Hi Dear Kai Fauth

Please see the attached file.

Springer Tracts in Modern Physics

Volume 141

Dear Kai,   Since the occupation probably of each site by any one of those chemical species is proportional with their stochiometric  numbers (there are no vacancies)     = x FA + (1-x) FB  similarly = xFA + (1-x) FB.  These expectation values are also reflect the fact that these sites are equally probable in the sense of mean for each species present in the compound. 

However, you can make this game little bit more interesting by introducing some correlation or weight factors!!!

But I should make the following remarked that this still doesn't  reflect the  genuine randomness since it ignores the statistical fluctuations. 

Well, yes, if I take the form factor to be the "stoichiometric average" at every site, then I get a regular lattice with average form factor. The structure factor then contains no information whatsoever on what is there (besides the lattice). If this approximation were close to the truth then there is little to learn from an experiment.

But I guess also that you are right, the statistical fluctuations should be what counts. How to assess their influence - I guess that's what I am asking for...

I cannot imagine that such problems have not been treated in the literature. I just don't know the terminology, context and keywords I'd best use for a search on this.

And again - I have a specific class of atomic configurations (substrate lattice, species A & B) in mind and am searching for the proper experiment to make statements on the statistics of the elemental distribution.

You may  handle this problem by using quasi-random generator while simulating the spectral line profiles during the summation operation associated with the individual contributions  coming from atomic scattering centers in each cells for a given Bragg's angle or  K- vector in reciprocal lattice space. 

What are you  going see in the simulated line profiles especially for high Braggs' angles?  The excessive  line broadening  similar to the  effects of temperature fluctuation on the line profiles. Line width will be proportional to the mean amplitude of the fluctuations.

Okay, thank you very much for the suggestion. Allow me some time to swallow that answer (have to translate into some kind of algorhithm... and only limited time density to do so :-)

hallo Kai if you say resonant then I would say don't rule out SEXAFS separating static disorder and dynamic disorder perhaps may be tricky (I don't know the details of your system).  But of course low temperature would help. In fact varying the temperature would also be interesting.   Surface system, metals - mobility?  But I would think that an average coordination number would be in and could be correlated at both edges? 

Hello David, actually yes, (S)EXAFS has come to my mind yesterday. We would have two different Lanthanide species (e.g. Er and DY) in a subsurface layer of a Si(111) (1x1) reconstruction. [M. Dähne has studied these {many} years ago by PES, though not with two different 4f elements at the same time]. Magnetism of these has never been looked at, afaik. In LEED (-IV) we get the same specimen quality when mixing elements compared to reconstructions with just one 4f element.

In LEED, though, the atomic form factors seem so similar that we haven't even tried analyzing potential differences between the cases. As for the (back-) scattering relevant for EXAFS, i don't know how much of a difference there will be between different lnthanide elements as neighbors. After all, those electrons which make the difference are pretty much hidden from the outside...  but that's just guessing on my side. I'll be happe to learn otherwise.

So, David, thanks for directing my thoughts that way. When is the next deadline for proposals at your place?

Hello Kai! Interesting question. The first idea for an experimental check would be a real-spce method like STM. However, in your case (similar elements, even sub-surface) this does not work.

An alternative (and fairly simple) experiment is XPD, x-ray photoelectron diffraction. At high energy, the core level electrons scatter predominantly at neighboring atoms with large scattering probability for small scattering angles ("forward focusing"). This allows you to map the neighboring atoms without coherence effects due to long-range order. This way and with the chemical sensitivity of XPS you can prove that your lattice is well ordered and that the environment looks the same for A- and B-atoms.

Beside there are of course interference effects showing up due to long range order. From a random lattice I would expect them to vanish completely. So, you could get both information in a single experiment. In order to visulaize I attached two papers from a monolayer of boron nitride on Ni(111). One paper (Auwärter) shows the "standard" XPD where you can see clear differences in the coherent parts of the pattern recorded from N and B core levels (due to different scattering factors). The second shows that you can do similar things with emission from the substrate (Muntwiler). The latter would be hard in your case because you need the pristine substrate as a reference and Si(111) 1x1 is not simple to prepare.

Concerning the structure factor, I am not an expert. I am not sure whether it would be possible to build a coherent sum over all scattering possibilities and then to factorize it into an atomic form factor and a structure factor. I guess (the so-called "half-educated" guess) that you would have to make an INcoherent sum over all lattice possibilities, i.e. to build all possible environments, calculate for the interference pattern for each of them and sum the intensities. Then increase the size of these domains (and thereby their number) and hope that everything converges :-)

Hallo Kai your right they are quite close in Z so this presumably also means a problem for XRD?  I cant say what the difference in the backscattering is without calculating, but I think your right small (perhaps there is a scattering resonance which could help to differentiate don't know).  However the chemistry I think of the two are quite different (however you say same reconstructions? and "those electrons that make all the difference are hidden from the outside" :-? ).  So looking at the local structure you might see differing distances (its complementary anyway).  What differences do you see in the I-V? 

You have missed the last one the next one is in June.  The expt is    Surface Science    so I am not certain of the set up and this would require detailed checking with beamline scientists etc. 

@Matthias:

Hi you!! My hope would actually be that usable formulae might have been worked out by theorists some 20, 30, 40 years ago, maybe in the context of disorder - order - transitions.

With XPD i see the problem that those changes which should matter are taking place within an atomic layer and I can't imagine up to what polar angle (away from grazing emission) you retain a sizeable sensitivity to that. But you're certainly right in that having the 'same'  XPD pattern would give credence to the assumption of equivalent lattice positions of the two species.

@David:

Actually, the Lanthanide elements are mostly very similar to one another in terms of chemical behaviour. That is due to the fact that they mainly differ in the number of (f) electrons in the open 4f shell, while the 'outer' electron configuration is (6s2 5d1) [in atomic nomenclature] is typically very similar across the series. Count on variations close to half filling (e.g. Eu), however.

The question is then, whether the change in local charge density (given by the variation in f electron number) provides enough of a contrast in backscattering at >= 50 eV above the resonance to give sufficient information on local order or preferential neighbor site occupation. Scattering resonances could help as you say, but I have no clue about those in the Lanthanides at this time.

@David on LEED-IV:

Actually the LEED-IV data of pure Er and Dy induced (1x1) reconstructions are so similar that we have not tried to analyze the differences. The comparison between experiment and simulation is 'semi-quantitative' anyhow in the sense that the relevant reliability (Pendry) factor is constructed such as to emphasize locations of extrema with respect to actual intensities. And contrary to the X-ray case, its necessarily a muliple scattering approach you have to implement, therefore there is no way to reverse engineer the differences between calculated and computed LEED-IV trajectories.

In the mixed (e.g. ~50% Er, ~50% Dy) case, upon hitting the coverage required for (1x1) reconstruction, LEED patterns and IV-curves again look very similar to the pure ones. Only by spectroscopy (e.g. AES) can we actually tell that both elements are present in the specimens. (For the time being, this is about all information we have but for the fact that from analyzing LEED-IV with respect to the RE "adsorption" site we obtain agreement with the existing structure models.)

I am getting the impression that diffuse scattering might hold a key to my question. Any good hints at what to read as an efficient way to get into this field would be appreciated.

Edit: have now ordered the book:

W. Schweika: "Disordered alloys: diffuse scattering and Monte Carlo simulations"

http://www.fisica.ufmg.br/~edmar/fisica/papers/PhD_ATA.pdf

@ Matthias on substrate XPD

One could maybe use H-terminated Si (111) as a reference. That shouldn't be too hard to do. Nevertheless, one would miss out the (different) structure of the top two Si atomic layers which form in the Lanthanide induced (1x1) reconstructions.

@ Tarik

I was unaware of thermal motion inducing line broadening. My state of knowledge is that it only affects intensities (Debye-Waller factor). So am I mistaken on that?!

Dear Kai,  In classical treatments, thermal fluctuations are assumed to be noncumulative. As a consequence, diffraction line widths remain sharp with increasing temperature, and only their intensity decreases as given by a Debye-Waller formula. On the other hand, layer thickness fluctuations are intrinsically cumulative which

affects the diffraction line widths and broadens them. This implies that a Debye-Waller formulation can only be applied to layer thickness fluctuation in very special

cases. It was  showed sometime ago [7] that very slight amounts (of the order of 7%) of layer thickness fluctuations can completely wash out superlattice peaks in crystalline(Pb)-amorphous(Ge) superlattices.

A third important consequence of layer thickness fluctuation induced broadening is the variation with lattice mismatch [8]. This is especially important for the field of metallic superlattices where in many cases superlattices are formed from lattice mismatched constituents.

Thanks, Tarik, for your explanations. Would you mind mentioning [7] and [8]?

Deatr Kai, In  our department  Prof. Dr. Amdulla Mekhrabov and his group (ex-member of the University of Moskov) have  doing extensive studies  on the diffusive scattering experimentally as well as theoretically last thirty years  ,and your  collaboration with him  might be fruitful for both parties.

[7] W. Sevenhans, M. Gijs, Y. Bruynseraede, H. Homma and 1.K. Schuller, Phys. Rev. B. 34 (1986) 5955.

[8] J.-R. Locquet, D. Neerinck, L. Stockman, Y. Bruynseraede and I.K. Schuller, Phys. Rev. B. 39 (1989) 13338.

Update: today I received the book I ordered via inter library loan.

W. Schweika: "Disordered alloys: diffuse scattering and Monte Carlo simulations"

Seems to be a good read! However, for surfaces and interfaces it points to a specific review, also treating diffuse scattering:

Dietrich & Haase, Phys. Rep. 260, 1 (1995)

Guess it might take some time to make my walk through this stuff.

Hi Dear Kai Fauth

Please see the attached file.

Springer Tracts in Modern Physics

Volume 141