction of G and the x′ and y′ axes are parallel to the longitude and

(4)

From these two examples, we find that Ky = Kx + Kz. This relation is generally valid for other practically used constraining fields. Therefore, we obtain Ky > Kx, Kz. When rotated by the rotation vector ω, the χ1χ2χ3 coordinates deviate from the laboratory xyz coordinates (Figure 1), resulting in the increase

Figure 1. Rotation of the χ1χ2χ3 coordinates by the rotation vector ω defined with respect to the laboratory xyz coordinates. At ω = 0, the χ1, χ2, and χ3 axes are parallel to the x, y, and z directions, respectively. Figure 3. Relationship between the old xyz coordinates and the new x′y′z′ coordinates.

in anisotropic magnetic energy, as expressed by eq 1. The anisotropic magnetic energy attains a minimal value at ω = 0, where χ1∥x, χ2∥y, and χ3∥z. The rotation by the vector ω is regarded as equivalent to the three consecutive rotations about the x, y, and z axes by the angles ωx, ωy, and ωz, respectively. All possible combinations of three rotations are equivalent because each rotation is infinitesimally small and hence commutable. The probability of finding the rotational state of χ1, χ2, and χ3 axes between ω and ω + dω with respect to the xyz coordinates is expressed by ⎛ −E(ω) ⎞ f (ω) dω = K −1 exp⎜ ⎟ dω ⎝ kBT ⎠

latitude lines, respectively (Figure 3). This transformation is expressed by a matrix R that is defined by ⎛ cos Θ cos Φ cos Θ sin Φ −sin Θ ⎞ ⎟ ⎜ R = ⎜−sin Φ cos Φ 0 ⎟⎟ ⎜ ⎝ sin Θ cos Φ sin Θ sin Φ cos Θ ⎠ −1

The matrix R satisfies R = R. Through this transformation, the rotation vector ω and the matrix C are transformed to ω′ ≡ t (ωx′,ωy′,ωz′) = Rω and C′ = RCtR, respectively. Accordingly, the magnetic energy can be expressed as follows in terms of the new coordinates:

(5)

with K being the normalization constant, expressed by K=

∭

⎛ −E(ω) ⎞ exp⎜ ⎟ dω ⎝ kBT ⎠

(7)

t

E(ω′) = tω′C′ω′

(6)

(8)

This energy is quadratic in ωx′, ωy′, and ωz′. The probability density for ω′ is expressed by

where kB is the Boltzmann constant, T is the temperature, and dω = dωx dωy dωz. The range of integration can be extended to −∞ < ωx, ωy, ωz < ∞ because the integrand is significant only in the vicinity of small |ω| values. By using this probability, we obtain the mean squared values as = kBT/(2Cx), = kBT/(2Cy), and = kBT/(2Cz).

⎛ −E(ω′) ⎞ f (ω′) = k−1 exp⎜ ⎟ ⎝ kBT ⎠

(9)

where k is the normalization constant. 1816

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The vector ω′ is decomposed into two vectors: the vector ω′∥ = t(0,0,ωz′), parallel to G, and the vector ω′⊥ = t(ωx′,ωy′,0), perpendicular to G. Since ω′∥ is parallel to G, it simply rotates G about its own axis and does not explicitly contribute to the displacement ΔG. On the other hand, ω′⊥ is explicitly related to ΔG such that ΔG = ω′⊥ × G. In order to obtain the probability density for ω′⊥, we integrate eq 9 over ω′∥ (= ωz′) to obtain f (ωx ′ , ωy ′) = k−1

∞

⎛

⎞

∫−∞ exp⎜⎝ −Ek(Tω′) ⎟⎠dωz′ B

Figure 4. Contour maps of E(α,β) = 2.87 × 10−21 J for various reciprocal lattice vectors designated by Θ and Φ (Figure 2). The angles α and β correspond to −ω′x and −ω′y, respectively. Hence α and β are parallel to the y′ and −x′, respectively (see Figure 3). The contour is calculated using the following parameters: χ1 = −1 × 10−6, χ2 = −1.7 × 10−6, χ3 = −2 × 10−6; V = 1 μm3; T = 300 K; bx = 1, by = 0.7; B = 10 T.

(10)

The probability density f(ωx′,ωy′) is expressed as k−1 exp[−E(ωx′,ωy′)/kBT], where the energy E(ωx′,ωy′) is quadratic in ωx′ and ωy′. From the relationship between ΔG′ and ω′⊥, we obtain ωy′ = ΔGx′/|G| ≡ −β and ωx′ = −ΔGy′/|G| ≡ −α, where the angles α and β represent the fluctuation in G with respect to the y′ and −x′ axes, respectively. By using these relations, we obtain the following probability density for α and β: ⎛ E (α , β ) ⎞ −1 f (α , β) = kαβ exp⎜ − ⎟ kBT ⎠ ⎝

because Ky > Kx and χ1 > χ2 > χ3. Thus, we have Cy > Cx. Another possible option could be to set Cx = Cz. From this equality, we obtain Kz/Kx = rχ [= (χ2 − χ3)/(χ1 − χ2)]. As reported previously,14 this is equivalent to (bx2 − by2)/by2 = rχ in the case of the amplitude-modulated elliptical magnetic field. It is also equivalent to 4(r − 1)/(2 + π − 2r + πr) = rχ in the case of the frequency-modulated elliptical magnetic field, as reported previously,20 where r ≡ ωq/ωs. We now discuss the case of a uniaxial constraining field. When the constraining field is uniaxial with respect to the z axis, such as a static field that is applied to the z axis or a magnetic field uniformly rotating in the xy plane, we can set Kz = 0 (i.e., Cz = 0) in eqs 13a and 13b. In such a case, we obtain

(11)

Here, kαβ is the normalization constant and E(α,β) is expressed by E(α , β) = aα 2 + bαβ + cβ 2

(12)

where a = M −1Cz[Cx + Cy + (Cx − Cy) cos 2Φ] b = − 2M −1Cz(Cx − Cy) cos Θ sin 2Φ c = 2M −1[CxCy sin 2 Θ + Cz cos2 Θ(Cy cos2 Φ + Cx sin 2 Φ)]

E (β ) =

(13a)

CxCy β 2 Cx − (Cx − Cy)sin 2 Φ

(14)

2

The mean square average is calculated as follows:

with M = 2(Cx sin 2 Θ cos2 Φ + Cy sin 2 Θ sin 2 Φ + Cz cos2 Θ)

2

=

(13b)

From eq 12, we find that the equation E(α,β) = const defines an ellipse whose principal axes are, in general, inclined with respect to the x′y′ coordinates. We find, from eq 13a, that if (i) Θ = π/2 (G is on the equator), (ii) Φ = 0 (G is in the χ1χ3 plane), or (iii) Φ = π/2 (G is in the χ2χ3 plane), the coefficient b in eq 12 vanishes, and in such a case, the energy is expressed by E(α,β) = aα2 + cβ2, indicating that the principal axes lie along the x′ and y′ axes. In Figure 4, the contour maps for E(α,β) = 2.87 × 10−21 J are shown for the above-mentioned three cases, where the following values are used: χ1 = −1 × 10−6, χ2 = −1.7 × 10−6, χ3 = −2 × 10−6; V = 1 μm3; T = 300 K; bx = 1, by = 0.7; and B = 10 T. The principal axes of the quadratic form of eq 12 are inclined with respect to the α (∥y′) and β (∥-x′) axes, except in the three cases that are described above. As an example, we show in Figure 4, the inclination of the principal axes for some representative distributions of the G vector, where the same physical values are used. The elongated ellipse of the fluctuation in G is disadvantageous when the X-ray analysis is performed because it causes the X-ray diffraction spots to become elongated, resulting in low resolution. The optimal dynamic fields should be those that transform all the ellipses to circles. This could, in principle, be achieved by setting a = c and b = 0 (eq 12). However, this condition is not realized; in order to obtain b = 0, the equality Cx = Cy should be satisfied, but this equality is not possible

[Cx − (Cx − Cy)sin 2 Φ] 2CxCy

kBT (15)

Here, i = S and R, denoting the application of static and uniform rotating fields, respectively. In the case of a field rotating in the xy plane, Cx and Cy are defined as Cx  CRx = (4 μ0)−1B2V(χ2 − χ3) and Cy  CRy = (4 μ0)−1B2V(χ1 − χ3) by setting Kx = Ky = 1/2 in eq 2. On the other hand, in the case of a static field applied in the z direction, Cx and Cy are defined as Cx  CSx = (2 μ0)−1B2V(χ1 − χ3) and Cy  CSy = (2 μ0)−1B2V(χ1 − χ2) by setting Kx = Ky = 1. It should be noted that in the static case, the directions of the magnetic axes are defined as χ1∥z, χ2∥x, and χ3∥y, as shown in Figure 5. The fluctuations and are related to the squares of the azimuthal half-widths HS2 and HR2 of the X-ray diffraction spots, and we obtain the following equations: HS2 fS

2

=

(χ2 − χ3 ) 1 − sin 2 ΦS χ1 − χ2 (χ1 − χ3 )(χ1 − χ2 )

(16a)

(χ1 − χ2 ) 1 + sin 2 ΦR χ1 − χ3 (χ1 − χ3 )(χ2 − χ3 )

(16b)

and HR 2 fR

2

=

where the factors f S and f R include a geometrical correction17 that is needed when the fluctuation in the reciprocal lattice vector G is projected onto the Ewald sphere. In addition, these 1817

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1.4 (rotating), while the value determined previously ranged from 1.4 (rotating) to 1.7 (static).18 We studied the orientation fluctuation of the reciprocal lattice vector G(hkl) of a biaxial microcrystal whose magnetic axes are three-dimensionally confined by a dynamic magnetic field. The distribution of fluctuation was obtained as a function of the magnetic susceptibilities χ1, χ2, and χ3 of the microcrystal and the applied dynamic magnetic field used. Most reciprocal lattice vectors exhibited an elliptical distribution, whose principal axes were inclined with respect to the longitude and latitude of the reciprocal lattice sphere. The condition on the applied field that can reduce the difference in the magnitude of distribution between different G(hkl) vectors was discussed. The distributions under a static field and uniformly rotating magnetic fields, considered as special cases of the dynamic magnetic field, were investigated to revise the previously reported method for the determination of the ratio of anisotropic magnetic susceptibilities. Because the anisotropic magnetic energy has been included rigorously in the present study, the estimation of the ratio of anisotropic magnetic susceptibilities becomes more reliable.

Figure 5. The definition of the angles ΘS and ΦS that are used to characterize the direction of the reciprocal lattice vector G with respect to the χ1χ2χ3 coordinates when a static magnetic field is concerned. The angles Θ and Φ, defined in Figure 2, are used for the other cases.

factors include the term kBTμ0/(VB2). It should be noted that the definition of ΦS differs from that of ΦR; the definition of ΦR is the same as that of Φ and is shown in Figure 2, but ΦS is defined in Figure 5. The plots of eqs 16a and 16b are shown in Figure 6. The value of rχ is determined from Figure 6, using the values of the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Maret, G.; Dransfeld, K. In Topics in Applied Physics; Herlach, F., Ed.; Springer: Berlin, 1985 Vol. 57, Chapter 4. (2) Yamaguchi, M.; Tanimoto, Y. Magneto-science; KodanshaSpringer: Tokyo, 2006. (3) Kimura, F.; Mizutani, K.; Mikami, B.; Kimura, T. Cryst. Growth Des. 2011, 11, 12. (4) Kimura, T.; Chang, C.; Kimura, F.; Maeyama, M. J. Appl. Crystallogr. 2009, 42, 535. (5) Kimura, F.; Kimura, T.; Oshima, W.; Maeyama, M.; Aburaya, K. J. Appl. Crystallogr. 2010, 43, 151. (6) Yamaguchi, M.; Ozawa, S.; Yamamoto, I. Jpn. J. Appl. Phys. 2010, 49, 080213. (7) Tanase, M.; Bauer, L. A.; Hultgren, A.; Silevitch, D. M.; Sun, L.; Reich, D. H.; Searson, P. C.; Meyer, G. J. Nano Lett. 2001, 1, 155. (8) Maret, G.; Schickfus, M. V.; Mayer, A.; Dransfeld, K. Phys. Rev. Lett. 1975, 35, 397. (9) Fujiwara, M.; Oki, E.; Hamada, M.; Tanimoto, Y.; Mukouda, I.; Shimomura, Y. J. Phys. Chem. A 2001, 105, 4383. (10) Wu, C. Y.; Sassa, K.; Iwai, K.; Asai, S. Mater. Lett. 2007, 61, 1567. (11) Akiyama, J.; Asano, H.; Iwai, K.; Asai, S. Mater. Trans. 2008, 49, 787. (12) Genoud, J.-Y.; Staines, M.; Mawdsley, A.; Manojlovic, V.; Quinton, W. Supercond. Sci. Technol. 1999, 12, 663. (13) Zhang, Y. D.; Budnick, J. I. Appl. Phys. Lett. 1997, 70, 1083. (14) Kimura, T.; Yoshino, M. Langmuir 2005, 21, 4805. (15) Nakatsuka, N.; Yasuda, H.; Nagira, T.; Yoshiya, M. J. Phys.: Conf. Ser. 2009, 165, 012021. (16) Horii, S.; Okamoto, N.; Aoki, K.; Haruta, M.; Shimoyama, J.-i.; Kishio, K. J. Appl. Phys. 2012, 112, 043913. (17) Kimura, T.; Song, G.; Matsumoto, K.; Fujita, K.; Kimura, F. Jpn. J. Appl. Phys. 2012, 51, 040202. (18) Song, G.; Matsumoto, K.; Fujita, K.; Kimura, F.; Kimura, T. Jpn. J. Appl. Phys. 2012, 51, 060203. (19) Goldstein, H., Classical Mechanics, 3rd ed.; Pearson Education, Inc.: Upper Saddle River, NJ, 2002, Chapter IV, Appendix A. (20) Kimura, T. Jpn. J. Appl. Phys. 2009, 48, 020217.

Figure 6. Determination of anisotropic diamagnetic susceptibilities under (a) static and (b) rotating magnetic fields. The angle ΘS is defined in Figure 5, and the definition of ΘR is the same as that of Θ, which is shown in Figure 2.

slope and the interception. The obtained expressions correspond to those derived in the previous paper (the inverse of eqs 4 and 9 in ref 17, respectively). The equations that were derived in the previous paper are similar to the presently derived equations. However, in general, the quantitative agreement between the previous and present results is not satisfactory and depends on the value of susceptibility. The main reason for this discrepancy may be attributed to the fact that in the previous calculation, the effect of ω′∥ was not considered, and hence, the anisotropic magnetic energy was poorly approximated. By using the newly derived values of the intensity of fluctuation, we re-estimated the experimental data reported previously for the anisotropic magnetic susceptibility of cellobiose. The value of rχ = (χ2 − χ3)/(χ1 − χ2) that is determined by the present scheme ranges from 1.1 (static) to 1818

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(21) Kimura, T.; Kimura, F.; Yoshino, M. Langmuir 2006, 22, 3464. (22) Yamaguchi, M.; Ozawa, S.; Yamamoto, I.; Kimura, T. Jpn. J. Appl. Phys. 2013, 52, 013003.

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