Implementation of axial divergence correction of Finger et al. in TOPAS

metty

In order to learn a bit more about the different axial divergence models
I implemented the model of Finger et al. on my own in TOPAS. I used
equation 6 of the original manuscript of Finger, Cox and Jephcoat
from 1994 and folded this equation into the peak shape by the
user_defined_convolution command in TOPAS 4.2

Equation 6 is defined as:

D(2phi, 2theta) = (L/[2*H*h(2phi)*cos(2phi)]) * W(2phi, 2theta)

where D(2phi, 2theta) is the diffraction profile,
L is the sample to detector distance, 2theta is the position
of the Bragg peak, H is the height of the detector opening,
h(2phi) is the vertical coordinate of the intersection between
diffraction cone and detector cyclinder and W(2phi, 2theta) is
a weighting function.

In my implemented correction, the correct weighting function is determined
by an if-function and all upper and lower limits are also automatically
determined by equations.

However, if I run a refinement with my implemented correction, the result
is always worse (Rwp and difference curve) compared to the integrated
Finger et al. model. I know that the axial divergence in TOPAS is in
principle treated by another set of equations than I use, but I always
thought that they are comparable to the original model proposed by
Finger et al.

Has somebody an idea how this can be explained? Are the treatments
in TOPAS and in the Finger et al. paper probably that much different?
Or is it the way how my correction is folded by the command
user_defined_convolution into the peak shapes?

Thank you for your answers and comments. :)

PS: Of course I used the obtained S and H parameter from the implemented
correction in my correction and vice versa, but it did not help.

rowlesmr

Could you post the actual code you're running?

That will help rule out any coding errors, and see if everything is properly implemented.

Matthew

alancoelho

Hi Metty,

See equation 17 in:

J. Res. Natl. Inst. Stand. Technol. 109, 1-25 (2004)

The axial divergence aberration has infinities at the Bragg angle. Thus there's no way to get user_defined_convolution to agree exactly with the internal computation. If you were to set a maximum for the infinity then you should get close.

To test if your aberration is correct (source, sample and receiving slit all being of the same length with no axial divergence in the primary beam) then it should look like the simple axial model which is well know; ie. it should look like:

Sqrt(e2 / e) - 1

where e = 2 Phi - 2 Th_Bragg; and e2 is as given in the paper. Note, the Finger, Cox and Jephcoat is of this form.

cheers
alan

alancoelho

Hi again Metty

The following INP file creates a test pattern when CREATE_ is defined (ie. un-comment the line).

'#define CREATE_

Defining and running with CREATE_ creates a test pattern using the internal Full axial model.

Without CREATE_ fits a user_defined convolution to the test pattern. The attached plot shows two cases for two different step sizes which shows reasonable fits. Note the infinity is avoided using esmall which is defined as follows:

prm esmall = If(Abs(e) < 1e-7, -0.00001, e).

'++++++++++++++++++++++++++++++++++++++++
'#define CREATE_

#ifdef CREATE_
iters 0
yobs_eqn aac.xy = 1;
min 12
max 16
del .001
Finger_et_al(12, 12)
Out_X_Ycalc(finger.xy)
#else
continue_after_convergence
xdd finger.xy

prm lr 13.84144`
prm ls 13.84144`
prm e1 = -0.5 (1/Tan(2 Th)) ((lr-ls)/(2 Rs))^2; : -0.00311
prm e2 = -0.5 (1/Tan(2 Th)) ((lr+ls)/(2 Rs))^2; : -0.00564
prm e = X Pi/180;
prm esmall = If(Abs(e) < 1e-7, -0.00001, e);

macro sqrt_(x) { Sqrt(Abs(x)) }

user_defined_convolution =
1000000
If (e < e2,
0,
sqrt_(e2 / esmall) - 1
);

min = e2 180/Pi;
max = 0;

x_calculation_step .005
#endif

lam la 1 lo 1.540596 lh 1e-5
Radius(200)

xo_Is
xo 15 I 100
'++++++++++++++++++++++++++++++++++++++++

djurdjijadzodan

Dear colleagues,

I am sorry for jumping in the conversation, but I have several additional basic questions regarding the axial divergence corrections that I couldn't have figured out myself by analyzing all the main TOPAS sources (I am sorry in advance if any of answers on my questions is provided in those sources in an indirect way, I prefer to still ask you for your opinions).

Question 1. Based on which criteria one decides if Simple_Axial_Model or Finger_et_al correction is "preferred" to be used?
In one of the Durham tutorials (https://topas.webspace.durham.ac.uk/pawley/) I could have seen the use of Simple_Axial_Model for XRD data collected on a conventional X-ray diffractometer. However, I have also found an example where Simple_Axial_Model was used for synchrotron data: (https://doi.org/10.4028/www.scientific.net/MSF.651.1, page 29).
In the publication https://doi.org/10.1107/S0021889894004218 it has been written that the Finger_et_al correction can be used for variety of geometries including both synchrotron instruments (of parallel geometry ) and conventional X-ray diffractometers (are diffractometers both with reflection and transmission geometries included here?). I feel that I am missing to understand something very fundamental here since I can't confidently distinguish when would be better to use the first and when the second correction?

Question 2. Regarding the "Finger_et_al (s2,h2)" correction:

a) Do s2 and h2 represent sample length and receiving slit length, respectively or half-sample length and half-receiving slit length, respectively? Based on the TOPAS manual I would say that "full" (and not half-) length values are to be used, but the illustration on page 1 of the publication https://doi.org/10.1107/S0021889894004218 has lead me to the doubt.

b) Do the two (sample length and receiving slit length) values always need to be refined to a same value, or it strictly depends from the diffractometer setup ? Usually I would see the correction in examples in the following form: Finger_et_al (12,12), or of course some other value could have been used instead of the value 12 (but the two values in the correction would be equal numbers). Additionally, in the NIST certificate for 1976a standard material (from 10th of April 2008), it was written that the two values of the "Finger_et_al" correction have been set to be "nearly equal" upon refinements (their data were collected on the Siemens D500 diffractometer with a monochromator and the authors provided the technical details about the setup, but they haven't explicitly stated the sample length and receiving slit values).
I assume that if one determines the sample length to be 12mm and the receiving slit length to be 16mm for the experimental setup at which they collect the data, those values are to be used as input parameters. However, I also assume that the "real" sample length value can in practice be different than initially "measured" and that after the refinement we could evaluate the output Finger_et_al (16,16) as a reasonable? But should one always tend to refine both values to the same number, or it is not necessary?

c) instead of receiving slit length value, could a receiving detector length value be used (if no receiving slit has been installed in the setup)?

Question 3. Regarding the "Simple_Axial_Model (c,v)" correction:

a) According to the topas manual the v value should represent the receiving slit length, can the detector length value be used instead? Additionally, I am also having the same concern as under 2a - what is the value that I should use as input, ""full" or half length value? More importantly, what is the value that one should expect after the refinement, can the value differ for more than 10percent of the initial input value?

I am asking this since I have tried using the value 14 in one of my examples (since the length of the detector was around 14mm and no slit that would limit the angular range of the Lynx-eye XE-T position sensitive detector was placed, but there were Soller slits in both incident, 4.1deg, and diffracted,4deg, beam paths) and after the refinement the value in the simple_axial_model was lower than 14 for more than 10percents of the initially used value (the output value was around 9). However, there were many other things that I could have done wrongly in that specific example (background, absorption corrections and so on), so I would like to know how one could estimate what simple axial value is reasonable to be get after refinement for each specific setup? Is it really the (half-) receiving or detector slit length value, or the inclusion of Soller slits (or some other diffractometer components) could have an additional impact on the value?

b) Does it make sense to try to refine the value in the Simple_Axial_Model only using the first few reflections (at low 2theta values) considering that effect of axial divergence in that region is expected to be large? Or it wouldn't be a good idea considering that the same effect is expected to be large at higher 2theta values as well (Cheary, Robert W., Alan A. Coelho, and James P. Cline. "Fundamental parameters line profile fitting in laboratory diffractometers." Journal of Research of the National Institute of Standards and Technology 109.1 (2004): 1.)?

I am sorry for the long post and thank you in advance on your time.

Kind regards,
Djurdjija

rowlesmr

Hi Djurdjija

A long question for the end of the year. I'll try and give at least some answers.

1:
The Finger et al model assumes that the incident beam is parallel in the axial direction, and that all axial divergence is due to the specimen and detector widths. This is often a valid assumption for synchrotron sources.
I've never (knowingly) used the simple model; I've always used the full axial model. Is there any reason why you don't want to use the full model?

2:
AFAIK the lengths specify the full length of the beam on the specimen and the full length of the recieving slit/detector, in the axial direction. Note that it is the length of the X-ray beam on the specimen, not the length of the specimen (assuming the specimen is completely bathed in the beam and centred in the instrument.).

The length on the specimen should be similar to the length of the filament in the X-ray tube. I normally set the values to standard/default values and don't touch them.

If you're using a pixel-type detector, then the pixels act as their own receiving slit, and the detector length = recieving slit length.

3:
If you're using a pixel-type detector, then the pixels act as their own receiving slit, and the detector length = recieving slit length.

I would leave the values at their actual measured values (not refined) and also use the full axial model to properly model axial divergence

Axial divergence should be refined (if you refine it) over the whole pattern, as the peaks start to broaden again as you get to higher angles.

Hope this is at least a start.

Matthew

djurdjijadzodan

Dear Matthew,

I am immeasurably grateful on your answers and sorry for the super long post. Since I am a novice in the field I am a bit insecure to making some of the initial TOPAS decisions.

1:
I wanted to test both simple and full axial models on the same datasets and compare the results (to the extend it is possible), because in the [1] I had read that FP had a limited success as a method of accurate lattice parameter determination (which is my main goal) - even though the authors then referred to a NIST 660a certificate where the FP approach had been, according to my understanding, successfully used (I must admit that I haven't fully understood why the authors wrote that). Maybe FP nowadays can be always used with no issues?

2:
How could we measure or calculate the length of the X-ray beam on the specimen if we for example know all the main parameters about the setup (primary and secundary radii, filament length, divergence angle, apertures of primary and secondary Soller slits, detector length...)?

Kind regards,
Djurdjija

[1] doi: 10.6028/jres.109.002 (page 24)

rowlesmr

I'm not really a measure-the-cell-edge-precisely person, but I do know that Jim Cline and friends at NIST use fundamnetal parameters (both TOPAS and in-house software) to certify their standards, so I'm guessing its OK?

You can measure the width of the beam at the specimen position using a fluorescent specimen; I think there are Gd-based materials that glow green when hit with X-rays. You can then use a camera or lay a ruler on the specimen and measure the width.

djurdjijadzodan

Dear Matthew,

Thank you so much once again. I will then proceed further with FP approach without unnecessary doubts.

Kind regards,
Djurdjija