# Estimation of lattice strain in alumina–zirconia nanocomposites by X-ray diffraction peak profile analysis

- 162 Downloads

## Abstract

Zirconia nanoparticles were synthesized by a solution combustion technique. Zirconia nanoparticles were grown inside alumina templates to control the crystal size. The alumina templates were characterized by pores of sizes of approximately 6–8 nm. X-ray diffraction line profile analysis using Williamson–Hall method and Warren and Averbach revealed that the alumina templates were characterized by negligible defect-related effect of lattice distortion. Rietveld structure refinement did not reveal any gross difference with the literature reported values for cell parameters ‘*a*’ and ‘*c*’ and fractional atomic coordinates *x* and *z* for Al and O atoms indicating no large-scale bond deformation. The template X-ray reflections in the nanocomposites are skewed in nature which indicates some distortion of the templates might have taken place. The distortion is, however, plastic in nature which is evident from the higher level of lattice distortion viz. 0.2% of lattice microstrain.

## Keywords

α-Al_{2}O

_{3}–ZrO

_{2}nanocomposite X-ray line profile analysis Microstructure Rietveld refinement

## Introduction

Stabilized zirconia (ZrO_{2}) ceramics finds application in diverse fields. The fully stabilized (cubic) zirconia (FSZ) is used for heating elements, oxygen sensors and fuel cell applications. Partially stabilized zirconia (PSZ) or tetragonal zirconia particles (TZP) (a mixture of cubic and tetragonal or single-phase tetragonal) find application in mechanical and structural applications like thermal barrier coatings and as toughening agents in many ceramic composites viz. zirconia-toughened alumina (ZTA) ceramics. Pure zirconia (ZrO_{2}) with cubic and tetragonal structures, respectively, exists only at high temperatures typically above 1200 °C below which only monoclinic phase exists. The cubic (fluorite) crystal structure of ZrO_{2} can be described as the Zr^{+4} ions forming a cubic close pack structure (FCC lattice) with the O_{2}− ions occupying all the tetrahedral holes [1].

With the decrease in temperature, ZrO_{2} undergoes a cubic to tetragonal (*c* → *t*) phase transition at around 2380 °C and a tetragonal to monoclinic (*t* → *m*) phase transition at around 1205 °C [2, 3]. At the cubic to tetragonal phase transition, the fluorite cubic structure distorts to the tetragonal structure with the tetragonal c-axis parallel to one cubic \(\left\langle {001} \right\rangle\) axes. On the other hand, the tetragonal to monoclinic phase transition is characterized by a martensitic phase transition accompanied by large hysteresis in the phase transition temperature, greater than 200 °C and the generation of large shear and volume elastic strains in the monoclinic phase. It has been observed that the martensitic transformation temperature for the *t* → *m* can be brought down significantly by adding additives of suitable characteristics which facilitate the stabilization of the tetragonal phase [4, 5].

Addition of appropriate dopant cations viz. Y^{3+}, Ca^{2+}, Mg^{2+}, Ce^{4+}, etc. is known to stabilize the tetragonal and cubic polymorphs. Phase stability at room temperature of such materials and their dependence on particle/crystallite size has been extensively studied by numerous researchers [6, 7, 8, 9, 10, 11]. Despite such effort, the fundamental issues concerning the atomistic origin of phase stability with a special reference to size stabilization and origin of lattice microstrains of *t*- and *c*-ZrO_{2} polymorphs are still unresolved especially in undoped zirconia ceramics. The charge compensating oxygen vacancies in doped ZrO_{2} hold the key to ionic conduction in these ceramics. The simultaneous presence of dopant cations and oxygen vacancies in large concentration means the local atomic environments in the stabilized materials are very different from the corresponding stoichiometric (*t* and *c* phases) phases. Hence, there is urgent need of characterizing the local atomic structure in the real space.

Recent advance in the nanostructure research indicates that valence of Zr ions decreases with decreasing particle size suggesting the occurrence of oxygen deficiency in tetragonal ZrO_{2−δ} (0 < *δ*<1) where *δ* is the vacancy concentration. This opens up a relevant question whether size-stabilized *t*-ZrO_{2} or even *c*-ZrO_{2} (formed as a result of more distortion of the oxygen sublattice) is actually an oxygen deficient one, and a detailed structural analysis (both long and short range) of ZrO_{2} nanoparticles may solve the controversy.

## Theoretical formulation

The study of lattice imperfections in crystalline materials is an important aspect of microstructural characterization. Among the currently used methods, X-ray diffraction line profile analysis (XRDLPA), in particular, describes the microstructure in terms of parameters such as coherent domain sizes or crystallite sizes, lattice microstrains, stacking fault probabilities, etc. Broadening of X-ray diffraction peaks is a consequence of both small crystallite size and lattice distortion therein. Methods have been developed over the past few decades to separate these two general effects along with other effects such as planar or twin faults [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. These, however, do not provide a direct and realistic physical picture of the microstructure in general. The concept of crystallite size is also superfluous. For example, nanocrystalline materials which are of current interest as technologically potential materials exhibit certain properties which are sensitive to the grain sizes and/or their distribution. The X-ray methods yield certain averages (area or volume averages) of coherent domain sizes conceived as ensemble of columns of unit cells in a direction perpendicular to the diffracting planes. These average column lengths do not necessarily conform to the average grain sizes measured from TEM. Further, it is often tedious to determine both the average grain size and their distribution representative of the whole sample using TEM.

### Integral breadth methods

*ω*) or the integral breadth (

*β*) is employed. Mathematically, the integral breadth is defined as,

*ω*/

*β*(=

*φ*, called the shape factor) by graphical method. de Keijser et al. [15] derived some empirical relations for the same.

#### Size–strain analysis

The size–strain analysis in terms of integral breadths is simple enough if either crystallite size broadening or microstrain broadening is present, and can be performed easily using either Scherrer equation or Wilson’s equation [13, 14, 15]. If both are present simultaneously, specific assumptions regarding the order dependence of broadening and\or analytical functional forms for size- and strain-broadened profiles are necessary.

*ε*is the strain and s is (2sin

*θ*/

*λ*).

Equation 4 is called Williamson–Hall equation; Eq. 5 is similar to the Halder and Wagner [18] approximation for the integral breadth of a Voigt function; Eq. 6 is valid for Gaussian assumption for both size- and strain-broadened profile. If *β* for two or more order of a same family of reflection is known then the crystallite size and strain may be obtained from the above equations.

### Fourier method

If both small crystallites and microstrain broaden the X-ray diffraction lines, a more detailed and rigorous analysis was developed by Warren and Averbach [12] through the developments of Bertaut [19] and combining Stokes’ [20] method of deconvolution based on Fourier representation of line profiles.

#### Warren and Averbach method

*K*is approximately a constant,

*A*and

*B*are the cosine and sine Fourier coefficients and

*L*is a distance perpendicular to the diffracting planes. In practice,

*L*takes specific discrete value

*n*Δ

*L*,

*n*is an integer and Δ

*L*is inversely proportional to the length of the measurement range in the reciprocal space, and

*A*

_{S}(

*L*) and the order-dependent strain Fourier coefficients

*A*

_{D}(

*L,s*), and the formal cosine Fourier coefficient is expressed as

*A*

_{S}(

*L*) and

*A*

_{D}(

*L,s*) separately, two orders of reflection have to be measured. In the original analysis of Warren and Averbach, the size–strain separation is done by taking the logarithm of Eq. 9 giving

*A*

_{S}(

*L*) and

*A*

_{D}(

*L,s*) are expressed, respectively, as

*N*

_{n}/

*N*and \(\left\langle {{ \cos }(2\pi Ls\varepsilon_{L} )} \right\rangle\) where

*N*

_{n}represents the number of columns

*n*cells long and

*N*is the total no. of cells in the sample. Substituting and taking approximation, Eq. 10 reduces to

For two orders of reflection, a plot of ln *A*(*L,s*) against *s*^{2} gives *N*_{n}/*N* from intercept and \(\left\langle {\varepsilon^{2}_{L} } \right\rangle\) from the slope. *N*_{n}/*N* in the limit *n* → 0 is defined as the average column length \(\left\langle D \right\rangle_{\text{eff}}\) .

### Rietveld method

The basic drawback of conventional Warren–Averbach method is the accurate description of the profile tails which is often impossible to determine in case of samples showing considerable peak overlap (viz. materials of lower symmetry). A generally accepted solution of this above shortcoming is to resort to the profile fitting methods. The primary objective of profile fitting is to fit a numerical function specifically referred to as profile shape function (PSF) to a measured diffraction line. Each PSF is typically described by three parameters and sometimes a fourth for a tunable PSF. They are (1) the line position 2*θ*_{0}, (2) peak or integrated intensity *I*_{0}, (3) line width expressed as FWHM. An optimizing algorithm is employed to adjust the PSF’s parameters until the differences between the observed and calculated lines are minimized. Early X-ray Rietveld studies [21, 22, 23, 24, 25, 26, 27] and integral breadth methods discarded the assumption of simple profile shape functions like the Gaussian or Cauchy (Lorentzian) for X-ray line profiles and established that tunable functions like Voigt, pseudo-Voigt and the Pearson VII functions are likely to be the better choice for X-ray line profiles. In the Rietveld method, the whole diffraction pattern is fitted simultaneously and it requires a detailed structural model (viz. site occupancies, fractional atomic coordinates, etc. apart from the unit cell dimensions).

## Experimental

### Synthesis of alumina templates and alumina–zirconia nanocomposite

It is well known that undoped tetragonal/cubic phase of ZrO_{2} can be stabilized at room temperature by controlling the size of the crystallites [28]. Thus, in order to synthesize stable ZrO_{2} nanoparticles which retain tetragonal/cubic crystal structure at room temperature, it is essential to prepare templates with pores less than 100 nm. α-Al_{2}O_{3} is one such inorganic template which has very limited solute solubility in zirconia and thus will not alter the local environment of the Zr atoms in the lattice. Analysis of crystal structure of such nanocomposites may give information regarding the oxygen vacancies if any, hence its implication on phase transformation and phase stabilization.

Solution combustion synthesis (SCS) (Aruna et al. and references therein [29]) is a quick and simple process for the synthesis of a variety of nano-sized materials. The process involves a combustion reaction in solution phase of different oxidizers (e.g. metal nitrates) and fuels (e.g. urea, glycine, hydrazides). The combustion reaction is usually initiated in a pre-heated furnace or on a hot plate at a much lower temperature. In a typical reaction, a homogeneous mixture of water, metal nitrate precursors and fuel is allowed to dehydrate leading to the exothermic decomposition of the fuel. The chemical energy released from the exothermic reaction between the metal nitrate and fuel appears as large amount of heat which is sufficient enough to raise the temperature of the system to higher value (> 1600 K) for a short duration of time. As a result of decomposition of the fuel and the nitrate, a sudden evolution of large volume of gas occurs. This makes large particles or agglomerates get disintegrated leading to the formation of materials, which can be easily crushed to obtain fine particles. Another important aspect of combustion synthesis is the formation of porous material which will serve the purpose of templates in the present study.

Metal nitrates were chosen as precursors to prepare the alumina templates as well as alumina–zirconia composites. Micron-sized α-Al_{2}O_{3} was prepared by the combustion of aluminium nitrate [Al(NO_{3})_{3}·9H_{2}O; LR grade, LOBA CHEMIE] and urea [NH_{2}CONH_{2}; GR grade, MERCK] mixture in the molar ratio 1:2.5 [30]. Redox mixture of aluminium nitrate and urea was dissolved in minimum quantity of double-distilled water, and the dish containing the solution was introduced into a furnace maintained at temperature 500 ± 10 °C. The solution boils, foams, ignites to burn with flame and produces voluminous foamy alumina powder. The as-prepared α-Al_{2}O_{3} sample was annealed at a temperature of 800 °C in a muffle furnace for 10 h to determine any change in the microstructural properties. The Al_{2}O_{3}–ZrO_{2} composites with a 50:50 ratio were prepared by combustion technique using aluminium nitrate–urea and zirconyl nitrate [ZrO(NO_{3})_{2}·H_{2}O; LR grade, LOBA CHEMIE]. The stoichiometry of metal nitrate and fuel is calculated using their total oxidizing and reducing valances in such a way that the equivalence ratio is unity and maximum energy is released during combustion.

### Sample characterization

The X-ray powder diffraction pattern of the as-prepared and annealed samples as well as that of the standard material (here fully recrystallized Si powder [31]) was taken at room temperature in a highly stabilized Philips PW1830 X-ray generator using a Ni-filtered Cu Kα radiation. A divergence slit of opening 1 deg in the primary beam and a receiving slit of width 0.1 mm in the secondary was used for data collection. The data were collected in a step-scan mode with a step size of 0.02 deg 2*θ* and the counting time of 10 s per step.

#### Data analysis

In the present study, we have adopted a modified Rietveld refinement procedure to obtain refined structural (atomic coordinates, thermal parameters, etc.) and microstructural parameters (coherent domain size/crystallite size, microstrain, etc.). The whole pattern fitting has been done with the help of the software MAUD [32]. Initial simulation of the diffraction pattern was carried out with the help of following structural information: (1) α-Al_{2}O_{3} phase with space group *R*\(\bar{3}\)*c* and lattice parameters *a *= 4.7607°A and *c *= 12.997°A according to the JCPDS card no. 88-0826 and (2) tetragonal ZrO_{2} with space group *P*4_{2}/*nmc* with lattice parameters *a *= 3.629°A and *c *= 5.197°A according to the JCPDS card no. 89-7710. The X-ray diffraction patterns of the standard material and the prepared composites were fitted with pseudo-Voigt function convoluted with a truncated asymmetric function. The background was fitted with a fourth-order polynomial. The instrumental function was determined by measuring the intensity across 10 peaks of fully recrystallized Si powder [31] having large crystallites and free from any defect broadening. The instrumental parameters (viz. three Caglioti parameters [21, 22, 33] for FWHM, two for Gaussian content of the profiles and two asymmetry parameters) were set as non-refinable with the values obtained from the fitting of the X-ray diffraction pattern of the standard material. The microstructural parameters such as coherent domain/crystallite size \(\left\langle {D_{\text{eff}} } \right\rangle\) (\(\left\langle {D_{\text{eff}} } \right\rangle\) is also defined as area-averaged effective crystallite size), rms microstrain \(\left\langle {\varepsilon^{2} } \right\rangle^{1/2}\), etc., along with lattice parameters and structural coordinates were held as refinable parameters during Rietveld refinement. The goodness-of-fitting is judged by the reliability parameters *R*_{wp}.

## Results and discussion

### Characterization of α-alumina templates

_{2}O

_{3}templates. Pores of sizes of approximately 6–8 nm are readily observed. The porous structure results from the escape of the gaseous products during combustion synthesis. It is thus clear that combustion synthesis is capable of producing templates for further synthesis of zirconia nanoparticles inside the templates and control of crystal size can be achieved.

_{2}O

_{3}particles with approximate sizes of 0.5 μm or less. Further, the particles have approximately hexagonal morphology. The X-ray diffraction pattern of the as-prepared sample is shown in Fig. 3. The X-ray diffraction lines are in conformity with PDF card no 88-0826 corresponding to α-Al

_{2}O

_{3}phase.

### Preliminary analysis: X-ray diffraction line profile fitting

The as-prepared α-Al_{2}O_{3} sample produces a diffraction pattern with large number of peaks. The profile fitting analysis was performed for [012] and [113] crystallographic directions which exhibited least degree of overlapping. A profile fitting methodology was used to determine the profile shape parameters and the hence the total integral breadth of the corresponding X-ray diffraction lines. The profile shape was assumed to be a pseudo-Voigt function numerically convoluted with a truncated exponential function [22]. In the fitting process, the Gaussian content (*η*), the half width at half maximum (*ω*), the peak height (*I*_{o}) and the Bragg 2*θ* position were simultaneously refined along with the parameters of the linearly varying background. The integral breadth of the profiles is computed according to the relation \(\beta = \omega \left[ {\pi \left( {1 - \eta} \right) +\eta \left( {\pi /\ln 2} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} } \right]\). The Cauchy and Gaussian component of the integral breadths of the ‘true’ specimen-broadened profile and the total integral breadth of the samples were obtained according to the formalism described by de Keijser et al. [15]. A similar analysis was performed for the annealed sample and the alumina–zirconia composite.

### Williamson–Hall plots

*β*

_{F}* (total integral breadth in sin

*θ*scale) =

*β*size +

*β*

_{strain}(s).

The Williamson–Hall plot for both the as-prepared and annealed sample shows some unusual features. The WH plot is neither constant with the increase in the diffraction vector (the case of pure size broadening), nor it increases linearly with the diffraction vector (the case of pure strain broadening). It is observed that the true integral breadth increases nonlinearly with the diffraction vector *d**. Linear fit produces very low crystallite size, and the values are different for the different regions of extrapolations assumed. This confirms further the observed nonlinearity of the Williamson–Hall plot. The few scattered points in the high s region could be attributed to inaccuracy in pattern decomposition from overlapping reflections. Annealing at 800 °C does not produce any appreciable change in the WH plot apart from slight increase in the crystallite size as evident from the intercept of the plot. However, the same alumina reflections show different behaviour in the composites and show lower crystallite as well as higher microstrain. Higher alumina reflections did not yield reliable data due to high degree of overlapping.

### Rietveld refinement

Result of Rietveld refinement for as-prepared alumina sample

Parameter | Value |
---|---|

Unit cell (Å) | |

| 4.762 (0.1) |

| 13.002 (0.6) |

| 120 (2) |

\(\left\langle {\varepsilon^{2} } \right\rangle^{1/2}\) (× 10 | 3.8 (4) |

Al | 0.3525 (1) |

O | 0.3094 (6) |

O site occupancy | 0.970 (5) |

| 18.157 |

| 14.410 |

GoF | 1.260 |

### Characterization of nanocrystalline zirconia in alumina templates (50:50)

### Microstructural characterization of zirconia particles and alumina templates

Results of mirostructural characterization of zirconia nanoparticles

Sample | (hkl) | \(\left\langle D \right\rangle_{v}\) (Å) | (× 10 | Refl | \(\left\langle D \right\rangle_{\text{eff}}\) (Å) | \(\left\langle {\varepsilon^{2} } \right\rangle^{1/2}\) (× 10 | \(\left\langle {D_{v} } \right\rangle\) (Å) |
---|---|---|---|---|---|---|---|

t-ZRO | 111 002 202 311 222 | 307 265 209 242 222 | 3.30 5.64 2.83 1.75 1.49 | [111] | 161 | 1.47 | 231 |

Results of microstructural characterization of alumina

Sample | Reflections | \(\left\langle D \right\rangle_{\text{eff}}\) (Å) | \(\left\langle {\varepsilon^{2} } \right\rangle^{1/2}\) (× 10 | \(\left\langle {D_{v} } \right\rangle\) (Å) |
---|---|---|---|---|

α-Al | 012 024 | 337 | 0.68 | 515 |

ZrO Using Si standard | 012 024 | 281 | 2.36 | 341 |

ZrO Using Al | 012 024 | 1796 | 2.40 | – |

Result of Rietveld refinement for as-prepared nZA sample

Parameter | Alumina | Zirconia |
---|---|---|

Unit cell (Å) | ||

| 4.765 (0.5) | 3.600 (0.9) |

| 13.013 (0.2) | 5.200 (1.2) |

| 66 (5) | 39 (1) |

\(\left\langle {\varepsilon^{2} } \right\rangle^{1/2}\) (× 10 | 13.8 (4) | 1.5 (10) |

Al | 0.3534 (4) | – |

O | 0.3053 (15) | 0.1999 (4) |

O site occupancy | 0.955 (13) | 0.854 (28) |

| 8.937 | |

| 7.291 | |

GoF | 1.226 |

Comparing the parameters of Tables 1 and 4, it is clear that much higher value of lattice microstrain and a lower value of crystallite size are obtained for the template alumina. It is hence clear that the zirconia particles has grown inside the pores of the alumina templates and the difference of thermal expansion coefficients of alumina and zirconia might have introduced residual stresses in the alumina matrix and may have thus resulted in higher values of lattice microstrain.

## Conclusion

Zirconia nanoparticles were synthesized by a solution combustion technique from stoichiometric amount of zirconyl nitrate and urea. The nitrate combustion route produces crystalline zirconia. Zirconia nanoparticles were grown inside alumina templates to control the crystal size. The alumina templates were characterized by pores of sizes of approximately 6-8 nm as observed from electron microscope photographs. The SEM micrographs show flake-like appearance of the α-Al_{2}O_{3} particles with approximate sizes of 0.5 μm or less. Further, the particles have approximately hexagonal morphology. The X-ray diffraction pattern of the as-prepared α-Al_{2}O_{3} sample shows crystalline nature. X-ray diffraction line profile analysis using Williamson–Hall method and more rigorous Fourier method of Warren and Averbach reveals that the alumina templates are characterized by negligible defect-related effect of lattice distortion. The amount of lattice microstrain ranges between 0.0004 and 0.0007 depending on the method of analysis. Rietveld structure refinement using Structure-Microstructure Refinement software MAUD did not reveal any gross difference with the literature reported values for cell parameters a and c and fractional atomic coordinates x and z for Al and O atoms indicating no large-scale bond deformation. The Williamson–Hall plot is nonlinear in nature. The zirconia nanoparticles inside the pores of alumina templates are characterized by tetragonal crystal structure which is a unique example of size-stabilized zirconia. The template X-ray reflections are skewed in nature which indicates some distortion of the templates might have taken place. The distortion is, however, plastic in nature which is evident from the higher level of lattice distortion viz 0.2% of lattice microstrain. It is hence clear that the zirconia particles have grown inside the pores of the alumina templates, and the difference of thermal expansion coefficients of alumina and zirconia might have introduced residual stresses in the alumina matrix. The size of zirconia nanoparticles is ~ 40 nm which is slightly higher than that required for size stabilization. Structure refinement reveals a large amount of oxygen deficiency. Thus, it may be argued that size-stabilized zirconia is an oxygen deficient form.

## Notes

## References

- 1.Kisi, E.H., Howard, C.J.: Crystal structures of zirconia phases and their inter-relation. Key Eng. Mat.
**153–154**, 1–36 (1998)CrossRefGoogle Scholar - 2.Shelef, M., McCabe, R.W.: Twenty-five years after introduction of automotive catalysts: What next? Catal. Today
**62**(1), 35–50 (2000)CrossRefGoogle Scholar - 3.Kašpar, J., Fornasiero, P., Hickey, N.: Automotive catalytic converters: current status and some perspectives. Catal. Today
**77**(4), 419–449 (2003)CrossRefGoogle Scholar - 4.El-Sharkawy, E.A., Khder, A.S., Ahmed, A.I.: Structural characterization and catalytic activity of molybdenum oxide supported zirconia catalysts. Micropor. Mesopor. Mater.
**102**(1–3), 128–137 (2007)CrossRefGoogle Scholar - 5.Chary, K.V.R., Reddy, K.R., Kishan, G., Niemantsverdriet, J.W., Mestl, G.: Structure and catalytic properties of molybdenum oxide catalysts supported on zirconia. J. Catal.
**226**(2), 283–291 (2004)CrossRefGoogle Scholar - 6.Porter, D.L., Heuer, A.H.: Microstructural development in MgO-partially stabilized zirconia (Mg-PSZ). J. Am. Ceram. Soc.
**62**(5–6), 298–305 (1979)CrossRefGoogle Scholar - 7.Garvie, R.C., Hannink, R.H.J., Urbani, C.: Fracture mechanics study of a transformation toughened zirconia alloy in the CaO–ZrO
_{2}system. Ceramurgia Int.**6**(1), 19–24 (1980)CrossRefGoogle Scholar - 8.Scott, H.G.: Phase relationships in the zirconia-yttria system. J. Mater. Sci.
**10**(9), 1527–1535 (1975)ADSCrossRefGoogle Scholar - 9.Toraya, H.: Effect of YO
_{1.5}dopant on unit-cell parameters of ZrO_{2}at low contents of YO_{1.5}. J. Am. Ceram. Soc.**72**(4), 662–664 (1989)CrossRefGoogle Scholar - 10.Sorensen, B.F., Kumar, A.N.: Fracture resistance of 8 mol% yttria stabilized zirconia. Bull. Mater. Sci.
**24**(2), 111–116 (2001)CrossRefGoogle Scholar - 11.Yashima, M., Sasaki, S., Kakihana, M., Yoshimura, M.: Raman scattering study of cubic-tetragonal phase transition in Zr
_{1−x}Ce_{x}O_{2}solid solution. J. Am. Ceram. Soc.**77**(4), 1067–1071 (1994)CrossRefGoogle Scholar - 12.Warren, B.E.: X-ray Diffraction. Addison-Wesley, New York (1969)Google Scholar
- 13.Klug, H.P., Alexander, L.E.: X-ray Diffraction Procedures. Wiley, New York (1974)zbMATHGoogle Scholar
- 14.Langford, J.I.: Rapid method for analysing the breadths of diffraction and spectral lines using the Voigt function. J. Appl. Cryst.
**11**(1), 10–14 (1978)CrossRefGoogle Scholar - 15.de Keijser, ThH, Langford, J.I., Mittemeijer, E.J., Vogels, A.B.P.: Use of the Voigt function in a single-line method for the analysis of X-ray diffraction line broadening. J. Appl. Cryst.
**15**(3), 308–314 (1982)CrossRefGoogle Scholar - 16.Schoening, F.R.L.: Strain and particle size values from X-ray line breadths. Acta. Cryst.
**18**, 975–976 (1965)CrossRefGoogle Scholar - 17.Williamson, G.K., Hall, W.H.: X-ray line broadening from filed aluminium and wolfram. Acta Metall.
**1**(1), 22–31 (1953)CrossRefGoogle Scholar - 18.Halder, N.C., Wagner, C.N.J.: Separation of particle size and lattice strain in integral breadth measurements. Acta Cryst.
**20**, 312–313 (1966)CrossRefGoogle Scholar - 19.Bertaut, E.F.: Debye–Scherrer lines and size distribution of Bragg domains in polycrystalline powders. Acta Cryst.
**3**, 14–18 (1950)CrossRefGoogle Scholar - 20.Stokes, A.R.: A numerical Fourier-analysis method for the correction of widths and shapes of lines on X-ray powder photographs. Proc. Phys. Soc. Lond.
**61**(4), 382–391 (1948)ADSCrossRefGoogle Scholar - 21.Enzo, S., Fagherazzi, G., Benedetti, A., Polizzi, S.: A profile-fitting procedure for analysis of broadened X-ray diffraction peaks. I. Methodology. J. Appl. Cryst.
**21**, 536–542 (1988)CrossRefGoogle Scholar - 22.Lutterotti, L., Scardi, P.: Simultaneous structure and size-strain refinement by the Rietveld method. J. Appl. Cryst.
**23**, 246–252 (1990)CrossRefGoogle Scholar - 23.Pawley, G.S.: Unit-cell refinement from powder diffraction scans. J. Appl. Cryst.
**14**, 357–361 (1981)CrossRefGoogle Scholar - 24.Thompson, P., Cox, D.E., Hastings, J.B.: Rietveld refinement of Debye–Scherrer synchrotron X-ray data from A1
_{2}O_{3}. J. Appl. Cryst.**20**, 79–83 (1987)CrossRefGoogle Scholar - 25.Young, R.A.: The Rietveld method. In: Young, R.A. (ed.) The Rietveld method, pp. 1–38. Oxford University Press, Washington DC (1996)Google Scholar
- 26.Hill, R.J.: Calculated X-ray powder diffraction data for phases encountered in lead/acid battery plates. J. Powd. Sour.
**9**(1), 55–71 (1983)ADSCrossRefGoogle Scholar - 27.Hill, R.J., Howard, C.J.: Quantitative phase analysis from neutron powder diffraction data using the Rietveid method. J. Appl. Cryst.
**20**, 467–474 (1987)CrossRefGoogle Scholar - 28.Garvie, R.C.: Stabilization of the tetragonal structure in zirconia microcrystals. J. Phys. Chem.
**82**(2), 218–224 (1978)CrossRefGoogle Scholar - 29.Aruna, S.T., Mukasyan, A.S.: Combustion synthesis and nanomaterials. Curr. Opin. Sol. State Mater. Sci.
**12**(3–4), 44–50 (2008)ADSCrossRefGoogle Scholar - 30.Kingsley, J.J., Patil, K.C.: A novel combustion process for the synthesis of fine particle α-alumina and related oxide materials. Mater. Lett.
**6**(11–12), 427–432 (1988)CrossRefGoogle Scholar - 31.van Berkum, J.G.M., Sprong, G.J.M., de Keijser, ThH, Delhez, R., Sonneveld, E.J.: The optimum standard specimen for X-ray diffraction line-profile analysis. Powd. Diffr.
**10**(2), 129–139 (1995)ADSCrossRefGoogle Scholar - 32.Lutterotti, L.: MAUD Version 2.71, http://maud.radiographema.com/(2016). Accessed 5 Nov 2017
- 33.Caglioti, G., Paoletti, A., Ricci, F.P.: Choice of collimators for a crystal spectrometer for neutron diffraction. Nucl. Instrum.
**3**(4), 223–228 (1958)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.