Intro-ish...

MLFSOM is not the first program written to simulate diffraction patterns
from protein crystals (XCADS ref?, Arndt & Wonacott ref?). Indeed
simulated diffraction patterns are as old as crystallography itself
since understanding the diffraction phenomenon requires the construction
of a model that can reproduce the data. The goal of MLFSOM is to not
only simulate the diffraction signal but also all the sources of error
that corrupt it. The so-called photon-counting error that limits the
accuracy of spot intensities is perhaps the best known, but other
factors such as the intrinsic noise of the detector (called read-out
noise in the case of CCD detectors), timing errors of the shutter,
instability in the x-ray beam, radiation damage and irregular spot
shapes also affect data quality. The relative importance of all these
sources of error is not immediately clear and no doubt depends on the
nature of the experiment, hence the need for a simulation. Evaluating
these sources of noise requires putting them on the same scale as the
signal as it appears on the actual measuring device used: the area
detector. For this reason, the simulation in MLFSOM is performed on an
absolute scale, and the inputs to the program are in real-world units.
For example, photon flux is provided in photons/s, crystal size is
provided in microns, radiation damage progression is determined by dose
in Gy, detector noise is provided in rms electrons/pixel, and so on.
Whenever possible, published values are used for these parameters. For
example, detector performance is taken from manufacturer's
specifications. Parameters where no published values are available (such
as the jitter in the shutter) were measured experimentally at the
Advanced Light Source beamline 8.3.1, but it is a simple matter to input
the characteristics of a different beamline if they are known.

**Program structure:**

The programmatic structure of MLFSOM is built upon two third-party
packages: the CCP4 suite (CCP4 1994), and FIT2D (Hammersley et al.
1996). The first stage is converting the input protein data bank (PDB)
file into structure factors and this is done by a script called
ano_sfall.com, which decorates the molecule with hydogens using the CCP4
program HADD and calculates structure factors using SFALL. The structure
factor file (mtz format) produced by this script will also contain
anomalous differences and a simplistic bulk solvent contribution
(described below). This or any other mtz formatted file may be input
into the second script: mlfsom.com. This script computes the spot
positions and intensities as well as the various background
contributions and noise levels and then writes an input file for FIT2D,
using various features of that program to plot Gaussians and other
shapes on a pixel field, applying noise and then outputting the pixel
field as binary integer data. The addition of an SMV-format (ref?)
header allows these simulated images to be input into standard data
processing programs.

**Disordered solvent contribution:**

In order for solvent flattening and model refinement to function
normally the disordered or "bulk" solvent must be taken into account.
This is implemented with a simplistic model where an electron density of
0.27 e-/Å^3^ is assigned to each voxel in the unit cell more than 2.7 Å
away from any atom listed in the PDB file. This is far simpler than the
"true" electron density distribution in a real crystal (Lounnas et al.
1994; Phillips & Pettitt 1995; Burling et al. 1996; Makarov et al. 2002;
Zhang et al. 2007 ref?), and indeed simpler than that implemented in
most popular refinement packages (Tronrud 1997; refmac ref,
phenix.refine ref?). Therefore, in this work, the systematic error of
the solvent model arises from lack of sophistication in this simulation.
However, the topic of this work centers on the feasibility of obtaining
an interpretable electron density map from SAD data and solvent
flattening, so this simple solvent model should suffice.

**Anomalous scattering:**

The anomalous contribution of each atom was calculated using the
structure factor of atomic core electrons. This was implemented using
the "Ano" atom type in the CCP4 program SFALL (CCP4 1994) which is
equivalent in real space to constructing a single Gaussian at each
atomic position with unit volume and width equivalent to the atomic
B-factor. This "unit" structure factor was then converted into the
anomalous and dispersive contributions of each atom using values of f'
and f" calculated by the program CROSSEC (Cromer 1983; Waasmaier &
Kirfel 1995 and references therein). The anomalous contribution of all
atoms in the PDB file was accounted for, including carbon, nitrogen and
oxygen. The summed anomalous signal was then added to or subtracted from
the total structure factor of the protein and solvent to form F^+^ and
F^-^, respectively using SFTOOLS (CCP4).

**Spot locations:**

The locations of diffraction spots on the detector were computed using
the coordinate system and matrix formalism described in Arndt and
Wonacott (1977) chapter 7. The real-space unit cell with edges a, b, c
and angles α, β, γ was converted into three reciprocal lattice vectors
**a\***,**b\*** and **c\*** in a right-handed x-y-z coordinate system
where x is the direction of the x-ray beam, z is the spindle axis[^1]
and y is "up"[^2]. The spindle is right handed in the sense that a
crystal rotation (Φ) of +90° will rotate a vector lying along the
positive x axis onto the positive y axis. Taking the popular "reference
orientation" where **a\*** lies perfectly along the x-axis and **b\***
lies perfectly within the x-y plane, the x,y,z components of the three
reciprocal lattice vectors were written as a 3 x 3 matrix called **B**:

$$B = \begin{bmatrix}
\text{a*} & 0 & 0 \\
\text{b*cosγ*} & \text{b*sinγ*} & 0 \\
\text{c*cosβ*} & \text{c*}\frac{\text{cosα*} - \text{cosβ*cosγ*}}{\text{sinγ*}} & \frac{\text{c*V}_{\text{cell}}}{\text{a*b*c*sinγ*}}
\end{bmatrix}$$

a\* = bc sinα/V~cell~

b\* = ca sinβ/V~cell~

c\* = ab sinγ/V~cell~

cos α\* = (cosβ cosγ-cosα)/(sinβ sinγ)

cos β\* = (cosγ cosα-cosβ)/(sinγ sinα)

cos γ\* = (cosα cosβ-cosγ)/(sinα sinβ)

V~cell~
=$2\text{abc}\sqrt{\text{sin}(\text{skew})\text{sin}(\text{skew-α})\text{sin}(\text{skew-β})\text{sin}(\text{skew-γ})}$

skew=(α+β+γ)/2

Where a\* is the length of the **a\*** vector, α\* is the angle between
**b\*** and **c\***, and so on. Other reference positions are possible,
but this convention has been historically popular and will be used
exclusively here. This **B**-matrix was then rotated using either a
unitary matrix **U**, or by specifying three "missetting" angles about
the x, y and z axes[^3] that generate a **U** matrix. The product **BU**
is called the **A** matrix, and represents the reciprocal lattice
setting in the lab frame when the spindle position Φ is at 0.
Multiplying the **A** matrix by a column vector of the Miller indices
{h,k,l} yields the x-y-z coordinates of the corresponding reciprocal
lattice point (relp), and rotating the spindle by an angle Φ will rotate
the relp by the same angle about the z axis[^4]. Diffraction occurs when
the relp intersects the surface of the Ewald sphere, which is nominally
centered at (x,y,z) = (-1/λ,0,0) and has radius 1/λ. By definition the
Ewald sphere surface must intersect the origin of reciprocal space, but
if the incident x-ray beam is not perfectly perpendicular to the
spindle, then the center of the Ewald sphere swings away from the x-axis
in the corresponding direction.

**Spot intensity:**

Once the total structure factors of the crystal have been computed, they
must be converted into intensities. The relationship between the
intensity of a fully-recorded spot and the square of the structure
factor is given by Darwin's formula (Darwin, 1914; Blundell & Johnson
1971).

\(1\)
$I(\text{hkl}) = I_{\text{beam}}r_{e^{2}}\frac{V_{\text{xtal}}}{V_{\text{cell}}} \cdot \frac{\lambda^{3}L}{\text{ωV}_{\text{cell}}}P \cdot A \cdot |F(\text{hkl})|^{2}$

Where:

I(hkl) - integrated spot intensity (in photons)

I~beam~ - intensity of the incident beam (photons/s/m^2^)

r~e~ - classical electron radius (2.818 x 10^-15^ m/electron)

V~xtal~ - volume of the crystal (in m^3^)

V~cell~ - volume of the crystal unit cell (in m^3^/cell)

λ - x-ray wavelength (in m)

ω - angular velocity of the crystal (radians/s)

L - Lorentz factor

P - polarization factor

A - x-ray transmittance of the path through the crystal to the spot

F(hkl) - structure factor (equivalent electrons/cell)

The structure factor is defined as the ratio of the scattering from one
unit cell to that of a single classical electron, so the classical
electron cross section (r~e~^2^) is included. The intensity will also be
proportional to the number of unit cells in the beam, which is given by
the crystal volume divided by one of the unit cell volume terms. This
relation holds regardless of the mosaic structure of the crystal
(Woolfson 1970). It should be noted that the units of the spot intensity
(photons/spot) are not the same as that of the incident beam
(photons/s/m^2^) nor those of the classical electron scattering
(photons/steradian). Despite this, all of these quantities remain
commonly referred to as "intensity", which can lead to a considerable
amount of confusion if the units are not given explicitly. The change of
units arises because the spot intensity must be integrated over the
reciprocal lattice point (hkl or "relp") as it moves through the Ewald
sphere and several geometric factors must be taken into account
(described below). Although it is unusual to express x-ray wavelengths
in meters, it is very important that values input into Equation 1 be
converted into the units indicated, or very large errors can result. For
example, taking typical values for a 2 Å hkl from a protein crystal[^5]
and typical beam characteristics[^6], we arrive at an integrated full
spot intensity of 27,251 photons, which we will see later is remarkably
consistent with experiment.

**Lorentz factor**

The Lorentz factor L in Equation 1 is always greater than one and is not
to be confused with its inverse[^7], the Lorentz correction L^-1^ which
data processing programs such as MOSFLM multiply by observed integrated
intensities to correct for the Lorentz factor. The Lorentz factor first
appeared in Debye (1914), and is the ratio of the speed of the relp in
reciprocal space to the component of its velocity vector that is normal
to the Ewald sphere surface. This factor reflects how different relps
will spend different amounts of time passing through the Bragg
condition, so this geometric correction is grouped with other geometric
factors in Equation 1 such as the angular velocity of the crystal. The
cube of the wavelength and one of the unit cell volume terms are also
geometric corrections since these determine the size of the relp in
reciprocal space, and hence the integration volume (Woolfson 1970). The
relationship between the Lorentz factor and the detector surface will
obviously depend on the camera geometry, but in the common case where
the crystal rotation axis is perpendicular to the x-ray beam, the
Lorentz factor is given by:

\(2\) $L = \frac{1}{\sqrt{\text{sin}^{2}\text{2θ} - \zeta^{2}}}$; ζ~┴~ =
cos 2θ Z~det~/D~xtf~

Where:

θ - Bragg Angle[^8]

ζ - λ**d\***∙$\hat{z}$ normalized projection of the relp vector onto the
rotation axis (z)

ζ~┴~ - ζ in terms of spot coordinates on a flat detector normal to the
incident beam

Z~det~ - coordinate of the diffraction spot on the detector along the
axis parallel to the rotation axis (relative to beam center in mm)

D~xtf~ - sample-to-detector distance along direct beam path (in mm)

The polar coordinate ζ is calculated by taking the z-component of the
relp vector[^9] **d\*** and multiplying it by the x-ray wavelength λ in
Å. Note that as **d\*** approaches the z-axis ζ approaches λ/d = 2sinθ
and L approaches infinity[^10]. The denominator of Equation 2 reaches
zero when **d\*** passes into the "blind region" where the rotating relp
never touches the Ewald sphere and the Lorentz factor becomes undefined.

In the also common case where the detector is a flat plane and normal to
the incident x-ray beam, ζ becomes ζ~┴~, which can be calculated from
the position of the spot the detector as shown. However, for arbitrary
detector positions, ζ must be computed from the spindle geometry, and in
the general case of the beam not perfectly normal to the rotation axis,
L must be calculated by taking the projection of the relp velocity
vector along the diffracted ray which was done in this work.

**Polarization factor:**

The polarization factor P is always less than one and accounts for
losses of scattering efficiency when the incident and scattered beam
polarization vectors do not line up. First described by Azaroff (1955)
and re-formulated for synchrotron radiation by Kahn et al. (1982), we
use here the expression given by Drenth (1999):

\(3\) P = (1 + cos^2^2θ - c~p~ cos 2ψ sin^2^2θ)/2

where:

P - polarization factor used in Equation 1

θ - Bragg angle

ψ - angle between the polarization E-vector[^11] and the projection of
the diffracted ray onto a plane normal to the incident beam[^12]

c~p~ = degree of polarization

The input "degree of polarization" c~p~ ranges from 1 to 0 to -1 as the
beam polarization varies from "horizontal" (along the rotation axis) to
unpolarized to "vertical" respectively. Note that the polarization
factor becomes zero for scattering directly down the polarization
axis[^13] and there is no scattering of x-rays in this direction, which
is why the rotation axis is usually designed to be coincident with the
polarization vector.

**Sample Absorption:**

The absorption factor A depends on the size, shape and atomic
composition of the crystal and any other objects in the path taken by
the x-rays on their way into the crystal and out to the spot of
interest. The profile of the beam is also a factor, and for full
accuracy photons from each point in the x-ray source must be ray-traced
to every accessible part of the crystal volume and from there out into
the spot. The average x-ray transmittance along all these paths is the
absorption factor:

\(4\)

$$A = \frac{1}{V_{\text{xtal}}I_{\text{beam}}}\iiint_{\text{xtal}}^{}{I_{\text{prof}}(y,z)\text{exp}( - \mu_{\text{air}}t_{\text{air}}(x,y,z) - \mu_{\text{xtal}}t_{\text{xtal}}(x,y,z) - \mu_{\text{loop}}t_{\text{loop}}(x,y,z) - \text{.}\text{.}\text{.})}\text{dxdydz}$$

Where:

A - absorption factor (transmittance)

V~xtal~ - volume of the crystal

I~beam~ - total intensity of the incident beam

I~prof~ - intensity of the beam profile at the coordinate 0,y,z

μ~x~ - attenuation coefficient of substance x; μ~x~^-1^ is the
attenuation length

t~x~ - component of total path taken by x-ray through substance x, via
crystal coordinate x,y,z

The value of μ~x~ may be obtained using the density of the material and
the x-ray cross section of the elemental composition of substance x
(NIST XCOM ref?), but there is no analytic solution for the integrated
absorption factor A for anything but the simplest shapes, such as a flat
slab-shaped crystal (International Tables C §6.3.3 p. 595). Even a
perfect sphere must be evaluated numerically but the sphere is a
convenient "average shape" for a protein crystal and look-up tables are
available for this integral (Dwiggins 1975). For the calculation here we
will consider a spherical crystal of radius R with uniform attenuation
coefficient μ and denote the total transmission of a beam diffracting at
angle 2θ simply as A = T~sphere~(2θ,μ,R). A convenient program for
accurate calculation of μ~xtal~ is RADDOSE (Murray et al. 2004), and for
a typical elemental composition of protein[^14] in a crystal that is 50%
water, μ~xtal~^-1^ = 3.6 mm and the value of A for a 2 Å spot through a
100 μm diameter crystal is 0.979, so A is usually a small correction.

**Average signal**

**The average Lorentz-Polarization factor:**

The average value of the product of the Lorentz and polarization factors
for measurable spots[^15] at a given resolution of interest can be
approximated to within 1% error by:

\(5\)
$\left\langle \text{LP} \right\rangle \approx \frac{1\text{.}\text{05} \cdot \pi}{\text{4 s λ}} - \frac{1}{2} = \frac{1\text{.}\text{05} \cdot \pi}{\text{4 sinθ}} - \frac{1}{2}$

Note that the average value of LP does not depend on the degree of
polarization, as the asymmetry of the polarization factor "averages out"
upon integration around a constant-resolution circle, and also
that$\left\langle \text{LP} \right\rangle$ tends to cancel one of the λ
terms in Equation (1), but not exactly.

**Average structure factor:**

Note that the unit cell volume occurs twice in Equation 1. One of these
is involved in the number of unit cells contributing to the spot
(V~xtal~/V~cell~), and the other is a geometric factor of integration
because the "size" of the reciprocal lattice point is inversely
proportional to V~cell~ (Woolfson 1970). However, the apparent
amplification from double-counting V~cell~ is effectively cancelled by
the \<F^2^\> term. This is because the average square structure factor
of a random distribution of atoms rapidly approaches the sum of the
individually-squared atomic structure factors, and the average
scattering from a protein beyond \~4 Å resolution can be approximated by
a random atomic distribution (Wilson 1949). Since the disordered solvent
contribution to Bragg peaks is weak at high angles, the number of atoms
contributing to a high-angle spot can be taken as the number of ordered
protein atoms in the unit cell:

(6a)
$N_{\text{cell}}\left\langle f_{a}(s)^{2} \right\rangle \simeq N_{C}f_{C}(s)^{2} + N_{N}f_{N}(s)^{2} + N_{O}f_{O}(s)^{2} + N_{H}f_{H}(s)^{2}\text{.}\text{.}\text{.}$

(6b)
$\left\langle F(s)^{2} \right\rangle \simeq \frac{V_{\text{cell}}}{V_{M}\left\langle M_{a} \right\rangle}\left\langle f_{a}(s)^{2} \right\rangle\text{exp}( - \text{2Bs}^{2})$

(6c) V~cell~ = n~symop~∙n~ASU~∙M~r~∙V~M~

Where:

N~cell~ total number of atoms in the unit cell (including hydrogen)

\<ƒ~a~(s)^2^\> number-averaged squared atomic structure factor of
protein (\~5 at s = 0)

N~Ee~ total number of atoms of element Ee

ƒ~Ee~(s) structure factor of element Ee

\<F(s)^2^\> average value of the squared structure factor of the crystal

n~symop~ number of symmetry operators in the space group

n~ASU~ number of protein molecules in the asymmetric unit

M~r~ molecular weight of the protein (Daltons or g/mol)

\<M~a~\> number-averaged protein atom mass (M~r~/N~protein~ \~7.1 g/mol)

s reciprocal scattering vector length (sin(θ)/λ)

B average (Wilson) B factor (in Å^2^)

V~cell~ volume of the unit cell (in Å^3^)

V~M~ Matthews's coefficient (Å^3^/Dalton)

Note that F and ƒ~a~ depend on s. Although one might expect that the
average structure factor and average atomic mass of a protein would be
closer to that of carbon or nitrogen, half of the atoms in protein are
hydrogen, and this brings down the number-averaged quantities. However,
counting hydrogen also brings down the number-averaged atomic mass and
the ratio ƒ~N~(s)^2^/14 is fairly equal to \<ƒ~a~(s)^2^\>/\<M~a~\>. So,
if only a rough approximation is desired the protein can be considered
to be made of nitrogen. Since ƒ~a~ and M~a~ are essentially constants
for protein[^16] and V~M~ also has a restricted range (Matthews 1968;
Kantardjieff 2003) Equation 6b reduces to \~ 0.2 V~cell~[^17] and this
is what tends to cancel one of the 1/V~cell~ terms in Equation 1. That
is, given two protein crystals with the same V~xtal~ (and Wilson B
factor) but one with V~cell~ twice that of the other, the average spot
intensity from the large unit cell crystal will be half of that from the
smaller unit cell crystal. This is because there are half as many unit
cells in the crystal and is also required by conservation of energy
since the total number of photons scattered by a fixed number of atoms
must be divided over the total number of spots.

**Angular velocity vs exposure time**

Note that the exposure time (t) does not appear explicitly in Equation
(1) because it is hidden in the rotation speed ω = ΔΦ/t where ΔΦ here is
the rotation covered during an exposure (in radians). What happens if
the crystal is not rotated during the exposure? Does the spot intensity
become infinite? Of course not, but in reality it does approach the
intensity of the incident beam as the mosaic spread approaches zero, the
crystal volume becomes large and the x-ray beam becomes perfectly
monochromatic and parallel. This limiting case is routinely achieved
with the perfect silicon crystals used in monochromators where nearly
100% of the incident beam is reflected. Equation (1) does not apply in
these cases as its derivation did not take into account the drop in
I~beam~ due to scattering as the beam moves through the crystal as well
as the possibility that the diffracted beam itself can act as a new
"incident" beam and re-reflect in other parts of the crystal. Such a
complete treatment is called the dynamical theory of diffraction, which
will not be considered here. Equation (1) is based on what is known as
the kinematical approximation to the dynamical theory and assumes that
the mosaic domains are small compared with the attenuation length[^18]
of the x-rays in the crystal and that the drop in the main beam
intensity due to diffraction is negligible, which is generally a very
good assumption for protein crystals (James 1962).

What value should we choose for ΔΦ? It cannot be smaller than the mosaic
spread if we are to fully-record the spot, but since we are interested
in collecting more than just one spot we must set ΔΦ to the full
rotation range of the data set and set t to the total accumulated
exposure time. Now, this means that many unique hkls will be observed
more than once so an accounting must be made of the extra signal
available from merging equivalent observations. For a crystal belonging
to a space group with n~symop~ symmetry operators a total of 2x n~symop~
equivalent spots[^19] of a typical unique hkl will be measured in a 360°
rotation, so, for simplicity, we will use 360° for ΔΦ and multiply the
spot intensity by 2x n~symop~.

\(7\) $\omega = \frac{2\pi}{2n_{\text{symop}}t_{\text{360}}}$

Where:

ω - effective angular velocity for the data set (radians/s)

2π - 360°

n~symop~ - number of symmetry operators in the space group

t~360~ - total accumulated exposure time of a complete data set

In practice, a data-collection strategy is often devised to take
advantage of reciprocal-space symmetry and collect a complete data set
with a shorter ΔΦ, but if a proportionally longer exposure time is used,
then the increased t "cancels" the reduced ΔΦ and the total accumulated
intensity of any given unique hkl will be the same as a 360° sweep. That
is, a strategized data set will contain fewer but proportionally
brighter spots. This does not mean data collection strategy is useless,
we will consider background noise later.

**Absorbed Dose:**

The amount of radiation damage is proportional to dose (Garman ref?),
which, in this case can also be calculated from the absorption
coefficient of a sphere since the fraction of incident photons that
interacted with atoms in the sphere is 1-T~sphere~(0,μ,R). Not all of
the stopped energy will be absorbed since some of it is scattered or in
some cases fluoresced away, and this can be accounted for by using the
mass energy absorption coefficient μ~en~ in place of the mass
attenuation coefficient μ (Seltzer 1993). To obtain dose we divide the
absorbed energy over the volume of the sphere V~xtal~=4/3πR^3^ and apply
the beam intensity over the cross-sectional area (π R^2^):

\(8\) D =
$\frac{\text{3 eE}_{\text{ph}}\text{ I}_{\text{beam}}\text{ t}}{\text{4 R ρ}}\left( 1 - T_{\text{sphere}}(0,\mu_{\text{en}}\text{,R}) \right)$

where:

D - dose in Gy (J/kg)

e - electron charge (1.6022 x 10^-19^ J/eV)

E~ph~ - photon energy (eV/photon)

I~beam~ - incident beam intensity (photons/s/m^2^)

t - exposure time (s)

ρ - density of sphere material (kg/m^3^ or g/L)

R - radius of the sphere (m)

μ~en~ - mass energy-absorption coefficient of sphere material (m^-1^)

Note that for a cube shaped crystal of the same width as our sphere, the
transmission is T~cube~ = exp(-μ~en~∙2R), which implies that the (1-T)
term approaches 2μ~en~∙R when μ~en~ is much larger than R and most of
the beam is transmitted[^20]. This means that the 1/R term in Equation
\# is effectively cancelled in most practical cases, but we will keep
this equation in its exact form and also continue to use the sphere
model for absorption since we must use the same shape to calculate the
absorption factor and we don't want the corners of a cube-shaped crystal
to complicate our analysis against resolution.

In general, dose calculations are not simple, and although a sphere is
the simplest possible shape, this expression comes with certain caveats.
For example, if R becomes large compared to 1/μ~en~ of the crystal
material, then some fraction of the photons scattered from the core will
be absorbed before escaping the sphere and some of the energy discounted
to scattering by Seltzer must be added back to the dose. A similar
correction must also be made for energy assumed lost to fluorescence if
R becomes large compared to 1/μ~en~ for the fluorescent photon energy.
Conversely, as R becomes comparable to the track length of a
photoelectron, an additional mechanism of energy escape becomes
available which was not taken into account by Seltzer, but has been
estimated by Nave and Hill (2004). Here we apply the CSDA[^21]
approximation to the electron track, and assume the fraction of the
photoelectron energy delivered to the crystal will be roughly the ratio
of the crystal radius to the track length. That is, half the
photoelectrons emitted from the surface of the crystal will travel
outward and deposit no energy, whereas the other half will traverse the
core and deposit twice as much energy as electrons emitted from the
center of the crystal, making the center the "average" emission point.
Energy from the photon that is not delivered to the photoelectron is
delivered to Auger and low-energy fluorescence processes, which have a
very short range in protein.

**Radiation damage:**

The radiochemical mechanism behind the fading of diffraction spots is
not presently clear, but it has been calibrated experimentally. Owen and
Garman (2006) tested the so-called "Henderson limit" (Henderson 1990)
and observed that the sum of all diffracted intensities (after
correcting for the LP factor) from ferritin crystals faded linearly with
dose and reached 50% after absorbing D~1/2~ = 43 (± 3) MGy. However,
this linear trend did not continue to 0% intensity at 86 MGy, and the
authors noted a resolution dependence to the intensity falloff in the
form of increasing Wilson B factors. Kmetko et. al. (2006) found that
the change in B factor was highly linear with dose, and the slope or
"coefficient of sensitivity" was 0.95 B/MGy for lysozyme and 1.34 B/MGy
for apoferritin. This would appear to be inconsistent with the Owen and
Garman (2006) observations since applying a B factor of 15 Å^2^ to
observed structure factors of apoferritin (PDB ID: 2CLU) reduces the
total intensity to 50%, but B = 15 Å^2^ corresponds to only 11 MGy, not
the 43 MGy required for Owen and Garman to see intensities drop by half.
The curve of total intensity vs. applied B-factor is also roughly an
exponential, it is not linear.

Howells et. al. (2005) performed a meta-analysis of twelve radiation
damage studies available at that time from five different imaging fields
spanning four orders of magnitude in resolution and six orders of
magnitude in dose and found a general trend D~1/2~ ≈ 10∙d MGy where d is
the feature size in Å. This relation is remarkably consistent with both
the Owen and Garman (2006) as well as the Kmetko et. al. (2006)
observations if the intensity of an individual spot fades exponentially
with dose:

(9a) I = I~0~ exp(-2 A∙s)

(9b) A = 2π∙ℓ P(u) = 1/(1+(2u/ℓ)^2^)

(9c) ℓ/D = H = 11x10^-3^ Å/MGy

where:

I - intensity after absorbing a dose D

I~0~ - intensity from an undamaged crystal

s - reciprocal scattering vector length (sinθ/λ = 1/(2d))

u - atomic displacement from ideal lattice position (Å)

P(u) - Probability of a given lattice displacement (u)

ℓ - full-width at half maximum of a Lorentzian distribution.

A - Lorentzian Debye-Waller coefficient (not to be confused with B)

B - canonical Debye-Waller coefficient or "temperature factor"
(8π^2^∙\<u^2^\>)

D - absorbed dose (MGy)

H - ratio of Lorentzian width ℓ to absorbed dose (11x10^-3^ Å/MGy)

Note that the Lorentzian Debye-Waller factor A is a coefficient for s,
whereas the more well-known Debye-Waller temperature factor B (8π^2^
\<u^2^\>) is a coefficient for s^2^. This name "A" is used here for
consistency with the B-factor but should not be confused with the
absorption factor "A" above, which is a transmittance.

Applying this resolution-dependent exponential decay using the value for
H derived from Howells *et al.* (2005) to observed structure factors
from apoferritin (PDB ID: 2CLU) results in a total intensity vs dose
curve that stays within 5% of a straight line until dropping to 50% at
42 MGy, which is well within the error bars proposed by Owen and Garman
(2006). If these same "A-factor" modified data are scaled against the
"zero-dose" data using SCALEIT (Howell & Smith 1992) as in Kmetko et.
al. (2006), then the refined relative B-factor vs dose curve stays
within 0.5 B of a straight line with slope 1.31 B/MGy up to \~130 MGy.
Since SCALEIT does not support an A-factor but rather refines an overall
scale and B-factor it must converge on the best compromise between these
two parameters to fit data modified by an A-factor. A similar A-factor
treatment of lysozyme data (to 1.4 Å resolution) fits a slope of 1.03
B/MGy, and both of these B/MGy slopes are in very good agreement with
the "coefficient of sensitivity" reported by Kmetko et. al. (2006), and
we can conclude that the equation derived by Howells et. al. is
consistent with the observations of these two subsequent papers.

So, what is an A factor? The impact of disorder on the diffraction
pattern is that one must multiply the structure factors of a perfect
lattice by the Fourier transform of the displacements of the atoms from
their ideal lattice points (u). If these displacements obey a Gaussian
distribution, then the A-factor must be zero, since the Fourier
transform of a Gaussian is another Gaussian and the Gaussian function
has the form F(s) = exp(-Bs^2^). However, other atomic displacement
distributions are possible, and the Fourier transform of a Lorentzian
distribution does have the form F(s) = exp(-As). Since s =0.5/d, the
broad-ranged 1/d dependence of damage limits pointed out by Howells et.
al. (2005) strongly suggests that the atomic displacement distribution
induced by radiation damage is Lorentzian. If it were a Gaussian, one
would expect a 1/d^2^ dependence.

How could so many random changes not result in a Gaussian distribution
as dictated by the Central Limit Theorem? The only way to avoid the
Central Limit Theorem is if the process generating the displacements is
not random, but rather dominated by a single mechanism. Although it is
impossible to be certain of this mechanism from the slope of a line on a
graph, the introduction of a point defect in a soft but incompressible
crystal lattice does result in a distribution of atomic displacements
that closely resembles a Lorentzian (not shown). That is, the defect
will pull its neighbors out of place, and they, in turn will pull their
neighbors out of place by a lesser amount. In general, the histogram of
the atomic displacements will depend on the relative spring constants
holding the lattice points together, but an incompressibility constraint
(springs along the diagonal lattice vectors) results in a broad-tailed
Lorentzian-like distribution. A simple example of this can be seen by
pushing a pointed object into a block of rubber.

Therefore, the general mechanism of resolution loss due to radiation
damage in frozen hydrated biological samples may well simply be
mechanical strain propagating out through the soft material from the
damage sites. The width of the Lorentzian distribution of atomic
displacements will be proportional to the force acting at the defect
site, and the value H has the reciprocal units of the Young's modulus.

In order to incorporate this exponential decay into our expression for
spot intensity, we must integrate intensity over some exposure time (t),
which might as well be infinity as we will consider the accumulation of
background scattering later. The total amount of photons that can be
extracted from a spot (I~max~) will correspond to some "effective"
exposure time (t~eff~) that would give a spot with intensity I~max~ if
there were no such thing as radiation damage (equation 1). This t~eff~
will be the integral of the time-dependent fractional reduction in spot
intensity due to radiation damage. Integrating our expression for dose
and converting photon energy into wavelength (eE~ph~ = hc/λ = J/photon):

(10a)
$\frac{I_{\text{max}}}{\frac{I}{t_{\text{eff}}}} = \int_{0}^{\infty}{\text{ exp}( - 4\pi D(t)\text{ H s})dt}$

(10b)
$t_{\text{eff}} = \frac{0\text{.}1 \cdot \lambda\text{ R}\rho}{3\pi\text{ h c I}_{\text{beam}}\text{ H s }(\text{1-T}_{\text{sphere}}(\text{0,}\mu_{\text{en}}\text{,R}))}$

Where:

t~eff~ - the "effective" exposure time that would give I~max~ in the
absence of radiation damage (seconds)

I~max~ - maximum possible spot intensity due to radiation damage limits
(photons)

I - spot intensity given by Equation 1 (no radiation damage)

D(t) - dose accumulated at time t (Gy)

H - Howells's coefficient (11 x 10^-3^ Å/MGy)

s - reciprocal scattering vector length (sinθ/λ in 1/Å)

0.1 - conversion from Å to m, g/cm^3^ to kg/m^3^ and MGy to Gy

λ - x-ray wavelength (Å)

h - Plank's constant (6.626 x 10^-34^ J∙s)

c - speed of light (299792458 m/s)

R - radius of the spherical crystal (m)

ρ - density of crystal (\~1.2 g/cm^3^)

I~beam~ - incident beam intensity (photons/s/m^2^)

μ~en~ - mass energy-absorption coefficient of sphere material (m^-1^)

substituting all these expressions into equation 1, we get:

(11)$I_{\text{max}} > (s) \simeq \frac{4 \cdot \text{10}^{\text{29}}\text{ r}_{e^{2}}{\rho R}^{4}\lambda^{4}}{9\pi\text{ hc H s n}_{\text{ASU}}\text{ M}_{r}V_{M^{2}}}\left( \frac{1\text{.}\text{05} \cdot \pi}{\text{4 s λ}} - \frac{1}{2} \right) \cdot \frac{T_{\text{sphere}}(\text{ 2θ ,μ,R})}{1 - T_{\text{sphere}}(\text{0,μ}_{\text{en}}\text{,R})} \cdot \frac{f_{a}(s) >^{2}\text{exp}( - \text{2Bs}^{2})}{M_{a} >}$

Where:

\<I ~max~ \> (s) - maximum possible average spot intensity (photons) at
"s"

s - sin(θ)/λ or 0.5/d if d is the d-spacing of interest (Å^-1^)

θ - Bragg angle

r~e~ - classical electron radius (2.818 x 10^-15^ m)

10^29^ - conversion from m^4^ to Å^4^, g/cm^3^ to kg/m^3^ and MGy to Gy

ρ - density of crystal (\~1.2 g/cm^3^)

R - radius of the spherical crystal (m)

λ - x-ray wavelength (Å)

h - Plank's constant (6.626 x 10^-34^ J∙s)

c - speed of light (299792458 m/s)

H - Howells's coefficient (11 x 10^-3^ Å/MGy)

n~ASU~ - number of proteins in the asymmetric unit

M~r~ - molecular weight of the protein (Daltons or g/mol)

V~M~ - Matthews's coefficient (\~2.4 Å^3^/Dalton)

\<ƒ~a~(s)^2^\>number-averaged squared protein atom structure factor
(electron units^2^)

\<M~a~\> - number-averaged atomic weight of a protein atom (\~7.1
Daltons)

B - average (Wilson) temperature factor (Å^2^)

μ - attenuation coefficient of sphere material (m^-1^)

μ~en~ - mass energy-absorption coefficient of sphere material (m^-1^)

Note that the incident beam intensity is missing from this equation
because by the maximum possible spot intensity was integrated out to an
infinite exposure time. Also note that one R in the R^4^ term is
effectively cancelled by the (1-T) term for small crystals and also that
one λ in the term is roughly cancelled (within \~10% in typical cases)
by the \<LP\> correction.

**Minimum useful crystal size:**

At this point we are now prepared to calculate the size of the smallest
protein crystal that can ever be expected to produce useful spots at a
desired resolution. For example, if we require a signal-to-noise ratio
of 2 in the outer resolution bin of say 2 Å, then the average spot in
the 2 Å bin must contain at least four photons[^22], and Equation (11)
predicts that with a 1 Å x-ray beam a 1 μm wide perfect[^23] lysozyme
crystal[^24] will scatter an average \<I~max~\> of four photons each
into 2 Å spots before they fade away completely. This limit is
independent of exposure time since the total integrated number of
photons diffracted will level off at infinite dose. We are also
presently only considering photon-counting statistics and ignoring all
other sources of noise, including background scattering.

Theoretically, however, background can be virtually eliminated with a
perfect crystal, a near-zero emittance beam focused on a noiseless
detector with very small pixels and very fine rotation steps. It is
doubtful that such a device will ever be built or that perfect protein
crystals will ever be common, and this makes the 1.0 μm wide lysozyme
crystal a fundamental limit. That is, unless some way is found to
mitigate global damage, a lysozyme crystal smaller than 1 μm will never
yield a complete data set to 2 A. Crystals with larger unit cells will
have to be bigger than 1 μm. For example, a lysozyme crystal with a
Wilson B factor of 20 Å^2^ must be 2.4 μm wide to produce 4-photon spots
in the 2 Å bin, and a 10 MDa asymmetric unit with V~M~ = 2.4 must be at
least 13.4 μm wide to produce 4-photon spots at 3.5 Å.

The wavelength dependence of the minimum crystal size is also

Note that these are minimum requirements. The reality of finite mosaic
spread and pixel size set a practical lower limit on how much background
scattering will contribute to a given spot measurement, and this means
that more than four Bragg-scattered photons will be required for a
desired signal-to-noise ratio.

**Background scatter:**

Scattering from air and other non-crystalline substances was calculated
by:

\(#\)
$I_{\text{bg}} = \Omega_{\text{pixel}} \cdot I_{\text{beam}} \cdot t \cdot r_{e^{2}}\frac{N_{A}\rho \cdot V}{M_{r}} \cdot P \cdot A \cdot f_{\text{bg}}(s)^{2}$

I~bg~ - intensity of background scattering (photons/pixel)

Ω~pixel~ - solid angle subtended by a pixel (steradians/pixel)

I~beam~ - intensity of the incident beam (photons/s/m^2^)

t - exposure time (s)

r~e~ - classical electron radius (2.818 x 10^-15^ m/electron)

N~A~ - Avogadro number (6.022 x 10^23^ molecules/mol)

ρ - density of the substance (g/cm^3^)

V - volume of substance exposed to the beam (cm^3^)

M~r~ - formula weight of the substance (g/mol)

P - polarization factor

A - absorption factor

f~bg~ - structure factor of the substance (electrons)

s - reciprocal scattering vector length (sin(θ)/λ)

Note that the N~A~ρV/M~r~ term is simply the number of molecules of the
substance in the beam. For monatomic gasses, f~bg~ is the atomic
structure factor, but for more complex substances such as air (a
diatomic gas mixture), liquids and amorphous solids the interference
between scattering from different atoms must be accounted for in f~bg~.
In general, f~bg~ is the convolution of atomic structure factors with
their respective pair distribution functions, but in practice it is
simplest to calibrate f~bg~ by measuring the scattering from a known
quantity of the substance of interest. For this simulation, calibrated
values of f~bg~ vs s for several substances such as helium, air, liquid
water and Paratone-N oil were stored in text files and a simulation run
specified the thickness, density, and formula weight of the substance
exposed to the x-ray beam to generate a background level on an absolute
scale. For example, a 35 mm air[^25] gap between the collimator and
beam-stop will scatter 434 photons/s into a low-angle 100 μm wide pixel
that is 180 mm away from the sample area[^26] when the flux is 10^12^
photons/s[^27].

**Diffuse scattering:**

Amorphous substances are not the only thing that contributes to
background scattering. An imperfect crystal will also contribute to
background since the number of photons scattered by the atoms in the
crystal is fixed, but the spot intensities depend on the B factor, and
these "extra photons" must go somewhere. In general, the diffuse
scattering from a crystal can contain information about the molecular
motions that lead to excursions from a perfect lattice, but no attempt
was made to simulate this here. Instead, only the first-order
(centrosymmetric) term of the diffuse scattering was implemented:

\(#\)
$I_{\text{ds}} = \Omega_{\text{pixel}} \cdot I_{\text{beam}} \cdot t \cdot r_{e^{2}}\frac{V_{\text{xtal}}}{V_{\text{ASU}}} \cdot P \cdot A \cdot \sum_{a}^{}{f_{a}(s)^{2}(\text{1-exp}( - \text{2As} - \text{2B}_{a}s^{2}))}$

I~ds~ - intensity of the crystal diffuse scattering (photons/pixel)

Ω~pixel~ - solid angle subtended by a pixel (steradians)

I~beam~ - intensity of the incident beam (photons/s/m^2^)

t - exposure time (s)

r~e~ - classical electron radius (2.818 x 10^-15^ m)

V~xtal~ - volume of the crystal (m^3^)

V~ASU~ - volume of the crystal asymmetric unit (m^3^)

P - polarization factor

A - absorption factor

∑ - is over each atom in the asymmetric unit

*f*~a~ - structure factor of atom a (electrons)

A - is the Debye A factor derived for radiation damage above

B~a~ - is the Debye-Waller temperature factor of atom a (Å^2^)

s - reciprocal scattering vector length (sin(θ)/λ)

Note that the diffuse scatter contributed by each atom corresponds to
the background scattering that atom would produce if it were a
monoatomic gas minus the photons that go into the Bragg spots, and the
intensity of this diffuse scatter increases as the d-spacing decreases,
making it more difficult to measure the weakening outer spots as the
average B factor increases.

**Compton scattering:**

In a similar way to the conservation of photons that leads to diffuse
scattering, the finite size of the atomic form factor implies that
photons scattered from electrons in the atom that do not constructively
interfere must also go somewhere. This is a fairly good approximation to
Compton scattering, which arises from all atoms (James 1947). The fine
structure of the inelastic scattering can contain information about the
energy level structure in the substance, but no attempt was made to
simulate this here. Instead, the following first-order approximation to
Compton scattering was used:

\(#\)
$I_{\text{Compton}} = \Omega_{\text{pixel}} \cdot I_{\text{beam}} \cdot t \cdot r_{e^{2}}\frac{V_{\text{xtal}}}{V_{\text{ASU}}}\frac{P \cdot A}{1 + \text{κ }(1 - \text{cos θ})^{2}}\sum_{a}^{}{Z_{a} - \left| f_{a}(s) \right|}$

I~ds~ - intensity of the crystal diffuse scattering (photons/pixel)

Ω~pixel~ - solid angle subtended by a pixel (steradians)

I~beam~ - intensity of the incident beam (photons/s/m^2^)

t - exposure time (s)

r~e~ - classical electron radius (2.818 x 10^-15^ m)

V~xtal~ - volume of the crystal (m^3^)

V~ASU~ - volume of the crystal asymmetric unit (m^3^)

P - polarization factor

A - absorption factor

κ - photon energy (keV) / 511 keV

∑ - is over each atom in the asymmetric unit

Z - atomic number of atom a (electrons)

*f*~a~ - structure factor of atom a (electrons)

s - reciprocal scattering vector length (sin(θ)/λ)

Note that the structure factor term is not squared here since this is
inelastic scattering and there is no constructive interference between
the scattering of different electrons. This equation accounts for
Compton scattering from atoms in the crystal, and not the disordered
atoms in the solvent channels and surrounding droplet. The Compton
scattering from these amorphous substances is already included in the
empirically calibrated value of f~bg~, provided the wavelength of the
simulation is similar to that used in the calibration.

**Spot shape:**

Equation 1 gives the integrated spot intensity in photons which is the
signal to be measured but the noise associated with that signal will
depend heavily on the shape of the spot as this determines how many
pixels are involved in the spot as well as how much one spot profile
resembles another[^28]. The shape of a diffraction spot is essentially
the convolution of six processes: beam profile, beam divergence,
spectral dispersion, crystal size and shape, crystal mosaic spread and
detector point-spread function. However, this is not a simple
convolution as it has certain geometric constraints. For example, parts
of the incident x-ray beam profile that do not intersect any part of the
crystal volume obviously do not contribute to diffraction spots, so the
crystal extent "cuts off" the beam profile, or *vice versa* if the beam
is smaller than the crystal. The convolution is further constrained by
Bragg's Law itself, which is never violated. For example, consider a
Laue diffraction pattern. Although the wavelength has a broad range this
will have no effect on spot shape if the mosaic spread of the crystal is
zero. This is because for a given d-spacing and crystal orientation
there will be one and only one sharply-defined wavelength for which the
Bragg condition is satisfied. This will change if the crystal rotates,
and a mosaic spread is equivalent to an isotropic rotation. This is why
a finite mosaic spread leads to radial streaking of Laue spots.

If all six of the above spot-broadening processes have "zero" width,
then the diffraction spot really is an incredibly sharp point, and as
the crystal rotates through the Bragg condition, all of the unit cells
will diffract simultaneously and instantaneously to produce a spot with
the integrated energy defined by Darwin's formula. This is an important
limiting case since improvements in beamline technology and crystal
quality can theoretically approach zero background. As this limit is
approached, radiation damage becomes the only factor limiting data
quality.

However, if the "width" of one of the above parameters is not zero, then
different parts of the crystal (mosaic domains) will diffract at
different times, giving rise to the so-called "rocking curve" as the
crystal is rotated (described below), but also giving rise to finite
spot size. Both of these factors determine how much background
scattering accumulates in the spot. We will now discuss each of these
effects in turn starting at the detector.

**Detector point-spread function**

The point-spread function (PSF) of any detector represents how the
signal from an absorbed photon spreads out in space during the detection
process, and is the spot shape that would be observed if the detector
were hit with an infinitely sharp x-ray beam. The shape of the PSF must
therefore convoluted with the "native" spot shape in order to reproduce
a realistic spot, and it was found in these studies that the PSF does
indeed dominate the visual appearance of spots. The PSF of a fiber optic
coupled CCD detector is not a Gaussian, but rather follows the equation
for the solid angle subtended by a pixel from a point source some
distance g above the pixel plane:

\(#\) PSF(r) = g/2π∙(r^2^+g^2^)^-3/2^

Where r is the distance in the pixel plane from the point directly under
the beam and g is \~30 μm for an ADSC Q315r detector. This value for g
was derived empirically by analyzing and averaging the pixels of 1800
spots generated by a \~20 μm x-ray beam[^29], and the above equation was
found to be accurate within experimental error out to \~2 mm from the
center of the spot, where the intensity had fallen five decades. This
PSF is very sharp, so care must be taken that it is properly integrated
over a pixel of interest. The analytic expression for the integral of
this PSF in Cartesian coordinates is:

\(#\)
$\iint_{}^{}{\text{PSF}(\text{x,y})\text{ dx dy} = \text{arctan}\left( \frac{\text{xy}}{g\sqrt{g^{2} + x^{2} + y^{2}}} \right)}$

and the integral of the PSF over a pixel is computed by summing the
value of this integral expression at the upper right[^30] and lower left
corners of the pixel and then subtracting the value at the upper left
and lower right corners. For example, if a sharp beam delivers 10^6^
photons to the detector, then 215 photons worth of signal will be
delivered to a 0.1 x 0.1 mm pixel located 1.0 mm in x and 0.5 mm in y
from the point of impact if the PSF characteristic distance g is 0.03
mm.

The physical meaning of g is not entirely clear, but it can be thought
of as an effective "scattering thickness" of the phosphor and taper.
That is, the visible light photons generated by an x-ray photon impact
will spread out uniformly and their intensity must obey the inverse
square law regardless of how much they are scattered by the medium[^31].
When these photons are grouped into pixels, the solid angle subtended by
the pixel determines how many photons enter it (the above equation).
However, in a fiber optic taper, the scattering length is not uniform
and photons will travel a much greater distance down the axis of the
taper (down the fibers) before being scattered than they will along the
other two axes (across the fibers). This means that the small amount of
scattering as the photons move down the taper will spread out a spot in
exactly the same way as a much thinner layer of powdered glass (with
thickness g). No doubt the phosphor itself also contributes to g, and it
may have some photon energy dependence, but this hypothesis was not
tested in this work.

**Interaction of an x-ray beam with a real detector:**

The sensitivity of a typical x-ray detector changes with incident photon
energy and angle of incidence. This is because the front face of most
any x-ray detector consists of some kind of protective window material,
followed by the x-ray sensitive layer, which has a finite thickness.
Some of the incoming photons will be lost in the front window and not
all of the photons that penetrate the window will be absorbed in the
x-ray sensitive layer. The fraction of the incoming beam that is
absorbed in the x-ray sensitive layer will depend on the absorption
coefficient (μ)[^32] of the materials used and the incidence angle.
Specifically, if the incoming beam deviates from perpendicular to the
detector surface by some angle β, then the effective thickness of both
the window and the x-ray sensitive layer increase by a factor 1/cosβ.
Note that in the common situation where the detector plane is
perpendicular to the direct beam path, β = 2θ where θ is the Bragg
angle, but we will use β here for generality.

In the case of most modern CCD detectors, the window is beryllium or
aluminized plastic and the x-ray sensitive layer is a finely powdered
Terbium-doped Gd~2~O~2~S phosphor on the order of 20-40 μm thick. Once
an x-ray photon is absorbed by a phosphor, a fixed fraction of its
energy[^33] is converted into visible light which is emitted in all
directions. Some of these visible light photons are picked up by the
"optical chain" of the detector[^34] and eventually a fraction of them
create storable electrons in the CCD chip. The number of CCD electrons
created for each incident x-ray photon is called the electro-optical
gain, and ranges from 5 to 7 in current product literature (ref?). Since
the initial flash of visible photons is isotropic and the losses in the
optical chain are fixed, the dependence of detector sensitivity on the
angle and energy of the incident x-ray photon are determined largely by
the thickness of the phosphor and window. The total electro-optical gain
is then given by:

\(#\) G~eo~ =
exp(-μ~w~t~w~/cosβ)(1-exp(-μ~phos~t~phos~/cosβ))∙E~ph~$\frac{T_{\text{bond}}\text{QE}_{\text{CCD}}}{W \cdot \text{TR}^{2}}$

where:

G~eo~ - electro-optical gain (CCD electrons/x-ray photon)

μ~x~ - absorption coefficient of window (w) or x-ray sensitive layer
(phos)

t~x~ - thickness of window (w) or x-ray sensitive layer (phos)

β - deviation of beam from normal incidence

E~ph~ - photon energy (eV)

W - energy required to produce a visible light photon (\~ 25 eV)

TR - fiber optic taper ratio (3.7)

T~bond~ - transmission of the phosphor-taper and taper-CCD bonds (\~70%)

QE~CCD~ - detective quantum efficiency of the CCD (\~25%)

The integer values placed in the image file ("area detector units" or
ADU) are the result of amplifying and digitizing the number of electrons
given by G~eo~∙photons. Usually, this amplifier gain is 3 or 4 electrons
per ADU, giving ADU = photons∙G~eo~/G~amp~. For example, consider a beam
containing 100 1 Å photons[^35] impacting a typical detector[^36] at
30°. On average, 96.1 of these photons will be absorbed by the phosphor,
which will generate 47,660 visible light photons. Using the bond
transmission and taper ratio indicated above, only 2,437 of these will
reach the CCD and produce 609 electrons of stored charge, giving a final
G~eo~ of 6.1 electrons/x-ray photon and a total of 152 ADU divided over
the spot if G~amp~ = 4. Although the photon losses are quite high, it is
important to note that the noise is still dominated by x-ray photon
counting statistics. That is, from experiment to experiment, the
intrinsic variability of a spot containing an average of 100 photons
will be 10%[^37], but the variability in producing an average of 47,660
photons is only 0.46% and the variability in generating 609 electrons is
4.05%. Propagating errors[^38], the total fractional noise is 10.8% or
the same signal-to-noise one would get with an average of 85.7 photons
incident upon a "perfect" detector, so the effective overall detective
quantum efficiency[^39] (DQE) is 85.7%. For a given optical chain
described by the above equation this DQE is independent of the average
number of incident photons, but there are other sources of noise such as
the CCD read-out noise and fractional errors in the pixel sensitivity
calibration that are not taken into account by this expression of the
DQE. Read noise and fractional noise will maximally affect very weak and
very strong spots respectively, and this has led to some
intensity-dependent definitions of the DQE, but these will not be used
here. Instead, the other sources of noise will be modeled explicitly and
what will be called "DQE" here is the combined effects of
photon-counting noise and electron-counting noise, which is more true to
the layman's concept of a DQE as the fraction of incident photons that
are "detected".

Note also that there is some optimal value[^40] of β for each photon
energy that maximizes G~eo~. Some additional energy dependence may arise
if visible photons generated at different depths in the phosphor layer
have different collection efficiencies, but this was not modeled here.

**Spot distortion at the detector:**

The finite depth of the detector also influences spot shape and spot
position. Consider a diffracted beam of width w hitting the detector
plane at angle β (where β = 0 at normal incidence). At the plane of the
detector surface the footprint of this beam will be w/cosβ long[^41] and
w wide[^42], and the increase in length 1/cosβ is called the obliquity
factor. If the x-rays are strongly absorbed by the phosphor, then
obliquity will define the spot shape and the centroid of the spot will
be the impact point of the center of the beam. But, if the x-rays are
highly penetrating and the phosphor layer is t~phos~ deep, then the beam
will continue to generate light as it passes through the phosphor, and
add another t~phos~tanβ to the radial spot size before exiting the
bottom of the phosphor layer. This is called the parallax factor and
both adds to the total length of the spot as well as shifts the centroid
of it. Normally, the intensity of the beam falls off as it moves through
the phosphor, so the effective depth is never the full t~phos~, but some
fraction of it: t~eff~. For this calculation, t~eff~ was assigned to the
depth at which half of the beam intensity had died away. The combined
obliquity and parallax (OP) effect is therefore given by:

ℓ~r~ = w~r~/cosβ + t~eff~ tanβ

ℓ~t~ = w~t~

t~eff~ = - cosβ / μ~phos~ ln((1+exp(-μ~phos~t~phos~/cosβ))/2)

Where:

ℓ~r,t~ - radial/tangential size of spot on detector

w~r,t~ - radial/tangential width of diffracted beam just before hitting
the detector

β - angle of incidence of the beam on the detector face (assumed in
radial direction)

μ~phos~ - mass energy absorption coefficient of phosphor

t~phos~ - phosphor thickness (μm)

t~eff~ - effective phosphor thickness (μm)

**Crystal size and beam profile:**

Consider now the origins of w, the width of the diffracted beam just
before it arrives at the detector. If the beam divergence, spectral
dispersion and mosaic spread are all zero, then w will not change as the
beam propagates from the crystal to the detector and the spot size will
be dominated by crystal and beam size effects. That is, the diffracted
beam originates in the volume of the crystal that is illuminated by the
incident beam and this volume will have the width of the incident
beam[^43] and the same thickness[^44] as the crystal. When viewed from
the detector at the spot location, this volume will still appear as wide
as the beam in the tangential direction but elongated in the radial
direction[^45]:

w~rx~ = ℓ~box~ cos2θ + ℓ~xtal~ sin2θ

w~tx~ = ℓ~box~

where:

w~rx~ - width of the beam in the radial direction due to the
crystal/beam size

w~tx~ - width of the beam in the tangential direction due to the
crystal/beam size

ℓ~box~ - linear size of the beam or crystal (whichever is smaller)

ℓ~xtal~ - linear depth of the crystal (along axis of incident beam)

θ - Bragg angle

In general, the radial size (but not the tangential size) of the
resulting spot on the detector surface will be longer still due to
obliquity and parallax, which was described above, but obliquity and
parallax depend on detector orientation, whereas the width of the beam
just before hitting the detector (w) does not.

**Beam Divergence:**

Since no real x-ray beam is perfectly collimated it must spread out to
some extent as it propagates through space. The rate of this spreading
out can be described as an angle γ which will be called "divergence"
here, but is also sometimes called "crossfire". As long as γ is a
relatively small angle, the increase in beam width after the beam
propagates another t meters through space will be tγ if γ is given in
radians[^46]. For example, if an x-ray beam with divergence 1
milliradian passes through a 0.1 mm hole, it will be 0.1 mm wide
immediately after the hole, but 1.1 mm wide one meter away from the
hole. This "hole" can also be a protein crystal and the diffracted ray
will "inherit" the divergence of the incident beam[^47]. That is,
consider a crystal with zero mosaic spread rotating in an incident beam
with divergence γ. The incident beam can be thought of as a collection
of zero-divergence beams that all impact the crystal at different
angles. For each of these "perfect" beams, the divergence of the
diffracted ray is also zero since Bragg's Law is satisfied at one and
only one angle of crystal rotation, and if the crystal is stationary no
more than one of these "perfect" beams will produce a diffracted ray.
However, as the crystal is rotated each of the "perfect" beams will
produce a diffracted ray in turn as each of them satisfies Bragg's Law,
and the distribution of diffracted angles is the same as the
distribution of incident angles. In fact, divergence can be thought of
as equivalent to a spindle rotation in a zero-divergence beam where the
detector rotates with the crystal. Since divergence is generally
different in the horizontal and vertical directions, they can each be
though of as a separate rotation and the effect of beam divergence on
spot shape is simply to add to the existing beam width to the width
induced by divergence:

w~h~ = t~air~ γ~h~

w~v~ = t~air~ γ~v~

where:

w~h~ - width of the beam in the horizontal direction due to divergence

w~v~ - width of the beam in the vertical direction due to divergence

t~air~ - distance from the crystal to the spot on the detector

γ~h~,γ~v~ - horizontal, vertical divergence, respectively (in radians)

Note that for the obliquity and parallax (OP) calculation, the angle of
incidence β is not generally directed in the horizontal or vertical, so
when adding the OP effects to the divergence effects the components of β
in the horizontal and vertical directions must be used to calculate the
OP component in the horizontal and vertical separately. A similar
decomposition of β may be needed to account for OP in the radial and
tangential directions if the detector plane is not normal to the
incident beam.

If the x-ray beam is focused on the detector, then the divergence will
be negative and the spot on the detector can indeed be smaller than the
crystal. But, for any real optical system the beam cannot have an
infinitely small focal spot and the product of

**Mosaic spread:**

Now consider the effect of mosaic spread on spot shape. By definition,
the mosaic spread is a rotation, and rotating the crystal cannot change
the d-spacing of a given reciprocal lattice point (relp). That is, the
relp is constrained to lie on the surface of a sphere of radius d\*
centered on the origin of reciprocal space, regardless of the mosaic
spread or any other crystal rotation. So, in the limit of "infinite"
mosaic spread the diffraction spot becomes a powder ring[^48]. However,
mosaic spread generally refers to small rotations so the range of a
particular relp due to mosaicity is a tiny patch of this sphere's
surface, which is called a spherical cap. Each point in the cap
represents a slight rotation relative to the "average" crystal
orientation, and the mosaic domains of the crystal can be represented as
a collection of points bounded by the cap. This means that the mosaic
spread has no effect on the radial size of the spots[^49] but only
broadens the spots in the tangential direction. This broadening must
still satisfy Bragg's Law, so the broadening induced by a (small) mosaic
spread is:

w~tm~ = t~air~ (2η) 2sinθ

where:

w~tm~ - width of the beam in the tangential direction due to mosaic
spread

2η - the mosaic spread (full width of angle distribution in radians)

the factor 2sinθ arises because the mosaic rotation is about the origin
of reciprocal space (amplified by the length of d\*), but the diffracted
ray evolves from the center of the Ewald sphere (de-amplified by the
length of λ\*), and Bragg's Law dictates that the d\* vector is a
fraction 2sinθ of the length of λ\*. The value 2η is kept separate here
because the symbol η is generally used to describe the radius of the
spherical cap, but we are interested in the full width of the mosaic
spread.

**Spectral dispersion and unit cell variation:**

No x-ray beam is perfectly monochromatic, and the range of available
wavelengths Δλ relative to the average wavelength λ is called the
spectral dispersion Δλ/λ. Dispersion has no effect on the tangential
spot size because changes in wavelength do not rotate the crystal, but
the increase in radial spot size due to spectral dispersion can be
obtained by differentiating Bragg's Law:

λ = 2 d sinθ and Δλ/Δθ = 2 d cosθ → Δθ = Δλ/λ tanθ

if θ is in radians, then the take-off angle of the spot (2θ) changes by
Δ(2θ) \~ 2 Δθ and:

w~rd~ = t~air~ Δ(2θ) = t~air~ Δλ/λ 2tanθ

where:

w~rd~ - width of the beam in the radial direction due to spectral
dispersion

Δλ/λ - fractional spectral dispersion (full width)

It should be noted that although mosaic spread formally cannot enlarge
spots in the radial direction, any variability of unit cell dimensions
will enlarge spots in the radial direction in exactly the same way as
spectral dispersion:

w~ra~ = t~air~ Δa/a 2tanθ

where :

w~ra~ - width of the beam in the radial direction due to unit cell
non-uniformity

Δa/a - fractional variation of unit cell edge a.

There is no "differential amplification" factor of λ\* vs d\* here as
there was for mosaic spread because Δλ/λ and Δa/a are fractional
quantities.

**The total spot shape:**

Combining all of these spot shape effects together requires convoluting
the spot shapes that would result from each effect separately (see
below), but this convolution is complicated by spot partiality. For
example, consider a horizontal beam divergence large enough to make a
fully-recorded spot 10 pixels wide but we position the crystal so that
the relp is equally divided between two images. On one image, only the
"left" half of this 10-pixel wide spot will appear because half of the
available angles in the beam did not satisfy Bragg's Law, but do on the
next image, where the "right" half of the spot appears. These two spots
obviously have different shapes, and similar cut-off problems exist for
every combination of mosaic rotation dispersion and divergence.
Furthermore, all of the above expressions for spot shape as well as the
expression for the rocking width are first-order approximations and only
apply for relatively small values of mosaic spread, dispersion and
divergence. For example, in the absence of dispersion and divergence and
mosaic spread above \~0.5° the spots start to form little arcs, and
eventually turn into powder rings when the mosaic spread becomes 180°.
For this simulation, these cut-off and nonlinearity problems of spot
shape were solved by breaking up divergence, dispersion and mosaic
spread into sub-ranges small enough for the above expressions to apply
and performing the convolution for each combination of sub-ranges
individually.

**Rocking Range:**

The reflecting range of each of these sub-beam and sub-crystal
combinations was calculated using the equations derived by Greenhough
and Helliwell (1982):

\(#\) $\text{ΔΦ}_{\eta} = \left( \text{2η} \right)\text{  L sin 2θ}$

\(#\) $\text{ΔΦ}_{\lambda} = \Delta\text{λ/λ 2 L sin}^{2}\text{ θ }$

\(#\)
$\text{ΔΦ}_{\text{γh}} = \gamma_{H}\sqrt{\left( L^{2}\text{sin}^{2}\text{2θ} - 1 \right)}$

\(#\) $\text{ΔΦ}_{\text{γv}} = \gamma_{V}$

\(#\)
$\text{ΔΦ} = \text{L sin 2θ}\left( \text{2η} + \Delta\text{λ/λ tan θ} \right) + \sqrt{\left( L^{2}\text{sin}^{2}\text{2θ} - 1 \right)\text{ γ}_{H^{2}} + \gamma_{V^{2}}}$

Where:

ΔΦ~x~ - reflecting range (full range in radians) due to the effect of
"x"

2η - mosaic spread (full width of rotation distribution in radians)

L - Lorentz factor

Δλ/λ - spectral dispersion (full width)

γ~H~ - horizontal[^50] divergence (full width in radians)

γ~V~ - vertical divergence (full width in radians)

θ - Bragg angle

All of the ranges in these equations are full widths, as opposed to half
widths. Note the value 2η has been used for the full width of the
crystal mosaic spread. This is because η usually appears in the
literature as the radius of the spherical cap traced out by the
orientations of the mosaic domains, but the full mosaic spread is of
interest here. The last equation is an expression for the convolution of
all these effects if they all have a "top-hat" shape, the mosaic spread
is isotropic and the divergence profile has an elliptical shape. If, for
example, the divergence has a rectangular shape[^51], then in cases
where the relp passes through the Ewald sphere in a corner-to-corner
orientation, the true rocking range will be as much as 30% higher than
that predicted by this last expression.

**Shape of the Rocking Curve:**

The last section dealt with determining the "start" and "end" values of
a Bragg reflection, which determines the limits of integration needed by
data processing programs, but the distribution of intensity between
these extremes is also important for determining accurate crystal
orientation. Although often taken as the integral of a disk or sphere
passing through the Ewald sphere, the exact shape of the rocking curve
is more complex and indeed is different for every reflection. We shall
begin as above by taking the influence of each effect in turn.

If all other parameters are negligible, the rocking curve due to a
top-hat distribution of spectral dispersion or either vertical or
horizontal divergence is itself a top-hat distribution. That is, the
integral of spot intensity plotted against Φ is zero up until the
"starting" Φ, unity at and after the "ending" Φ and forms a straight
line between these two points. For example, in the case of zero mosaic
spread, zero dispersion and zero horizontal divergence one can consider
a finite vertical divergence to be a "fan" of evenly-spaced and
zero-divergence beams impinging on the crystal. In the case of vertical
divergence[^52] the "fan" has the same axis as the Φ rotation for every
spot and the effect is obvious: as the crystal is rotated, each of these
beams will satisfy the Bragg condition in turn, contribute an equal
amount of x-rays to the spot and their influence will be evenly divided
over the Φ rotation. Hence, the rocking curve has a top-hat shape. An
equivalent geometric construction is to consider the relp to be an
evenly-spaced series of "sub-relps" forming a line in reciprocal space
centered on the "average" relp position. For a drawing, see Arndt and
Wonacott (1977). The same model can be made of the horizontal
divergence. In this case, the row of sub-relps is not aligned with the Φ
rotation, but they are still evenly spaced and must pass through the
Ewald sphere one at a time. The spacing between the sub-relps in terms
of Φ rotation will depend on the orientation of this line in reciprocal
space, but as long as the full rocking range is small enough for the
Ewald sphere surface to be considered flat, the spacing must be even and
the incremental intensity per unit ΔΦ will be a constant. A similar
construction can be made for spectral dispersion: an equivalent model to
constructing a range of equally-spaced Ewald spheres[^53] is to break up
each relp into a radial distribution of sub-relps. These, again form an
evenly-spaced line of sub-relps that pass through the Ewald sphere at a
constant rate.

Since the only difference between the shape of all these top-hat rocking
curves when plotted against Φ is the distsance between Φ~start~ and
Φ~end~, it is convenient to normalize the progression into the rocking
curve (Φ-Φ~start~) by dividing it by ΔΦ to yield ε. Once this is done,
the plot of "partiality" p[^54] against ε is simply a line with slope
equal to one:

\(#\) $p_{\text{line}}(\varepsilon) = \varepsilon$

Where:

p~line~ - cumulative fraction of full intensity recorded

ε - (Φ-Φ~start~)/ΔΦ normalized coordinate of rocking curve

The rocking curve due to mosaic spread, however, is not as simple since
the spherical cap is a two-dimensional object in reciprocal space, but
the rocking curve is still the integral of the shape of the relp as it
passes through the Ewald sphere. Provided the mosaic spread is isotropic
and small enough for the Ewald sphere to be considered flat, this
rocking curve will be the integral of a disk:

\(#\)
$p_{\text{disk}}(\varepsilon) = 2\varepsilon\sqrt{1 - 4\varepsilon^{2}} + \frac{\text{arcsin}2\varepsilon}{\pi} + \frac{1}{2}$

Where:

p~disk~ - partiality of a disk-shaped relp

Note that both p and ε are normalized quantities that range from 0 to 1.
Each spot will have a different ΔΦ, and this will "stretch" the rocking
curve when plotted against Φ, but normalizing any mosaicity-only rocking
curve using ΔΦ~η~ from Equation (#) will yield a curve that follows
Equation (#). However, this is only true when the divergence and
dispersion are zero. If either are significant, then not only does ΔΦ
increase, but the shape of the rocking curve will lie somewhere between
equations (#) and (#). Take for example a finite horizontal divergence
and a finite mosaic spread. In this case, instead of a row of points the
relp will be a stack of disks, forming an oblique cylinder in reciprocal
space and the rocking curve will be the integral of this volume as it
passes through the Ewald sphere. The exact shape will therefore depend
on the angle made by this oblique cylinder with the Ewald sphere
surface, so it is easy to see how each spot essentially has a unique
rocking curve shape. Formally, the total rocking curve is the
convolution of the curves arising from the mosaic spread, divergence and
dispersion. If all these effects have similar ΔΦ, then the total rocking
curve is reasonably approximated by that of a sphere passing through the
Ewald sphere, but this assumption breaks down if one effect dominates.

**Gaussian kernels in spot shape:**

The convolution of two arbitrary functions is a computationally
expensive process, but the convolution of two Gaussians can be performed
analytically by taking the quadrature sum of the two parent Gaussian
widths. This is because any convolution is a product in Fourier space
and the Fourier transform of a Gaussian is another Gaussian with a width
that is the inverse of the width in real space. Since the width term is
in the denominator of the exponent the real space width of the
convolution is the square root of the sum of the squares of the two
real-space Gaussian widths. For two-dimensional Gaussians with different
widths in different directions, the convolution is a Gaussian whose
covariance matrix is the sum of the covariance matrices of the two
convoluted Gaussians:

a = cos^2^(θ)w~r~^2^+sin^2^(θ)w~t~^2^

b = -sin(2θ)w~r~^2^ +sin(2θ)w~t~^2^

c = sin^2^(θ)w~r~^2^+cos^2^(θ)w~t~^2^

G(x,y) = exp(-(ax^2^ + bxy + cy^2^))

G~1~⊗ G~2~ = exp(-((a~1~+a~2~)x^2^ + (b~1~+b~2~)xy + (c~1~+c~2~)y^2^))

w~r~ = ((a+c - sqrt(a^2^+b^2^+c^2^-2ac))/2)^½^

w~t~ = ((a+c + sqrt(a^2^+b^2^+c^2^-2ac))/2)^½^

θ = ArcTan(-b/(a-c))/2

where subscripts denote the two Gaussians to be convoluted, w~r~ and
w~t~ are the major and minor widths of a Gaussian, respectively, θ is
the angle between the major axis and the x axis, and ⊗ is the
convolution operator. Note that upon reducing to the one-dimensional
case (θ=0;w~t~=0), the value of w~r~ for the convolution reduces to the
quadrature sum of w~r~ of the two convoluted Gaussians.

The value of the above convolution algorithm in computing spot shapes
lies in the fact that most any function can be approximated by a sum of
Gaussians, and the convolution of a sum is the sum of the convolutions
of each element of the sum. For example, a top-hat function[^55] can be
approximated by a row of evenly-spaced Gaussians that all have the same
height and width. Moreover, the row of Gaussians is a very good
approximation to a top-hat function convoluted with a Gaussian. For
purposes of spot shape calculation all that is required is that the
total idealized intensity laid down on the image be equal to the total
integrated spot intensity given by Darwin's formula and that the
deviations from the ideal spot shape be small when compared to
photon-counting noise and the size of a pixel.

Take for example an x-ray beam divergence with a top-hat shape. If the
divergence was high enough to spread a given spot over three pixels,
then the incident beam was divided into three independent beams, each
with one third of the flux, one third of the divergence, and with a
Gaussian profile. One beam took the same direction as the original beam,
and the outer two deviated by 1/3 of the divergence in either direction.
This algorithm places the outer edges of the row of Gaussians at the
same position as the edges of the original top-hat, which represents the
least-squares best fit of a train of three Gaussians to a top-hat
function. Continuing the example, if the spectral dispersion of the beam
was large enough to spread the Bragg spots over more than one pixel,
then each of the three divergence "sub-beams" was subsequently broken up
into more sub-beams, each with different photon energy. In a similar
way, large mosaic spreads were broken down into "sub-crystals" with
different orientations. The spots produced by each combination of
sub-beam and sub-crystal were then calculated using Bragg's Law, and the
resulting "sub-spots" all plotted separately and summing to the total
integrated intensity of the full Bragg reflection. This algorithm can
lead to a combinatorial explosion of sub-spots, especially for spots
that lie close to the so-called "blind region" near the rotation axis in
reciprocal space. Such spots are usually rejected by integration
software, so this simulation was set up to neglect spots that broke down
into more than 10,000 sub-spots.

[^1]: Considered to be horizontal

[^2]: Opposing gravity

[^3]: Applied in the order z, y then x in keeping with the convention
    used in MOSFLM

[^4]: Although the reciprocal unit cell angles are not the same as the
    real cell angles, any rotation in real space corresponds to an
    identical rotation in reciprocal space.

[^5]: 100 μm wide crystal (sphere), F = 170 electrons, 50 Å cell edge, A
    = 96%

[^6]: 10^12^ ph/s, 100 μm wide beam (round), 1 Å wavelength, rotating at
    1°/s, L = 2.2, P = 0.92

[^7]: Nor with the "Lorentz factor" in the theory of relativity

[^8]: Half the angle between the direct beam path and the diffracted ray

[^9]: Which has length d\* = 1/d where d is the d-spacing

[^10]: 2sinθ \> sin2θ for 0° \< θ \< 90°

[^11]: Generally the z-axis defined here

[^12]: Generally the y-z plane defined here

[^13]: 2θ = 90° and ψ = 0°

[^14]: C~31.8~N~8.56~H~49.8~O~9.54~S~0.249~, protein density 1.35
    g/cm^3^, 1 Å x-rays

[^15]: Located outside the "blind region"

[^16]: \<*f*~a~\> ≈ Boron, M~a~ = 7.1

[^17]: for V~M~ = 2.5, and s equivalent to a Bragg spacing of 2.5 Å

[^18]: Absorption is one component of attenuation, scattering is another
    mechanism of attenuation

[^19]: Including I+ and I- as separate observations

[^20]: The limit of 1-exp(-x) as x → 0 is x

[^21]: Continuous slowing down approximation

[^22]: I/σ = 4/sqrt(4) = 2

[^23]: No mosaic spread and Wilson B = 0.

[^24]: M~r~ = 14.3 kDa, n~ASU~ = 1, V~M~ = 2.0

[^25]: Density: 1.2 mg/cm^3^, M~r~ \~ 28 g/mol, *f*~bg~ \~ 14 electrons
    inside 10 Å resolution

[^26]: Ω \~ (100 μm/180 mm)^2^

[^27]: Regardless of beam size since the volume of illuminated air is
    proportional to beam size

[^28]: the basic assumption of profile fitting

[^29]: Using a 15 μm round hole in a tantalum pinhole 85 mm from the
    detector surface with the beam divergence stopped down to 50 x 50
    μrad.

[^30]: Most positive values of x and y

[^31]: as long as the medium is uniform and absorption is negligible

[^32]: which depends on photon energy

[^33]: usually about 10% or one visible photon per \~25 eV of absorbed
    energy

[^34]: usually a fiber optic taper

[^35]: E~ph~ = 12398 eV

[^36]: t~w~ = 13 μm; μ~w~^-1^ = 610 μm ; t~phos~ = 40 μm; μ~phos~^-1^ =
    11 μm

[^37]: Signal = 100; noise = √100

[^38]: (σ~total~/I~total~)^2^ = (σ~x-ray~/N~x-ray~)^2^ +
    (σ~vis~/N~vis~)^2^ + (σ~electron~/N~electron~)^2^

[^39]: fraction of incident photons "detected"; DQE = (I/σ)~out~ /
    (I/σ)~in~

[^40]: 43.7° in the example case

[^41]: In the radial direction: that of the plane containing β

[^42]: In the "tangential" direction: normal to the beam and contained
    within the detector surface plane

[^43]: or the width of the crystal, whichever is smaller

[^44]: along the direction of the incident beam

[^45]: along the line containing the spot and the direct beam

[^46]: The so-called "small angle approximation"

[^47]: although it propagates in a new direction

[^48]: The intersection of the Ewald sphere with the d\* sphere

[^49]: In a monochromatic beam, powder rings are sharp

[^50]: along spindle axis

[^51]: As is the case if the divergence is controlled by orthogonal
    flat-bladed slits

[^52]: Defined here as normal to the rotation axis

[^53]: That is, with radii equally spaced between the maximum and
    minimum values of λ

[^54]: The fraction of the final full intensity recorded

[^55]: Also known as the "rectangular function" or "boxcar" function