Integral breadth & the double Voigt method

rowlesmr

Hi all

I'm going through some things for a workshop, and I'm trying to get my head around integral breadth.

I had understood that the integral breadth of a peak is the width of a rectangle with the same height and area of the peak.

Reading through Balzar's chapter in Defect and Microstructure Analysis by Diffraction, b is defined as "integral breadth in units of Å[sup]-1[/sup]", where b is given as b(2q)cosq[sub]0[/sub]/l.

I'm happy with q[sub]0[/sub] being the position of the peak we're looking at, but what is b(2q)? Is it the FWHM?

rowlesmr

I think I've figured it out now:

b = 1/cs

where

FWHM = l/(cs cosq)

b(2q) is the FWHM in radians.

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Which leads me to another question about the accuracy of the functions in Topas.

Using the function

Lvol = 1 / IB_from_CS(csG, csL);

and csG = 120 nm, csL=150 nm, Topas gives me an Lvol of 60.68347 nm.

If I do a manual calculation (as outlined in Balzar's book chapter and Balzar & Popovic), with csG = 120 nm, csL=150 nm, you can calculated that:

bG=1/120, bL=1/150, k=bL/(sqrt(Pi) bG)=0.45135,

Which gives b=bG*exp(-k^2)/erfc(k)=0.01299,

For Lvol = 1/b= 76.98102 nm.

What is the source of the discrepancy?

As far as I can tell, the calculation path is the same, except for the "Voigt_Integral_Breadth_GL" function in Topas, which I don't know how it calculate b.

de

Dear Matthew

Lets start by calculating the IB of CSG and CSL.
IB_G(rad)=1/(CS_G*sqrt(ln(2)/pi)=0.0887056 (If you look in the manual u will see it in deg. Chap 6.3.2)

IB_L(rad)=1/(CS_L*(2/pi))=0.01047198

k=IB_L/(sqrt(Pi)*IB_G)=0.66604

IB_V=IB_G*exp(-(k^2)/erfc(k)=0.01644098

LVolIB=1/IB_V=60.8236243

The minor discrepancy in the LVolIB 60.68 vs 60.82 is probably hidden somewhere in the Voigt_Integral_Breadth_GL.

Just a sidenote LVolIB is the volume weighted apparent crystallite size.
The surface or area weighted apparent crystallite size is LsurfIB=CS_L/pi or 1/(2*IB_L).
In order to relate the LVolIB value to a volume weighted sphere diameter you have to multiply LVolIB*(4/3)
For the surface or area weighted sphere diameter you have to multiply LsurfIB by (3/2).

Regards
Dominique

rowlesmr

Sorry about these posts. The server was timing out and posting edits as new posts.

rowlesmr

Hi Dominique

That shows that Topas is self-consistent: Tech ref correctly describes what TOPAS is doing. It doesn't explain the existence of the constant multipliers in the formulae for IB_G and IB_L.

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Ah.

In Balzar's size/strain round robin paper [1], he cites Langford [2], who says that the ratio of the integral breadth to the FHWM for Gaussian and Lorentzian peaks is (1/2)*sqrt(Pi/ln2) and Pi/2, respectively.

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Which then leads me to ask again: What is b(2q) if it isn't the FWHM?

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[1] Balzar, D., N. Audebrand, M. R. Daymond, A. Fitch, A. Hewat, J. I. Langford, A. Le Bail et al. 2004. "Size–Strain Line-Broadening Analysis of the Ceria Round-Robin Sample." Journal of Applied Crystallography 37 (6): 911-924. doi: 10.1107/s0021889804022551.
[2] Langford, J. I. 1978. "A Rapid Method for Analysing the Breadths of Diffraction and Spectral Lines Using the Voigt Function." Journal of Applied Crystallography 11 (1): 10-14. doi: 10.1107/s0021889878012601.
[3] Balzar, D., and H. Ledbetter. 1993. "Voigt-Function Modeling in Fourier Analysis of Size- and Strain-Broadened X-Ray Diffraction Peaks." Journal of Applied Crystallography 26 (1): 97-103. doi: 10.1107/S0021889892008987.

de

Can you give direct reference to the b(2q) ? Was not able to find it.

However what you seem to refer to is the conversion from s=1/d=d* units to degree 2Theta.
E.g. LVolIB=1/IB_V; IB_V in rad s units .
or LVolIB=lambda/IB_V*cos(theta); IB_V in rad °2Theta.

rowlesmr

Take Balzar, D. & Popović, S. (1996). J. Appl. Crystallogr. 29, 16-23. for example

b is defined (p. 17, line 16) as "b(2q) cos q/l.

b(2q) is not explicitly defined, but:

Eqn (1) gives the Scherrer eqn in the form: D = l/[b[sub]S[/sub](2q) cos q], where D is the domain size in the direction of the diffraction vector, and the subscript "S" denotes broadening due to size.

Combining these two expressions gives D=1/b[sub]S[/sub].

This infers that b(2q) is the FWHM in radians. But as has been shown above, it isn't.

So, what is b(2q)?

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Langford (Langford, J. I. (1978). J. Appl. Crystallogr. 11, 10-14.) explicitly talks about integral breadth (b) and FWHM (2w; he calls it the "half-width"). He also introduces the concept of a "form factor": 2w/b, which is the constants g1 and l1 in the TOPAS tech ref.

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The best answer I can come up with is that b(2q) is the form-factor-corrected width of the peak (in radians).

ie

G[sub]S[/sub]=l/[D cosq] --> b[sub]S[/sub](2q) = (l/[D cosq]) * (1/f)

G[sub]D[/sub]=e tanq --> b[sub]D[/sub](2q) = e tanq * (1/f)

where G is the FWHM, f is the form factor correction for either Gaussian or Lorentzian, subscripts S/D are size/distortion(strain).

This is at least consistent with the TOPAS Tech Ref.

de

beta is commonly the integral breadth (line 15).
In line 16: beta(s)=beta(2theta)*cos(theta)/lambda is just the conversion from (2theta) to (s=1/d=d*) units.
Equation 1 gives
LvolIB=lamda/(beta(2theta)*cos(theta)) for the integral breadth in (2theta) and
LvolIB=1/beta(s) for the integral breadth in (s).

so "b(2q)" is from my understanding the integral breadth in rad 2theta.

Not too sure where the confusion about b(2q) comes from. Probably there is a misunderstanding concerning the "Scherrer equation". The "Scherrer equation" is not exclusive to FWHM. In general for a measure of breadth b (rad,2theta) the apparent crystallite size e=lambda/(b*cos(theta)) or in (s) again e=1/b (b(rad,s)). b can be FWHM, IB, the slope of the variance...
All these measures will provide different so called apparent crystallite sizes for instance LVolIB, LVolFWHM... It has been discussed in great detail in the classic paper of Langford and Wilson (JAC 11, 102-113, 1978) how these apparent crystallite sizes are related to physical size parameters of different objects.
Concerning the IB and slope of the variance there is a direct physical relation (see Langford and Wilson). The apparent crystallite size from the FWMH (2w) has no direct physical significance and has to be interpreted from the physical line profile. Therefore this technique is to be discouraged.

rowlesmr

Ta a lot.

I sort of figured it out in my previous post, but didn't know it: the form-factor-corrected peak width _is_ the integral breadth in radians (or degrees, if you're a masochist).

The cos(q)/l comes from integrating the bragg equation (1/d = 2 sin(q)/l) to get an area.

Int(1/d dq) = -2 (cosq)/l.

So, the conversion of an integral breadth in radians to one in Å[sup]-1[/sup] involves multiplying b(2q) by (cosq)/l. The minus sign is lost because the area must be positive, and the factor of 2 goes away because the integral breadth in radians is in 2q.

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My confusion arose from it not actually being defined anywhere in several papers that took great pains to define everything.

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Ta muchly

de

Actually I think it is simply:

s=2*sin(theta)/lambda; theta=2theta/2=x/2

d/dx(2*sin(x/2)/lambda)=cos(x/2)/lambda

Anyway greets
Dominique

rowlesmr

That makes sense too...