The document discusses diffractometers and errors that can occur when using them to measure diffraction patterns. It describes common sources of error such as misalignment, use of a flat specimen, absorption in the specimen, and displacement of the specimen. It explains how errors can be reduced using analytical methods like extrapolating the lattice parameter or resolving diffraction peaks. Specifically, it outlines Cohen's analytical method which minimizes random errors by fitting sin^2θ values to linear equations to determine the true lattice parameter.

1. Diffractometer
Presented by
Sehrish Inam
Advanced Experimental Techniques In Physics
Course Incharge:
Ghulam Mustafa Khan
NED university of
Engineering and
Technology

2. Table of contents
O Historical background
O Diffractometer and error correction
O Worked example for error correction
O Conclusion

3. Historical Background
of Diffractometer

4. DEBYE SCHERRER Powder camera
• Powder x-ray diffraction in accordance with
Braggs law to produce one diffracted beam
• It shows diffraction lines and holes for incident
and transmitted beam
• A very small amount of powdered material is
sealed into the fine capillary
• The specimen placed and aligned to be the
centre of camera through collimator

5. DEBYE SCHERRER Method
O Method of studying the structure of finely
crystalline substance using x-ray diffraction
O Narrow parallel beam monochromatic x-rays
upon falling onto crystalline samples
O And reflected by the crystallite that make up
the sample
O Produce no of coaxial that having one
common axis
O Diffracting cone

6. THE SCHERRER
EQUATION
O D=K*λ/BCosϴ
We use this formula to fine diameter
D = diameter of the crystallite forming film
λ=wavelength
B=FWHM full width half maximum
ϴ=is the Braggs angle

7. Fundamental principal of x-ray
powder diffraction(XRD)
O X-ray diffraction is based on constructive
interference of monochromatic x-rays and
crystalline sample
O X-ray generated by cathode ray tube ,filtered
generated monochromatic x-ray
O Correct orientation of crystal ,Braggs angle
satisfy Braggs equation

8. DEBYE SCHERRER
CAMERA

9. Error in the measurement of theta
• Film shrinkage
• Incorrect camera radius
• Off -centering of specimen
• Absorption in specimen
• Divergence of the beam

10. How error can be corrected?
O By applying extrapolation function for
achieving high precision
O Nelson-Riley function It has greater range of
linearity, Only few angle present
More precision then bradley-function
O At larger angle error will be minimize

11. Advantages of DBSC Camera
•Simplest camera type ,simple to use
•Need very little sample are at low cost
•Film is reasonably good storage medium
•Does not required expansive medium

13. Diffractometer Errors
and correction
Presented by Sehrish Inam

14. Diffractometer

16. Diffractometer
Advantages
O To obtain an x-ray diffraction pattern.
O The data can be stored on floppy disks and
easily retrieved.
O Produce high-quality versions of the diffraction
patterns using, for example, laser printers.
O Data can also be directly processed by using
computer software to obtain information about
the structure, lattice parameters, lattice strain,
crystallite size of the specimen.

17. Diffractometer
Errors
1. Misalignment of the instrument.
2. Use of a flat specimen
3. Absorption in the specimen

18. Diffractometer
Errors
4. Displacement of the specimen from the
diffractometer is largest single source of
error and causes an error in d given by

19. Diffractometer
Errors
5.Vertical divergence of the incident
beam. This error is minimized, with loss
of intensity, by decreasing the vertical
opening of the receiving slit.

20. Diffractometer
Reduction of Error
Extrapolate the lattice parameter against
Cos2ϴ/Sinϴ for displacement of the
specimen from the diffractometer and
against cos2ϴ for the error of flat
specimen and absorption in the specimen.

21. Diffractometer
Reduction of Error
OWhen a peak is resolved into α1 and α2
,will have two lattice parameter points
for each hkl value.
OThe resolution of the peak into α1 and
α2 components can be achieved by
simply enlarging the 2ϴ scale.

22. Diffractometer
Increment in Number of Diffraction Peaks
ODecreasing the wavelength of the radiation
used
OUsing both the K α1 and K α2 components,
the angular separation between α1 and α2
increases with increasing value of ϴ.
O Modern diffractometers use
monochromators that are aligned to diffract
only the Kα component.

23. Diffractometer
OThe presence of Kβ peaks in a pattern can
usually be revealed by calculation, since if a
certain set of planes reflect Kβ radiation at an
angle ϴ they must also reflect Kα radiation at
an angle ϴβ they must also reflect Kα radiation
at an angle ϴα and one angle may be calculated
from the other from the relation

24. Diffractometer
Owhere λ2
kα/ λ2
kβ has the value 1.226 for Cu
K radiation.
ODraw straight line by using the least-
squares method.
OThe random errors involved in measuring
the peak positions are responsible for the
deviation of the various points from the
extrapolation line.

25. Analytical method

26. Random errors are chance errors, such as those involved in
measuring the position of the diffraction peak. These errors may
be positive or negative, and they do not vary in a regular manner
with some particular parameter, say the Bragg angle θ.
ANALYTICAL METHOD:
An analytical method that minimizes the random errors in a
reproducible and objective manner has been proposed by Cohen.
and it can be used to calculate the lattice parameters precisely for
cubic and noncubic systems.
We know that
𝜆 = 2𝑑 𝑠𝑖𝑛𝜃
Squaring the Bragg equation and taking logarithms, we get
log 𝑠𝑖𝑛2
𝜃 = log
𝜆2
4
− 2 log 𝑑

27. Differentiate above equation.
Δ𝑠𝑖𝑛2
𝜃
𝑠𝑖𝑛2 𝜃
= −2
Δ𝑑
𝑑
if we assume that the combined systematic errors take the form
Δ𝑑
𝑑
= 𝐾𝑐𝑜𝑠2 𝜃
By combining above two equations, we get
Δ𝑠𝑖𝑛2
𝜃 = −2𝐾𝑐𝑜𝑠2
𝜃𝑠𝑖𝑛2
𝜃
Δ𝑠𝑖𝑛2 𝜃 = 𝐷𝑠𝑖𝑛22𝜃
where D is a new constant. (This equation is valid only when the
𝑐𝑜𝑠2
𝜃 extrapolation function is valid).
The true value of 𝑠𝑖𝑛2 𝜃 for any diffraction peak is
𝑠𝑖𝑛2 𝜃 𝑡𝑟𝑢𝑒 =
𝜆2
4𝑎02
(ℎ2 + 𝑘2 + 𝑙2)

28. where ao is the true value of the lattice parameter we wish
to obtain. But
𝑠𝑖𝑛2 𝜃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 − 𝑠𝑖𝑛2 𝜃 𝑡𝑟𝑢𝑒 = Δ𝑠𝑖𝑛2 𝜃
𝑠𝑖𝑛2
𝜃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 −
𝜆2
4𝑎02
ℎ2
+ 𝑘2
+ 𝑙2
= 𝐷𝑠𝑖𝑛2
2𝜃
𝑠𝑖𝑛2 𝜃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 = Aα + 𝐶𝛿
The parameter D is called the "drift" constant, and is a fixed
quantity for any diffraction pattern but differs from one
pattern to another. Best precision is achieved when the
value of D is as small as possible.
According to the theory of least squares, the best values of
the coefficients A and C are those for which the sum of the
squares of the random observational errors is a minimum;
i.e.
𝑒2 = 𝑎 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 = [Aα + 𝐶𝛿 − 𝑠𝑖𝑛2 𝜃 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 ]2

29. A pair of normal equations can be obtained by
differentiating above equation with respect to A and C
and equating them to zero. Thus,
𝛼 𝑠𝑖𝑛2 𝜃 = 𝐴 𝛼2 + 𝑐 𝛼𝛿
𝛿 𝑠𝑖𝑛2 𝜃 = 𝐴 𝛼𝛿 + 𝑐 𝛿2
By solving these equations we determine A, and from
this value of A the true lattice parameter 𝑎0 can be
determined.