Structural Methods
Bremmstrahlung
-Emitted when an incoming electron interacts with the nuclear force field, results in a deceleration and change in direction -if electron is slowed down, it will exit the force field/material with less energy; the excess energy is liberated as an x-ray -the excess energy is expressed as X-rays within a spectrum of wavelengths, as the acceleration is not consistent
glide planes
-combo of translation parallel to mirror plane with subsequent reflection across the mirror -translation is either along a cell edge or a face diagonal -reflection = changes hand -designated by a, b, c, n, or d depending on translation -for a, b, c: glide is half axial lenght -for n: glide is half the face diagonal -for d: diamond d glides occur for centered lattices and correspond to 1/4 along a diagonal translation -glide operation inverts chirality; screw axis does not -enantiopure molecules cannot crystalize in a space group that has a glide plane
solving the structure
-determine space groups and make selection that is compatible with both Z and molecular symmetry
Traditional R-value
-discrepancy index or conventional residual, R -want it to be as small as possible -if Fobservered = Fcalculated, then it is a perfect structure
absolute structure
-general term to encompass situations which correspond to determination of absolute configuration (or conformation), or of polar-axis direction or resolving the ambiguity of enantiomorphic pairs or of axis direction (From slides) -"The spatial arrangement of the atoms of a physically identified noncentrosymmetric crystal and its description by way of unit-cell dimensions, space group and representative coordinates of all atoms" -for non-centrosymmetric structures, correct "absolute structure" must be determined
asymmetric unit
-smallest part of a crystal from which the complete structure can be obtained from space group symmetry operations (including translations) -differs from the UC in that the UC is smallest part of crystal that is required to build up the lattice by translations only
X-ray Diffraction
-the scattering of X-rays by the regularly spaced atoms of a crystal, useful in obtaining information about the structure of the crystal -x-rays are generated when electrons give up some of their energy when they interact with electrons or nucleus of an atom
thermal parameters
-thermal motion of an atom is represented by an ellipsoid within which the electrons of the atom have a certain probability of being found (typically 50%) -each atom in the molecule should exhibit a "motion" (although this cannot be distinguished from disorder) that is understandable -atom vibration should NOT be along the bond vector but perpendicular (atom moves up and down, doesnt stretch) -central atoms should not be moving as much -a larger ellipsoid could indicate a superposition of the different possible locations of the atoms
phase problem
-we don't measure F (which contains the amplitude and the phase); we measure intensity -therefore, we cannot measure everything needed -this results in the phase problem, because we cannot know the phase (which contain the bulk of the structural information) when we do the experiments -phases can take any value for non-centrosymmetric structures, but must be 0 or 180 degrees for a centrosymmetric structure -remember the duck/cat slides: phases are more important than intensity and direct amplitude in solving these problems and determining structure
triclinic space groups
P1 P1bar
monoclinic groups
P2 P21 C2 Pm Pc Cm Cc P2/m P21/m C2/m P2/c P21/c C2/c
cubic
a = b = c; α = β = γ = 90° 4 C3 axes in Td P, F, I
rhombohedral
a = b = c; α = β = γ ≠ 90° 1 C3 axis P
primitive cells (P)
lattice points only at its corners (smallest unit cells)
main limitation of x-ray diffraction
main limitation with respect to intensity is the amount of heat produced
what are the intensities dependent upon
nature and position of the atoms concerned
path length difference and phase difference in 2D
path length difference = lambda(hx + ky) phase difference = 2pi(hx + ky)
mirror plane
reflection sigma -sigma v = mirror plane is parallel to principal axis of rotation -sigma h = perpendicular to principal axis of rotation -sigma d = bisects the angle between 2 C2 axes, and is a subset of sigma v
n-fold rotation axis
rotation by 2pi/n Cn
/
used before m if it is perpendicular, as in 2/m = C2h point group
volume of crystal lattice
v = abc[1-cos^2(alpha)-cos^2(beta)-cos^2(gamma) + 2cos(alpha)cos(beta)cos(gamma)]^1/2
Data Collection
-a couple of reflections are selected that will act as check reflections (measured every 100 reflections) -these reflections check diffractometer stability, compound stability, crystal movement -amount of data required depends upon crystal symmetry: -triclinic: l(hkl) = l(-h,-k,-l) so only half a sphere is unique, or four octants (0-k-l to hkl) -for monoclinic: only two octants need to be collected (unique b 00-l to hkl or -h00 to hkl) -orthorhombic: one octant needed (000 to hkl) -if cell is centered = even less data is required to be collected because there are systematic absences (odd reflections dont exist!)
accuracy vs precision
-accuracy: indicates how close a measurement is to the accepted (true) value -precision: indicates how reproducible a measurement is (ie number of sig figs)
Powder X-ray diffraction
-analyzes microcrystalline samples that represent multiple crystalline formations -allows for material identification, as well as determining how pure the sample is -gives info only about the lattice parameters -pattern is rings
structure refinement
-atomic parameters are varied systematically so as to give the best agreement of the observed structure factor amplitudes with those calculated for the propsed structure |Fobserved| and |Fcalculated| -procedure used is that of the method of least squares (which minimizes the sum of the squares of the observed values and the calculated values) -an electron density difference map is then calculated to indicate the positions of other atoms, and the process is repeated -process is iterative: as more atoms are located, calculated phases better resemble those for the total structure, thereby allowing more atoms (and lighter atoms) to be located
weighted R value
-based not only on F but also on the sigma of F as well -reflections with a higher sigma(F) will be downweighted from those with lower sigma(F) values = allows you to refine the data by assigning more significant reflections more weight -weighted R values tend to be larger -since it is based on more info, it is the better value on which to base the accuracy of the structure determination
unit cell
-basic building block of a crystal and is that unit which when repeated infinitely in 3D gives the crystal; characterized by cell lengths a, b, and c (corresponding to the x, y, and z directions), and interaxial angels alpha, beta, gamma -can have many different shapes -NOT the smallest unique part of a crystal from which the entire crystal may be built due to symmetry; the way in which contents of unit cell may be arranged (asymmetric unit) causes this
miller indices
-h, k, l -if the plane cuts the axes at x/a, y/b, z/c, the Miller indices are: h = 1/x, k = 1/y, I = 1/z -this corresponds to number of times a set of planes would cut a unit cell axis -if any Miller indices have fractions, then all values are multiplied to convert indices to lowest integer -not that planes that run parallel to an axis intercept it at infinity = miller index = 0
Anomalous Dispersion
-if an atom in a structure absorbs (even moderately) the X-rays used, then a phase change occurs and intensity is therefore changed = Friedel's law no longer holds true -in these cases, correct absolute structure may be determined by seeing which hand gives the best agreement with Fcalc
standard deviation
-indicates variability of data = how similar the individual values are -low = high precision -crystallographers use +- 3sigma (encompasses ~whole Gaussian) to determine whether something is statistically significant
Four-Circle Diffractometer
-involves measuring intensities of several thousand hkl reflections -Eulerian cradle consists of four circles that act together to orient the crystal such that all possible reflecting planes may be examined automatically
axial photographs
-lattice is verified by recording axial rotation photographs, which should indicate both the correct axis length and also the correct symmetry examples: -triclinic: a, b, and c axes do NOT have mirror symmetry -monoclinic: b axis has mirror symmetry -orthorhombic: a, b, and c axes have mirror symmetry
limitations of electron diffraction
-limited to small molecules that exist in gas phase -not suited for location of H atoms since a peak in the radial distribution curve is a function of the atomic numbers of the pair of atoms involved -molecular vibration may lead to proposed geometries that are different to the ground state (distance between atoms seems short than it is = "shrinkage effect")
artificially lowering the R value
-most common way to reduce R value is to omit weak reflections -this is an accepted practice by most since all crystals exhibit weak reflections at higher diffraction angles -an intensity cutoff with I>3sigma(I) means that only reflections with intensities greater than three times their estimated deviation (sigma) will be used in the calculations
F^2 refinement
-most crystal structures are refined against F -for well-behaved structures the geometrical parameters and their estimated standard deviations are almost identical for refinement based on all Fsigma^2 values and for refinement against F ignoring data with Fsigma less than 3sigma(Fsigma) -this is best way to refine data, but R value will be greater
centering
-moves atom into center -example: C2: 1) 2-fold rotation 2) moves atom to center; 2-fold rotation
structure factor for neutron diff and x ray diff
-neutron: diffraction event is symmetric = no preferred direction -x-ray diffraction: interference of rays occurs from same atom, preferred direction is direction in which light is traveling
unit cell determination
-once crystal has been centered, rotation photograph is obtained, in which crystal is rotated about its phi axis -lattice is verified by recording axial rotation photographs, which should indicate both the correct axis length and correct symmetry = can determine unit cell
n-fold screw axis
-rotation by 360/n and a translation parallel to axis by fraction of r/n of unit cell -21 = 1/2 translation -32 = 2/3 translation -rotation = same hand
incorrect space groups
-since centrosymmetric and non-centrosymmetric space group pairs have same systematic absences, it is not possible to distinguish them on this basis alone -try to determine if the structure is consistent with a possible mirror plane in the centrosymmetric structure: 1. does the molecule possess any molecular symmetry that may also be crystallographic? 2. are pairs of atom coordinates such that they may be related by some symmetry operator? 3. are bond lengths and angles acceptable? 4. are the shapes of the thermal ellipsoids reasonable?
Resolution
-the ability to separate two peaks, not the distance between them -do this by collecting more data at other (higher) angles; however, scattering at high angle values is limited by crystal quality -resolution is typically of the order of 1 angstrom for a normal small molecule experiment
Patterson Method
-used prior to efficient computers to overcome the phase problem -only requires knowledge of intensities -if any two atoms in the unit cell are separated by a vector (u,v,w), then there will be a peak in the 3D Patterson map at (u,v,w) -height of this peak is approximately proportional to ZiZj = very useful in determining position of a heavy atom because a single heavy atoms will have a high intensity -since heavy atoms can often be located from the patterson map, the incorporation of heavy atoms into crystals is often used to help solve structure of proteins, for example
estimated standard deviations (esd or sigma)
-used to indicate whether differences in two measurements, such as bond lengths, are significant -basic assumption: differences in observed quantities from true values are only due to random errors -chemists use a 3sigma criterion to determine whether measured bond lengths are, or aren't, significantly different; but this is only valid if there is NO systematic error in the data, such as disorder or absorption problems -esd's of non-hydrogen atoms are underestimated -esd's of heavy-atom positions are less reliable than those of light-atom positions -esd's of cell parameters are grossly underestimated by a factor of 5 for cell lengths and 2.5 for cell angles
x-ray scattering power
-x-rays are scattered by interaction with the electrons of an atom: incident radiation forces the electrons to oscillate and thus radiate -scattering power increases with atomic number -therefore, it is difficult to locate light atoms (H) in the presence of heavy atoms -increase in diffraction angle = decrease in scattering power
axes of order
1, 2, 3, 4, 6
Bragg's Law
Bragg's law (nλ = 2dsinθ) details when constructive interference will occur. Crystals are composed of uniformly positioned atoms that can diffract x-rays, which is also known as a reflection. As a crystal lattice contains many different planes that can diffract x-rays, the reflections can interfere. Whether or not an x-ray reflection can be visible is determined by Bragg's law, which states that constructive interference will only occur when the path lengths traveled by the X-rays are an integer multiple of the wavelength, as seen above in the equation. -therefore, the molecular structure may be determined
two types of x-ray diffraction
Bremsstrahlung, K-shell emission
intensity equation
I = |F|^2
P212121
P = primitive 21 = 2-fold screw axis along x 21 = 2-fold screw axis along y 21= 2-fold screw axis along z
Pnma
P = primitive n = n-glide along (y, z) diagonal and reflect perpendicular to x m = mirror plane perpendicular to y a = a glide with reflection perpendicular to z
Pca21
P = primitive c = c-glide with reflection perpendicular to x-axis a = a-glide with reflection perpendicular to y-axis 21 = 2-fold screw axis along z
x-ray vs neutron diffraction bond lengths
X-ray diffraction: the position of the electron is given; however, electron density of the hydrogen atom's single electron will be shifted toward the heavier atom to which the hydrogen is bonded = bond length appears shorter than it is Neutron diffraction, the position of the nucleus is given instead, resulting in an inconsistency in the measured bond lengths.
hexagonal
a = b ≠ c; α = β = 90°; γ = 120° 1 C6 axis P
tetragonal
a = b ≠ c; α = γ = β = 90° 1 C4 axis P, I
monoclinic
a ≠ b ≠ c; α = γ = 90°; β ≠ 90° 1 C2 axis P, C
orthorhombic
a ≠ b ≠ c; α = γ = β = 90° 3 C2 axes P, C, F, I
triclinic
a ≠ b ≠ c; α ≠ β ≠ γ no essential symmetries P
body centered cell (I)
also has a lattice point at its center
the crystal lattice
any crystal may be regarded as being built up by the continuing 3D translational repetition of some basic structural pattern, which may be an atom, a molecule, or even a complex assembly of molecules
area of peak calculation
area = (ZiZjNij)/Rij Zi, Zj = atomic number Nij = number of times distance occurs Rij = distance between two atoms
when there are both mirror planes and a horizontal plane
as in D6h: write 6/mmm
Electron Diffraction
basic experimental features for a typical electron diffraction experiment include an electron source, an electron gun, electromagnetic lenses, a gas sample, and a recording device. In a basic electron diffraction experiment, a constant stream of electrons is shot out of the electron source before passing through the electromagnetic lenses. Meanwhile, the gas being studied is shot out of the gas sample nozzle. The electron stream then passes through the gas, which causes the electrons to be diffracted. The recording device then records the final location of the electrons. This allows us to determine how significantly the gas caused the electrons to be scattered.
stationary single crystal
because of Bragg relationship, a stationary single crystal gives very few observable reflections; in order to generate complete diffraction pattern it is necessary to rotate the crystal in x-ray beam
K-shell emission
bombarding electrons cause electrons from the inner shells of atoms of the metal target to be ejected. To fill these now-vacant spots, electrons drop down from higher energy levels. This results in sharply defined characteristic x-rays and sharp peaks of specific wavelengths
Extended X-ray Absorption Fine-Structure (EXAFS)
can be used to determine: -bond lengths with an accuracy of 0.02A -coordination numbers to one atom in four or five -atomic numbers to within one row of the PT
what type of symmetry operation gives rise to systematic absences in the hkl set of reflections?
centering (i think? page 32 of notes)
how does X-ray diffraction work
electrons are generated by passing a high current through a wire filament which are then accelerated to a high velocity by applying a potential of 50kV
what type of symmetry operation gives rise to systematic absences in the 0kl, h0l, and hk0 sets of reflections?
glide planes
side-centered cell (A, B, or C)
has lattice points at the centers of two opposite faces
what does raw data of an x-ray diffraction experiment comprise of
hkl reflections associated intensities, I
comma symbol
implies you have inverted chirality => generated the enantiomer
Friedel's Law
in the absence of "anomalous scattering," the magnitude of F(hkl) is equal to that of F(-h,-k,-l) I(hkl) = I(-h,-k,-l) Phase(hkl) = phase(-h,-k,-l)
intensity and amplitude relationship
intensity of scattered radiation is proportional to square of amplitude: I = |F|^2
face-centered cell (F)
lattice points at the centers of its six faces
systematic absence
occur only when there is some symmetry element with a translational component within unit cell = a screw axis, glide plane, or a non-primitive cell (centering) -these absences occur because diffraction maxima are only observed if n(wavelength) = 2dsintheta -for (h, k) planes, if h+k = 2n = even, then the reflection may be seen (but won't necessarily be seen; dependent upon other factors, such as intensity) -if h+k = (2n+1) = odd => no reflection -translational component of unit cell effectively changes lattice spacing, which can change whether or not we can see reflection -this allows us to back-track and see what other elements are involved
what does the magnitude of |F| depend on?
only on the relative positions of the atoms in the unit cell
X-ray diffraction: position, intensity
position of reflection: informs us about lattice spacing (d) intensity of reflection: informs us about positions and types of atoms in the cell; tells us about the nature of the atoms
what type of symmetry operation gives rise to systematic absences in the h00, 0k0, and 00l sets of reflections?
screw axes
goodness of fit
should be close to but greater than 1 should never be less than 1 because that means your model is better than the actual structure
neutron diffraction
since neutrons do not carry a net electric charge, they are not scattered by electrons, but by interactions with nuclei scattering does not change dramatically with atomic number especially useful for the location of light atoms, especially H atoms neutrons scatter equally in all direction because nucleus is so small possible to distinguish a site of deutariation Isotopes contain different numbers of neutrons. As neutron diffraction is caused by interactions of the neutrons with the nuclei, the changes in the number of neutrons results in changes to the scattering length. Therefore, isotopes directly influence scattering power. The scattering power is constant with diffraction angle. However, neutron scattering does end up falling off at higher angles because of thermal motion. much larger crystals are required; sources of neutrons are not common; nuclear force has short range = need direct hit
Structure factor
the x-radiation that is scattered by one unit cell of a structure in any direction in which there is a diffraction maximum has a particular combination of amplitude and phase; this is the structure factor, F or F(hkl)
calculate "Z," number of molecules
unit cell volume / 20 (approximation for volume of Carbon) / number of non-hydrogen atoms
structure factor
x-radiation that is scattered by one unit cell of a structure in any direction in which there is a diffraction maximum has a particular combination of amplitude and phase = structure factor F
