The minimum crystal size needed for a complete diffraction data set

A formula for absolute scattering power is derived to include spot fading arising from radiation damage and the crystal volume needed to collect diffraction data to a given resolution is calculated.

regions. The "grazing relp" border is a circle, and it is illustrative to consider moving the relp circle in Fig. 1 up the z axis until it intersects the Ewald sphere at just one point. At this grazing point, Bragg's Law: is satisfied, and so the height (h) of this circle above the z = 0 plane must be the radius of the Ewald sphere λ* times the sin of the take-off angle of the spot (2θ): (A2) h = λ*sin 2θ Now, the area in either of the blind regions is a section of the surface of a sphere that is cut off by a plane at z = ±h. This shape is known as a spherical cap, and the area of a spherical cap from a sphere of radius r that was cut at height h is given by: There are two such caps, and we are interested in the area left over after they are cut off (observable relps) relative to the original area of the sphere: The denominator of (B6) was solved as Equation (A6), and is simply half of the accessible surface area of the constant-resolution sphere, which has radius λ/d or 2sinθ, and accessible fraction f obs = cosθ: Now, all the points that lie on or very near the relp circle pictured in Fig. 1 will have the same L and P factors, so we must "weight" the LP of each contact point on the Ewald sphere surface by the circumference of the relp circle. That is, we define the area element in Equation (B6) with a re-casting of Equation (A3) where we can compute the area of the constant-resolution sphere above the relp circle in Fig. 1 by substituting the radius of this sphere (λ/d = 2sinθ) for "r" and ζ from Equation (B2) for "h": We now employ the indefinite integral: And substitute this into the definite integral: These simulations were repeated using the full expression for the polarization factor with different values for the degree of polarization ( ), but the numerical results were identical to = 0 (not shown), and we saw no need to repeat the derivation using the full polarization factor expression.

Appendix C: Spot-fading integral
Here we assume that the average spot intensity (photons/spot) at a given resolution fades exponentially, but begin with a slightly different representation of Equation (13) than in the text: Note the use of angle brackets 〈 〉 to denote the average in a given resolution bin, and that here we use the term "intensity rate" to refer to a spot intensity (photons/spot) divided by the exposure time used to record it (seconds). This is because accurate spot fading experiments must record the decay curve by sampling the same spots over and over again using per-observation exposure times that are short relative to the damage limit (T DL ). That is, when sampling a changing signal, the sampling time must be short enough so that the signal does not decay appreciably during a given sample. This is equivalent to requiring that the photons/spot per unit time (intensity rate) is constant for a given exposure. So, effectively, spot fading experiments measure changes in intensity rate. Here we use a lower case "i" to differentiate an intensity rate (photons/spot/s) from an integrated spot intensity (photons/spot), which we will continue to denote with a capital "I", and also replace the dose D reso with DR× t so that DR represents the time- Since we are not considering the accumulation of background counts, T DL could be chosen to be infinity and 〈I〉 DL would then truly account for every last photon that will fall into a spot before it fades away completely, but in practice the damage limit is usually declared at a point where 〈i〉 (T DL ) is not zero, as discussed in §2.11. To account for potentially variable damage limit criteria, we define the "decay fraction" at the end of data collection as: is actually a series of "mini" data sets with exposure time t DS . As long as t DS is very small when compared to the damage limit (T DL ), the spot intensities (photons/spot) in the first "mini" data set (I ND ) will be "undamaged" and given by Darwin's formula (Equation (1)).
The exact I ND measured will be proportional to t DS , as can easily be seen by substituting ω eff from Equation (10)  I ND -spot intensity (photons/spot) from the first "mini" data set measured quickly enough to have suffered no radiation damage effects.
It is readily apparent that the quotient I ND /t DS (photons/spot/s) is an intensity rate, as is the average value 〈I〉 ND /t DS . Formally, the right hand side of Equation (C5) is independent of time, but as the reality of radiation damage progresses the same t DS will record spot intensities that fade (on average) according to the exponential decay of Equation (C1).
Therefore, the starting value of this decay curve is 〈I〉 ND /t DS , and we may substitute this intensity rate for 〈i〉 (0) in the above Equations: Now the sum of all the equivalent observations in all the mini data sets up to a given accumulated exposure time is given by Equation (C2), and even if a single data set were collected with total exposure time T DL , the average number of photons that eventually contribute to a spot (〈I〉 DL ) is simply the integral of the decay over time (Equation (C4)).
We now substitute 〈I〉 ND /t DS for 〈i〉 (0) in Equation (C4), as well as D en /t from Equation (11) for the dose rate (DR), convert the photon energy into wavelength (q e E ph = hc/λ = J/photon), apply the Nave-Hill fraction f NH from Equation (12) and scale to convenient units. We arrive at Equation (14)