X-Ray Diffraction
Light is composed of electromagnetic radiation with an electric component that modulates
versus time perpendicular to the direction it is travelling
(the magnetic component is also perpendicular to the electric, but it is not causing diffraction)
wavelength
Force
Electric field
of light
Time
The electric component will interact with charged species (causing the modulating force)
In an atom, because the electrons are ~2000 times lighter than a proton the electrons will be
displaced by interaction with light, therefore an X-ray diffraction is caused by diffraction of
electrons in the sample
Ultimately, X-ray diffraction will indicate the position of electrons (not nuclei)
(Because diffraction will only be resolved if the wavelength of light used is smaller than the
distance between objects, X-ray diffraction can indicate position of electrons) [C-C bond for example is 1.54 Å, visible light is ~4000 Å, X-ray light used is ~0.7 Å]
56 X-Ray Diffraction
Consider electrons at spots A and B in parallel planes separated by a distance d
θ
A
θ
d
B
When a X-ray beam of light hits the plane at an angle θ, the electrons will vibrate due to
interaction with the electric field, and as vibrating charges will radiate light at the same
reflected path θ
A parallel beam of light can be diffracted by electron B in the other plane
The two diffracted beams will be in phase only if the correct relationship between the
wavelength of light used (λ) and the distance between the spots (d) is an integer number
2d sin θ = nλ
Bragg’s law
57 X-Ray Diffraction
Instead of the distance between two electrons (or two molecules), a single crystal is used in order to have a regular array of the diffracting units to allow the
diffracted light to be bright enough enough to measure
Second unit cell axis
Incoming X-ray
Diffraction
lines
Rotate 90˚
Single crystal
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One axis in
unit cell
detector
The regular array of the crystalline lattice will cause diffraction peaks on the detector when
the regular array diffracts in a reinforcing manner, depending upon the wavelength of light
used, the distance between the sample and the detector and the repeat size of the regular array The distance between the peaks will therefore be related to the unit cell dimension (the regular array within the crystal)
By rotating the crystal in the X-ray beam, different planes of diffraction can be detected in three dimensions
58 X-Ray Diffraction
The regular pattern of the diffraction peaks therefore is an indication of the dimension of the
repeat unit in the crystal (the unit cell)
The spacing of the rows and columns indicates the unit cell
The electron distribution within a unit cell however is related to the relative intensity of individual spots
Due to how the electrons are distributed within the unit cell, the diffracted lines can have either constructive or destructive interference
All spots would only have the ~ the same intensity if the electrons are equally distributed
within the unit cell, how they reinforce however indicates the unequal distribution
Symmetrical
distribution of
electrons within
unit cell
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Unsymmetrical
distribution of
electrons within
unit cell
With computer programs, it is possible to determine the electron density within a unit cell
(and hence where are electrons in a bond) by measuring the relative intensity of a large
number of diffraction peaks
59 X-Ray Diffraction
When a single crystal structure is determined what is actually obtained is the relative
placement of electron density in a unit cell
Consider the molecule Celebixanthone
OH O
OCH3
O
OH
OH
*observe electron distribution
(nuclei position is assumed)
Oxygen atoms can be
distinguished from carbon due
to extra electron density
*electrons appear as spherical balls
surrounding each atom
(electrons are NOT between atoms)
Stout, et. al., Tetrahedron, 1963, 19, 667-676
Contour interval: 1 e / A3
60 X-Ray Diffraction
Does this mean Lewis and Valence Bond Theory is wrong?
What is a bond and why do the atoms remain attached if no electron density is between the atoms?
Need to consider Electron Deformation Density (sometimes called “electron difference density”)
Consider how much the spherical atoms are distorted when forming a bond by subtracting the
electron density from a pure symmetrically oriented sphere
F
F
N C
C N
F
F
Still seems like spherical balls for atoms
Appears differently with electron
deformation density
Contour interval: 0.1 e / A3
Can now see electrons between bonds
(and lone pairs on nitrogen)
Hirshfeld, Acta. Cryst., 1984, B40, 484-492 Dunitz, et. al., Helv. Chim. Acta, 1983, 66, 123-133
61 X-Ray Diffraction
While we can now be comfortable with our Valence Bond Theory (and Lewis can sleep comfortably!) it still seems like this is a small effect and 2 electrons are
not held between bonds as Lewis proposed but are mainly near each atom of a bond
Consider tetraphenylbutatriene to see some issues
C C C C
Make slice and rotate
Deformation density for each
phenyl looks promising
Density is not symmetrical,
looks like a π bond from
overlapping p orbitals!
Adjacent π bonds are
orthogonal to each other!
C1-C2
C2-C3
Problem is that the electron
density in “bond” can be
integrated
62 L. Leiserowitz, et. al., J. Am. Chem. Soc., 1975, 97, 5627-5628
X-Ray Diffraction
There are many gratifying results from electron deformation density maps for a qualitative
verification of Lewis dot structures (and Valence Bond Theory in general): electrons are
located between bonded atoms, when π bonds are formed between atoms the electron density
is not symmetric about the internuclear axis and lone pairs of electrons can be detected
A problem however is the amount of electron density between bonded atoms
Valence Bond Theory would predict there are 2 electrons involved in a single bond, 4 electrons in a double bond and 6 electrons in a triple bond
(and 3 electrons in a bond that “resonates” between a single and double bond)
The electron density in the
“bonding” region can be
quantified by integrating the
deformation map
C1-C2
Electron 0.22
density
C2-C3
Ph-Ph
C1-Ph
0.30
0.20
0.11
The amount of electron density in a “bonding” region is
significantly less than predicted by Lewis
L. Leiserowitz, et. al., J. Am. Chem. Soc., 1975, 97, 5627-5628
63 X-Ray Diffraction
The amount of electron density in a bond was determined from a number of high quality
electron deformation density maps
Actual density in a bond
determined by experiment is
~ 5% of predicted Lewis dot
structures
0.3
Charge
(electrons)
0.2
Tetraphenylbutatriene
Other structures
0.1
C C C CC C
1.2
1.3
1.4
C C
1.5
1.6
Bond Length (Å)
Density determined from electron deformation density maps is ~0.1 electrons for a single
bond (not 2!), 0.2 electrons for a double bond and 0.3 electrons for a triple bond
L. Leiserowitz, et. al., J. Am. Chem. Soc., 1977, 99, 6106-6107
64 X-Ray Diffraction
What can electron deformation maps indicate about strained compounds?
Where is the electron density located in a strained system?
Consider the highly strained cyclopropane
H
H
H
H
H
H
The electron density
truly does form
“bent” bonds,
bonding electron
density is not along
internuclear axis
Valence Bond Theory therefore does do a very good qualitative job describing bonding, it lacks in providing some quantitative ability for both bonding and also reactivity
For a better quantitative job, Molecular Orbital Theory is useful
D. Nijveldt and A. Vos, Acta Cryst., 1988, B44, 296-307
65