Instrumental resolution file using TCHZ

andreasdrejer

Hi all

I have a powder pattern of a CeO2 standard and would like obtain the instrumental contribution due to peak broadening.
I looked through the TOPAS tutorials and did tutorial 21: http://community.dur.ac.uk/john.evans/topas_workshop/tutorial_ceo2_sizestrain.htm

Here a TCHZ peak type is used and all parameters are refined:
TCHZ_Peak_Type(pku,-0.00031`,pkv,-0.00014`,pkw, -0.00078`,pkx, 0.00080`,pky, 0.00339`,pkz, 0.00309`)

In the book "Rietveld Refinement - Practical Powder Diffraction Pattern Analysis using TOPAS" by R. Dinnebier, A. Lienewber and J. Evans, the parameters in TCHZ is defined as:
U, V, W and Z are the Gaussian part and X and Y are the Lorentzian part.
In addition U and X relates to strain, whereas Z and Y are related to domain size.

What then confuses me, is that it is stated in the book that Z is a combination between U and W and therefore Z should not be refined at the same time as U and W without additional constraints. But in the tutorial all of the parameters are refined at the same time. Also a standard is regarded as not contributing to size effects, so why is Z and Y refined? It is stated in the book that details about the constraint can be seen in the TOPAS manual, but here I find nothing that describes this.

1. My question is then: When I refine my CeO2 data to determine the instrumental peak broadening, should I then not refine on Z as the same time as U and W? Or should I introduce a constraint before?
I know the peak shapes differ from instrument to instrument, but could anyone maybe elaborate on what a nice-to-do and where should I be careful when refining the parameters in the TCHZ model?

2. In addition I am also very curious about how to determine the Gaussian (lg) and Lorentzian (lh) half widths in the emission profile for my instrument. As far as I understood these are independent on the profile parameters, but would they be refined before or after the profile parameters? Or would one in general just use some standard values? And lastly how to evaluate if their refined values makes sense?

Best regards
Andreas Drejer

rowlesmr

Andreas

2: If you're using a standard tube, then just use the emission profiles that come with topas.

If you're doing something funky, like a Johannson monochromator, then you'll need to measure it.

If you use a piece of a 100 silicon wafer, and measure the highest angle you're able to get to, then that profile is the emission profile: no size or microstrain effects, just the emission profile + slits and detector smearing.

They are not parameters you would refine during normal use. Refine them against a standard (eg 100 Si), if you must, and then fix them and forget about them. I don't know how you would go about finding out how realistic they are, apart from finding a sample that produces narrower peaks than it should; then you know you're wrong.

1: see S2.5.5 as well. If refining CeO2 as an instrument standard, then you should fix Z and Y to zero, as the size broadening is assumed to be zero. Having said that, its an instrument standard, so the exact values of the numbers don't matter. When you're refining peak shape on your unknowns, the CeO2 values will be fixed, and so don't count.

When refining your unknown, only allow U, Z, X, and Y to refine. They should always be larger than the values set be CeO2.

Matthew

andreasdrejer

Thanks for the answer Matthew.

I have both normal in-house data (Cu source, no monochromator with beta filter) and synchrotron data with a single wavelength. So far I have just used default settings from the TOPAS book. For in-house instrument I use:
lam
ymin_on_ymax 0.0001
la 2 lo 1.5405909 lh 1
la 1 lo 1.544399 lh 1

And for the synchrotron data I have used:
lam
ymin_on_ymax 0.0001
la 1.0 lo 0.20721 lh 0.1

I tried to refine on lh for my standard (on both models above), but it didn't change much, so I guess that the values I use are decent, and therefore keep them as they are.

For the TCHZ peak type, for my unknown would you then use the U, Z, X and Y from CeO2 + some parameter (forced to be positive) for each of the parameters?
And why only U, Z, X and Y (I guess that you mean that W and V will just have to be fixed values obtained from the CeO2 refinement).

I think that my general confusion regarding the profile parameters arises because I have seen many different uses of it and it makes me wonder if there just are several ways of using them, or if many people actually makes mistakes.

- Andreas

johnsoevans

Thanks for the original question. It made me rethink a couple of things from 15 years ago!

I agree with everything Matthew says. The philosophy in this tutorial is that you "don't care" what function is used for the instrumental profile. Anything that fits the sharp standard is fine. Here a TCHZ PV does a good enough job.
In other cases you can throw in any function you like to fit the sharp data. Not everybody will like this approach.

Once you have determined the instrumental function you keep it fixed with your real sample, then just convolute on additional terms to describe the sample.

With these sharp data I think you do get a slightly better fit with u refining, so it's left on in the tutorial. You'll probably find that the analysis of the broad data doesn't change if you use u=0 when determining the instrumental function.

You asked about the u, v, w, x, z relationship. If you just play with equation 2.66 in the book (with sin^2 th+cos^2 th = 1 or similar) I think you should come to:

new_u = old_u +old_z
new_w = old_w + old_z

As such the following two peak shapes (using the tutorial data) are equivalent:

TCHZ_Peak_Type(pku,-0.000312716637,pkv,-0.000135284736,pkw, -0.000778312576,pkz, 0.00080161585,pkx, 0.00333150637,pky, 0.00309665661)
TCHZ_Peak_Type(pku, 0.000477324636`,pkv,-0.000133003347`,pkw, 2.32467957e-05`,!pkz, 0.00,pkx, 0.00339197454`,pky, 0.00308823283`)

If I was a politician or a second-hand car salesman I might say that the tutorial doesn't tell you what parameters to refine, it just gives you some possible values for starting peak shape parameters -:). I agree though that it could be misleading. I've therefore updated the tutorial slightly.

The reason behind the vague tutorial wording is actually something silly/historic that I forgot to update. There was a period >~ 15 years ago when there were a couple of different macros floating around for TCHZ peak shapes. One was ordered alphabetically like gsas (u, v, w, x, y, z). The more sensible and current one puts the G/L terms together (u, v, w, z, x, y). During one of our early Rietveld schools there was a mixture of versions around and it was easiest to get students to refine all terms as their values weren't relevant to the problem. I should have spotted and updated this ages ago!

rowlesmr

Hi Andreas

I, too, agree with everything John said!

.

Your choices of emission profile aren't wrong. I don't normally use the TCHZ model, so I would use the CuKa5-Berger emission profile with FP. My synchrotron profiles look like yours, but the Australian Synchrotron PD beamline is a bit more Gauss than Lorentz.

As to why only U, Z, X, and Y. It comes from the functional dependence. The Gaussian FWHM ~ sqrt(U * tan^2(Th) + Z/cos^2(Th)), and we know that microstrain is ~tanTh and crystallite size is ~1/cosTh, so U and Z are accounted for. The same for X and Y in the lorentzian case. V and W have no such physical interpretation, so we wave them away as solely instrumental (IIRC I think V should be strictly negative, but that is going from a very old memory).

Yes. You'd use UVWXYZ fixed from CeO2, and then refine positive values for UZXY for your sample which add to the CeO2 values. You'd use these values to do any calculations wrt size/strain.

Hope this makes sense.

Matthew

andreasdrejer

Hi Matthew and John

Thanks for both of your replies and valuable inputs! I've tried out different strategies with the peak shape and it makes a lot more sense to me now.

- Andreas

djurdjijadzodan

Dear Andreas, Matthew and John,

I have had exactly the same dilemmas as Andreas regarding refining the parameters in the TCHZ peak type macro and thanks to your discussion I have narrowed down the number of my questions.

In the TOPAS software version that I use, the TCHZ peak type macro is being displayed in the following way: TCHZ_Peak_Type(pku, 0.00039,pkv, -0.00221,pkw, -0.00146,!pkz, 0.0000,pky, 0.00957,!pkxz, 0.0000) and the equations defining the macro (from the "topas.inc") are:

macro TCHZ_Peak_Type(u, v, w, z, x, y)
{
local #m_unique tch_p_l = x Tan(Th) + y / Cos(Th);
local #m_unique tch_p_g = Sqrt( Abs( u Tan(Th)2 + v Tan(Th) + w + z / Cos(Th)2) );
local #m_unique tch_p =
(
tch_p_g5 +
2.69269 tch_p_g4 tch_p_l +
2.42843 tch_p_g3 tch_p_l2 +
4.47163 tch_p_g2 tch_p_l3 +
0.07842 tch_p_g tch_p_l4 +
tch_p_l5
)0.2;
local #m_unique tch_q = tch_p_l / tch_p;
peak_type pv
pv_lor = 1.36603 tch_q - 0.47719 tch_q2 + 0.1116 tch_q3;
pv_fwhm = tch_p;
}
macro TCHZ_Peak_Type(u, uv, v, vv, w, wv, z, zv, x, xv, y, yv)
{
#m_argu u
#m_argu v
#m_argu w
#m_argu z
#m_argu x
#m_argu y
If_Prm_Eqn_Rpt(u, uv, min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4)
If_Prm_Eqn_Rpt(v, vv, min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4)
If_Prm_Eqn_Rpt(w, wv, min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4)
If_Prm_Eqn_Rpt(z, zv, min = Max(-1, Val-.1); max = Min(2, Val+.1); del 1.0e-4)
If_Prm_Eqn_Rpt(x, xv, min = Max(0.0001, Val-.1); max = Min(2, Val+.1); del 1.0e-4 )
If_Prm_Eqn_Rpt(y, yv, min = Max(0.0001, Val-.1); max = Min(2, Val+.1); del 1.0e-4 )
TCHZ_Peak_Type(CeV(u, uv), CeV(v, vv), CeV(w, wv), CeV(z, zv), CeV(x, xv), CeV(y, yv))

Question 1. In the TCHZ macro that is, upon request, being automatically displayed in the JEdit software, the pkx term seems to "contain" (beside the x parameter) also the z parameter. However, according to the .inc file, the parameters x and z seem to be defined separately (if I am not mistaken?). I am trying to understand if the two parameters (x and z) are "coupled" in the displayed "pkxz" term of the TCHZ macro in the TOPAS version that I use in order to be able to decide if I need to initially set (and fix) the pkxz (beside the pkz) parameter to zero when refining the TCHZ parameters for XRD pattern of a standard of interest?

Question 2. In the book "Rietveld Refinement - Practical Powder Diffraction Pattern Analysis using TOPAS" by R. Dinnebier, A. Lienewber and J. Evans it is written that the z parameter shouldn't be refined together with the u and w parameters without additional constrains. What are those additional constrains in practice? I also haven't managed to find the answer in any TOPAS source available online so far.

I was thinking to refine first the v and w parameters for XRD pattern of a standard of interest (to refine both v and w parameters at the same time for example), then to fix the retrieved v and w values (which could be considered as "solely instrumental") while refining one by one the u, y , x parameters and at the end perhaps also the z parameter (if z parameter ever really need to be refined)? Actually, in some cases the TOPAS software instructs me to refine certain parameters together (and give me no opportunity to refine the parameters one by one). I assume it is because the parameters are correlated so I don't find it as a problem, but I still wonder how to proceed further with the refinements after refining the (u), v and w parameters (is the order at which those parameters are being refined important for the TOPAS software)?

Question 3. Regarding the signs in front of the u,v,w,x,y and z parameters, If I have understood your discussion well, considering the way at which (equations) the terms in the TCHZ macro in the TOPAS software are defined, we don't need to "force" the v parameter to be negative and the u and w parameters to be positive at the end of refinements (as mentioned to be needed for the case of u, v and w parameters in the "original" Caglioti function, the page 208 in the following publication: 10.6028/jres.120.013), but to accept any signs in front of all the parameters (including also the x, y and z parameter) that the software will display after refinement cycles (provided that the calculated pattern will match to a "satisfying degree" to the observed pattern)?

Question 4. Is there an unique combination of parameters' values in the TCHZ macro that provide the best fit for each example, or there could be more than one combination of parameters that provide the "satisfying or best" fit?

I am sorry for the long post and thank you in advance if you will have time to try to help me.

Kind regards,
Djurdjija

johnsoevans

Hi Djurdjija:

The "pkxz" is just a typo in the jedit macros you have. My fault - sorry! The reason for the typo is described above! The parameter names don't actually affect the refinement so the typo doesn't matter, but it's logical to give names u, v, w, z, x, y (or just use @ symbols). The menus currently online should give you sensible parameter names by default.
There's a little bit on how z correlates with u and w in the YouTube video around 1h 11 minutes in. I would typically refine u, v, w, x and y as a starting point and keep z fixed at 0.0 as it correlates 100% with u and w (for why see above). Normally you can refine them all at the same time. If parameters are highly correlated (but not 100%) it doesn't really make sense to fix one then refine the other. If you're after quantitiative information from peak shapes you could follow process in the CeO2 tutorial. This uses an empirical TCHz peak shape for the instrumental contribution then convolutions to describe size/strain for the sample.
If I'm just using the TCHz peak shape as an empirical function I normally don't worry about signs provided the fit is good.
The u,v,w,x,y parameters are correlated so you can sometimes find that quite different combinations of values will give a reasonable fit to the data. In general though there is usually a single combination that will be best for a given data set. It is possible to get stuck in a local minimum so you could try a little simulated annealing on the values using "val_on_continue" language. The other trick is to reset values to ones that you think are sensible for your instrument and see if you converge back to the same minimum. For example, I know that if I get a negative value of u or w resetting them both to small positive values and re-refining will often improve the fit. I would typically reset parameters to e.g. (0.001, 0.001, 0.005, 0.0, 0.03, 0.001).

Hope that's some help.

djurdjijadzodan

Dear John,

Thanks so much for the extensive explanations, I now feel really comfortable to make the next refinement decisions.

Kind regards,
Djurdjija

djurdjijadzodan

Dear John,

I am sorry for additional questions.

I have just realized also that in the TCHZ macro the parameters x and y are listed as 5. and 6. parameters, respectively while in the TCHZ line automatically displayed in JEdit upon request the parameters x and y are listed as 6. and 5. parameters, respectively. I assume the right order is the one listed among the equations in the topas. inc meaning that the following should be correct:
TCHZ_Peak_Type(@ pku, some value,@ pkv, -another value,@ pkw, a third value, ! pkz, 0.0000, pkx, @ forth value, @ pky, fifth value)?
In the publication: Wu, E., E. Mac A. Gray, and E. H. Kisi. "Modelling dislocation-induced anisotropic line broadening in Rietveld refinements using a Voigt function. I. General principles." Journal of applied crystallography 31.3 (1998): 356-362 (page 359) it has been written that: the U, V and W parameters are instrumental coefficients. However, we already discussed (your book as the reference) that actually U parameter (together with X parameter) relate also to microstrain effects. I then concluded that it must be that in practice there is no any sample (standard) that really has 0 strain contribution to the peak profile and hence we decide to exclude the U parameter from the list of "purely" instrumental parameters in the TCHZ line, but to be on the safe side always consider only v and w parameters as the only instrumental parameters (which we determine using a standard and then afterwards fix those while refining XRD pattern of sample of interest). However in the NIST certificate of lanthanum hexaboride powder (660a) It has been stated that the powder displayed no strain broadening (the reference: Freiman, S. W., and N. M. Trahey. "NIST Certificate, SRM 660a Lanthanum Hexaboride Powder, Line Position and Line Shape Standard for Powder Diffraction." (2000)). Do we in that case consider U parameter also as solely instrumental parameter?

Kind regards,
Djurdjija

djurdjijadzodan

P.S. It seems that I have found myself answer on my second question in the publication: 10.6028/jres.120.013 where on the page 194 has been written that synchrotron radiation must be used to detect microstructural broadening in the 660a standard. I have also realized that even if there would be really 0 strain effect in any standard, we would probably still need to refine the U parameter for the XRD pattern of the sample of interest in order to include the strain effect of the sample (meaning that we still keep fixed only the v and w parameters determined from the standard) - as discussed above.

I am however still wondering if for XRD pattern of our sample of interest I could simply keep the v and w parameters fixed to previously determined values on a standard and all other parameters in the TCHZ line (except the z) let to be refined with the following code lines (that would take into account size and strain broadening of the sample):

CS_L (@, value)
CS_G (@, value)
Strain_L (@, value)
Strain_G (@, value)

(or
LVol_FWHM_CS_G_L(1, 4489.51014, 0.89, 5424.90744, !csgc, 10000, !cslc, 10000)
e0_from_Strain( 0.00004, !sgc, 0.0001, !slc, 0.0001)
Or the lastly two lines of code can be used (as you showed in one of your you tube video) only without the TCHZ line while using the Full Axial model instead?)

I wish to analyze our XRD data both using the empirical and fundamental parameters approach, hence I am trying to delimit the issues above.

rowlesmr

This may say things you already know, but I'm just doing it for completeness.

The TCHZ model is a parameterisation to create pseudo-voigt peaks from the Gaussian parameters UVWZ and Lorentzian parameters XY. AFAIK, the weights used to combine the Gaussian and Lorentzian componenets are completely emperical.

The Gaussian value is given as Sqrt( Abs( u Tan(Th)^2 + v Tan(Th) + w + z / Cos(Th)^2) ). This is an extension of the Cagliotti equation to explicitly put in a parameter with a 1/cos relationship to deal with crystallite size broadening. u is a microstrain parameter. Technically you can use the identity tan²(Th) + 1 = sec²(Th) to get at crystallite size, but it requires constraining parameters in an unstable fashion. In this implementation, v and w can be considered "instrumental" parameters, as they don't have any direct specimen-related analogue.

The Lorentzian value is given as x Tan(Th) + y / Cos(Th), where x is related to microstrain, and y to crystallite size. There are no solely "instrumental" parameters here.

If you are going fully empirical, then you would refine the model on your standard data, letting all the parameters (UVWZXY) refine to whatever they want to be. (I seem to recall that V must be negative, and the remainder must be positive, but that was some derivation I did back during my PhD.)

When refining your unknowns, you can let U, Z, X, and Y refine to give you flexibilty in the gaussian and Lorentzian components of crystallite size and microstrain, but the values must be constrained to always be equal to or larger than that set by your standard. If they want to be smaller, then you need a better standard (or a worse specimen).

If you want to combine TCHZ and the LVol_FWHM_CS_G_L/e0_from_Strain macros, then fix all UVWZXY values to those refined from the standard and then allow the LVol_FWHM_CS_G_L/e0_from_Strain values to be whatever they want to be.

HTH

Matthew

rowlesmr

Hi again Djurdjija

Answers for your specific questions 1 & 2:

1:
If in doubt, topas.inc is the final arbiter (or at least the actual macro that you use, which, in your case is in topas.inc)
If in doubt, have a look at topas.log after you've run a refinement; it will contain your input file converted to the final topas language and you can see how all of the macros have been expanded.

2:
Yes, Wu et al do explicitly say "U, V and W are instrumental coefficients." just under eqn 15. But this is tempered by their statement at eqn 17: "So SG and SL are the strain coefficients for the Gaussian and Lorentzian components, respectively". They then use (U+SG) as the coefficient for tan²(Th) in the Gausian equation and just SL as the coefficient for tanTh in the lorentzian equation. Here, Wu etal are explicitly differentiating between the instrumental and specimen compoents of line broadening; cf eqn 17, whereas in the TCHZ macro, there is no attempt to do so.*

* You can do it yourself, though:

'refined from a standard and then fixed for the unknown specimen. Assumes the instrument profile is pseudovoigt.
prm !u_inst  0.0078235917
prm !v_inst -0.0344718162
prm !w_inst  0.0100832168
prm !z_inst  0.0
prm !x_inst  0.0005273577
prm !y_inst  0.0011070738

'refined values of u, z, x, & y which are in addition to the instrument broadening from above
' You can see that z and x are essentially zero, so these components have no difference to instrumental broadening.
prm u_spec  0.0085672335`_0.000266914282 min 0
prm z_spec  4.19458693e-09`_9.19572085e-05_LIMIT_MIN_0 min 0
prm x_spec  5.2482109e-08`_0.00160301416_LIMIT_MIN_0 min 0
prm y_spec  0.109568168`_0.000613288454 min 0

'Now combine the two values to get the peak profile you want.
prm !u_total =u_inst + u_spec; :  0.0163908252`_0.000266914282
prm !v_total =v_inst; : -0.0344718162
prm !w_total =w_inst; :  0.0100832168
prm !z_total =z_inst + z_spec; :  4.19458693e-09`_9.19572085e-05
prm !x_total =x_inst + x_spec; :  0.000527410182`_0.00160301416
prm !y_total =y_inst + y_spec; :  0.110675242`_0.000613288454

'and then put the parameters into the macro call
TCHZ_Peak_Type(u_total, v_total, w_total, z_total, x_total, y_total)

djurdjijadzodan

Dear Matthew,

Thank you so much, everything makes a perfect sense to me now and I have no any additional doubt anymore.

Wish you a nice holiday.

Kind regards,
Djurdjija

townstone

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