It is true that fwhm of xrd peaks increases with external pressure even within the hydrostatic limit. This is because of the strain (micro or macro) created in the lattice due to the applied pressure. This is responsible for the increasing fwhm. Generally this fwhm varies in a systematic way in the hydrostatic limit. If you are out side the hydrostatic limit, then this variation will be anisotropic in nature. As the peaks in the powder diffraction consist of scattering from many individual crystalites, each crystallite will see a different pressure and hence the 2theta value of each peak will have an uncertainity of +-d2theta. This is responsible for the anisotropic increase in fwhm in the nonhydrostatic region.

Dear Irshad,

it sounds convincing.

You may attach a reference dealing with that subject in more detail.

Many thanks in advance.

Thanks Irshad for this explanation.

I think within the hydrostatic range, the strain is nothing but the reduction in lattice parameter resulting in shifting of peak position towards higher 2theta values. But how this affects the FWHM within the hydrostatic or quasihydrostatic range? What for single crystal XRD ?

What is the subject material? Please post your XRD data for best feedback.

Increase in FWHM generally indicates increase in defect density. It would be interesting to examine the change in the shape of the profile as well to get more clues as to the type of defects creating the "broadening".

When pressure is applied to a solid, the atoms are pushed out of their thermodynamic equilibrium crystallographic sites, not in a symmetric form, given places to a solid where the atoms occupies positions near the equilibrium. This is why the diffraction peaks are broad with respect to a solid in equilibrium. This broadering can be suprimed if the solid is heated (see graph attached for heating of CuInSe2 after a high pressure cycle obtained under synchrotron radiation in a high pressure- high temperature cell)

@ Prof Gerhard Martens

Thanks for your comment sir. Right now i do not have a proper reference to validate my comment. i hope it may be available in any of the articles dealing with the high pressure experimental technique . if i get one i will definitely post it here.

@ Subodha Sahoo

apart from the usual decrease in lattice parameters in hydrostatic limit, the uniaxial compression and microstarain devoloped with incresing pressure could also result in the increse in fwhm. As Prof. Grima Gallardo pointed out , different atoms residing in a particular plane can move in an asymmetric form, which could also results in the broadening. As i am not familier with single crystals, i really do not know how this fwhm of single crystals varies as a function of pressure. certainly some of the above discussed aspects may be valid there also.

Thanks Prof. P. Grima Gallardo and Irshad.

"When pressure is applied to a solid, the atoms are pushed out of their thermodynamic equilibrium crystallographic sites, not in a symmetric form, given places to a solid where the atoms occupies positions near the equilibrium."

Once the pressure (load) is relesed, there would be two types of strains that would remain in the lattice, "elastic" and "plastic". Generally, these are the residual stress/strains that continue to deform the lattice relative to the "equilibrium" state. Whatever that means? Conventionally, the elastic strain is the component in the continuum model that distorts the lattice without discontinuities (uniform dilatation) in the lattice while the plastic strain component results from discontinuities (defects) in the lattice and the related local expansion or contraction of the lattice. Yes?

I think, P. Grima Gallardo is describing in situ (synchrotron radiation in a high pressure- high temperature cell) examination with XRD while Subodha Sahoo is discussing ex situ examinations using conventional XRD instrumentation. Two completely different paradigms. Please verify these assumptions about in situ and ex situ. It is not clear if Subodha is observing the increase in FWHM as the pressure is applied or after it has been applied and released.

BTW how many (hkl) reflections did you monitor while observing changes in FWHM?

My objective for these comments is not to be contentious but to clarify the concept in my mind with the help of the expertise in the RG community. Thinking out loud!

Thanks Sir Ravi Ananth,

Its insitu synchtron based ADXRD.

And I found the broadening of XRD peaks during the increasing pressure cycle and it is quite reversible upon complete decompression cycle. You can have a look at the data till 9.2 GPa as shown in this proceeding.

Conference Paper High Pressure Structural and Vibrational Properties of Cu 3 VS 4

Nice work Subodha! I'll follow with keen interest. As will I follow the answers to this question. Thanks!

What is typically the temporal resolution of the XRD data acquired for such in situ measurements on the synchrotron?

I'd certainly be curious about the relationship of FWHM and (hkl) as well under such loading conditions.

Haven't recieved the full transcript yet. Mind posting a typical plot of FWHM vs. Pressure profile?

"reduction in lattice parameter resulting in shifting of peak position towards higher 2theta values" - All (hkl)'s?

"solid where the atoms occupies positions near the equilibrium" - increase in entropy? Strained lattice?

"different atoms residing in a particular plane can move in an asymmetric form, which could also results in the broadening" - How is the Bragg profile affected? Symmetric, asymmetric, below FWHM profile change, etc.? I'm very interested in this aspect of the Bragg profile and its intense correlation with specimen voxel Nano structure.

Any published XRD references in this subject would be deeply appreciated.

The response of various scientists contains the answer to the question raised. Grima Gallardo has given an example of line widths observed in high pressure x-ray diffraction experiments. I would like to add some of my thoughts on this topic and briefly describe the formalism that allows a mathematical formulation of nonhydrostatic stresses and the resulting line shift and line broadening.

Hydrostatic pressure:

We first discuss the case of hydrostatic pressure. As we all understand, hydrostatic pressure is a stress system that contains no shear stress component. Typical examples of this are the pressures exerted by fluids (liquid or gas). In the case under discussion, the pressure is exerted on a polycrystalline aggregate. One would expect that the crystallites in the sample will undergo hydrostatic compression that will produce strains that conform to the symmetry of the crystal system. The set of planes (hkl) of all crystallites will experience equal strains. This will result in a shift of diffraction lines but no broadening of the lines. An experimental study on polycrystalline solids in a diamond anvil cell (DAC) using helium pressure medium [1] showed no significant broadening under hydrostatic pressure up to about 70 GPa. As the pressure was gradually increased above 70 GPa, the pressure tended to become nonhydrostatic. This resulted in the broadening of the diffraction lines. The broadening of the lines is a sensitive method to detect the onset of the nonhydrostatic stresses in any experiment. This article also gives nice figures that help visualization of what happens to the crystallites under nonhydrostatic stresses. Note that we used a term ‘nonhydrostatic pressure’. We discuss the nature of nonhydrostatic stresses and point out that there exists a rigorous mathematical description of such stresses.

Nonhydrostatic Pressure:

It is important to understand these terms before we talk about their effects on the diffraction lines. We consider a specific case of a polycrystalline sample compressed in a DAC to discuss the nonhydrostatic stress state. This is because it is easy to formulate mathematically the nonhydrostatic stress sate for an actual experimental setup. The solid sample deforms plastically and begins to flow outwards in radial direction when the high enough load is applied on the DAC. At some thickness, determined by the applied average stress and the mechanical properties of the sample, the flow ceases and time-independent stress state is set up in the sample. The stress state in the sample is highly complex; it varies from crystallite to crystallite both in magnitude and direction. The magnitude of the stress lies between a maximum and a minimum. In turn, it produces strains of varying magnitude in each crystallite. A mathematical description of the stresses (strain) is possible through the various moments of stresses (strains) about the mean [2]. The average of the stresses (mean stress) in the crystallites along a line represents the macro-stress. Note that macro-stresses and macro-strains are tensor quantities. The macro-stresses cause the diffraction line to shift. The variance of stresses (second moment about the mean) represents the spread of stresses (strains) about the mean value. The square root of variance is a measure of micro-strains. The diffraction lines from the crystallites with varying strains are spread out (broadened) about the mean position.

Where does the pressure figure in this formulation? Let us now consider only the macro-stresses. There exists a stress gradient in radial direction, the stress being highest at the center and decreasing rapidly away from the center. We assume the x-ray beam used in the experiment is small enough to probe only a small volume of the sample at the center, which coincides with the position of peak stress in a well-aligned DAC setup. The stress possesses axial symmetry about the line perpendicular to the anvil face and passing through the center of the anvil. The axial stress component, s3, is larger than the radial component, s1. The difference between the two, (s3 – s1) = t, is a measure of the strength of the solid and the mean normal stress (s3 + 2S1)/3 is the equivalent hydrostatic pressure. For a general formulation see ref. 3. When we talk about pressure in a DAC under nonhydrostatic condition, it is the mean normal stress, (s3+2S1)/3. This pressure can reach a few hundred GPa in a x-ray diffraction experiment with a DAC. For diffraction experiments on Mo to pressures up to 400 GPa see ref. 4.

The plastic deformation brings about many physical changes in the sample. The nature of the change depends on whether the sample is ductile or brittle. In case of ductile materials, the generation of defects (point, line and planar) is the dominant factor. In case of brittle materials, the breaking down (‘comminution’) of the crystallites during compression dominates. However, even brittle materials acquire appreciable degree of ductility at high pressures and can develop defects typical of ductile materials. Effectively, these processes reduce the crystallites size (coherently scattering domain size) of the sample. Thus, the crystallites size decreases with increasing nonhydrostatic pressure. This contributes to line broadening. Typical study on ductile and brittle material can be found in ref. 5 and ref. 6, respectively. Example of NaCl is interesting as it is brittle at one atmosphere but flows easily undergoing plastic deformation under moderate pressures [7].

When we talk about line broadening under pressure, it is implied that the pressure is nonhydrostatic in the sense discussed above. The line broadening arises due to two factors: (1) Reduction of crystallite size and (2) appearance of the micro-stains in the crystallites under increasing nonhydrostatic pressure.

Line-width decrease under high pressure

A change in the polycrystalline sample, not considered above, that is caused by plastic deformation is the development of texture (preferred orientation). In case of nanocrystalline Au, the width of (111) diffraction line shows a decrease with increase in pressure [8]. This has been attributed the development of texture. Further, the measured crystallite size of 20 nm n-Au is found to decrease as the pressure is increased but it begins to increase above 40 GPa [Fig. 4 of ref. 9]. Thus, the sharpening of the line appears to be related to the grain growth that is caused by texture.

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