Analysis of Powder Diffraction Patterns and Vegard`s Law

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Materials Science and Engineering 405/572 


Analysis of Powder Diffraction Patterns and Vegard`s Law


            Dr. Who’s timey wimey is getting overly wibbly wobbly and the TARDIS is missing his targets.  He believes that overlapping dimensions have caused a ferroelectric in the TARDIS mainframe to fail and that it has experienced a solid-state reaction during travel.  Visual inspection shows that the ceramic is cracked, and The Doctor believes that the

-Ba-Sr solid solution is varying across the piece.  He has taken three powder samples from the failed ferroelectric;

-one of which he has a good guess of the composition. 

-Barium and strontium form a solid solution in the perovskite structure as: BaxSr1-xTiO3

-Literature shows that where x=0.4-0.8 the solid solution of BaxSr1-xTiO3 is cubic. 

-If x is varied across the ferroelectric there will be a lattice parameter mismatch, which may cause mechanical failure. 

-The Doctor needs you to determine the lattice parameter and composition of each sample via Rietveld refinements on each data set. 

-Vegard’s law will be applied to the refined lattice parameters to determine the solid solution composition for each sample. 


  • Create a new instrument parameter file with the provided LaB6 data.
  • Perform a Rietveld refinement on each of the following data sets of BaxSr1-xTiO3 where the Ba:Sr ratio is:
    • 10:90   (BST_1090.xrdml)
    • Mystery A (UnknownA.xrdml)
    • Mystery B (hint: has higher Ba)  (UnknownB.xrdml)

The known samples were identified using HighScore.  The ratios provided are meant to be a starting guess.  Use either BaTiO3 or SrTiO3 as a starting model, add in a partial occupancy of the other part by adding another atom into the structure. The .CIF files provided for SrTiO3 (MyBaseFileName_23076_SrTiO3.cif) and BaTiO3 (MyBaseFileName_67518_BaTiO3) differ slightly. Enter phase peak and phase data manually. 

  • Plot the lattice parameters as a function of composition, be sure to include error bars of 3σ for each refined lattice parameter.
  • Use Vegard’s law to determine the solid solution composition of each sample. 

Laboratory Write Up:

Since this is only a mini-lab you are not required to write up a report, however we

will hone your results and discussion skills through a series of short answers. Some may ask

you for plots, tables, or diagrams with a written observation of trends or inconsistencies

(Results), while others may ask you to answer theoretical questions or to connect the dots of

your observations (Discussion).

For this report you need to answer/report the following. Simply create a document

which has these in order and labelled. You will still need to follow the proper formatting

(figure captions, table titles etc.). When you are discussing refining the pattern, remember

that you are modifying a calculated pattern generated from complex equations handled by

GSAS, the measured pattern doesn’t change, and incorrect usage will be marked off. This

should effectively look like a standard lab report, but with less in it:


Briefly discuss (keep it short- introduce these things but don’t go into all of the detail) :

  • Vegard’s law, what parameters will be refined on and used
  • The perovskite Ba,SrTiO3- its structure and applications as a material.  What is a ferroelectric.   Provide an image of the structure from CrystalMaker. 


  1. Abstract: Write an abstract that summarizes the entire experiment, succinctly. 4-5 sentences.
  2. Introduction: Unlike a normal mini-lab, you will need to provide a short introduction to Vegard’s law (what parameters will relate to it in GSAS?) and the perovskite Ba,SrTiO3 (structure and applications, what class of material is it? (not that it is a ceramic)). Equations and crystal structure drawings are necessary.


Experimental: You may assume that the data was obtained with the normal set-up used for all of the other laboratories; however, do NOT use your own parameter file (.prm) when going through GSAS, there should be a .prm file in the data provided. You do not need to write an experimental section.

  1. Results and Discussion:  Provide (one) table with all refined parameters for LaB6 and all four samples (with error).  Are these results expected? Why or why not?  Explain/discuss any poor fits or odd results. 
  2. Results and Discussion:  Discuss the crystallographic information used in the starting models.  What starting parameters (lattice parameter and occupancy) did you use and why?   
  3. Results and Discussion:  Provide a table where the composition of each solid solution is calculated using Vegard’s law. 

Address and discuss any deviations between refined composition (via occupancies) and composition from Vegard’s law. 

  1. Results and Discussion: Plot refined data for 4 samples overlaid with normalized intensity vs 2θ.  This should demonstrate a shift in peaks with changing composition.  Discuss how this relates to lattice parameter, why it changes with varied x, and how it resulted in failure of the ferroelectric.  Index all peaks.  Provide an inset if needed to make the shift clear. 
  2. Results and Discussion:  Graph of lattice parameter vs. composition with a fit line and error bars as 3σ.  Also include on this graph at least 4 points from ICSD of BaxSr1-xTiO3 that cover the range of your compositions. 
  • Discuss error and GOF
  • Reference all starting models
  1. Conclusion: Present a sufficient conclusion of your results, what do they imply for the Doctor? 
  1. References: Standard Section

Overview:  You are provided with The CIFs for BaTiO3 and SrTiO3 to use as starting models for GSASII refinements. Full refinements will provide composition via occupancy, lattice parameter, etc.  In this lab you will use the refined lattice parameter in Vegard’s law to calculate the solid solution composition of four samples and you will compare these results with refinements.


Price: £199

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