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Hello Oliver,
when you fit the values from ICA Tab 6.1.1.1 with gnuplot, the values
of C and N become much more comparable. c(C) = 0.017 and especially
c(N) = 0.025 > 0!!!
for C:
Final set of parameters Asymptotic Standard Error
======================= ==========================
a1 = 0.604126 +/- 0.02326 (3.85%)
a2 = 2.63343 +/- 0.03321 (1.261%)
a3 = 1.52123 +/- 0.03528 (2.319%)
a4 = 1.2211 +/- 0.02225 (1.822%)
b1 = 0.185807 +/- 0.00629 (3.385%)
b2 = 14.6332 +/- 0.1355 (0.9263%)
b3 = 41.6948 +/- 0.5345 (1.282%)
b4 = 0.717984 +/- 0.01251 (1.743%)
c = 0.0171359 +/- 0.002045 (11.93%)
for N:
Final set of parameters Asymptotic Standard Error
======================= ==========================
a1 = 0.723788 +/- 0.04334 (5.988%)
a2 = 3.24589 +/- 0.04074 (1.255%)
a3 = 1.90049 +/- 0.04422 (2.327%)
a4 = 1.10071 +/- 0.0413 (3.752%)
b1 = 0.157345 +/- 0.007552 (4.8%)
b2 = 10.106 +/- 0.1041 (1.03%)
b3 = 30.0211 +/- 0.3946 (1.314%)
b4 = 0.567116 +/- 0.01914 (3.376%)
c = 0.0252303 +/- 0.003284 (13.01%)
In 1967, Mann only calculated to sin \theta/lambda = 0, ... 1.5, and
their tabulated values do indeed fit decently within that range, but
not out to 6A.
I thought this was notworthy, and I am curious which values for these
constants refinement programs use nowadays. Maybe George, Garib,
Pavel, and Gerard may want to comment?
Cheers,
Tim
On 09/18/2012 10:11 AM, Oliver Einsle wrote:
> Hi there,
>
> I was just pointed to this thread and should comment on the
> discussion, as actually made the plots for this paper. James has
> clarified the issue much better than I could have, and indeed the
> calculations will fail for larger Bragg angles if you do not assume
> a reasonable B-factor (I used B=10 for the plots).
>
> Doug Rees has pointed out at the time that for large theta the
> c-term of the Cromer/Mann approximation becomes dominant, and this
> is where chaos comes in, as the Cromer/Mann parameters are only
> derived from a fit to the actual HF-calculation. They are numbers
> without physical meaning, which becomes particularly obvious if you
> compare the parameters for C and N:
>
>
> C: 2.3100 20.8439 1.0200 10.2075 1.5886 0.5687 0.8650
> 51.6512 0.2156 N: 12.2126 0.0057 3.1322 9.8933 2.0125
> 28.9975 1.1663 0.5826 -11.5290
>
> The scattering factors for these are reasonably similar, but the
> c-values are entirely different. The B-factor dampens this out and
> this is an essential point.
>
>
>
> For clarity: I made the plots using Waterloo Maple with the
> following code:
>
> restart; SF :=Matrix(17,9,readdata("scatter.dat",float,9));
>
> biso := 10; e := 1; AFF :=
> (e)->(SF[e,1]*exp(-SF[e,2]*s^2)+SF[e,3]*exp(-SF[e,4]*s^2)
> +SF[e,5]*exp(-SF[e,6]*s^2)+SF[e,7]*exp(-SF[e,8]*s^2)
> +SF[e,9])*exp(-biso*s^2/4);
>
> H := AFF(1); C := AFF(2); N := AFF(3); Ox :=
> AFF(4); S := AFF(5); Fe := AFF(6); Fe2 := AFF(7); Fe3 :=
> AFF(8); Cu := AFF(9); Cu1 := AFF(10); Cu2 := AFF(11); Mo
> := AFF(12); Mo4 := AFF(13); Mo5 := AFF(14); Mo6 := AFF(15);
>
> // Plot scattering factors
>
> plot([C,N,Fe,S], s=0..1);
>
>
> // Figure 1:
>
> rho0 := (r) -> Int((4*Pi*s^2)*Fe2*sin(2*Pi*s*r)/(2*Pi*s*r),
> s=0..1/dmax); dmax := 1.0; plot (rho0, -5..5);
>
>
> // Figure 1 (inset): Electron Density Profile
>
> rho := (r,f)
> ->(Int((4*Pi*s^2)*f*sin(2*Pi*s*r)/(2*Pi*s*r),s=0..1/dmax));
> cofactor:= 9*rho(3.3,S) + 6*rho(2.0,Fe2) + 1*rho(3.49,Mo6) +
> 1*rho(3.51,Fe3); plot(cofactor, dmax=0.5..3.5);
>
>
> The file scatter.dat is simply a collection of some form factors,
> courtesy of atomsf.lib (see attachment).
>
>
>
> Cheers,
>
> Oliver.
>
>
>
> Am 9/17/12 11:24 AM schrieb "Tim Gruene" unter
> <t...@shelx.uni-ac.gwdg.de>:
>
> Dear James et al.,
>
> so to summarise, the answer to Niu's question is that he must add
> a factor of e^(-Bs^2) to the formula of Cromer/Mann and then adjust
> the value of B until it matches the inset. Given that you claim
> rho=0.025e/A^3 (I assume for 1/dmax approx. 0) for B=12 and the
> inset shows a value of about 0.6, a somewhat higher B-value should
> work.
>
> Cheers, Tim
>
> On 09/17/2012 08:32 AM, James Holton wrote:
>>>> Yes, the constant term in the "5-Gaussian" structure factor
>>>> tables does become annoying when you try to plot electron
>>>> density in real space, but only if you try to make the B
>>>> factor zero. If the B factors are ~12 (like they are in
>>>> 1m1n), then the electron density 2.0 A from an Fe atom is not
>>>> -0.2 e-/A^3, it is 0.025 e-/A^3. This is only 1% of the
>>>> electron density at the center of a nitrogen atom with the
>>>> same B factor.
>>>>
>>>> But if you do set the B factor to zero, then the electron
>>>> density at the center of any atom (using the 5-Gaussian
>>>> model) is infinity. To put it in gnuplot-ish, the structure
>>>> factor of Fe (in reciprocal space) can be plotted with this
>>>> function:
>>>>
>>>> Fe_sf(s)=Fe_a1*exp(-Fe_b1*s*s)+Fe_a2*exp(-Fe_b2*s*s)+Fe_a3*exp(-Fe_b3*s*s
>>>>
>>>>
)+Fe_a4*exp(-Fe_b4*s*s)+Fe_c
>>>>
>>>>
>>>> where: Fe_c = 1.036900; Fe_a1 = 11.769500; Fe_a2 = 7.357300;
>>>> Fe_a3 = 3.522200; Fe_a4 = 2.304500; Fe_b1 = 4.761100; Fe_b2 =
>>>> 0.307200; Fe_b3 = 15.353500; Fe_b4 = 76.880501; and "s" is
>>>> sin(theta)/lambda
>>>>
>>>> applying a B factor is then just multiplication by
>>>> exp(-B*s*s)
>>>>
>>>>
>>>> Since the terms are all Gaussians, the inverse Fourier
>>>> transform can actually be done analytically, giving the
>>>> real-space version, or the expression for electron density vs
>>>> distance from the nucleus (r):
>>>>
>>>> Fe_ff(r,B) = \
>>>> +Fe_a1*(4*pi/(Fe_b1+B))**1.5*safexp(-4*pi**2/(Fe_b1+B)*r*r)
>>>> \ +Fe_a2*(4*pi/(Fe_b2+B))**1.5*safexp(-4*pi**2/(Fe_b2+B)*r*r)
>>>> \ +Fe_a3*(4*pi/(Fe_b3+B))**1.5*safexp(-4*pi**2/(Fe_b3+B)*r*r)
>>>> \ +Fe_a4*(4*pi/(Fe_b4+B))**1.5*safexp(-4*pi**2/(Fe_b4+B)*r*r)
>>>> \ +Fe_c *(4*pi/(B))**1.5*safexp(-4*pi**2/(B)*r*r);
>>>>
>>>> Where here applying a B factor requires folding it into each
>>>> Gaussian term. Notice how the Fe_c term blows up as B->0?
>>>> This is where most of the series-termination effects come
>>>> from. If you want the above equations for other atoms, you
>>>> can get them from here:
>>>> http://bl831.als.lbl.gov/~jamesh/pickup/all_atomsf.gnuplot
>>>> http://bl831.als.lbl.gov/~jamesh/pickup/all_atomff.gnuplot
>>>>
>>>> This "infinitely sharp spike problem" seems to have led some
>>>> people to conclude that a zero B factor is non-physical, but
>>>> nothing could be further from the truth! The scattering from
>>>> mono-atomic gasses is an excellent example of how one can
>>>> observe the B=0 structure factor. In fact, gas scattering
>>>> is how the quantum mechanical self-consistent field
>>>> calculations of electron clouds around atoms was
>>>> experimentally verified. Does this mean that there really is
>>>> an infinitely sharp "spike" in the middle of every atom? Of
>>>> course not. But there is a "very" sharp spike.
>>>>
>>>> So, the problem of "infinite density" at the nucleus is
>>>> really just an artifact of the 5-Gaussian formalism.
>>>> Strictly speaking, the "5-Gaussian" structure factor
>>>> representation you find in ${CLIBD}/atomsf.lib (or Table
>>>> 6.1.1.4 in the International Tables volume C) is nothing more
>>>> than a curve fit to the "true" values listed in ITC volume C
>>>> tables 6.1.1.1 (neutral atoms) and 6.1.1.3 (ions). These
>>>> latter tables are the Fourier transform of the "true"
>>>> electron density distribution around a particular atom/ion
>>>> obtained from quantum mechanical self-consistent field
>>>> calculations (like those of Cromer, Mann and many others).
>>>>
>>>> The important thing to realize is that the fit was done in
>>>> _reciprocal_ space, and if you look carefully at tables
>>>> 6.1.1.1 and 6.1.1.3, you can see that even at REALLY high
>>>> angle (sin(theta)/lambda = 6, or 0.083 A resolution) there is
>>>> still significant elastic scattering from the heavier atoms.
>>>> The purpose of the "constant term" in the 5-Gaussian
>>>> representation is to try and capture this high-angle "tail",
>>>> and for the really heavy atoms this can be more than 5
>>>> electron equivalents. In real space, this is equivalent to
>>>> saying that about 5 electrons are located within at least
>>>> ~0.03 A of the nucleus. That's a very short distance, but it
>>>> is also not zero. This is because the first few shells of
>>>> electrons around things like a Uranium nucleus actually are
>>>> very small and dense. How, then, can we have any hope of
>>>> modelling heavy atoms properly without using a map grid
>>>> sampling of 0.01A ? Easy! The B factors are never zero.
>>>>
>>>> Even for a truly infinitely sharp peak (aka a single
>>>> electron), it doesn't take much of a B factor to spread it
>>>> out to a reasonable size. For example, applying a B factor of
>>>> 9 to a point charge will give it a full-width-half max (FWHM)
>>>> of 0.8 A, the same as the "diameter" of a carbon atom. A
>>>> carbon atom with B=12 has FWHM = 1.1 A, the same as a "point"
>>>> charge with B=16. Carbon at B=80 and a point with B=93 both
>>>> have FWHM = 2.6 A. As the B factor becomes larger and
>>>> larger, it tends to dominate the atomic shape (looks like a
>>>> single Gaussian). This is why it is so hard to assign atom
>>>> types from density alone. In fact, with B=80, a Uranium atom
>>>> at 1/100th occupancy is essentially indistinguishable from a
>>>> hydrogen atom. That is, even a modest B factor pretty much
>>>> "washes out" any sharp features the atoms might have.
>>>> Sometimes I wonder why we bother with "form factors" at all,
>>>> since at modest resolutions all we really need is Z (the
>>>> atomic number) and the B factor. But, then again, I suppose
>>>> it doesn't hurt either.
>>>>
>>>>
>>>> So, what does this have to do with series termination?
>>>> Series termination arises in the inverse Fourier transform
>>>> (making a map from structure factors). Technically, the
>>>> "tails" of a Gaussian never reach zero, so any sort of
>>>> "resolution cutoff" always introduces some error into the
>>>> electron density calculation. That is, if you create an
>>>> arbitrary electron-density map, convert it into structure
>>>> factors and then "fft" it back, you do _not_ get the same map
>>>> that you started with! How much do they differ? Depends on
>>>> the RMS value of the high-angle structure factors that have
>>>> been cut off (Parseval's theorem). The "infinitely sharp
>>>> spike" problem exacerbates this, because the B=0 structure
>>>> factors do not tend toward zero as fast as a Gaussian with
>>>> the "atomic width" would.
>>>>
>>>> So, for a given resolution, when does the B factor get "too
>>>> sharp"? Well, for "protein" atoms, the following B factors
>>>> will introduce an rms error in the electron density map equal
>>>> to about 5% of the peak height of the atoms when the data are
>>>> cut to the following resolution: d B 1.0 <5 1.5 8 2.0 27
>>>> 2.5 45 3.0 65 3.5 86 4.0 >99
>>>>
>>>> smaller B factors than this will introduce more than 5% error
>>>> at each of these resolutions. Now, of course, one is often
>>>> not nearly as concerned with the average error in the map as
>>>> you are with the error at a particular point of interest, but
>>>> the above numbers can serve as a rough guide. If you want to
>>>> see the series-termination error at a particular point in the
>>>> map, you will have to calculate the "true" map of your model
>>>> (using a program like SFALL), and then run the map back and
>>>> forth through the Fourier transform and resolution cutoff
>>>> (such as with SFALL and FFT). You can then use MAPMAN or
>>>> Chimera to probe the electron density at the point of
>>>> interest.
>>>>
>>>> But, to answer the OP's question, I would not recommend
>>>> trying to do fancy map interpretation to identify a mystery
>>>> atom. Instead, just refine the occupancy of the mystery atom
>>>> and see where that goes. Perhaps jiggling the rest of the
>>>> molecule with "kick maps" to see how stable the occupancy is.
>>>> Since refinement only does forward-FFTs, it is formally
>>>> insensitive to series termination errors. It is only map
>>>> calculation where series termination can become a problem.
>>>>
>>>> Thanks to Garib for clearing up that last point for me!
>>>>
>>>> -James Holton MAD Scientist
>>>>
>>>>
>>>> On 9/15/2012 3:12 AM, Tim Gruene wrote: Dear Ian,
>>>>
>>>> provided that f(s) is given by the formula in the Cromer/Mann
>>>> article, which I believe we have agreed on, the inset of
>>>> Fig.1 of the Science article we are talking about is claimed
>>>> to be the graph of the function g, which I added as pdf to
>>>> this email for better readability.
>>>>
>>>> Irrespective of what has been plotted in any other article
>>>> meantioned throughout this thread, this claim is incorrect,
>>>> given a_i, b_i, c > 0.
>>>>
>>>> I am sure you can figure this out yourself. My argument was
>>>> not involving mathematical programs but only one-dimensional
>>>> calculus.
>>>>
>>>> Cheers, Tim
>>>>
>>>> On 09/14/2012 04:46 PM, Ian Tickle wrote:
>>>>>>> On 14 September 2012 15:15, Tim Gruene
>>>>>>> <t...@shelx.uni-ac.gwdg.de> wrote:
>>>>>>>> -----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1
>>>>>>>>
>>>>>>>> Hello Ian,
>>>>>>>>
>>>>>>>> your article describes f(s) as sum of four Gaussians,
>>>>>>>> which is not the same f(s) from Cromer's and Mann's
>>>>>>>> paper and the one used both by Niu and me. Here, f(s)
>>>>>>>> contains a constant, as I pointed out to in my
>>>>>>>> response, which makes the integral oscillate between
>>>>>>>> plus and minus infinity as the upper integral border
>>>>>>>> (called 1/dmax in the article Niu refers to) goes to
>>>>>>>> infinity).
>>>>>>>>
>>>>>>>> Maybe you can shed some light on why your article
>>>>>>>> uses a different f(s) than Cromer/Mann. This
>>>>>>>> explanation might be the answer to Nius question, I
>>>>>>>> reckon, and feed my curiosity, too.
>>>>>>> Tim & Niu, oops yes a small slip in the paper there, it
>>>>>>> should have read "4 Gaussians + constant term": this is
>>>>>>> clear from the ITC reference given and the
>>>>>>> $CLIBD/atomsf.lib table referred to. In practice it's
>>>>>>> actually rendered as a sum of 5 Gaussians after you
>>>>>>> multiply the f(s) and atomic Biso factor terms, so
>>>>>>> unless Biso = 0 (very unphysical!) there is actually no
>>>>>>> constant term. My integral for rho(r) certainly
>>>>>>> doesn't oscillate between plus and minus infinity as
>>>>>>> d_min -> zero. If yours does then I suspect that
>>>>>>> either the Biso term was forgotten or if not then a bug
>>>>>>> in the integration routine (e.g. can it handle properly
>>>>>>> the point at r = 0 where the standard formula for the
>>>>>>> density gives 0/0?). I used QUADPACK
>>>>>>> (http://people.sc.fsu.edu/~jburkardt/f_src/quadpack/quadpack.html)
>>>>>>>
>>>>>>>
which seems pretty good at taking care of such singularities
>>>>>>> (assuming of course that the integral does actually
>>>>>>> converge).
>>>>>>>
>>>>>>> Cheers
>>>>>>>
>>>>>>> -- Ian
>>>>>>>
>>>> -- - -- Dr Tim Gruene Institut fuer anorganische Chemie
>>>> Tammannstr. 4 D-37077 Goettingen
>>>>
>>>> GPG Key ID = A46BEE1A
>>>>
>>>>
>
- --
- --
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen
GPG Key ID = A46BEE1A
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