The only way to produce a heavier element is to put energy into the system, something which can only take place in a supernova. So since our Earth contains elements heavier than iron, we must assume that we are made of the ash of a previous generation of stars. Or as Carl Sagan was famous for saying, “We are made of star stuff.”
This actually makes that radiometric dating of those old stars interesting. They could not have been first generation since they contained very heavy elements.
We can also measure the age of a globular cluster of stars by looking at the luminosity of the brightest stars. We can do this because we understand the way burn rates, mass, age and brightness are related to each other. By this technique the oldest globular clusters have been dated at 11.5 ± 1.3 billion years.
What has been encouraging about this whole field is that independent measurements are converging upon similar estimates of the age of the universe. It is true that these results are dependent on models of stars and nucleosynthesis, but those models are becoming more robust each year. They also calculate more than star ages. They predict the relative abundance of all of the elements and isotopes in the universe, in good agreement with what is observed.
One last technique for establishing the age of the universe is to look at the rate of expansion. As we will see in a later chapter, we can measure the distances to galaxies and the rate at which they are moving and extrapolate back to the moment of the Big Bang.
All of these techniques: radiometric dating of stars, luminosity of stars in globular clusters, nucleosynthesis and the abundance of elements and finally the expansion of the universe all agree 13.7 billion years as the age of the universe. I actually find it amazing that the Earth has been here for a lot of that time, about a third of the life of the cosmos.
***
This is a good time to pause and look at where we have been in these last two chapters. At one end of our time scale are the lifetimes of exotic, subnuclear particles such as the delta particle or the Z boson. Their lifetimes are measured in yactoseconds, 10−24 s. This is about the time it takes for light to cross a proton. The slowest thing we have seen, the event that has taken the longest time, is the evolution of the universe itself. The age of the universe is 13.7 billion years or 1017 s. It takes light about that long, to within an order of magnitude, to cross the universe that we can see. It is not a simple coincidence that time, size and the speed of light are connected like this. For something to evolve, no matter what its size is, its parts have got to interact with each other, and no interaction is faster than light.
In the next four chapters we will look at the size of an array of objects and structures, much as in Chapter 2 we looked at the size of biological organisms, and we will try to find some organizing principle. We will try to be aware of the way things interact and forces cause structure. The world is not just a haphazard jumble of objects. There is order; we just need to see it.
10
Down to Atoms
Poems can have meters and one may judge an event with a metric, but neither of these has much to do with a platinum–iridium bar in Paris. One can meter out punishments or rewards or rations or even electricity. We can learn to keep the beat or meter of a song with the aid of a metronome. Yet none of these has anything to do with a hundred-thousandths of the distance between the Earth’s equator and pole.
The word meter (or metre) and metric are both based on the Greek word metron, to measure. We often think of the metric system as a system based on multiples of ten or a thousand or a million, but the word itself only tells us that it is a system for measurements. Its official name is Le Système International d’Unités or, in English, The International System of Units, often shortened to just SI. The system could have been many things. But its most important feature is that it is recognized internationally. It could have been based upon the mile, the hour, or the calorie. The basic length unit could have been set by the height of a student at MIT named Smoot. In fact the bridge next to MIT that connects Cambridge and Boston has been measured as “364.4 Smoots, plus or minus 1 ear.” But that unit is hard to reproduce.
Today the meter is defined as the distance over which light can travel in 1/299,792,458 of a second, and a second is defined as the amount of time it takes for 9,192,631,770 oscillations of a Cs-133 atom. This may seem like a convoluted way of defining our most basic unit of measurement. But it meets the original criteria that the enlightened thinkers of the French revolution recognized as really important. A meter could be created by anyone anywhere. If we were in radio communications with aliens from another galaxy we could describe our meter and they could produce a meter stick of exactly the right length. But a combination of light and cesium clocks is not how the meter started.
***
Back in Chapter 1 we briefly introduced the meter and its origins. It was designed to be 1/10,000,000 of the distance from the equator to the pole. It was an inspirational goal, a majestic plan. It embodied some of the ideals of the Enlightenment, it was universal and it belonged to all mankind. So Delambre and François measured part of a meridian in France and the meter was established. On December 10, 1799 that meter was inscribed upon a platinum bar that was deposited in the National Archives of France.
The meter was inspired by the 1/10,000,000 of a quadrant, but the real meter was a metal bar. Within a few years the accuracy of the Delambre and François survey was in doubt, but the bar remained.
By the middle of the nineteenth century the need for an international measurement standard became more and more apparent as machinery and trade increased. So in the 1870s France hosted a series of meetings and conferences that established the meter as the international standard. Even the US was a signer of that original treaty. It is true that the inch, foot and mile still dominate in the US, but did you ever notice that an inch is exactly 2.54 centimeters. In fact the inch is defined in terms of the meter.
At the September 1889 meeting of the General Conference on Weights and Measurement (CGPM) a new meter was established. This one was made of a platinum–iridium alloy. The original meter bar was showing its age. Its ends had signs of wear after years of being handled when copies were made to be sent out across the globe. At this time the methods of using the standard bar were also more tightly defined. The bar was only a meter when it was at 0°C, at the freezing point of water.
By 1927 there were a few additional caveats for an acceptable measurement. A standard air pressure was established and it was specified that the bar must be supported in a prescribed manner. These refinements may not seem very important, but what they tell us is that there was a rising ability to make precise measurements, as well as a need for greater accuracy. Yet still the meter was based on a metal bar in Paris.
The standard changed radically on 20 October 1960. The CGPM redefined the meter as 1,650,763.73 wavelengths of light in a vacuum from the 2p10 to 5d5 transition of krypton-86 atoms, a jump between two atomic orbits. This is a nine-digit number, which means that the meter was defined to within about ten atoms. But more importantly, the meter no longer resides solely in Paris. The meter had been freed from that metal bar. Anyone, anywhere could make an identical meter stick.
That, however, is not the end of the story. On 21 October 1983, the meter was abolished. The meter is no longer considered a primary unit of measurement. A meter is now derived from the second and the speed of light. In 1983 distance could be measured to 1 part in a billion (109) whereas clocks were approaching one part in one hundred trillion (1014). In fact clocks are now a hundred times better than they were then.
With the speed of light defined as 299,792,458 m/s exactly, it takes light exactly 1/299,792,458 seconds to travel 1 m. Ideally the meter would now be as accurate as clocks: one part in 1016. Inreality it may not make sense to think about the length of an object in units smaller than atoms.
Actually in 2002 there was one more caveat added to the definition of the meter. The meter is a proper length. Proper length is a concept that comes
out of relativity and the 2002 convention says that a meter is only well defined over short distances where the effects of general relativity are negligible. In other words, if you are measuring an object next to a black hole, or a length that spans a galaxy, you should not expect many meaningful digits of accuracy.
The standard meter bars produced in the 1880s are really objects of beauty: metallic sculptures. The bars are a bit over a meter long with marks inscribed upon their sides. By modern measurements those marks are placed with a precision of 10−7 meters. The bars are made of platinum–iridium, a silvery colored metal, and its cross section is a modified X, a shape now called a Tresca, after Henri Tresca, who suggested its use because of its rigidity (see Figure 10.1). These days such bars are museum pieces, retired since 1960. They are no longer doing what a meter should be doing. Meters are meant to measure things by their nature and name.
Figure 10.1 The cross-section of a standard meter bar from the 1880s. The design was proposed by Henri Tresca and is very stable.
Before we can measure things, especially small things, we will need to subdivide our meter into smaller units. We can use simple geometry to subdivide our meter stick into ten subdivisions called decimeters. Geometry will also allow us to divide decimeters into ten centimeters (10−2 m) and each centimeter into ten millimeters (10−3 m). A centimeter is about the size of a periwinkle, the Etruscan pygmy shrew, or the bumblebee humming bird, which we met in Chapter 2. It is also the minimum size of a warm-blooded, or homeothermic, animal.
Equipped with a meter stick marked in centimeters and millimeters we can now measure some amazingly small things, including the thickness of paper, foil and hair. Once we know the thickness of a hair, we can then measure the wavelength of light and the size of oil molecules, an amazing accomplishment for a macroscopic stick.
***
We start with paper. If you turn up the edge of the piece of paper you are reading this on and hold it next to a meter stick all you can really see is that it is much smaller than that smallest division; it is smaller than a millimeter. But we can do better if we are a bit cleverer. Close the book for a moment and measure the thickness of all the pages, not including the cover. Now all you need to do is take that measurement and divide it by the number of pieces of paper in the book. An easy way of finding that number is to take the last page number and divide by two because pages are printed on both sides. The first draft of this book was written in a notebook 1.5 cm thick with 150 sheets of paper, so each sheet of paper is 10−4 m thick; that is, a tenth of a millimeter or 100µ m.
A bit more of a challenge is the thickness of cooking foil. The outside of the box claims that it contains 30 m2, and my meter sticks tells me it is 30 cm wide, so the roll of foil must be 100 m long. If I look at the end of the roll I can estimate that each wrap is about 10 cm long, and so there must be about 1000 wraps. My meter stick can also measure the thickness of the foil on the roll at about a centimeter. So if 1 cm is 1000 layers of foil, then 1 layer is about 10−5 m. This is also the diameter of an average cell.
We have just measured something the size of a cell with a meter stick marked to only millimeters, and that measurement is really pretty good. Of course the reason for our success is that we actually measured a large collection of paper or foil and then did a bit of division. This is a technique we will continue to use for smaller and smaller objects.
If we could stack hair like we can paper or foil we could use the same technique to measure a hair. But hair tends not to stack so simply. However, there is a technique that we can use and which is often attributed to Isaac Newton. Start with a long hair—from portraits I have seen of Newton his was very long. Now wrap that hair in a nice neat coil around a pencil, counting the number of turns as you go. Make sure each winding is snug up against the previous winding. If you have dark brown, almost black hair you will find that you have about ten windings for every millimeter of pencil. The more windings you make, the better your measurement. Ten windings per millimeter means the hair is 10−4 m thick, or 100µ m. As we saw in Chapter 2, hair ranges from about 50µ m for light blond hair to 150µ m for thick black hair.
***
Now we would like to measure the size of a light wave, which is really small. We can measure it relative to a hair and, since we know the size of a hair in meters, we will know the size of the light wave. We can do this because light is a wave and follows the same rules as sound or water waves. Waves have two important properties for our measurement; they spread out around corners, and when two sets of waves meet they add up or interfere with each other.
Sound clearly spreads around edges and corners. That is the reason you can hear something you cannot see, especially outside where there are few walls for sound reflection. Also if you watch an ocean wave pass a jetty or breakwater, it too spreads out and can roll towards the corners of the harbor. We do not notice light doing this, but it does. One of the big differences between light waves and sound or water waves is their size. Water waves can measure tens of meters from peak to peak. Sound waves range from a meter to a millimeter. Light waves are about 10−7 m, a tenth of a micrometer or a thousandth of a hair’s breadth. In order for a wave to wrap around a corner, that corner must be sharp compared to the size of the wave. So a razor blade, or a hair can deflect light that passes by it.
The other wave phenomenon that we need is interference. When two waves meet each other they can combine constructively or destructively. Constructive means that the peaks line up in the same place, at the same time, and add together to form a bigger wave. Destructive means that the peak of one wave lines up with the trough of the other wave and cancel each other. This phenomenon can be dramatic with sound waves. If you have two speakers set a meter or so apart and playing a monotone, you can move your head around and hear places where the sound is stronger or weaker. The distance between louder, constructive nodes and quieter, destructive nodes is related to the wavelength. Real music is made of all sorts of waves and so the nodes for every frequency are scrambled all over the place so you generally do not notice interference in normal circumstances.
Water waves will of course do the same thing. If a wave passes an island or a large rock the waves will wrap around the island and meet each other at the back. There are places where the waves add and other places where they cancel. The way they join is called an interference pattern and its shape depends upon the size of the island and the wavelength of the ocean waves.
Light does the same thing. White light, like music, has a lot of frequencies. But if you take a monochromatic light, like a laser pointer, and shine it on a single hair, most of the light will go by the hair unaffected. A small part of it will be bent by the hair, and then interfere with the light going around the other side of the hair. On the wall you will see a strong spot where most of the light went, but also a series of spots. The spacing of that series depends upon the color or wavelength of the light and the thickness of the hair.
Interference is such an importance phenomenon that I will describe it with one more analogy. If a precision marching band goes around a street corner it generally performs something called a wheel (see Figure 10.2). In order for the rows of marchers to stay in line the band members near the corner slow down. Now let us imagine that the band members can only march at one speed, for our band is part of a light beam and light has only one speed. After the turn the rows will be messed up because the outside marcher went farther. However it is possible that marchers in the outside columns might line up with members of a different row if the extra distance they went was a wavelength, or a row gap. Actually if the extra distance is one, or two, or three, or more row gaps they will line up. But if it is a half, or one and a half, they will not.
Figure 10.2 A marching band performs a wheel and demonstrates interference. (A) A marching band performs a wheel. (B) A marching band with constant speed. After the corner, columns 1 and 3 are in phase and constructive. Columns 1 and 2 are out of phase and destructive.
F
igure 10.3 Laser light on a single hair shows an interference pattern. When a red laser is shone on a single hair most of the light goes straight (center spot), but some of it will form an interference pattern, which is seen as spots next to the center one. If it is about 10 m from the hair to the wall, the spots are spread out by 5–10 cm, depending upon the hair and laser.
Back to our light and hair. If we shine a laser on a single hair, most of the light passes by the hair and forms a bright spot on a distant wall. But some of the light will be deflected by the hair (see Figure 10.3). When photons from the right-hand side of the hair and photons from the left-hand side reach the wall they can form an interference pattern. The first spot on the side of the major spot is where light from one side of the hair traveled exactly one wavelength farther than light from the other side. The second spot means that the light’s path length was different by two wavelengths. That means it is just a bit of geometry to measure the wavelength of this light. We find that red light has a wavelength of about 6.7 × 10−7 m, blue light about 4.7 × 10−7 m.
***
From the meter stick we measured the size of a hair, from a hair we can measure the wavelength of light. Now, knowing the size of light, we can measure the thickness of some molecules.
We have all seen puddles on roads with a very thin film of oil. Sometimes we view them as unsightly pollution, other times as beautiful swirls of color. What is happening is that sunlight is shining on that film and some of that light is reflecting off the surface and towards the eye. The rest of the light passes through the oil film to the surface of the water where again some of it is reflected up towards the eye. Light off the oil’s surface and light off the water’s surface travel slightly different distances. If the distance is a wavelength the light combines constructively and we see it brightly.
How Big is Big and How Small is Small Page 17