The number of reflections from a crystal measured up to a $Q_{max} = 4\pi \sin{\theta}/\lambda$ is dependent on the symmetry of the space group and the size of the unit cell. You will have a larger number of reflections if the symmetry is low and also if the unit cell has a larger volume. But one has to be careful about the unit cell size. The convention unit cell is not the cell we must take into account because it is not always the smallest possible cell. For comparison of the number of reflections we must chose the unit cell of minimum possible volume. For example for crystals with rhombohedral space group we usually chose a hexagonal cell that is not the cell with minimum possible volume. What actually is deciding is the number density of the reciprocal space and the volume of the Ewald sphere. You can calculate the number of reflections up to a certain Q (what I guess you call resolution according to X-ray crystallography convention) by simple formula given in the International Tables for Crystallography vol 2 (old edition).

I am assuming this example is from the same protein. The smaller the unit cell will have less relections and the larger cell will have more. While resolution will change the number of reflections for a given unit cell, as you have seen, you will still have more reflections with a larger unit cell. More atoms are represented in the in the larger ASU. Recall that reflections are in inverse reciprical space.

The number of reflections from a crystal measured up to a $Q_{max} = 4\pi \sin{\theta}/\lambda$ is dependent on the symmetry of the space group and the size of the unit cell. You will have a larger number of reflections if the symmetry is low and also if the unit cell has a larger volume. But one has to be careful about the unit cell size. The convention unit cell is not the cell we must take into account because it is not always the smallest possible cell. For comparison of the number of reflections we must chose the unit cell of minimum possible volume. For example for crystals with rhombohedral space group we usually chose a hexagonal cell that is not the cell with minimum possible volume. What actually is deciding is the number density of the reciprocal space and the volume of the Ewald sphere. You can calculate the number of reflections up to a certain Q (what I guess you call resolution according to X-ray crystallography convention) by simple formula given in the International Tables for Crystallography vol 2 (old edition).

The number of reflections depends on 2 factors: The resolution and the volume of the asymmetric unit. In point group 6 , there are 6 a.u.s; in 622, there 12; in 222 there are 4 if the space group is primtivie or 8 if it is centred (C/F/I). First calculate the volume of the unit cell, then divide by the number of aus in that space group. If the number of of copies goes up, the volume goes up, and the number of reflections goes up. 3 copies in the hexagonal cell v 1 copy in orthorhombic, means there will be 3 times as many reflections in hexagonal, IF the resolution is the same. But then, the number of reflections will change v resolution in a cubed manner, because the Ewald sphere VOLUME grows or shrinks: At 1A, you will have a number U of reflections; at 2A this number will be smaller by 8 times (2*2*2); at 3 A this number will be smaller by 27 times (3*3*3). So the ratio between hexagonal and ortho will be (res orth/res hex)**3 for 1 copy, then adjust by the number of copies.

I hope this makes sense. Good Luck.

As a rule of thumb, every non-hydrogen takes about 18 cubic angstroms of space. In protein structures there are solvent regions to be treated differently, but for small-molecule crystals you can divide the unit cell volume by 18 and get a pretty good idea how many atoms the cell may contain. Depending on real-space symmetry, not all of those atoms in the unit cell are unique; the number of independent atoms depends on the Point Group.

The number of reflections depends on the size of the unit cell (big cell = many reflections, smaller cell = fewer reflections to a given resolution). The higher the resolution, the more reflections we can observe in a diffraction experiment. Just as with real space, not all of the reflections in reciprocal space are unique; the number of independent reflections depends on the Laue Group.

Because of the atom-to-volume relationship mentioned above, one expects about 80 independent reflections per independent non-hydrogen atom per cubic angstrom of resolution for centrosymmetric structures, and 40 independent reflections per independent atom and cubic angstrom for non-centrosymmetric structures (that is not taking into account anomalous scattering) . In this context it is important to take lattice types into consideration, just as Tapan explained.

What may be more interesting than the absolute number of reflections or atoms is the data-to-parameter ratio of a refinement. For small-molecule structures, the IUCr recommends a ratio if 10:1 for centrosymmetric structures, which corresponds to approximately 0.84 angstroms of resolution. The ratio can be improved by using constraints (remove parameters from refinement) or restraints (add non-diffraction data to refine against). Protein crystals rarely diffract to 0.84 Angstroms, however thanks to the large amount of solvent, the number of parameters is only roughly half for the same unit cell volume. Still, typically, restraints are heavily used in protein structure refinements to improve the data-to-parameter ratio and, thus, to increase the stability of the refinement.

You mention the presence of several crystallographically independent copies of the molecule (Z'>1). This is a great opportunity to apply similarity restraints on bond lengths and angles and/or NCS-restraints.

What Peter says about using restraints to reduce the number of parameters refined is true. However, molecules may be crystallographically independent because their position in the cell precludes a crystallographic symmetry operation from relating one to the other, or because the molecules have unique conformations. Thus restrains must be applied in a way that allows for conformational differences while reducing the number of parameters. This is routinely done in proteins where you want to determine the conformation of the side chains while maintaining reasonable bond lengths and angles (i.e. only the torsion angles are refined). This is similar to the way powder data can be used to refine a structure. IN SHELX this can be done using the SAME restraint that will keep 1,2 and 1,3 distances the same (i.e. bond lengths and angles).

Resolution of a datas et is the lowest value of d(hkl)=lambda/2sin(theta) that

you have collected. The number of reflections in full reciprocal sphere for

a given resolution is proportional to volume of the unit cell. If there are more

atoms in the asymmetric unit, it will occupy more volume and correspondingly

will have more unique reflections also for a given resolution.

The term more 'reflections per asymmetric unit' is not representative of the

quality of data, instead more 'reflections/parameter ratio' is the one which

tells of better resolution.

The number of observable reflections depends on the observable reciprocal space volume. This can be straightforwardly computed from the Ewald construction for the Bragg Law in the reciprocal space, i.e. by resorting to the concept of limiting sphere. Being 2/lambda the radius of the limiting spgere, its volume amounts to

(4 / 3) pi (2 / lambda)^3

As by definition a primitive cell contains 1 node, the volume associated to each reciprocal lattice node is V*, the volume of the primitive reciprocal cell. Therefore, the number of dicrete nodes that can be found within the limiting sphere is just the ratio between the volume of the limiting sphere and the volume of 1 node - that is

N = (1 / V*) (4 / 3) pi (2 / lambda)^3

But 1/V* = V, the volume of the primitive real cell. So

N = V (4 / 3) pi (2 / lambda)^3

This result is completely general and can be used to compute the maximum number of measurable reflections at a given lambda. When you collect data up to a resolution of 1 / d = 2 sin(theta) / lambda < 2 / lambda, you can as well compute the number of expected reflections at that resolution by simply substituting the 2/lambda factor with the actual value or your highest reciprocal lattice vector, that is

N = V (4 / 3) pi [2 sin/(theta) / lambda]^3

Obviously, this formula does not take into acount the occurrence of systematic extinctions - nevertheless, their presence also brings information on your crystal structure, so they should be considered as 'data' as well. Moreover, not all the accessible data at a given resolution are independent, of course - if you want just an estimate of the symmetry-independent (non-redundant) data, you should simply scale N by the appropriate factor depending on the symmetry of your Laue group (e.g. 1/8 for an orthorhombic lattice).

Hope this will be useful to you.

You guys are all awesome for sharing. This subject was hard to grasp in the past. I'm getting better at it. Thanks. I'll follow.

π µ ω θ β α λ δ Δ ρ° - Greek symbols for cut/paste - from Word. I cheat and create my text on Word before posting when possible.

Leonardo Lo Presti! What is "real cell"?

N = V.(4/3).π.[2sinθ/λ]^3 - correct?

Resolution! In XRD this usually means the smallest angle resolvable (precisely) and smallest detector pixel size available? Based on optics and slit systems used (i.e., beam conditioning), with XRD it is possible to resolve mm, to um, to nm, to femto-meters using various detectors, optics & techniques such WAXS, SAXS, XRR, Rocking Curve and so on. The rocking curve method for instance allows us to measure precisely crystallographic orientation deviations in the fractions of Arc Sec range.

How do I calculate the number of expected reflections from a Si (111), SiC-6H (0001), LaSrAlO4 (001), GaAs(001), or GaSb(001) single crystal each of known surface orientation with Cu K α 1? From 3° -180° for example.

I usually find them all by groping through the reciprocal space in real time with a Bragg XRD Microscope:

http://www.youtube.com/watch?v=dFCQS8oUyT0

[email protected]

"Leonardo Lo Presti! What is "real cell"?"

Just the (primitive) unit cell in the real (not reciprocal) space.