Dear Dirk,
The factor of 2 comes from the fact that the diameter of a sphere is
twice its radius. The radius of the limiting sphere for data to a certain
resolution in reciprocal space is d_star_max. If you sample the electron
density at points distant by delta from each other, you periodise the
transform of the continuous density at that resolution by a reciprocal
lattice of size 1/delta. If you want to avoid aliasing, i.e. corruption of
one copy of your data in its sphere of radius d_star_max by the data in a
translate of that sphere by 1/delta, you must ensure that 1/delta is larger
than 2*d_star_max (the diameter of the limiting sphere. In other words,
delta must be less than (1/2)*(1/d_star_max), which is your Shannon/Nyquist
criterion, since 1/d_star_max is your d_min or "resolution".
With best wishes,
Gerard.
--
On Fri, Apr 15, 2011 at 03:11:41PM +0200, Dirk Kostrewa wrote:
> Dear Ian,
>
> oh, yes, thank you - you are absolutely right! I really confused the
> sampling of the molecular transform with the sampling of the electron
> density in the unit cell! Sometimes I don't see the wood for the trees!
>
> Let me then shift my question from the sampling of the molecular transform
> to the sampling of the electron density within the unit cell. For the
> 1-dimensional case, this is discretely sampled at a/h for resolution d,
> which is still 1x sampling and not 2x sampling, as required according to
> Nyquist-Shannon. Where is my error in reasoning, here?
>
> Best regards,
>
> Dirk.
>
> Am 15.04.11 14:25, schrieb Ian Tickle:
>> Hi Dirk
>>
>> I think you're confusing the sampling of the molecular transform with
>> the sampling of the electron density. You say "In the 1-dimensional
>> crystal, we sample the continuous molecular transform at discrete
>> reciprocal lattice points according to the von Laue condition, S*a =
>> h". In fact the sampling of the molecular transform has nothing to do
>> with h, it's sampled at points separated by a* = 1/a in the 1-D case.
>>
>> Cheers
>>
>> -- Ian
>>
>> On Fri, Apr 15, 2011 at 12:20 PM, Dirk Kostrewa
>> <[log in to unmask]> wrote:
>>> Dear colleagues,
>>>
>>> I just stumbled across a simple question and a seeming paradox for me in
>>> crystallography, that puzzles me. Maybe, it is also interesting for you.
>>>
>>> The simple question is: is the discrete sampling of the continuous
>>> molecular
>>> Fourier transform imposed by the crystal lattice sufficient to get the
>>> desired information at a given resolution?
>>>
>>> From my old lectures in Biophysics at the University, I know it has been
>>> theoretically proven, but I don't recall the argument, anymore. I looked
>>> into a couple of crystallography books and I couldn't find the answer in
>>> any
>>> of those. Maybe, you can help me out.
>>>
>>> Let's do a simple gedankenexperiment/thought experiment in the
>>> 1-dimensional
>>> crystal case with unit cell length a, and desired information at
>>> resolution
>>> d.
>>>
>>> According to Braggs law, the resolution for a first order reflection
>>> (n=1)
>>> is:
>>>
>>> 1/d = 2*sin(theta)/lambda
>>>
>>> with 2*sin(theta)/lambda being the length of the scattering vector |S|,
>>> which gives:
>>>
>>> 1/d = |S|
>>>
>>> In the 1-dimensional crystal, we sample the continuous molecular
>>> transform
>>> at discrete reciprocal lattice points according to the von Laue
>>> condition,
>>> S*a = h, which gives |S| = h/a here. In other words, the unit cell with
>>> length a is subdivided into h evenly spaced crystallographic planes with
>>> distance d = a/h.
>>>
>>> Now, the discrete sampling by the crystallographic planes a/h is only 1x
>>> the
>>> resolution d. According to the Nyquist-Shannon sampling theorem in
>>> Fourier
>>> transformation, in order to get a desired information at a given
>>> frequency,
>>> we would need a discrete sampling frequency of *twice* that frequency
>>> (the
>>> Nyquist frequency).
>>>
>>> In crystallography, this Nyquist frequency is also used, for instance, in
>>> the calculation of electron density maps on a discrete grid, where the
>>> grid
>>> spacing for an electron density map at resolution d should be<= d/2. For
>>> calculating that electron density map by Fourier transformation, all
>>> coefficients from -h to +h would be used, which gives twice the number of
>>> Fourier coefficients, but the underlying sampling of the unit cell along
>>> a
>>> with maximum index |h| is still only a/h!
>>>
>>> This leads to my seeming paradox: according to Braggs law and the von
>>> Laue
>>> conditions, I get the information at resolution d already with a 1x
>>> sampling
>>> a/h, but according to the Nyquist-Shannon sampling theory, I would need a
>>> 2x
>>> sampling a/(2h).
>>>
>>> So what is the argument again, that the sampling of the continuous
>>> molecular
>>> transform imposed by the crystal lattice is sufficient to get the desired
>>> information at a given resolution?
>>>
>>> I would be very grateful for your help!
>>>
>>> Best regards,
>>>
>>> Dirk.
>>>
>>> --
>>>
>>> *******************************************************
>>> Dirk Kostrewa
>>> Gene Center Munich, A5.07
>>> Department of Biochemistry
>>> Ludwig-Maximilians-Universität München
>>> Feodor-Lynen-Str. 25
>>> D-81377 Munich
>>> Germany
>>> Phone: +49-89-2180-76845
>>> Fax: +49-89-2180-76999
>>> E-mail: [log in to unmask]
>>> WWW: www.genzentrum.lmu.de
>>> *******************************************************
>>>
>
> --
>
> *******************************************************
> Dirk Kostrewa
> Gene Center Munich, A5.07
> Department of Biochemistry
> Ludwig-Maximilians-Universität München
> Feodor-Lynen-Str. 25
> D-81377 Munich
> Germany
> Phone: +49-89-2180-76845
> Fax: +49-89-2180-76999
> E-mail: [log in to unmask]
> WWW: www.genzentrum.lmu.de
> *******************************************************
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