Why we can't see an atomic nucleus

 kw: speculative musings, calculations, photon energies, resolution

Looking at a small, old photograph made with a Brownie box camera in about 1950, I asked my father, "Can we take a picture of this picture and blow it up real big, so it is clearer?" He said, "The grain of the film would be magnified, not the details in the original scene." A few years later, using a slightly-better-than-a-toy microscope, I designed an "expander", a device with a couple of lenses that attached above the eyepiece to enlarge the image even more. I wanted to see things at 2,000X, and the microscope was limited to 400X (which is pretty good, actually). To my surprise, while the image was indeed five times as large, no more detail was visible. It was just fuzzier.

With more time and study I learned that the maximum practical magnification of an optical microscope is 800X, for someone with normal vision. Using higher magnification (up to 1,600X is available with many higher quality microscopes) makes it a little easier to see the details that are barely visible at 800X, but doesn't bring more details into view. Why is this? It comes down to the nature of light. The basic concept is that you need a "probe" smaller than the finest detail you want to see. To see bacteria, light is good enough, but to see smaller details, something smaller is needed. An atom is 10,000 times smaller than a small bacterial cell, and atomic nuclei are 100 million times smaller. We'll go step by step.

The visible spectrum is a narrow range of wavelengths of light, and you may have read that the range is from 400 nanometers (nm) at the blue end to 700 nm at the red end. These numbers are rounded off; for most people, the deepest violet (or "bluest blue") that is visible has a wavelength near 380 nm, and the "reddest red" visible is at about 750 nm. Our color vision is most sensitive to green light with a wavelength near 555 nm. This is used to calculate the highest practical magnification of microscopes. Using a bluish (sky blue, not deep blue) filter with an optical microscope reduces the fuzziness induced by longer wavelength light: red and orange are not eliminated, but reduced, making the image a bit clearer.

Articles abound regarding the ultimate resolution of microscope (and telescope) optics. Without getting into lots of equations, we can jump to the conclusion that a practical limit for most optical systems is about equal to half the wavelength of the illumination. Thus, in visible light, objects smaller than about 277 nm, or 1/3600th of a millimeter, cannot be discerned.

A second critical factor is the resolution of the unaided eye ("naked eye"). The definition of 20/20 vision is based on optical resolution of one arcminute, or 1/60th of a degree. This is sometimes stated as the ability to see lines spaced 1/10th mm apart at "reading distance"; if you do some figuring, that means "reading distance" is about 13.5 inches or 340 mm. I just held a book where I normally do for reading and measured 14 inches, so that's about right.

The ratio of 1/3600 to 1/10 is 360. How, then, can a microscope have useful magnification as high as 800X? Most microscope objective lenses are "stand-off" lenses, which sit a little farther from the subject than the diameter of their front lens. Special lens arrangements with a front lens at least twice as large as the lens-to-subject distance, used with oil between the lens and the object, can more than double the resolution, so the effective ratio gets into the range of 720 to 800. This is adequate to see bacteria such as E. coli, which are about 1/1000 mm in diameter and 2-3 thousandths of a mm long. At 800X, they would appear 0.8 mm in diameter and around 2 mm long "at reading distance". This is as far as an optical microscope can take us.

My original question was about seeing an atomic nucleus. First let's take a step in that direction and consider seeing atoms. A typical atom has a size of one or two tenths of a nanometer. That's about 1/10,000th the size of an E. coli bacterial cell. What kind of light can see that? To see atoms we need a wavelength of a tenth of a nanometer. Or less. Less is better. What kind of light does that?

To go further, we need an equation that relates photon energy to its wavelength. The standard form of this equation is

E = hc/λ

The symbols are

• E is photon (or other particle) energy
• h is Planck's constant (an extremely small number)
• c is the speed of light (an extremely large number)
• λ is the wavelength

Combining h and c in units consistent with energy in electron-Volts and wavelength in nm, this equation simplifies to

E = 1239.8/λ, and its converse, λ=1239.8/E

For our use, we can round 1239.8 to 1240. The electron-Volt, or eV, is a useful energy unit for dealing with light and with accelerated particles, such as the electrons we will consider momentarily. We find that blue light photons at 400 nm have an energy of 3.1 eV and red light photons at 700 nm have an energy of 1.77 eV. That is enough energy to stimulate the molecules in the retina of the eye without doing damage. Shorter wavelengths, with higher energy, such as ultraviolet, are damaging, which is a good reason to wear sunglasses outside.

What is the energy of a photon with a wavelength of 0.1 nm? It is 12,400 eV, often written 12.4 KeV. These are X-rays. The trouble with X-rays is that they are very hard to focus. Fortunately, because wave-particle duality applies to matter particles as well as to photons, the electron microscope was invented. Electrons can be easily turned into a beam and focused. Over time, electron microscopes were improved to the point that atoms can now be imaged directly.

This image was made with an electron microscope using an electron energy of 300,000 eV, or a wavelength of about 0.004 nm, or 4 picometers (pm). It shows a grain boundary between two silicon crystals. A technical detail: this is not a scanning electron microscope, which relies on electrons bouncing off the subject. This is transmission electron microscopy, or TEM, with the subject cut and polished to extreme thinness. The electrons that do bounce off scatter in all directions, and also knock electrons loose from the silicon, which makes for a messy image, but electrons that pass through ("transmit") form a good image, so TEM is best for this application.

On your screen the shortest distance between atoms should appear to be a few mm. Now consider: the nuclei of these atoms are 10,000 times smaller. What kind of probe can reach inside an atom to see the nucleus?

The wavelength of the electrons in this instrument, as noted, is 4 pm, while the size of an atomic nucleus is measured in femtometers (fm); 4 pm = 4,000 fm. That's much too big. A particle with a wavelength of 1 fm has to be very energetic indeed: 1,240,000,000 eV, or 1.24 GeV (1 GeV is one billion eV). The electron microscope that made the image above cost a couple of million dollars and is five feet high. Much of that size is needed to keep the very high voltage from sparking over and shorting out! Not to mention probably killing the operator. A billion eV is like lightning; it can jump for miles! A different kind of apparatus is needed.

Scientists probe the nucleus with large particle accelerators that cost many millions, or billions, of dollars, euros, whatever. They seldom use electrons. The electron is a light particle, and boosting it to energy greater than a billion eV is hard. The maximum seems to be 50 GeV. Using heavier particles, usually protons, works better. However, as the image at the top of this post implies, smacking a really high energy particle against a nucleus knocks off all kinds of stuff. That is really not conducive to making a "photograph" of a nucleus. Big accelerators such as the Large Hadron Collider in France and Switzerland have huge detectors to gather the spray of "stuff" that results from smacking nuclei really hard, with particles accelerated to extreme energies: 6.5 TeV (Trillion eV) so far (A detail: both probes and targets are accelerated, in opposite directions, so the combined collision energy is 13 TeV).

This all illustrates a critical point: The smaller the things you want to see, the more energy is needed to do so, and the more it will cost. To see really small things, right down to the level of the Planck length, which is about 1/6th of trillionth of a trillionth of a trillionth of a meter, would require enough energy to create another Universe. I don't think it is wise to play with such energies!