This paper reports on a simple synthesis and characterization of highly birefringent vaterite microspheres, which are composed of 20–30 nm sized nanocrystalls. Scanning electron microscopy shows a quite disordered assembly of nanocrystals within the microspheres. However, using optical tweezers, the effective birefringence of the microspheres was measured to be Δn=0.06, which compares to Δn=0.1 of vaterite single crystals. This suggests a very high orientation of the nanocrystals within the microspheres. A hyperbolic model of the direction of the optical axis throughout the vaterite spherulite best fits the experimental data. Results from polarized light microscopy further confirm the hyperbolic model.
©2009 Optical Society of America
Since the first demonstration of optical tweezers , there has been little advance in the particles used for trapping. The polystyrene and silica microspheres used in initial experiments turned out to be very suitable for making precise force measurements [2, 3]. In fact, similar particles are still used in more recent experiments concerned with studies of the force–velocity relationship of a biological motor and for adhesion and bonding strength [4, 5]. Yet there is room for further development, such as choosing a particle that allows for increased trap strength. Creating a probe particle with an antireflection coating will increase the trap strength as a higher core refractive index can be used . Another improvement is to introduce an extra degree of control of the particle. This can be achieved by selecting a highly birefringent particle so that spin angular momentum from a circularly polarized light beam can be transferred to it .
There are many interesting and potentially useful applications of the rotation of vaterite particles in optical traps. Their high birefringence results in large torques acting to spin or align the particles, and the torque can be measured optically . This immediately opens the path to many possible applications, such as torque spectroscopy of single molecules [9, 10, 11] or molecular motors [12, 13]. Since rotation of a particle in a fluid results in flow of the fluid around the particle, vaterite particles can be used as micropumps , or, when combined with torque measurement, as sensors to measure the properties of the surrounding fluid [15, 16, 17]. Such particles can also be used as “handles” for the manipulation of microscopic objects such as cells which can be difficult to rotate or align directly . They are also a suitable basis for the development of improved methods for the application of controlled torques [19, 20].
Vaterite is a natural occurring calcium carbonate crystal . Although it is rare in nature due to its instability, it can be artificially synthesized, . It is the least stable of three crystalline polymorphs of calcium carbonate, the other two being calcite and aragonite. Vaterite has a hexagonal crystal structure, with unit cell parameters a=7.169 Å and c=16.98 Å and Z=12 [23, 24]. Vaterite crystals typically form polycrystalline spherical particles that are microns in diameter, often referred to as spherulites. It is currently unclear whether the process of formation of these vaterite spherulites is the result of nano-aggregation or classical spherulitic crystal growth [25, 26]. However it is known that the spherulites are comprised of single crystal subunits that arrange with some degree of order that gives the spherulite an overall birefringence . A “sheaf of wheat” structure within spherulites was proposed by Morse et al. . One can imagine this structure as a bundle of fibers tied together at the center so that the ends fan out and as the number of fibers increases, and the fibers themselves grow, the ends close off to form a sphere. The growth begins with a rod structure and progresses to a dumbbell shape, finally closing to form a sphere [29, 30, 31]. Although vaterite spherulites have been produced before [25, 26, 24, 32], their birefringence hasn’t been quantified yet. Measurements that clearly demonstrate a very strong birefringence of these particles, and a detailed discussion about the ordered assembly of nanocrystals within the vaterite microspheres are presented in this paper.
Vaterite is generally synthesized through precipitation in a saturated aqueous solution containing calcium and carbonate ions. Various methods are used to favour the growth of vaterite over the other two calcium carbonate polymorphs, such as the presence of sulphate ions [22, 33] and temperature control . Slow diffusion of ions tends to favour relatively large crystals , while vigorous agitation has been used to grow spherulites [25, 26, 35]. Various additives have been used to promote the growth of vaterite and affect its morphology, such as organic additives for biomimetic growth,  double hydrophilic block copolymers for mesocrystal growth [37, 38], polyacrylic acid [39, 40] and polycarboxylic acid . For our experiments we have produced vaterite spheres via precipitation in a mixture of K2CO3 and CaCl2 solutions in a similar fashion to previously reported methods [22, 42, 43, 15]. However, we used a seeded growth method, resulting in higher yield and improved size uniformity compared to the previous method developed in our lab . The microspheres produced were typically 2 to 5µm in diameter . The preference for vaterite formation is likely to be due to the ionic activity product lying between the solubility product for amorphous calcium carbonate and the solubility product for vaterite [25, 45].
2. Experimental section
Our production method can be broken down into two distinct parts; the first involving agitation, the second is a seeding technique. The first part is based on a previously published procedure . In a typical synthesis aqueous solutions of CaCl2, K2CO3 and MgSO4 were prepared, all having a molarity of 0.1 M. 1.5mL of CaCl2 was pipetted into a 5mL plastic vial, followed by 60µL of MgSO4 and then 90 µL of K2CO3. The solution was agitated by vigorous pipetting of the solutions. To stabilise the products of the reaction, a few drops of Agepon (a wetting agent made by Agfa) were added to the solution.
The second part involves the use of seeds to act as nucleation sites to promote crystal growth, which is a common technique and has previously been used for growth of vaterite spherulites [25, 39]. The method requires the same three solutions used in the first part of the method. 5mL of CaCl2 is added to a 10mL plastic vial, followed by 1.5mL of MgSO4. After gentle mixing of the solution, 1mL of K2CO3 is added, followed by a drop of seed solution. The solution is then gently mixed and allowed to stand undisturbed for 5 minutes, after which Agepon is added to halt the reaction and stabilize the crystal products.
The seed solution used is ideally one that contains monodisperse, spherical vaterite particles only. The product of this seeding tends to have a higher yield of the desired particles than the original seed solution. This means that the product of seeding can then be used as a seed solution itself to produce a new “generation” of vaterite products. With each new generation, the yield of vaterite spheres increases. This method allows production of large enough quantities of vaterite spherulites for experimentation and characterization.
X-ray diffraction (XRD) was used to determine the abundance of the different calcium carbonate polymorphs in our samples and to estimate crystallite sizes. The diffraction patterns were measured using CuKα x-ray radiation in a Bruker Axs diffractometer.
2.2. Electron microscopy
Scanning Electron Microscopy (SEM) images of platinum coated samples were collected on a JEOL 6300. Crushed vaterite spherulites were prepared by grinding the ethanol suspension with a mortar and pestle before putting it on the sample holder. Bright field TEM images were recorded on a JEOL 2010 using an operating voltage of 200 kV. Samples were prepared by crushing vaterite microspheres and also by embedding microspheres in resin and then using a microtome to produce a thin slice for viewing.
2.3. Laser tweezers
A laser beam was focussed to a diffraction-limited spot by high numerical aperture (NA=1.3) microscope objective. The free-space wavelength of the beam was 1064 nm, with 250mW available at the focus.
Since the torque is due to spin angular momentum, it can be measured optically by determining the Stokes parameters of the transmitted beam . Since the beam is fully polarized, it is sufficient to measure the power of the left circularly polarized component, PL, and the right circularly polarized component, PR. Since the left circular component carries +h̄ angular momentum per photon, and the right component -h̄, the total angular momentum flux is (PL-PR)/ω, where ω is the optical angular frequency.
With a left circularly polarized trapping beam, and no particle in the trap, the angular momentum flux will be P/ω, where P is the power of the beam at the focus. The change in the angular momentum flux due to the presence of a birefringent particle is the rate at which angular momentum is transferred to the particle, that is, the optical torque. The power of the left and right circular components can be measured by using a λ/4 plate followed by a polarizing beam splitter as a circular polarization beam splitter. A photodetector can be used to measure the power in each of the outputs of the beamsplitter. Further details of the experimental setup and the technique have been described in [8, 16].
The experimental data in Fig. 3(a) was fitted, using a finite difference frequency domain (FDFD) T-matrix method . The distribution of the optical axis of vaterite (Fig. 3(b)) was derived from a hyperbolic cylindrical coordinate system .
The Köhler illumination images in Fig. 4(b) were produced using a modeling method that we had developed for mesoscale microscopy. The method combines the T-matrix for the vaterite  (or any given particle) with the T-matrix for the imaging system, which is calculated using the far field point matching scheme from , incorporating methods to account for the microscope objective and cross polarizer system. Since we want to simulate what is seen by the objective, the fields outside the captured angles are treated as zero in the point matching scheme. The cross polarizer was simulated by applying the appropriate operator to the scattered field to filter out one of the transverse electric field components.
3. Results and discussion
There are several techniques that can be used for the characterization of vaterite spherulites. These can be divided into those which help to understand the structure of the material and those which give clear information on the optical properties.
X-ray diffraction (XRD) was used to examine the crystal structure. A typical pattern from the crystals is shown in Fig. 1. It displays strong peaks indicative of vaterite with an average crystal size of approximately 20–30 nm , which was calculated using the Debye–Scherrer formula . The vaterite peaks are labeled with V, showing (004), (110), (112), (114), (211), (008), (300), (304) and (118) diffraction peaks from left to right. The XRD spectrum also shows the presence of aragonite and calcite polymorphs (unlabeled peaks in Fig. 1). However, vaterite seems to be the dominant phase component in a typical sample.
Electron microscopy was conducted to investigate the surface and internal structure of the vaterite spherulites. Figure 2 shows scanning electron microscopy (SEM) and transmission electron microscopy (TEM) images of vaterite microspheres. It can be seen that the vaterite particles are almost perfectly spherical and have quite a smooth surface with particle diameters ranging from 1.5 to 2µm (figures 2A,B). Closer inspection reveals a granular outer surface on the particles, with a granule size of approximately 30 nm (Fig. 2(B)). An example of a vaterite particle that has been cracked in half is shown in Fig. 2(C). A similar granular structure as present on the surface is observed in the interior of the vaterite particle. Another feature of the interior is the presence of grooves that, in general, radiate from the centre of the particle giving the appearance of a radial fibre-like structure.
Two TEM micrographs of vaterite particles that have been set in a resin and sliced into layers thinner than 100 nm using a microtome, are shown in figures 2D,E. From these images it seems that either the vaterite particles are very porous, or that some of the material within the particle has fallen out of the resin during the slicing process. In fact, it seems likely that both have occurred, as the granular structure observed in the SEM images is consistent with porous particles. Both TEM images (figures 2D,E) also show that these vaterite spherulites consist of sub-units 20–30 nm in size, which agrees well with the SEM and XRD results. We ascertained that these sub-units are indeed crystalline by conducting TEM on a crushed sample of vaterite particles, shown in Fig. 2(F). The crushed sample consisted of fragments of vaterite particles that were thin enough at the edges to allow imaging with TEM. We observed Moiré patterns within the individual sub-units, which are characteristic of crystal planes in a crystalline structure.
The vaterite spherulites have also been characterized by optical methods, which are of particular interest, because the optical properties determine the behavior of a vaterite particle in an optical trap. In fact, the optical trap itself allows us to characterize the birefringence of individual vaterite particles. The particles can be trapped and rotated using circularly polarized light and at the same time the applied torque can be measured optically. The details of the experimental setup and the technique are briefly described in the method section and also in more detail in [8, 16]. We measured the change in polarization caused by vaterite spherulites in the trap, for particle diameters ranging from 2–11 µm. The results for this measurement are shown in Fig. 3. The data was fitted using the relation,
where Δσs is the change in circular polarization caused by the vaterite particle, Dhalf is the diameter of the vaterite particle that corresponds to a half wave plate and d is the vaterite particle’s actual diameter. The degree of circular polarization is defined as σs=(PL-PR)/P, where PL and PR are the left and right circular powers, and P=PL-PR is the total power . Δσs is the difference between the circular polarization with and without a vaterite sphere in the trap. For circularly polarized light, incident on a wave-plate, a quarter-wave retardation (i.e., a λ/4 plate) gives Δσs=1, and half-wave retardation gives the maximum possible value Δσs=2. Intermediate retardations give intermediate values of Δσs (Fig. 3).
From the fit, D half=11.3µm, which can then be used to calculate the effective birefringence of the vaterite particles,
where λ is the wavelength of the trapping laser. This compares to Δn=0.1 for the vaterite crystal itself and to 0.087 for vaterite fibers [52, 27]. The result from equation 2 assumes that the vaterite spherulite has a uniform birefringence, or put another way, a unidirectional optical axis throughout the volume of the particle. From XRD and SEM results we can see that these spherulites are composed of vaterite nanocrystals, which seem to have assembled in a rather disordered manner. However, the high measured effective birefringence of these particles suggests a very ordered arrangement of the crystalline subunits. In comparison a quartz single crystal is far less birefringent (Δn=0.009) than polycrystalline vaterite spheres. This allows more torque to be applied to the vaterite microspheres in tweezer experiments.
We have addressed both the issues of non-uniform birefringence and the small size of the particles by using a finite difference frequency domain (FDFD) T-matrix method to calculate the change in circular polarization, a method which has previously been described in . Figure 3(a) shows the change in circular polarization due to vaterite particles of different diameters. The dashed curve is a fit to the data, which suggests that a vaterite with a diameter of 11.3µm will behave as a half wave plate. The dotted line shows the expected behaviour if the vaterite particle had a birefringence of 0.1 and an optical axis that was uni-directional throughout its volume. The solid curve is the prediction by the FDFD T-matrix hybrid model, which assumes a hyperbolic distribution of the direction of the optical axis throughout the particle (Fig. 3(b) and best fits the experimental data.
A second optical technique that we have used to characterize the vaterite particles is polarization microscopy. By adding polarizers to our microscope we can view the optically trapped vaterite particle between crossed polarizers, which gives us an image of the birefringence of the vaterite particle as seen by the illumination light. Since vaterite is positive uniaxial, a trapped sphere will orient with its effective optic axis parallel to the electric field vector of the trapping beam. Using a half-wave plate in the trapping beam, the trapped vaterite particle can be rotated to any angle about the beam-axis. Vaterite particles with 10 µm in diameter are shown in Fig. 4(a), rotated by 7.5 degree intervals through 180 degrees. Köhler illumination images for a 6.6λ vaterite particle in Fig. 4(b) were produced using a modeling method that we had developed for mesoscale microscopy. The method combines the T-matrix for the vaterite  (or any particle) with the T-matrix for the imaging system. The sequence of patterns calculated (Fig. 4(b)), is in agreement with the experimental results, shown in Fig. 4(a). The Köhler illumination is further described in the method section. It should be noted that white light was used in the cross polarization experiment whereas the calculation was based on monochromatic light, with λ being the wavelength of the illumination light. The agreement of experiment and theory strongly supports the hyperbolic model for the vaterite birefringence.
The value for the effective birefringence allows us to calculate the behavior of a vaterite particle in a typical optical trap (optical power=250mW, wavelength=1064 nm and the surrounding fluid is water). In particular we are interested in the effect of particle diameter on the optical torque and the viscous drag torque, as shown in the top plot of Fig. 5. The drag torque is for a sphere in water rotating at 50 Hz. The intersection of the two curves indicates that a vaterite sphere, 3.9µm in diameter, will rotate at 50 Hz when trapped with 250mW of optical power. The radius cubed dependence of the viscous drag means that vaterite particles rotate much more slowly as the particle size increases (bottom plot in Fig. 5), despite the optical torque increasing with radius. It leads to the conclusion that if fast rotation rates are required for an experiment, then small particles should be used. Note that these calculations use a geometric optics model that will not be very accurate for particles less than a couple of microns in diameter. The advantage of having a spherical particle is that the rotational viscous drag torque can be easily calculated, as was done for Fig. 5.
In conclusion, we report here for the first time the production and characterization of extremely birefringent vaterite microspheres. The effective birefringence of the microspheres was measured to be Δn=0.06, which is considerably higher than the birefringence of quartz single crystals. A hyperbolic model of the direction of the optical axis throughout the vaterite spherulite best agrees with the experimental data. Based on their high birefringence these particles are being used in fields such as microrheology, microfluidics and the micromanipulation of single biological molecules.
We wish to acknowledge the help of the Centre for Microanalysis and Microscopy (CMM) at the University of Queensland. This project was supported by the Australian Research Council.
References and links
4. D. E. Smith, S. J. Tans, S. B. Smith, S. Grimes, D. L. Anderson, and C. Bustamante, “The bacteriophage theta 29 portal motor can package DNA against a larger internal force,” Nature 413, 748–752 ( 2001). [CrossRef] [PubMed]
5. G. Knöner, B. E. Rolfe, J. H. Campbell, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Mechanics of cellular adhesion to artificial artery templates,” Biophys. J. 91, 3085–3096 ( 2006). [CrossRef] [PubMed]
6. Y. Hu, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Antireflection coating for improved optical trapping,” J. Appl. Phys. 103, 093119 ( 2008). [CrossRef]
7. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 ( 1998). [CrossRef]
8. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical measurement of microscopic torques,” J. Mod. Opt. 48, 405–413 ( 2001).
9. C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Mater. 4, 223–225 ( 2007). [CrossRef]
10. K. C. Neuman, T. Lionnet, and J.-F. Allemand, “Single-molecule micromanipulation techniques,” Annu. Rev. Mater. Res. 37, 33–67 ( 2007). [CrossRef]
16. S. J.W. Parkin, G. Knöner, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Picoliter viscometry using optically rotated particles,” Phys. Rev. E 76, 041507 ( 2007). [CrossRef]
18. C.-K. Sun, Y.-C. Huang, P. C. Cheng, H.-C. Liu, and B.-L. Lin, “Cell manipulation by use of diamond microparticles as handles of optical tweezers,” J. Opt. Soc. Am. B 18, 1483–1489 ( 2001). [CrossRef]
19. K. D. Wulff, D. G. Cole, and R. L. Clark, “Controlled rotation of birefringent particles in an optical trap,” Appl. Opt. .
20. M. Funk, S. J. Parkin, A. B. Stilgoe, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Constant power optical tweezers with controllable torque,” Opt. Lett. 34, 139–141 ( 2009). [CrossRef] [PubMed]
21. J. D. C. McConnell, “Vaterite from Ballycraigy, Larne, Northern Ireland,” Mineral. Mag. 32, 535–544 ( 1960). [CrossRef]
22. J. Johnston, H. E. Merwin, and E. D. Williamson, “The several forms of calcium carbonate,” Am. J. Sci. 41, 473–512 ( 1916). [CrossRef]
23. S. R. Kamhi, “On the structure of vaterite, CaCO3,” Acta Crystallogr. 16, 770–772 ( 1963). [CrossRef]
24. L. Dupont, F. Portemer, and M. Figlarz, “Synthesis and study of a well crystallized CaCO3 vaterite showing a new habitus,” J. Mater. Chem. 7, 797–800 ( 1997). [CrossRef]
25. J.-P. Andreassen and M. J. Hounslow, “Growth and aggregation of vaterite in seeded-batch experiments,” American Institute of Chemical Engineers Journal 50, 2772–2782 ( 2004). [CrossRef]
26. J.-P. Andreassen, “Formation mechanism and morphology in precipitation of vaterite — nano-aggregation or crystal growth?” J. Cryst. Growth 274, 256–264 ( 2005). [CrossRef]
27. J. D. H. Donnay and G. Donnay, “Optical determination of water content in spherulitic vaterite,” Acta Crystallogr. 22, 312–314 ( 1967). [CrossRef]
28. H. W. Morse and J. D. H. Donnay, “Optics and structure of three-dimensional spherulites,” Am. Mineral. 21, 391–426 ( 1936).
29. S. L. Tracy, D. A. Williams, and H. M. Jennings, “The growth of calcite spherulites from solution II. Kinetics of formation,” J. Cryst. Growth 193, 382–388 ( 1998). [CrossRef]
30. F. C. Meldrum and S. T. Hyde, “Morphological influence of magnesium and organic additives on the precipitation of calcite,” J. Cryst. Growth 231, 544–558 ( 2001). [CrossRef]
31. H. Cölfen and L. Qi, “A systematic examination of the morphogenesis of calcium carbonate in the presence of a double-hydrophilic block copolymer,” Chemistry —A European Journal 7, 106–116 ( 2001). [CrossRef]
32. C. Rodriguez-Navarro, C. Jimenez-Lopez, A. Rodriguez-Navarro, M. T. Gonzalez-Muñoz, and M. Rodriguez-Gallego, “Bacterially mediated mineralization of vaterite,” Geochim. Cosmochim. Ac. 71, 1197–1213 ( 2007). [CrossRef]
33. D. D. P. Davies and G. R. Heal, “Polymorph transition kinetics by DTA,” J. Therm. Anal. 13, 473–487 ( 1978). [CrossRef]
34. R. E. Gibson, R.W. G. Wyckoff, and H. E. Merwin, “Vaterite and µ-calcium carbonate,” Am. J. Sci. 10, 325–333 ( 1925). [CrossRef]
35. O. Söhnel and J. W. Mullin, “Precipitation of calcium carbonate,” J. Cryst. Growth 60, 239–250 ( 1982). [CrossRef]
36. S. Mann, B. R. Heywood, S. Rajam, and J. D. Birchall, “Controlled crystallisation of CaCO3 under stearic acid monolayers,” Nature 334, 692–695 ( 1988). [CrossRef]
37. L. Qi, J. Li, and J. Ma, “Biomimetic morphogenesis of calcium carbonate in mixed solutions of surfactants and double-hydrophilic block copolymers,” Adv. Mater. 14, 300–303 ( 2002). [CrossRef]
38. J. Rudloff, M. Antonietti, H. Cölfen, J. Pretula, K. Kaluzynski, and S. Penczek, “Double-hydrophilic block copolymers with monophosphate ester moieties as crystal growth modifiers of CaCO3,” Macromol. Chem. Physics 203, 627–635 ( 2002). [CrossRef]
39. M. Donnet, P. Bowen, N. Jongen, J. Lemaître, and H. Hofmann, “Use of seeds to control precipitation of calcium carbonate and determination of seed nature,” Langmuir 21, 100–108 ( 2005). [CrossRef]
41. J. Rieger, T. Frechen, G. Cox, W. Heckmann, C. Schmidt, and J. Thieme, “Precursor structures in the crystallization/precipitation processes of caco3 and control of particle formation by polyelectrolytes,” Faraday Discuss. 136, 265–277 ( 2007). [CrossRef] [PubMed]
42. D. Kralj, L. Brečević, and J. Kontrec, “Vaterite growth and dissolution in aqueous solution III. Kinetics of transformation,” J. Cryst. Growth 177, 248–257 ( 1997). [CrossRef]
43. J. Schlomach, K. Quarch, and M. Kind, “Investigation of precipitation of calcium carbonate at high supersaturations,” Chem. Eng. Technol. 29, 215–220 ( 2006). [CrossRef]
44. R. Vogel, M. Persson, C. Feng, S. J. Parkin, T. A. Nieminen, B. Wood, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Synthesis and surface modification of birefringent vaterite microspheres,” Langmuir 25, 11672–11679 ( 2009). [CrossRef] [PubMed]
45. D. Kralj, L. Brečević, and A. E. Nielsen, “Vaterite growth and dissolution in aqueous solution I. Kinetics of crystal growth,” J. Cryst. Growth 104, 793–800 ( 1990). [CrossRef]
46. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electron. J. Differ. Equ. Conf. 04, 37–50 ( 2000).
47. V. L. Loke, T. A. Nieminen, S. J. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “FDFD/T-matrix hybrid method,” J. Quant. Spectrosc. Radiat. Transfer 106, 274–284 ( 2007). [CrossRef]
48. P. Moon and D. E. Spencer, Field Theory Handbook (Springer-Verlag, Berlin, 1971). [CrossRef]
49. V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “T-matrix calculation via discrete-dipole approximation, point matching and exploiting symmetry,” J. Quant. Spectrosc. Radiat. Transfer 110, 1460–1471 ( 2009). [CrossRef]
50. A. L. Patterson, “The Scherrer formula for x-ray particle size determination,” Phys. Rev. 56, 978–982 ( 1939). [CrossRef]
51. H. P. Klug and L. E. Alexander, X-ray Diffraction Procedures (J. Wiley & Sons, Inc., New York, 1954).
52. H. J. Meyer, “Bildung und Morphologie des Vaterits,” Z. Kristallogr. 121, 220–242 ( 1965). [CrossRef]