Orientation control of micro-domains in anisotropic laser ceramics

Yoichi Sato; Jun Akiyama; Takunori Taira

doi:10.1364/OME.3.000829

1. Introduction

Since transparent ceramic materials demonstrated their superiority as laser gain media to single crystalline materials [1], many useful researches have made possibility of ceramic laser gain media extended. Ceramic laser gain media that consist of micro-domains made of a single-crystalline domain show following advantages: higher fracture toughness [2], higher productivity of a large-sized media [3], higher controllability in composition [4], higher ability for composite structures [5], and higher doping concentration of luminous ions [1]. As the result of many investigations, ceramic laser gain media have become very useful candidates for advanced lasers. However, traditional ceramic techniques that have produced many kinds of useful laser ceramics cannot synthesize anisotropic laser ceramics with micro-domains made of a material in non-cubic crystal system. Because of the existing the optical scattering due to the difference in refractive index between randomly oriented micro-domains in anisotropic ceramics, it is impossible for light to go straight on inside anisotropic ceramics. Even though there are many anisotropic laser gain media made of crystals including orthovanadate [6] and apatite [7], people who appreciated anisotropic laser gain media had been keeping away from the convenience of ceramic gain media.

In order to obtain transparent ceramics composed of anisotropic laser gain media, the crystal orientations of micro-domains in ceramics should be aligned identically. Many technologies for the micro-domain control have been investigated by developments in the giant micro-photonics [8], and all of them are developed from one principle that is expressed by the Gibbs free energy of k-th domain G_k as

d G_{k} = - S_{k} d T + μ_{k} d N_{k} - V_{k} γ_{g} d (\frac{1}{r_{k}}) + V_{k} \sum_{i j} σ_{i j} d ε_{i j} - P_{k} \cdot d E - M_{k} \cdot d B,

where T, γ_g, σ_ij, ε_ij, E, B, V_k, S_k, μ_k, N_k, r_k, P_k, and M_k are temperature, boundary energy of the grain, stress tensor, strain tensor, external electric field, applied magnetic flux density, volume, entropy, chemical potential, molecular number, curvature radius of the surface, dielectric polarization, and magnetization of k-th domain, respectively. The first and second terms of the right side in Eq. (1) express the crystallization from liquid phase in crystal growth. The second term also generates the motive force of the grain growth by the solid-state reaction [9]. The third term represents the grain growth known as Ostwald ripening [10]. Fourth, fifth, and sixth terms of the right side in Eq. (1) describe the orientation control of the micro-domain by the applied stress [11], the external electric field [12], and magnetic field [13], respectively.

While recent progress of magnetic anisotropy engineering enables the orientation control of micro-domains in micro-domains made of a magnetic medium [14], unfortunately many kinds of useful anisotropic laser gain media are non-magnetic materials. The orientation control process for non-magnetic material requires an ultra-high magnetic field supplied by a superconducting magnet, thus it is very difficult to use the benefit of laser ceramics that can provide a large aperture. Authors discovered that the rare-earth ions doped into non-magnetic laser host materials as luminous ions strongly enhance the magnetic anisotropy [8], which gives the solution for the difficulties in the fabrication of anisotropic laser ceramics. Highly transparent anisotropic laser ceramics were synthesized by use of an electromagnet [15], and finally authors established the first laser oscillation by use of anisotropic laser ceramics [16].

The laser performance in our anisotropic laser ceramics has not fully demonstrated, and authors consider further improvement in the quality of anisotropic laser ceramics. Therefore, we have to clarify process rules based on fundamentals of the micro-domain structural control, and have to qualify current anisotropic laser ceramics. In this work the theoretical studies on the magnetic anisotropy and the kinetics of the orientation control for micro-domain in laser ceramics are discussed, and according to these principles authors evaluated the quality of the orientation control in anisotropic laser ceramics.

2. Theory

2.1 Magnetostatic potential of the anisotropic micro-domain under magnetic field

Figure 1 shows the conceptual diagram for the orientation control of anisotropic laser ceramics. The external magnetic field applied along the control axis generates the magnetic torque T_k onto the k-th anisotropic micro-domain that reduces the angle θ_k between the control axis and the direction of easy magnetization axis in k-th micro-domain. φ_k is the angle describing a precession movement of the micro-domain, and is free from T_k, and fluctuates by the thermal disturbance depending on the product of T and Boltzmann’s constant k. By use of the magnetic susceptibility tensor χ of the media, M_k is given by

M_{k} = \frac{1}{μ} V_{k} χ \cdot B,

where μ is the magnetic permeability in vacuum. In the case of a crystal belonging to the uni-axial crystal system, χ is a diagonal tensor with two independent elements. The crystal direction with larger element χ_e is called as easy magnetization axis, of and the direction with smaller element χ_h is difficult magnetization axis. When one of crystal axes is easy magnetization axis and other two axes are hard magnetization axes, the easy magnetization axis can be aligned along applied static magnetic field [8].

Fig. 1 Conceptual diagram for the orientation control of anisotropic laser ceramics. Angle θ and φ are the angle between the control axis and the direction of easy magnetization axis and the precession angle of the micro-domain, respectively.

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From Eqs. (1)-(2) magnetostatic energy of k-th domain U_k under |B| = B is expressed by

U_{k} = \int_{B} d G_{k} = - \frac{1}{μ} V_{k} \int_{0}^{B} χ \cdot B \cdot d B = - \frac{V B^{2}}{2 μ} (χ_{e} - Δ χ \sin^{2} θ_{k}),

where Δχ is the magnetic anisotropy equal to χ_e-χ_h.

2.2 Figure of merit for the orientation control

The degree of the crystal orientation can be evaluated quantitatively in terms of the Lotgering factor f [17]. This factor can be calculated from I_r(hkl) and I_s(hkl) that are the peak intensity corresponding to (hkl) plane in x-ray diffraction (XRD) patterns from surfaces of an uncontrolled reference with random orientation and the sample for evaluation, respectively. f shows the value from 0 for randomly oriented samples to 1 for samples with perfect alignment of domain orientation, thus f is normalized by the branching ratio ρ_i(hkl) defined by

ρ_{i} (h k l) = I_{i} (h k l) / \sum_{h^{'}, k^{'}, l^{'}} I_{i} (h^{'} k^{'} l^{'}),

where the subscript i indicates “r” or “s”. By use of ρ_i(hkl), f is given by

f = \frac{\sum_{(h k l) \in S} ρ_{s} (h k l) - \sum_{(h k l) \in S} ρ_{r} (h k l)}{1 - \sum_{(h k l) \in S} ρ_{r} (h k l)},

where S is a certain subset of (hkl)-planes. If we want to know the degree of alignment along c-axis, subsets of (hk0) or (00l) are useful for S. However, because Lorentz factor in XRD patterns depends on the crystal structure, Lotgering factor is not related directly to the distribution of the orientation angle θ between the control direction and the controlled crystal axis.

2.3 Distribution function for the orientation angle under thermal equilibrium

The distribution function in thermal equilibrium f_eq(θ, φ, V, B) for N_V micro-domains with volume of V under magnetic flux density B expressed by Boltzmann distribution as

f_{e q} (θ, ϕ, V, B) d Ω = N_{V} \frac{\exp [- U (θ, ϕ, V, B) / k T] d Ω}{\int \exp [- U (θ, ϕ, V, B) / k T] d Ω^{'}},

where Ω is a solid angle and equals to sinθdθdφ. By use of the simple approximation, f_eq(θ, φ, V, B)dΩ can be given by

f_{e q} (θ, ϕ, V, B) d Ω \approx \frac{N_{V}}{4 π} \frac{2 B^{2} + B_{\min}^{2}}{B_{\min}^{2}} {(\cos θ)}^{2 B^{2} / B_{\min}^{2}} d Ω,

where B_min is the minimum magnetic flux density for orientation control which is defined so that the anisotropy in magnetostatic potential is equal to kT. B_min is given by

B_{\min} = \sqrt{\frac{2 μ k T}{V Δ χ}} .

In the case of diffusion phenomena, the distribution function is approximated by Gaussian in general. The approximation at the derivation Eq. (7) is convenient to calculate, especially by use of polar coordinates, and it is comparable to Gaussian distribution in the order of approximation.

2.4 Improvement of the orientation distribution by preferential growth

Preferential grain growth can be described by the third term in Eq. (1). With the assumption that the primary particles in the power compact have sphere-shape, Gibbs free energy G during initial condition in sintering process is given by

d G = - \sum_{k} γ_{g} V_{k} d (\frac{1}{r_{k}}) \approx \frac{4 π}{3} γ_{g} \sum_{k} r_{k} d r_{k},

with the bounding condition of the conservation in total volume as

\sum_{k} r_{k}^{2} d r_{k} = 0 .

Here we consider the interaction between 2 grains. In this case Eq. (9) can be simplified to

d G = \frac{4 π}{3} γ_{g} (1 - \frac{r_{1}}{r_{2}}) r_{1} d r_{1} .

Equation (11) shows the preferential grain growth in larger micro-domains, where smaller grains tend to reduce their volume and larger grains to increase their volume. It also indicates that grain growth will stop after almost uniform grain size can be established.

Fortunately, it is easier to control the orientation of micro-domains with larger domain size by slip-casting under magnetic field. Taking into consideration that the third power of a diameter of micro-domain D is proportional to its volume, we can find that the distribution function f_eq for micro-domains of D larger than the mean diameter of all micro-domains in slurry D_m is drastically improved. Even under B_min^m that is minimum magnetic flux density for micro-domains with D_m is given by

f_{e q} (θ, ϕ, V, B_{\min}^{m}) d Ω \approx \frac{N_{V}}{4 π} (\frac{2 D^{3}}{D_{m}^{3}} + 1) {(\cos θ)}^{2 {(D / D_{m})}^{3}} d Ω .

Equation (12) shows that micro-domains with larger domain size can be well aligned as a core particle in preferential grain growth if we can realize the appropriate distribution of the micro-domain size in the slurry under slip casting process.

In traditional ceramic processes the abnormal grain growth during sintering due to the non-uniformity of the particle size in powder compact has been considered to be not appropriate for the grain growth of transparent ceramics, because this process prevent to vanish residual pores within the specimen as described in Fig. 2(b) [18]. However, from the viewpoint of orientation control of micro-domains in anisotropic ceramics this abnormal grain growth can be a key technology as shown in Fig. 2(c). Though the normal grain growth keeps a distribution of orientation in powder compact during sintering as shown in Fig. 2(a), in abnormal grain growth the crystal orientation of larger particles tend to survive preferentially. By means of the control of the sintering process where the generation of residual pores is suppressed, we can realize well-aligned anisotropic ceramics via this preferential grain growth.

Fig. 2 Conceptual chart for the improvement of the orientation distribution in anisotropic laser ceramics by preferential grain growth following the orientation control via slip casting under magnetic field.

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2.5 Relation between Lotgering factor and orientation distribution

From the distribution function of the crystal orientation that is given by Eqs. (7) or (12) in fabricated anisotropic ceramics, we can estimate the direct relationship between Lotgering factor f and the distribution of the crystal orientation. X-ray diffraction intensity I_r(hkl) is proportional to the amount of the micro-domains that has (hkl)-plane perpendicular to incident X-ray. The angle between c-axis and control axis θ(hkl) of these micro-domains are expressed by

θ (h k l) = {Tan}^{- 1} \frac{c l}{a \sqrt{h^{2} + k^{2} - h k δ}},

where a, c, and δ is a lattice constant for two independent crystal axes in uni-axial crystals, and the index of crystal structure which is 1 for hexagonal crystals and 0 for tetragonal crystals. By use of θ(hkl) the branching ratio ρ_s(hkl) for the orientation controlled anisotropic ceramics can be expected to be comparable to

ρ_{s} (h k l) \approx \frac{f_{e q} [θ (h k l), V, B] I_{r} (h k l)}{\sum_{h^{'}, k^{'}, l^{'}} f_{e q} [θ (h k l), V, B] I_{r} (h^{'} k^{'} l^{'})} .

Finally, the theoretical relation between distribution function and Lotgering factor is given by

f = \frac{(\sum_{(h k l) \in S} f_{e q} [θ (h k l), V, B] I_{r} (h k l) / \sum_{h^{'}, k^{'}, l^{'}} f_{e q} [θ (h^{'} k^{'} l^{'}), V, B] I_{r} (h^{'} k^{'} l^{'})) - \sum_{(h k l) \in S} ρ_{r} (h k l)}{1 - \sum_{(h k l) \in S} ρ_{r} (h k l)} .

We can estimate also the distribution probability P_eq of micro-domains in slurry under thermal equilibrium from Eq. (7). By use of a certain θ₀ below π/2, P_eq[0 ≤ θ ≤ θ₀] for the distribution probability of the crystal orientation within the range of θ from 0 to θ₀ can be calculated as

\begin{matrix} P_{e q} [0 \leq θ \leq θ_{0}] = \frac{2 π}{N} \int_{0}^{θ_{0}} f_{e q} (θ, ϕ) \sin θ d θ + \frac{2 π}{N} \int_{π - θ_{0}}^{π} f_{e q} (θ, ϕ) \sin θ d θ \\ = 1 - {(\cos θ_{0})}^{\frac{2 B^{2} + B_{\min}^{2}}{B_{\min}^{2}}} . \end{matrix}

3. Experimental setup and results

3.1 Fabrication of transparent Nd:FAP ceramics

Although fluorapatite (FAP) has been expected as an anisotropic laser host crystal for laser fusion drivers [7], the growth of a boule with large size is quite difficult. Therefore, we select FAP for the demonstration of anisotropic laser ceramics. Slurry containing Nd:FAP particles and distilled water were mixed with a deflocculant, and the resulting mixture was mechanically ground to convert aggregate particles to single crystals. The slurry containing grounded particles was then poured into a porous mold, and slip casting was carried out in a horizontal static magnetic field of 1.4 tesla at room temperature. This magnetic field was generated by electric magnet (JER-3XG, JEOL). After casting, the samples were pre-sintered for 2 hours within 1600 °C in air. Post-sintering was carried out by use of hot isostatic pressing under 1600 °C for 1 hour at 190 MPa in Ar. Figure 3 shows a flowchart of the fabrication process for anisotropic laser ceramics and a photograph of polished a-cut Nd:FAP ceramics. The doping concentration of Nd³⁺ in these ceramics was 2at.% which examined by X-ray fluorescence analyzer (JSX-3400RII, JEOL).

Fig. 3 Flowchart of the process for orientation control at fabrication of anisotropic laser ceramics (a), and the photograph of synthesized 2.0 at.% Nd:FAP ceramics (b).

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3.2 XRD observation of Nd:FAP ceramics

Figure 4 shows XRD patterns that were observed by diffractometer (RINT-UltimaIII, Rigaku). Evaluated samples were randomly oriented Nd:FAP raw powder (Fig. 4(a)), Nd:FAP ceramics-1 (Fig. 4(b)), and Nd:FAP ceramics-2 (Fig. 4(c)). Nd:FAP ceramics-1 and -2 were synthesized by the same process in different production lot with optical scattering loss below 1.5 cm⁻¹. XRD patterns of ceramic specimens were detected from the surface perpendicular to the direction of applied magnetic field. All these diffraction peaks can be assigned to the standard data in ICDD - #00-015-0876 card for flourapatite.

Fig. 4 X-ray diffraction patterns of raw powder of Nd:FAP (a), Nd:FAP ceramic sample-1 (b), and Nd:FAP ceramic sample-2 (c). Here diffraction angle means the angle between the incident direction and the diffracted direction of X-ray.

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Compared to powder data, the diffraction patterns from Nd:FAP ceramics have narrow peaks. The X-ray diffractions of Nd:FAP raw powder, Nd:FAP ceramics-1, and Nd:FAP ceramics-2 from (300)-plane at the diffraction angle of 33.1 degree have FWHM of 0.23, 0.07, and 0.11 degrees. Therefore, FWHMs in XRD patterns suggest that the crystal quality of each micro-domains in Nd:FAP ceramics-1 is higher than those in Nd:FAP ceramics-2. Even though FWHM in Nd:FAP ceramics-2 is wider than Nd:FAP ceramics-1, Nd:FAP ceramics-2 can perform laser oscillation [16]. On the contrary, we cannot observe laser oscillation by use of Nd:FAP ceramics-1. This fact indicates in Nd:FAP ceramics-1 there are not the problem of crystal quality itself but the problem due to the orientation control.

4. Discussions

4.1 Lotgering factor of Nd:FAP ceramics

In ICDD - #00-015-0876 card there are 35 peaks in XRD pattern of fluoroapatite within the range of the angle between the incident direction and the diffracted direction of X-ray from 10 to 60 degree. Among these peaks only 10 peaks are the diffraction from the surface perpendicular to the direction of easy magnetization (c-axis): (100), (110), (200), (210), (300), (310), (400), (320), (410), and (330). By use of XRD pattern from Nd:FAP raw powder as a reference, Lotgering factors of Nd:FAP ceramics-1 and Nd:FAP ceramics-2 are calculated to be 0.96 and 0.93, respectively. Lotgering factor suggests that the quality of Nd:FAP ceramics-1 as anisotropic laser ceramics is higher than that of Nd:FAP ceramics-2.

According to the experimental result in sect 3.2, the higher Lotgering factor of Nd:FAP ceramics-1 suggests the existence of a problem in orientation control which cannot be interpreted by Lotgering factor.

4.2 Calculation of distribution probability

As discussed in sect. 4.1, it is necessary to make deeper investgations on the orentation distribution in anisotropic laser ceramics than the Lotgering method. Here we confirm the nature in the distribution of crystal orientation described by Eqs. (7) and (12). Figure 5 shows P_eq for primary particles aligned within the range of θ from 0 to θ₀ that is given by Eq. (16). Even though we are satisfied by the rough alignment within ± 30 degree, 4 times larger magnetic field than B_min is required to realize this situation without preferential grain growth. If we want to obtain powder compact where more than 95% of primary particles are aligned within ± 1 degree, more than 99 B_min is required, which considered to be more than 100 tesla of magnetic flux density from our previous report [19]. Therefore, nearly perfect alignment of easy magnetization axis of primary particles seems to be almost impossible by means of only slip-casting under magnetic field.

Fig. 5 Probability of the orientation of easy magnetization axis in primary particles well aligned within the range of 0 ≤ θ ≤ θ₀ in slurry under thermal equilibrium.

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In general, for obtaining highly transparent ceramics, raw powders for green body (powder compacts) should be consisted of primary particles where each of them has single crystalline micro-domain with the size of less than several hundred nm in diameter. Moreover, obtained transparent ceramics are composed of micro-domains of more than several-μm. This situation indicates that core particles in grain growth should be less than 1% of primary particles peptized in the slurry under slip casting. If preferential grain growth can be well performed, the nearly perfect alignment of anisotropic laser ceramics can be realized by the precise orientation control of only less than 1% micro-domains among a plenty of primary particles in slurry by the slip-casting under magnetic field.Figure 6 shows the probability of the distribution of crystal orientation in micro-domains, which depends on the particle size of D that is larger than D_m under B_min^m. Even if the size distribution of raw powder is quite controlled to be homogeneous, 99% of micro-domains with only 4 times larger size than mean particles can be aligned within ± 15 degree, where only 10% of mean particles are aligned within ± 15 degree. In the case of core micro-domains with 25 times larger diameter than mean particle, more than 99% of these core micro-domains can be aligned within the range of ± 1 degree along c-axis. This calculation indicates that nearly perfect orientation control is capable if we can make the preferential grain growth performed well.

Fig. 6 Probability for the orientation of core particles with size of D in preferential grain growth of powder compact formed by the slip-casting under B_min for mean particles.

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4.3 Relation between distribution of the orientation and Lorgering factor in Nd:FAP ceramics

We can examine the relation between Lotgering factor and the distribution of crystal orientation in the powder compact that is formed by the slip-casting under magnetic field by Eq. (15), from XRD pattern of raw powder in Fig. 4(a) and the lattice constant of fluoroapatite [20]. As shown in Fig. 7, higher external magnetic field brings the higher Lotgering factor. Moreover, from Eq. (16) also the distribution probability in the powder compact can be estimated. For examples, when Lotgering factor equals 0.44 (slip-casted under B_min), 99% of micro-domains have the angle between c-axis and control axis within ± 78 degree. On the contrary, 99% of micro-domains have the angle between c-axis and control axis within ± 53 degree inside the powder compact with Lotgering factor of 0.62 (slip-casted under 2 B_min).

Fig. 7 X-ray diffraction pattern of raw powder of Nd:FAP (line) and the calculated peak intensity of the diffraction from the orientation controlled powder compact made of Nd:FAP (marker).

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If the orientation distribution is assumed to be invariant through the sintering process, distribution for the crystal orientation of micro domains in anisotropic laser ceramics can be evaluated directly from Lotgering factor. Inside Nd:FAP ceramcs-1 that has Lotgering factor 0.96, 99% of micro-domains are aligned within ± 24 degree (or 80% within ± 14 degree) under this assumption. Similarly inside Nd:FAP ceramcs-2 that has Lotgering factor 0.93, 99% of micro-domains should be aligned within ± 27 degree (or 80% within ± 16 degree).

4.4 Distribution of the crystal orientation in Nd:FAP ceramics

Equation (5) indicates that the information about each diffraction peak is vanished during the calculation of Lotgering factor. In order to pick up the information of detailed behavior of each micro-domains that has various angle between c-axis and control axis, we evaluated change in branching ratio in XRD patterns during sintering process. From Eqs. (4) and (7), the ratio between ρ_s(hkl) and ρ_r(hkl) is given by

\frac{ρ_{s} (h k l)}{ρ_{r} (h k l)} \approx {(\cos θ)}^{2 B^{2} / B_{\min}^{2}} \frac{\sum_{h^{'}, k^{'}, l^{'}} I_{r} (h^{'} k^{'} l^{'})}{\sum_{h^{'}, k^{'}, l^{'}} {(\cos θ^{'})}^{2 B^{2} / B_{\min}^{2}} I_{r} (h^{'} k^{'} l^{'})} \propto \exp [- \frac{B^{2}}{B_{\min}^{2}} θ^{2} (h k l)] .

Although Eq. (17) indicates that the ratio between ρ_s(hkl) and ρ_r(hkl) has a Gaussian dependence on θ, this ratio in sintered anisotropic laser ceramics has an exponential dependence as shown in Fig. 8. While various causes can be expected about this difference: the influence due to low-angle grain boundary, the size distribution in micro-domains, and so on. However, the orientation distribution changes certainly before and after sintering process, which strongly suggest the existence of the improvement of orientation control due to preferential grain growth.

Fig. 8 Ratio between branching of intensity in the X-ray diffraction from the transparent Nd:FAP ceramics after sintering. Dashed lines are the fitting to the exponential function. Ratios at θ = 0 are represented by the average value of 10 diffraction peaks.

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Because the orientation distribution in anisotropic laser ceramics changes during sintering process, the real distribution of crystal orientation in micro-domains should be determined by such as the inverse pole mapping from electron backscatter diffraction (EBSD) method. However, the evaluation of the distribution in the orientation by use of Lotgering factor discussed in sect 4.3 is still useful from a viewpoint of the easy estimation based on the simple kinetics of orientation control.

Figure 8 shows that the ratio of branching ratio at larger θ in Nd:FAP ceramics-1 is smaller than Nd:FAP ceramics-2. However, some diffraction peaks with a certain θ still remain considerably in Nd:FAP ceramics-1. For example, the diffraction peak of (201) from domains with θ of 19.9 degree did not reduced by sintering. On the contrary, the ratio of branching ratio in Nd:FAP ceramics-2 shows clear dependence on θ, which indicates the fluctuation of θ in Nd:FAP ceramics-2 is well controlled by the kinetics of orientation control discussed in this work. The large deviation of the ratio in Nd:FAP ceramics-1 suggest the problems in the uniformity of the orientation control, such as insufficient pulverization under slurry treatments.

5. Summary

Although crystal orientations of micro-domains have to be well-aligned uniformly in order to obtain transparent anisotropic ceramics, it is almost impossible to realize nearly perfect alignment by means of only slip-casting under magnetic field. However, even if only core partcles for preferential grain growth in powder compacts are well aligned under magnetic field, we can improve their alignment during sintering process accompanied with the preferential grain growth. Therefore, it is important to find the orientation distribution of primary particles in the slurry under thermal equilibrium condition. Authors produced this distribution function in the numerical form, and evaluated anisotropic laser ceramics by use of this distribution function. Moreover, detailed XRD analysis suggests the improvement in the orientation distribution due to the preferential grain growth experimentally. In future, optimizing of the fabrication process according to this work will bring the further improvement in the laser efficiency of anisotropic laser ceramics.

Appendix A: Equation of the motion under magnetic field

The magnetic torque T_k can be derived by differential of the magnetostatic energy of k-th domain U_k with angle and is given by

T_{k} = - e_{ϕ} \frac{\partial U_{k}}{\partial θ} + e_{θ} \frac{1}{\sin θ} \frac{\partial U_{k}}{\partial ϕ} = - \frac{V_{k} Δ χ}{2 μ} B^{2} \sin 2 θ_{k} e_{ϕ},

where e_i is the unit vector for the axis of coordinate-i.

If the orientation control is processed by the applying magnetic field onto the micro-domain in slurry during slip-casting, each micro-domain receives both the damping due to the viscous resistance and the Langevin force R_k due to Brownian motion by slurry molecules. Consequently, the equation of motion for the anisotropic micro-domain is given by

I_{k} \frac{d^{2} θ_{k}}{d t^{2}} + 6 η V_{k} \frac{d θ_{k}}{d t} = - \frac{V_{k} Δ χ}{2 μ} B^{2} \sin 2 θ_{k} + R_{k} (t),

where I_k, η, and t are the moment of inertia of k-th micro-domain, viscosity of the slurry, and time, respectively. Because R_k is a random force, θ_k cannot be determined as a unique value in a certain time. In other words, θ_k and φ_k can be estimated stochastically from the distribution function f(θ, φ, V, B, t) for the orientation of micro-domain with volume V. This fact indicates that in principle the alignment of crystal orientation is not perfect. Therefore, it is important to reduce the deviation of θ in order to improve the optical quality of anisotropic laser ceramics.

In the orientation control processed by slip-casting of the slurry where micro-crystals made of raw materials are peptized under static magnetic field, the distribution function f(θ, φ, V, B, t) for the orientation of micro-domain with volume V is stabilized to the thermal equilibrium within a finite time τ. In the normal fabrication process where R_k can be treated as a white noise and the inertial force is enough small to ignore comparing η, Eq. (18) gives τ as

τ = \frac{6 μ η}{Δ χ B^{2}} .

Appendix B: Derivation of Eq. (7)

By use of Eqs. (3) and (8), Boltzmann factor of magnetic field is given by

\exp [- \frac{U (θ, ϕ, V, B)}{k T}] = \exp (\frac{V B^{2}}{2 μ k T} χ_{e}) {[\exp (- \sin^{2} θ)]}^{B^{2} / B_{\min}^{2}} .

Under the condition that Eq. (19) can be linearly approximated, Eq. (21) is expressed by Taylor’s expansion as

\exp [- \frac{U (θ, ϕ, V, B)}{k T}] \approx \exp (\frac{V B^{2}}{2 μ k T} χ_{e}) {(1 - \sin^{2} θ)}^{B^{2} / B_{\min}^{2}} .

From Eqs. (6) and (22), f_eq(θ, φ, V, B)dΩ can be given by

f_{e q} (θ, ϕ, V, B) d Ω \approx N_{V} \frac{{(1 - \sin^{2} θ)}^{B^{2} / B_{\min}^{2}} d Ω}{\int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \sin θ {(1 - \sin^{2} θ)}^{B^{2} / B_{\min}^{2}}} = N_{V} \frac{{(\cos θ)}^{2 B^{2} / B_{\min}^{2}} d Ω}{2 π \int_{- 1}^{1} x^{2 B^{2} / B_{\min}^{2}} d x} .

Therefore, we can obtain Eq. (7).

Acknowledgments

This work was partially supported by Genesis Research Institute, and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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Orientation control of micro-domains in anisotropic laser ceramics

Abstract

1. Introduction

2. Theory

2.1 Magnetostatic potential of the anisotropic micro-domain under magnetic field

2.2 Figure of merit for the orientation control

2.3 Distribution function for the orientation angle under thermal equilibrium

2.4 Improvement of the orientation distribution by preferential growth

2.5 Relation between Lotgering factor and orientation distribution

3. Experimental setup and results

3.1 Fabrication of transparent Nd:FAP ceramics

3.2 XRD observation of Nd:FAP ceramics

4. Discussions

4.1 Lotgering factor of Nd:FAP ceramics

4.2 Calculation of distribution probability

4.3 Relation between distribution of the orientation and Lorgering factor in Nd:FAP ceramics

4.4 Distribution of the crystal orientation in Nd:FAP ceramics

5. Summary

Appendix A: Equation of the motion under magnetic field

Appendix B: Derivation of Eq. (7)

Acknowledgments

References and links

Cited By

Figures (8)

Equations (23)

Optical Materials Express