Quantitative Phase Analysis

Wenjea J. Tseng Department of Materials Science and Engineering

National Chung Hsing University URL: http://audi.nchu.edu.tw/~wenjea/

Email: wenjea@dragon.nchu.edu.tw

Chapter 12

Chapter 12

Introduction Diffraction of x-rays by crystals has been used in quantitative phase analysis for years. By 1936, internal standard method was in use. Use of various quantitative methods, however, requires impeccable technique and careful calibration. Characteristics of using the x-ray diffraction instead of other spectral analyzers include: (1) The x-ray method reveals chemical phase composition instead

of just elemental information (note: EDS reveals elemental info). (2) The x-ray method examines samples in a comparatively larger

volume when compares to selective regions such as by microscopy.

(3) The uncertainty from the quantitative compositional x-ray determination falls in a range of about 1%; therefore, use of other analyses in parallel with the x-ray method is recommended in order for precise determination.

Chapter 12

Contents 1) Quantitative Composition (or Phase) Analysis by XRD

2) Quantitative X-ray Microanalysis: Element Analysis by Energy

Dispersive Spectroscopy (EDS)

v

Chapter 12

Concept of Quantitative XRD Intensity of diffraction pattern of a particular phase in a mixture of phases depends on the concentration of that phase in the mixture.

β

α

β

α

cc

II

∝ I: Intensity C: concentration in volume

Unfortunately, the relation between intensity and concentration is NOT generally linear, because the diffracted intensity depends markedly on the absorption coefficient of the mixture and this itself varies with the concentration. In simplified form, the above equation becomes

β

α

β

α

α

β

β

α

β

ββ

α

αα

β

α

µµ

µ

µccK

cc

KK

cK

cK

II

⋅≈⋅⋅==

Both are µmixture.

Chapter 12

Methods of Quantitative XRD There are three main methods available for the quantitative analysis:

1) External standard method

2) Direct comparison method

3) Internal standard method

Chapter 12

External standard method The weight ratio of one component, i.e., wα, from a two-phase mixture (α +β) can be determined from intensity comparison of the α−phase in the mixture, i.e., Iα, and the pure α−phase, i.e., Iαp. The pure phase serves as an external standard, and its diffraction pattern is performed in a separate measurement.

( )( ) ( )[ ] ( )ββαα

αα

α

α

ρµρµρµρµ

////

+−=

ww

II

p

The above equation also works for the β phase if a separate XRD is carried out for the pure β phase. In addition, the relation reduces to a linear dependence only when the mass absorption coefficients of α and β are identical.

αα

α wII

p

=

Chapter 12

The mass absorption coefficients of quartz and cristobalite identical.

Note that the determination of Iα and Iαp must be carried out under identical condition for the external standard method. In addition, a confirmation test is recommended by conducting XRD separately using the α and β phases of the calculated weight ratios.

Chapter 12

Direct comparison method This method is widely used since no requirement of the pure phase is necessary. For a binary mixture as an example, the fundamental equation for the diffraction intensity is

+

=

−

µθθθ

νπµ

πλ

2cossin2cos11

432

2

2

22

22

423 M

hkloo

hklepF

me

rAII

The above equation can be simplified as

µ2KRIhkl =

Mhkl

oo

epFR

me

rAIK

22

22

2

2

423

cossin2cos11

432

−

+

=

=

θθθ

ν

πµ

πλ

where

K is a constant, independent of the kind and amount of the diffracting substance, and R depends on θ, hkl, and the kind of substance.

Chapter 12

Direct comparison method (cont.) The intensity ratio of a binary mixture, Iα/Iβ, is then

ββ

αα

ββ

αα

β

α

µ

µcRcR

cKR

cKR

II

mixture

mixture ==

2

2

The volumetric concentration ratio, cα/cβ, can hence be determined from the measured ratio, Iα/Iβ, and a calculated R values. Once cα/cβ is found, the value of cα or cβ can be calculated from

1=+ βα cc

Chapter 12

Remarks In choosing diffraction lines to calculate the chemical composition, overlapping or closely adjacent lines from different phases must be avoided even though deconvolution can be made for the overlapped lines but the uncertainties involved warrant choosing of well-resolved lines. (See examples on next pages)

Chapter 12

Example 1 Ni-V steel containing austenite (γ) and martensite (α).

Peak overlap. Not chosen even though high diffraction intensity.

Can be used for the γ/α determination.

Chapter 12

Example 2 Two polymorph forms of TiO2 crystals: Anatase and Rutile.

Anatase

O2- Ti4+

Tetragonal structure

Rutile

Tetragonal ditetragonal structure

Chapter 12

Example 2 (cont.) Use of the direct comparison method of XRD to estimate

quantitatively the anatase-rutile weight ratio from mixtures.

R. A. Spurr and H. Myers, Quantitative Analysis of Anatase-Rutile Mixtures with an X-ray Diffractometer, Anal. Chem., 29 (1957) 760–762.

A: anatase R: rutile A (101) / R (110)

Chapter 12

Example 2 (cont.) Using the experimentally determined function derived by Spurr and Myers, we have estimated the anatase-rutile weight ratio of hollow TiO2 micro-spheres synthesized by a novel “template-implantation route”.

W. J. Tseng, P.-S. Chao, Synthesis and Photocatalysis of TiO2 Hollow Spheres by a Facile Template-Implantation Route, Ceram. Int., 39 [4] 3779-3787 (2013).

Chapter 12

Internal standard method In this method, a diffraction line from the phase being determined is compared with a line from a standard mixed with the sample in known proportions. The internal standard method is therefore restricted to samples in powder form. By mixing a known amount of a standard substance (S) with the original sample, the amount of phase A in the original sample can be determined by measuring the integrated intensities of XRD pattern through

AS

A KwII

=

IA: Intensity of phase A in diffraction pattern, IS: Intensity of standard in the same diffraction pattern, K: constant, WA: The weight fraction of phase A in the original sample.

Chapter 12

Internal standard method (cont.) The intensity ratio (IA/IS) is hence a linear function of WA. A calibration curve can be prepared from measurements on a set of synthetic (mixture) samples containing known concentration of A and the standard. Once the calibration curve is established, the concentration of A in an unknown sample is obtained by measuring the ratio IA/IS for the unknown sample and the same proportion of standard as was used in the calibration.

Chapter 12

Example The internal standard method has been widely used for the measurement of the quartz content in industrial dusts, using fluorite (CaF2) as the internal standard. Figure below shows a calibration curve.

Chapter 12

Practical Difficulties in Quantitative XRD Analysis

Various factors can cause great difficulty in quantitative analysis because they cause observed intensities to depart widely from the theoretical. 1) Preferred orientation. May use “averaging intensities” and/or

“averaging orientations” to try to minimize the orientation effect and may get satisfactory “reasonable” result. Use of powder samples in finely ground particle size is recommended.

2) Microabsorption. Use of powder samples in finely ground particle size is recommended.

3) Extinction. Mostly occur in single crystals. Use of powder samples in finely ground particle size is recommended.

Chapter 12

Contents 1) Quantitative Composition (or Phase) Analysis by XRD

2) Quantitative X-ray Microanalysis: Element Analysis by Energy

Dispersive Spectroscopy (EDS) v

Chapter 12

Energy Dispersive Spectrometer Energy-dispersive X-ray spectroscopy (EDS, EDX, or XEDS), sometimes called energy dispersive X-ray analysis (EDXA) or energy dispersive X-ray microanalysis (EDXMA), is an analytical technique used for the elemental analysis or chemical characterization of a sample in micrometer regions.

Example

One can obtain elemental information of about 1 µm3 regions from samples.

Chapter 12

Accuracy of EDS Accuracy of EDS spectrum can be affected by various factors. Many elements will have overlapping peaks (e.g., Ti Kβ and V Kα, Mn Kβ and Fe Kα). The accuracy of the spectrum can also be affected by the nature of the sample. X-rays can be generated by any atom in the sample that is sufficiently excited by the incoming beam. These X-rays are emitted in any direction, and so they may not all escape the sample. The likelihood of an X-ray escaping the specimen, and thus being available to detect and measure, depends on the energy of the X-ray and the amount and density of material it has to pass through. This can result in reduced accuracy in inhomogeneous and rough samples. Elemental accuracy in a range as low as 100 ppm can be determined from a sample volume as small as 10-12 cm3 with proper standard and calibration. Typical elemental accuracy is about 1%.

Chapter 12

Concept of Quantitative EDS Measured intensity ratio when compared to standard intensity of a particular element in a mixture is directly proportional to the concentration of that element in the mixture.

dardsi

i

dardsi

i

dardsi

i

dardsi

i

IIk

cc

orI

Ic

c

tan,tan,

tan,tan,

=

∝

k depends critically on atomic number (Z), absorption (A), and x-ray fluorescence (F) of the element.

Chapter 12

ZAF Technique As proposed by Castaing in 1951, the x-ray intensity from element i is measured by the same x-ray detector from both sample and standard in which the electron probe current and the angle between the sample surface and the direction of the measured x-ray (i.e., the take-off angle shown in previous ppt.) are held constant. Therefore, To a first approximation, k may be assumed equal for both the sample and standard. If the standard is pure element i, then

dardsi

i

dardsi

i

IIk

cc

tan,tan,

=

dardsi

ii I

Ictan,

= is the intensity measured in pure element.

dardsiI tan,

Chapter 12

ZAF Technique (cont.) Therefore, what the analyst measures is the relative x-ray intensity ratio between the elements of interest in the sample and the same elements in the standard. Both sample and standard are examined under identical experimental conditions. For better accuracy, k must be taken into account and may not be assumed equal in the sample and in standard; therefore, Where Ci is the weight fraction of the element of interest in the sample; Zi is the so-called atomic number effect; Ai is the absorption effect; and Fi is the fluorescence effect.

ii kZAFc )(=

Chapter 12

Example An EDS-ZAF analysis at different regions near the SiC@SiO2 core@shell composite and steel interface.

T.-T. Tseng et al., “Refractory Filler Sands of Core-Shell Composite Structure for the Taphole Nozzle in Slide-Gate System of Steel Ladles,” Ceramics International, 38 [2] (2012) 967-971.

Chapter 12

Summary Chemical compositions including phase structure and elemental composition can be determined by x-ray analysis quantitatively. The accuracy generally falls in a range of ±1% with standard as a calibration. Even thought the accuracy is limited, phase structure can be obtained and provides a useful information in many scientific and industrial applications.