Preface ....................................................... xiv
Notes for the Reader ......................................... xvii
PART ONE Fundamentals
1 On the Scattering of Electromagnetic Radiation by Atoms
and Molecules ................................................ 3
1.1 What is electromagnetic radiation? ...................... 4
1.2 Atoms are electrically polarized by electromagnetic
radiation ............................................... 7
1.3 Oscillating dipoles emit electromagnetic radiation ...... 8
1.4 The electrons in atoms and molecules scatter X-rays as
though they were unbound ............................... 10
1.5 The scattering of X-rays by molecules depends on
atomic positions ....................................... 11
1.6 Radiation detectors measure energy, not field
strength ............................................... 14
1.7 If the radiation being scattered is unpolarized, the
polarization correction depends only on scattering
angle .................................................. 14
1.8 The coherence length of the radiation used in
scattered experiments affects the accuracy with which
la can be measured ..................................... 15
1.9 Measurement accuracy also depends on transverse
coherence length ....................................... 16
Problems .................................................... 18
Appendix 1.1 Exponential notation, complex numbers,
and Argand diagrams .................................... 18
Appendix 1.2 The polarization correction for unpolarized
radiation .............................................. 19
2 Molecular Scattering and Fourier Transforms ................. 22
2.1 F(S) is a function of three angular variables .......... 23
2.2 Fourier series are a useful way to represent
structures ............................................. 24
2.3 In the limit of d = ∞, the Fourier series becomes
the Fourier transformation ............................. 26
2.4 The Great Experiment ................................... 28
2.5 The shift theorem leads to a simple expression for
the scattering of molecules ............................ 29
2.6 The scaling theorem: Big things in real space are
small things in reciprocal space ....................... 31
2.7 The square wave and the Dirac delta function ........... 32
2.8 Multiplication in real and reciprocal space: The
convolution theorem .................................... 34
2.9 Instrument transfer functions and convolutions ......... 37
2.10 The autocorrelation theorem ............................ 39
2.11 Rayleighs theorem ...................................... 40
Problems .................................................... 42
3 Scattering by Condensed Phases .............................. 44
3.1 The forward scatter from macroscopic samples is 90°
out of phase with respect to the radiation that
induces it ............................................. 44
3.2 Scattering alters the phase of all the radiation that
passes through a transparent sample .................... 47
3.3 Phase changes are indistinguishable from velocity
changes ................................................ 48
3.4 Polarizabilities do not have to be real numbers ........ 49
3.5 Atomic polarization effects are small .................. 50
3.6 The frequency dependence of polarizabilities can be
addressed classically .................................. 50
3.7 When the imaginary part of a is large, energy is
absorbed ............................................... 53
3.8 The refractive index of substances for X-rays is less
than 1.0 ............................................... 54
3.9 The wavelength dependences of the processes that
control light and X-ray polarizabilities are
different .............................................. 55
3.10 On the frequency dependence of atomic scattering
factors for X-rays ..................................... 56
3.11 Real X-ray absorption and dispersion spectra do not
look the way classical theory predicts ................. 57
3.12 The imaginary component of f can be determined by
measuring mass absorption coefficients ................. 59
3.13 Scattering can be described using scattering lengths
and cross sections ..................................... 60
3.14 Neutron scattering can be used to study molecular
structure .............................................. 61
3.15 Electrons are strongly scattered by atoms and
molecules .............................................. 64
3.16 Electrons are scattered inelastically by atoms ......... 65
Problems .................................................... 65
Appendix 3.1 Forward scatter from a thin slab .............. 66
Appendix 3.2 A classical model for the motion of electrons
in the presence of electromagnetic radiation ............ 67
Appendix 3.3 Energy absorption and the imaginary part of a .. 68
PART TWO Crystallography
4 On the Diffraction of X-rays by Crystals .................... 73
4.1 The Fourier transform of a row of delta functions is
a row of delta functions ............................... 74
4.2 Sampling in reciprocal space corresponds to
replication in real space (and vice versa) ............. 75
4.3 Crystals can be described as convolutions of
molecules with lattices ................................ 76
4.4 Lattices "amplify" Fourier transforms .................. 77
4.5 The Nyquist theorem tells you how often to sample
functions when computing Fourier transforms ............ 78
4.6 Lattices divide space into unit cells .................. 80
4.7 The minimal element of structure in any unit cell is
its asymmetric unit .................................... 83
4.8 The transform of a three-dimensional lattice is
nonzero only at points in reciprocal space that obey
the von Laue equations ................................. 84
4.9 The Fourier transforms of crystals are usually
written using unit cell vectors as the coordinate
system ................................................. 86
4.10 Bragg s law provides a second way to describe
crystalline diffraction patterns ....................... 87
4.11 Von Laue s integers are Miller indices ................. 89
4.12 Ewald s construction provides a simple tool for
understanding crystal diffraction ...................... 91
Problems .................................................... 92
Appendix 4.1 The Bravais lattices ........................... 93
Appendix 4.2 On the relationship between unit cells in
real space and unit cells in reciprocal space .......... 95
5 On the Appearance of Crystalline Diffraction Patterns ....... 96
5.1 Diffraction data are collected from macromolecular
crystals using the oscillation method .................. 96
5.2 Measured intensities must be corrected for systematic
error .................................................. 99
5.3 Radiation damage kills crystals ....................... 100
5.4 Diffraction patterns tend to be centrosymmetric ....... 100
5.5 Anomalous diffraction can provide useful information
about the chemical identities of atoms in electron
density maps .......................................... 102
5.6 Anomalous diffraction effects can be used to
determine the absolute hand of chiral molecules ....... 102
5.7 Crystal symmetry results in reciprocal space
symmetry .............................................. 103
5.8 Real crystals are not perfecdy ordered ................ 106
5.9 Disorder weakens Bragg reflections .................... 106
5.10 Disorder makes crystals scatter in directions that
are not allowed by von Laue s equations ............... 109
5.11 Thermal diffuse scatter need not be isotropic ......... 110
5.12 Average B-factors can be determined directly from
diffraction data ...................................... 113
5.13 Most crystals are mosaic .............................. 114
5.14 A single crystal structure can reveal the
alternative conformations of a macromolecule that is
polymorphic ........................................... 115
Problems ................................................... 116
Appendix 5.1 Debye-Waller factors and diffuse
scattering ............................................ 117
Appendix 5.2 Correlated motions and diffuse scatter in
one dimension ......................................... 120
Appendix 5.3 Random walks in two dimensions ............... 121
6 Solving the Phase Problem .................................. 123
6.1 The phases of reflections are measured by comparing
them to a standard .................................... 123
6.2 Macromolecular diffraction patterns can be phased by
adding heavy atoms to srystalls ....................... 125
6.3 The number of high-Z atoms per unit cell needed for
phasing is small ...................................... 125
6.4 The heavy atom isomorphous replacement strategy for
phasing requires the comparison of intensities
measured from different crystals ...................... 126
6.5 Anomalous data can also provide phase information ..... 128
6.6 Patterson functions display the interatomic
distances and directions of a crystal ................. 130
6.7 Macromolecular crystal structures cannot be solved
using Patterson functions alone ....................... 132
6.8 Heavy atom sites in derivatized crystals can be
located using difference Pattersons ................... 132
6.9 Atomic coordinates can be deduced from the Harker
sections of the Patterson functions ................... 133
6.10 Multiple-wavelength anomalous diffraction combines
anomalous and heavy atom phase determination in
a single experiment ................................... 135
6.11 Experimental error complicates the experimental
determination of phases ............................... 137
6.12 Experimental phase data specify phase probability
distributions ......................................... 139
6.13 The impact of phase errors on electron density maps
can be controlled ..................................... 141
6.14 The likelihood that the experiments done to phase a
diffraction pattern have produced reliable data can
be assessed statistically ............................. 143
6.15 Diffraction patterns can be phased by molecular
replacement ........................................... 144
6.16 Molecular replacement searches can be divided into a
rotational part and a translations part ............... 145
Problems ................................................... 146
Appendix 6.1 Heavy atom difference Pattersons and
anomalous difference Pattersons ....................... 147
7 Electron Density Maps and Molecular Structures ............. 150
7.1 Experimental electron density maps display the
variation in electron density within the unit cell
with respect to the average ........................... 150
7.2 Electron density maps are contoured in units of
sigma ................................................. 151
7.3 The point-to-point resolution of an electron density
map is roughly 1/|S|max ............................... 151
7.4 How high is high enough? .............................. 155
7.5 Macromolecular electron density maps having
resolutions worse than ~3.5 Ǻ are difficult to
interpret chemically .................................. 156
7.6 Solvent may be visible in macromolecular electron
density maps .......................................... 157
7.7 Initial models must be refined ........................ 158
7.8 R-factors are used to measure the consistency of
molecular models with measured diffraction data ....... 159
7.9 Free-R is useful tool for validating refinements ...... 161
7.11 Regions ehere models do not correspond to electron
density maps can be identified using difference
electron density maps ................................. 163
7.12 Experimental electron density maps can be improved
by phase modification arfd extension .................. 164
7.13 Solvent flattening and the Nyquist theorem ............ 166
7.14 Let the buyer beware .................................. 167
Problems ................................................... 170
Appendix 7.1 The inverse Fourier transform of the
spherical aperture function ........................... 170
Appendix 7.2 Estimating the R-factor of crystal
structures that are perfect nonsense .................. 172
Appendix 7.3 The difference Fourier ....................... 173
PART THREE Noncrystallographic Diffraction
8 Diffraction from Noncrystalline Samples .................... 179
8.1 X-ray microscopy can be done without lenses ........... 179
8.2 The Fourier transform of a projection is a central
section ............................................... 180
8.3 Continuous transforms can be inverted by solvent
flattening ............................................ 182
8.4 Can the structures of macromolecules be solved at
atomic resolution by X-ray imaging? ................... 183
8.5 Solution-scattering patterns provide rotationally
averaged scattering data .............................. 185
8.6 Solution-scattering experiments determine length
distributions and vice versa .......................... 186
8.7 Molecular weights and radii of gyration are easily
extracted from solution-scattering profiles ........... 188
8.8 Solution-scattering profiles are strongly affected
by the scattering length densities/electron
densities of solvents ................................. 191
8.9 Shape scatter dominates most solution-scattering
curves ................................................ 192
8.10 Measured radii of gyration are contrast-dependent ..... 194
8.11 Contrast variation experiments are more easily done
using neutron radiation than X-rays ................... 195
8.12 It is surprisingly difficult to compute solution-
scattering curves ..................................... 196
8.13 Useful models for the shapes of macromolecules can
be derived from solution-scattering curves ............ 198
8.14 The experimental apparatus for small angle
scattering resembles that used for coherent
diffractive imaging ................................... 199
8.15 Molecular weights and radii of gyration can be
measured by light scattering .......................... 201
Problems ................................................... 203
Further Reading ............................................ 204
Appendix 8.1 Derivation of the Debye equation .............. 204
Appendix 8.2 Small angle scatter and the radius of
gyration ............................................... 205
PART FOUR Optical microscopy
9 Image Formation Using Lenses ............................... 209
9.1 Magnified images of small objects can be produced
two different ways .................................... 209
9.2 The direction propagation of light can change at
index of refraction boundaries ........................ 211
9.3 In geometrical optics, waves are replaced by rays ..... 212
9.4 Curved glass surfaces can focus light ................. 213
9.5 The lens law .......................................... 213
9.6 The lens law is valid for paraxial skew rays .......... 216
9.7 Focusing lenses produce images ........................ 217
9.8 Lens performance is limited by aberration ............. 219
9.9 Spherical aberration is the most important of the
Seidel aberrations .................................... 220
9.10 There are four other Seidel aberrations ............... 222
9.11 Parallel bundles of rays focus in the back focal
plane of ideal lenses ................................. 223
9.12 The light wave in the back focal plane is the
Fourier transform of the light wave at the object
plane ................................................. 224
9.13 As light travels from the back focal plane to the
image plane it gets Fourier transformed again ......... 226
9.14 The images of points have the functional form Ji
(ж)/л: ................................................ 227
9.15 Rayleigh s criterion is used to estimate the
resolution of microscopes ............................. 229
9.16 Microscopes display depth of field and depth of
focus ................................................. 231
Problems ................................................... 233
Further Reading ............................................ 234
Appendix 9.1 Derivation of the Lens Law .................... 234
Appendix 9.2 Optical path length and spherical aberration .. 238
Appendix 9.3 The Fourier transform of a circular aperture .. 241
10 The Light Microscope ....................................... 244
10.1 Incoherent light is the illumination of choice
for ordinary microscopy ............................... 244
10.2 Incoherent image formation is easily described in
Fourier terms ......................................... 245
10.3 The Fourier transform of the square of any function
is the autocorrelation function of its Fourier
transform ............................................. 248
10.4 Light absorption accounts for much of the contrast
in ordinary microscope images ......................... 249
10.5 Fluorescence plays an increasingly important role in
biological microscopy ................................. 250
10.6 Light scattering contributes to image contrast ........ 252
10.7 Dark field illumination can be used to image objects
that scatter light .................................... 253
10.8 Phase objects can be visualized using phase contrast
microscopy ............................................ 254
10.9 Confocal microscopy ................................... 256
10.10 The point spread function of a confocal microscope
is the product of two objective lens point spread
functions ............................................ 257
10.11 Three-dimensional images can be recovered from two-
dimensional microscopic images ....................... 260
10.12 Light microscopes can resolve points that are
closer together than the Rayleigh limit .............. 262
Problems ................................................... 267
Further Reading ............................................ 268
Appendix 10.1 The distribution of light energy
on-axis, and near-focus ............................... 268
PART FIVE Electron Microscopy
11 Lenses that focus Electrons ................................ 273
11.1 Biological electron microscopy is done using two
different kinds of EMs ................................ 274
11.2 The magnetic field of a one-turn coil can focus
electrons ............................................. 276
11.3 The lenses in EMs are solenoids ....................... 279
11.4 Magnetic lenses have focal lengths .................... 280
11.5 The optical properties of EMs can be worked out
using quantum mechanics ............................... 281
11.6 Aberration seriously degrades the performance of
magnetic lenses ....................................... 282
11.7 Spherical aberration limits the resolution of
magnetic lenses ....................................... 283
11.8 The resolution of the images produced by lenses that
have large spherical aberration coefficients can be
improved by reducing their limiting apertures ......... 284
11.9 Electron microscopes have large depths of field and
focus ................................................. 285
11.10 Focal length variation and spherical aberration are
important determinants of the contrast transfer
functions of EMs ..................................... 286
11.11 Chromatic aberration is a focal length effect ........ 287
11.12 Chromatic aberration suppresses the high-resolution
features of images ................................... 288
11.13 Chromatic aberration has many sources in EMs ......... 288
11.14 The Scherzer resolution of an image and its
information limit are not the same ................... 290
11.15 Both the chromatic and spherical aberration of
magnetic lenses can be corrected instrumentally ...... 291
Problems ................................................... 292
Further Reading ............................................ 292
Appendix 11.1 On the Chromatic Aberration of Magnetic
Lenses ............................................... 292
12 Image Formation in the Electron Microscope ................. 294
12.1 Aperture and phase effects account for most of the
contrast in EM images ................................. 294
12.2 Aperture contrast is best understood using cross
sections .............................................. 295
12.3 There is little structural information in high angle
electron scatter ...................................... 296
12.4 Aperture contrast images have bright backgrounds and
low resolutions ....................................... 298
12.5 EM stains enhance aperture contrast ................... 299
12.6 Inelastic scattering damages specimens ................ 301
12.7 The phases of electron waves change as they pass
through objects ....................................... 301
12.8 TEMs are naturally phase contrast microscopes ......... 303
12.9 Underfocused images are better than in-focus images ... 304
12.10 CTFs have a big impact on high-resolution EM images .. 306
12.11 The CTFs relevant to an EM image can be determined
after the fact ....................................... 307
12.12 CTFs can be examined using optical diffractometers ... 309
12.13 CTF effects can be reversed .......................... 309
12.14 The transverse coherence of the electron beam
affects image quality ................................ 312
12.15 Partial coherence suppresses image detail ............ 313
12.16 Source brilliance set a practical upper limit on
transverse coherence ................................. 314
12.17 The sources used in electron guns differ
significantly in brilliance .......................... 315
12.18 Users can control the transverse coherence lengths
of EM beams .......................................... 315
Problems ................................................... 316
Further Reading ............................................ 317
Appendix 12.1 The effects of thermal motions on molecular
scattering profiles ................................... 317
Appendix 12.2 The images of weak-phase objects ............. 319
13 Electron Microscopy in Three Dimensions .................... 320
13.1 Three-dimensional reconstructions can be done in
reciprocal space ...................................... 321
13.2 Interpretable images cannot be obtained by direct
inversion of central section data sets ................ 322
13.3 If the transform of an object is known on a lattice
of appropriate dimensions, its transform can be
evaluated anywhere in reciprocal space ................ 324
13.4 В is a product of sine functions ...................... 326
13.5 F can be estimated by matrix inversion ................ 327
13.6 Error propagation determines resolution ............... 328
13.7 Tilt data sets are best described using cylindrical
polar coordinates ..................................... 329
13.8 Tilt data can be reduced one plane at a time .......... 330
13.9 Only a finite number of Gn values that must be taken
into account in a tomographic reconstruction .......... 332
13.10 Symmetry reduces the number of images required to
reconstruct the structure of an object to any given
resolution ........................................... 334
13.11 Symmetry determines the orders of the Bessel
functions that contribute to each layer plane in
a helical diffraction pattern ........................ 334
13.12 At modest resolutions, a single Bessel order may
account for the intensity observed on each plane in
a helical diffraction pattern ........................ 338
13.13 The images used for single particle reconstructions
often have very low signal-to-noise ratios ........... 339
13.14 Image orientations can be determined by the
random-conical tilt method ........................... 340
13.15 Common lines can be used to determine image
orientations ......................................... 342
13.16 Orientation refinement is an important part of most
single particle reconstructions ...................... 343
13.17 It is easy to validate reconstructions and to
estimate their resolutions ........................... 344
Further Reading ............................................ 349
Appendix T3.1 Solving of least squares problems ............ 349
Appendix 13.2 Error propagation in linear systems .......... 350
Appendix 13.3 The Fourier transform in cylindrical
coordinates .......................................... 352
Index ......................................................... 355
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