This was a singularly unfortunate choice of words on Dyson's part because it made the theory seem so improbable as to be impossible. Anyone who hears of it in that form is guaranteed to dismiss it as nonsense. The problem with this characterization of the sense of smell is not only that it alludes to a scientific instrument but to a single instrument. If we consider the mechanism by which the human eye works we see that an essential feature is that there are multiple units involved: the red, green and blue cones and the gray scale rods. And with these rather simple units the human eye is able to detect and distinguish the frequencies of the electromagnetic vibration. The human eye is a spectrometer but operates on completely different principles than the scientific instrument. What comes out of the material below is the understanding that color is not a property of the light, it is the result of the relative response of the cones of the eye to the electromagnetic radiation.

Color Vision

The retina of the eye contains tiny organs called cones which contain chemicals which are selectively sensitive to light radiation. There are three types of cones which have different sensitivities to different frequencies of light. The frequency of radiation is inversely related to its wavelength. It is easier to visualize a wavelength than a frequency so wavelength will be used in what follows. Wavelength is expressed in micrometers; i.e., in millionth of a meter.

The graph below shows the sensitivity of the three types of cones as a function of wavelength.

The obvious name for the cones that are most sensitive to so-called red light is red cones and likewise for the green and blue cones. It is not that red cones are sensitive only to red light; it is that their sensitivity is greatest for red light.

The efficiency curves for the red and green cones cross (i.e., are exactly equal) for radiation of wavelength about 0.56 μm. When the eye sees 0.56 μm radiation it stimulates the red and green cones about equally. The visual perception of near equal stimulation of the red and green cones is yellowness. The radiation of wavelength 0.56 μm is not yellow itself anymore than microwave radiation or radio waves have a color. Yellowness comes from equal stimulation of the red and green cones. Light including radiation of equal intensity at the 0.58 μm and 0.54 μm wavelength, the wavelengths of maximum efficiency for the red and green cones, would also be perceived as yellow light. There would be some stimulation of the blue cones by the 0.54 μm radiation which would lighten the yellowness of the perception. But generally any combination of light waves that give equal stimulation of the red and green cones and negligible stimulation of the blue cones will be perceived as yellow.

White color comes from equal stimulation of all three cones. Violet comes from equal stimulation of the red and blue cones. This has to come from light composed of longer wavelengths to stimulate the red cones without stimulating the green cones and shorter wavelength light to stimulate the blue cones but not the green cones. So there can be single wavelength light that appears to be yellow but there cannot be single wavelength light that appears to be violet.

The efficiency curves for the cone receptors are shown in the above diagram as going to zero, but they likely taper off asymptotically to zero like a Gaussian curve. The reason for saying this is that very high intensity light from a laser emitting infrared light is perceived as being deep ruby red. The infrared does not have a color but its intensity is so great that it stimulates the red cones in the tail of the efficiency curve where the efficiency of the perception is small but nonzero.

The are millions of colors that the eye can perceive. This means that each type of cones need only decern about a hundred of different levels of intensity. The result of this relatively simple mechanism of color differentiation is a sensitivity to wavelength and frequency that rivals that of a scientific spectrometer.

Tonal Hearing

Now consider human hearing. The organs of the inner ear detect and distinguish the the different frequencies of sound waves quite well. There is no esoteric physics involved. All that is involved is the phenomenon of resonance. A physical structure such as a stringed musical instrument has resonance frequencies such that if it is stimulated at those frequencies it will vibrate in response. Resonance occurs when the wavelength of the stimulus is an integral fraction of the wavelength of resonator; e.g., the stimulus wavelength is one half or one fourth of the resonance wavelength.

If a continous sound of one frequency was impinging upon a set of resonators there could be one which has the same wavelength as the sound. The resonators need be nothing more than strings of different lengths under tension or pipes of different lengths each with one end closed. The device with of the same resonance wavelength as the sound wasve would resonate and its vibration could be noted.

In the case of the inner ear a continuum of resonators is created by having a tapered channel. When the stimulus sound wave reaches a point in the channel where the width of the channel is an integral multiple of the wavelength of the sound wave then resonance occurs and the physical movement created by resonance is transferred to the minute hairs in the channel walls thus stimulating the auditory nerve. It is all beautifully simple but quite effective, beautifully effective.

In the schematic diagram above, the sound beam enters the channel at the lower left hand corner of the channel. It travels across the channel is reflected from the wall. It recrosses the channel only to be reflected. As the beam travels back and forth across the channel to the right the channel width decreases. At some point the path distance across the channel may match the wavelength of the sound and resonance occurs.

In the inner ear the tapering channel is coiled into a spiral for compactness but the function remains the same.

The eye and the ear provide two examples of human organs which measure in some sense frequencies. The ear responds to mechanical vibrations in the air. This is not a sophisticated problem, but the eye's response to electromagnetic radiation is something that would seem to be implausible if not impossible but it does so.

The Olfactory Receptors

Linda Buck of Columbia University discovered the smell receptors in about 191. They lie on a thumbnail-sized patch of tissue in the upper nasal passage. The mechanism for these receptor's operation has not yet been established but there is general agreement that they are what produces the sensation of smell. It is also not known how many different types of these receptors there are. This would be a crucial factor in determining the number of different odors humans can perceive.

Molecular Vibrations

The calculation of the frequencies of the vibrations of an actual molecule is a complex and difficult computation. It is worthwhile to consider first highly simplified models of molecules. In the image below is depicted a model of chlorine, Cl₂. The two spheres, representing the chlorine atoms, are connected by an elongated bond which can stretch or bend.

The atoms have equal masses of m and the bond has elasticity k, the elongation per unit of force applied. Physical analysis establishes an equation for the motion of the spheres under the elastic force of the bond. This equation, a differential equation, has a solution for the back-and-forth motion of the spheres (stretching of the bond) which is cyclical. The frequency ν of this motion is equal to the square root of the ratio of k to m.

ν = √k/m = (k/m)^1/2
 

Thus as the mass of the atoms increases the frequency goes down but if the springiness of the bond increases so does the frequency of the oscillation. But if the mass is doubled the frequency becomes not one half of what it was but instead about 71% (the square root of one half) of what it was.

In the case of diatomic molecules such as Cl₂, H₂, O₂ and N₂ there is only one mode of vibration, the stretching of the bond. For molecules of more than two atomics there are more modes of vibration. The simple formula for the number of modes of vibration of a molecule of n atoms is

3n-5 for a linear molecules
3n-6 for nonlinear molecules.
 

The n atoms each have 3 degrees of freedom for their positioning in 3D space. The molecule is viewed from the perspective of its center of mass. The specifying of the location and orientation of the molecule uses up 3 degrees of freedom in the location of the center of mass and 3 degrees of freedom in the angles of orientation of the molecule. Thus in general the degrees of freedom and modes of vibration within the molecule is 3n-6. But if the atoms all lie along a straight line it does not matter what angle the molecule has about that line, hence the degrees of freedom are 3n-5.

Any diatomic molecule is automatically a linear molecule. The degrees of freedom and number of modes of vibration of a diatomic molecule are 3(2)-5=1. Thus the only mode of vibration of a diatomic molecule is the stretching/contracting of the bond.

The water molecule H₂O has the bonds of the two hydrogen making an angle of about 107°

The degrees of freedom for the water molecule is 3(3)-6=3. These are: 1. The symmetric stretching of the bonds 2. The asymmetrical stretching of the bonds 3. The scissoring of the bond angle.

A carbon dioxide molecule has the three atoms in a line. Therefore its degrees of freedom are 3(3)-5=4.

The modes of vibration are the symmetrical and asymmetrical stretching and then two modes of bending vibration. The frequencies of vibration of the two modes of bending are equal.

The molecule depicted below is an ammonia molecule without any representation of the bonds between the three hydrogen atoms (in blue) and the nitrogen atom (in orange).

This molecule has all the motions associated with the stretching of the bonds plus it has combinations of these motions and scissoring of the bond angle. The number of modes of vibration of the ammonia molecule is 3(4)-6=6.

Below is the depiction of the methane molecule CH₄.

The hydrogen atoms (in blue) form the corners of a tetrahedral pyramid with the carbon atom (black) at the center of the pyramid. The methane structure also has a rich set of modes of vibration.

(To be continued.)

Enantiomers

Enantiomers are two structures having the same parts and same linkages between the parts but are not identical in the same way that a left hand and right hand are not identical. Below are depicted to molecules which are enantiomers of each other.

If one molecule is twirled around so the yellow and green atoms have the same alignment then the red atom for one is in front whereas the red atom for the other is behind. If the red atoms are held fixed and one molecule is rotated so the yellow and green atoms match then one molecule has the blue atom at the top and the other has it at the bottom.

Frequency, Wavelength and Wave Number

A wave phenomenon, such as a sound wave, goes through a cycle. In the case of a sound wave it is a fluctuation in pressure. From a starting point in which the pressure is equal to the background pressure it increases to level above the background pressure then decreases back to background pressure. But this is not a complete cycle. There is the second phase in which the pressure decreases to a level below background pressure before rising back up to background pressure. This is the complete cycle. The number of times per second that the pressure goes through the complete cycle is called the frequency of the sound wave. Below is shown a graph of the deviations in pressure over time for a sound wave.

The note of A on the musical scale has a frequency of 440 cycles per second. The wavelength of that note is 75.3 cm. The product of the frequency of a wave and its wavelength is equal to the speed of that wave.

The speed of sound varies with temperature and pressure. Under the standard conditions of 32° F (0° C) and 14.7 psi (100,000 pascals) its speed is 331.3 meters per second (741 miles per hour). The variation in pressure involved in sound is truly miniscule, about 0.1 of a pascal compared to a background pressure of 100,000 pascals.

Below are shown the cases of three different waves with different frequencies and wavelength and the fact that there is an inverse relationship between frequency and wavelength.

Either frequency or wavelength can be used to quantify a wave phenomenon, but another measure is also used, the wave number. The wavenumber is simply the reciprocal of the wavelength, or equivalently the frequency divided by the wave speed. In the case of the note of A on the musical scale its wavenumber is 0.0133 cm^-1. In the case of sound, wave number is not an especially convenient measure, and neither is it for electromagnetic radiation. Yellow light has a wavelength of about 5.5×10^-5 m and hence its wavenumber is 1.82×10⁶ m. But for molecular vibrations wave number is a convenient way to describe wave motion. The wavenumbers for molecular vibrations are in the range of 100 cm^-1 to 10,000 cm^-1.

(To be continued.)

Luca Turin's Theory of Smell

Luca Turin's theory reduced to its most basic nature is that the receptors in the nose respond to the different fundamental vibrations of a molecule and that produces the sensation of smell. There are most likely a number of different types of receptors responding to different ranges of vibrations. With only three types of receptors in the retina of the eye millions of colors can be distinguished. In the nose there may be significantly more than three types. Luca Turin characterizes the smell receptors as having overlapping ranges but that would not necessarily have to be the case. There could be uncovered ranges and molecules having vibrations only in such uncovered ranges would have no smell. Some substances such as the noble gases of helium, neon and argon are mono-atomic and therefore have no internal structural vibrations. Under Turin's theory they would have no smell. There are other molecules such as the diatomic molecules of oxygen O₂ and nitrogen N₂ which have a internal structural vibration but no smell. The vibrations of smell-less could have frequencies outside of the ranges covered by the smell receptors.

The alternative to Turin's theory is the Shape Theory of Smell; i.e. the smell of a molecule is determined by its shape. The one seemingly decisive bit of evidence for Shape Theory was that there are some enantiomers which have identical vibrations but different smells. An enantiomer of a molecule is one that has the same atoms but differs in shape the same way a left hand differs from a right hand. Luca Turin points out that the vast majority of enantiomers have the same smell. But in one group, the carvones, the enantiomers smell differently. This difference needs to be explained.

Turin's theory of smell does provide for a role for the shape of molecules. The receptors of a particular type probably can only accept molecules within a limited range of sizes and shapes. Molecules that are simply too big would not have a smell no matter what vibrations they have and this is found to be true. Molecules whose shape makes it difficult to fit into the receptors would have a weak smell. Turin's vibration theory has nothing to say about why some substances have an intense smell and others a weak smell. He acknowledges that the shapes of molecules would affect the intensities of their odors.

While Turin has made no speculations on this matter, the different smells for some enantiomers could be accounted for by there being different receptors that are compatible with different enantiomers. Let us say there are receptors of type A and B and molecule M will fit into A but not B but its enantiomer M' will fit into B but not A. The vibrations of M and M' are the same but if receptor A responds to the frequencies of M it is a different signal to the brain than if receptor B responds to the same frequency of M'. For other molecules the enantiomers may fit into both A and B and so they would have the same smell.

There is many bits of evidence for Turin's vibration theory of smell but two are notably outstanding. Most organic molecules contain hydrogen atoms as well as carbon atoms. Hydrogen exists in three isomeric forms. The simple hydrogen atom consists of a nucleus contain one proton and a shell surrounding the nucleus consisting of one electron. The proton has a mass about 1800 times larger than the electron so most of the mass is in the nucleus. The size of the hydrogen atom is determined by the electron shell. A second form of the hydrogen atom, called deuterium, has a neutral particle, the neutron, in the nucleus as well as the proton. The size and shape of the deuterium atom is virtually identical to that of the simple hydrogen atom, but the mass is about twice as large.

If the simple hydrogen atoms in a molecule are replaced by deuterium atoms then the molecule's shape is unaffected but its vibration frequencies are reduced substantially, roughly by a factor equal to the square root of (1/2). When deuteriated molecules were synthesized it was found that their smell differed from that of the ordinary version. There is nothing in Shape Theory that can account for that difference.

The second definitive bit of evidence for vibration theory is that Turin found two molecules with the same vibration but different shapes that smell the same. Turin noted that sulfur compounds have a distinctive unpleasant smell. Turin believed that that distinctive character came from the sulfur-hydrogen bond which has a wave number of 2500 cm^-1. He found a boron compound that had that same vibration and Lo and behold it had the same sulfurous smell. The shape of the boron compound was nothing like the sulfur compound's so Shape Theory would have a difficult time explaining the identical smell.

(To be continued.)