May 1991
AT&T Bell Labs 600 Mountain Ave.
Murray Hill N.J. 07974
 Copyright © 1991 AT&T All Rights Reserved

"The Earth's Atmosphere looks like a convex lens. Why doesn't the sun focus on the ground?"

         Your intuition that the earth's atmosphere forms a lens which can focus light is correct, but fortunately it's a weak lens and there is no danger of being fried like ants under a magnifying glass.


         A glass sphere has a focal length which is about the same size as the radius of the sphere (try a hand lens or plastic globe to check). The focal length depends on the index of refraction of the globe. If the earth were entirely made of very clean water, the sun would nearly focus on the opposite side of the planet. However, the index of refraction of air  is smaller than water by about their ratio in densities,  which is about a factor of 1000. Thus, the focal length of  the "air lens" formed by our atmosphere is about 1000 earth radii, or a few million miles. Since the atmosphere is only a few hundred miles thick, the focusing effect is hardly noticeable on the ground.

       At sunrise or sunset when the sun is near the horizon,  there is a small flattening of the shape of the sun due to atmospheric lensing. The lensing also shifts the apparent timing of the sunrise and sunset by a minute or so, compared to an earth with no atmosphere.

" What is the Atomic Structure of a Noble Gas?"


     (Note: This question was asked by a giggle of girls who  left a number and no name. It turned out to be Domino's  Pizza. Anyway, the question [as well as the pizza] wasn't  bad, so..) 

     The six gases, helium, neon, argon, krypton, xenon and  radon, are referred to as "noble" gases because they are  chemically so inert. Noble gases comprise a very tiny fraction of the atmosphere of Earth and are usually obtained by separating them from air or from the gases released from subterranean sources such as oil wells. That they had any  reactivity at all was only recently demonstrated in 1962 by N. Bartlett (who did much of his work at Bell Labs)-when he  isolated a red solid containing xenon, platinum, and  fluorine. As we shall see below, the unique chemistry of the  noble gases is particularly useful in lasers and semiconductor fabrication technology. 

     In order to understand the atomic structure of a noble gas, we must first lay a little groundwork for understanding  atomic structure in general. Atoms are composed of three  basic particles: electrons, protons, and neutrons.  Electrically neutral atoms have equal numbers of electrons and protons. This number is called the "atomic number" and  is the number that one sees associated with each element in  the periodic table of the elements. It is the electrons that determine the chemistry of an atom. For reasons rooted in  quantum mechanics, electrons in atoms group together into  "orbital sets" (we'll discuss quantum mechanics more fully  in an upcoming newsletter). These orbital sets have a number of special properties, one of which is that they can contain up to a maximum number of electrons. Another property is that these orbital sets have a ranking in energy. The lowest energy orbital set is called the 1s ( pronounced "one-s"),  the next orbital set is called the 2s, which is followed by the 2p orbital set and so on. S orbital sets have the  property that they can hold only up to two electrons and p orbital sets can hold up to six electrons. Incidentally, for  those of you who have already learned about orbitals, you  may notice that the p orbital set is made up of three  individual p orbitals. Each of these p orbitals contains a  maximum of two electrons and there are three p orbitals in  the set, thus the total allowed number of six electrons in the p orbital set. There are other orbital sets, called d and f, which can hold up to ten and fourteen electrons,  respectively. So, if you look at your periodic table, one of the noble gases is neon, designated by 10Ne. The "10" is its  atomic number so that we know immediately that neon has ten  electrons. How do we know which orbital sets these ten electrons go into? The answer is that the electrons fill up  orbital sets to the maximum allowable number, starting with  the lowest energy orbital set first. The lowest energy  orbital set is the 1s which holds two electrons, the next  lowest energy orbital set is the 2s which holds another two  electrons, then the next lowest energy orbital set is the 2p  which gets its full measure of six electrons. Notice that  for neon, all three of these orbital sets are filled to the max. This is precisely why neon is chemically so stable. You can think of chemical reactivity as the quest for an atom to maximally fill its orbital sets. Atoms without a completed shell of electrons try to steal electrons from other atoms in a chemical reaction. Atomic neon starts out with fully filled orbital sets and doesn't need to find more electrons. In fact, we will need to pull out an electron from one of those filled orbital sets to make neon react. This requires lots of added energy which is why neon is usually chemically stable.

 Noble gases have a number of uses because they are chemically so stable. For example, they are the gases which are put into fluorescent lamps. Even passing an electric current through them in these lamps does not cause them to react with any of the other materials in the glass or fluorescent coating of the lamp. However, chemists have found some conditions under which noble gases are chemically reactive. Usually, these reactions involve the elements fluorine and chlorine which are among the most highly reactive elements that we know. ( Incidentally, do not confuse fluorine with the fluoride in your toothpaste-fluoride is an ionic form of fluorine which is chemically much more stable. Likewise, the chlorine in a swimming pool is also in a form which is chemically stable). An example of a compound between a noble gas and fluorine is xenon difluoride (its chemical compound symbol is XeF2) which forms colorless crystals at room temperature. Xenon difluoride is highly unstable and will break up into fluorine and xenon atoms upon heating. This makes it a useful compound in semiconductor processing where fluorine is used to etch silicon in the in the long process to make integrated circuit chips. Fluorine is very reactive and the gas is hard to store and handle. Xenon difluoride is more convenient to work with because it comes as crystals that are somewhat less reactive. (Note that the term "reactivity" for fluorine versus xenon difluoride is a matter of degree-compare the reactivity of the wooden part of a match in burning to the phosphorus/sulfur coated tip!).  So, during chip processing, xenon difluoride is put into a chamber with the silicon, and heated to break it up into its constituent atoms to start the reaction between the fluorine atoms and the silicon wafer. For a more advanced treatise on the chemistry of noble gases (which includes quite a bit of history), consult F. A. Cotton and G. Wilkinson's classic treatise: "Advanced Inorganic Chemistry" (Interscience Publishers, New York).

"What happens when an electron collides with another electron? "

     Before considering what happens when two electrons collide, its worth considering a simpler, classical problem. What happens when two bubbles in a glass of water collide?

     As a bubble moves through the water, it pulls along some of the adjacent liquid. When two bubbles are on a collision course, they don't actually have to touch to scatter. Instead, the moving fields of water around each bubble will first touch, deflecting the bubbles from their path. The details depend on their relative speed. When the bubbles move slowly they just bounce off each other. At higher speeds they can hit, send off shock waves (sound) into the liquid, and then continue slowly on their way. Some of the energy from the collision might be turned into heat. This heat could boil the water, creating new bubbles, which in turn scatter.

     There are many similarities between electron and bubble collisions. Electrons, unlike protons, neutrons or baseballs, are believed to be "point particles", with a radius of zero, and thus can never actually "hit". Like bubbles, however, electrons drag a field around them as they move. This is their electric field, originating from their electric charge. At low speed they simply scatter off each other (like charges repel). The actual scattering angles and velocities are determined by conservation of energy and momentum. At slightly higher velocities they can collide and give off light, instead of sound, as was the case with bubbles. Usually, the light is in the form of two photons, going in opposite directions. This makes it easier to conserve momentum. [We shake electrons back and forth every day to generate "light". In a walkie-talkie, electrons shake back and forth in the antenna. This produces radio waves, which are made of photons of light which are at a wavelength which cannot be seen. In a microwave oven, higher energy photons capable of heating food are made in a device called a "klystron". Shake electrons even faster, and you get visible light. Faster still, you get X-rays like those at the dentist's office.]

     At the highest energies new particles can be produced. By Einstein's famous equation for the equivalence of mass and energy, E=mc2 you'll need enormous energies (e.g. a room sized, million volt accelerator ) to create enough mass to make a new particle. You also can't start the collision with two electrons, and end up with four, since this would violate conservation of charge. But, with enough energy you can create pairs of electrons and their anti-particles, positrons. Anything goes, providing you satisfy all conservation laws, including conservation of energy, linear momentum, angular momentum, and charge.

I'm making a bridge out of a paper tube. What is the stiffest tube with the least weight?

     In a bridge, you are trying to maximize strength while reducing cost. Thin tubes are used not so much to save weight, but as one way to reduce the cost of materials. Since in most bridges the weight of the road is transmitted by steel support cables to columns, we'll concentrate on what makes a strong column.

     There are two contributions to a column's strength. The first is the intrinsic strength of the column material, and the second is the way the material is arranged. Now, most building materials are strong when squeezed ( known as compression) but not when pulled (known as tension). For example, a plug of dried mud is hard to compress, but easily breaks when pulled. Other materials are strong when pulled, but not when pushed. For example, a steel wire is strong in tension, but collapses when pushed. This is not because the material is intrinsically weak in compression (after all, steel is pretty tough) but simply because anything long and thin buckles sideways when pushed sideways. Generally, materials are intrinsically stronger in compression than in tension.

     Compressional strengths can be very high. When you walk on concrete, wood or metal floors no footprints are left behind. This means the yield strength of flooring is greater than your weight divided by the area your foot touches (about 150lbs/(2 in) 2~40 pounds/square inch [p.s.i]). However, a high heeled shoe can concentrate all that weight in a 1/4 inch circle, for a pressure of 3000 psi, enough to leave marks on a wooden floor. That's why high heeled shoes and track cleats are kept out of the gym.

   Still, walking on wood is not like walking on peanut butter; the wood will dent until the air between the cell walls is compressed, but your foot won't squeeze through to the next floor. Steel has a yield point of about 200,000 psi, wood about 5000 psi, and paper (which is really wood with no air pockets) somewhere in between. So, if we want to make a wooden column to support 150 lbs all we need is a rod about 1/8" in diameter (if you were building a bridge, you might want to include a safety factor of two). However, if we make a column one foot high and try to stand on the rod, the slightest sideways force will cause it to buckle and collapse. What can be done to make a column a foot tall, using no more wood than the 1/8 inch rod?

     The trick is to provide sideways support to stop the buckling before it gets out of hand. One way is to tie some very fine strings to the rod, and pull in the direction opposite the buckling. But, this only begs the question since you still have to attach the strings to some solid object. You could divide the 1/8 inch rod into three 1/16" rods (which have the equivalent volume) and connect them with strings into a triangular tower. Strings will stop the rods from bowing out by becoming taut, but won't stop the rods from bowing in to the center of the tower. The best thing is to replace the string with short wood rods, since short rods are stiff in tension and compression.

     You can try a simple experiment with a sheet of paper. The paper is weak by itself, but folded in three and taped along one edge it forms a triangular column. Place a book carefully on the open top end of the column. Like the strings, if the edges try to bow out the side walls become taut and prevent further bowing. If the walls try to bow in, the walls are placed in compression. But, any small sideways force (like your finger) or asymmetry in construction will greatly reduce its strength. If we make the walls narrower, they become stiffer (remember, a 1/8" rod is flexible only because it is long; a 1/8'' cube is stiff). Thus, you might want to use many tiny triangular columns glued together. Each one keeps the walls narrow, and they all combine to prevent bowing. This structure is, of course, nothing more than corrugated cardboard.

     If you try to span a ditch with a tube, the strongest tube for a given weight has the largest diameter. As you walk along the tube, it bows down into the ditch. This places the top edge of the tube under compression, and the bottom edge in tension. Providing the tube doesn't buckle, the tube gets stronger by the square of the radius (e.g. a two foot diameter tube is four times as stiff as a one foot tube. To keep the weight constant, the wall thickness is reduced accordingly). Internal supports to prevent buckling greatly improve the strength of a tube bridge.

     Nature values economy as much as bridge builders. Bundles of thin walled tubes are used all the time to impart strength without costing too much energy during growth. The structure of trees (which use the hollow columns between cell walls to transport liquids as well as stiffen), bee's honeycombs and insect exoskeletons are all examples of nature's tendency to be efficient in the use of materials.

     You might want to:

  1. Have a contest to build the lightest, and strongest one foot cube pedestal made of typewriting paper.
  2. Measure the strength of tubes of different materials and wall thickness. Try bamboo, tall grasses, wet or dry wood, wasp nests (we suggest empty nests), plastic, .. Look for trends in the data. Good collaborative project for a physics and biology class.
  3. Make a (temporary) dam across a brook. Measure the yield strength of wet and dry dirt, dirt with sticks mixed in, plastic, and so on before you begin. Find out why beaver dams have one shape, and the Hoover dam, another.
  4. Use air to make a bridge out of weak material. Why is an air mattress weak, but stiff when inflated with air (and air is about as soft a material to strengthen a bridge as possible).
  5. There is an old magic trick where you try to jam a paper straw through an apple. Normally the straw folds up like an accordion. However, if you put your thumb over the end when ramming the apple, it goes through cleanly. Why? 

Weightless school bus

         Joanna Cole writes a wonderful series of science adventure stories for children, involving Ms. Frazzle and  her  magical  school bus. In her Lost in the Solar System story, the  children are launched into outer space. The question "Why do people feel weightless in space?" is answered by "Gravity gives objects weight. Without a large mass nearby, such as a planet, there is no gravity to pull objects down, so they do not have weight". A true statement, but incomplete. After   all, the shuttle astronauts are weightless, and they fly only 100 miles above the earth. Why then are people weightless in space?

What are imaginary numbers and what are their uses?

         Imaginary numbers are not mythical creatures, but are based on multiples of the square root of negative one, sqrt(-1), which is simply written as i. The square root of any negative number can be expressed as some multiple of sqrt(-1), e.g. sqrt(-9)=sqrt(9)*sqrt(-1)=3i. Like the number "0", a placeholder for "nothing", i took a long time to understand and find its place as a useful mathematical tool. The need for i arises when you want a mathematical operation on a number to  generate numbers of the same kind. For example, if we add any two integers we get another integer. When an operation  on a number generates only numbers of the same kind, the operation is called "closed". Clearly, most of life is concerned with "closed" operations; when we add two sheep to three sheep we get five sheep, not a bunch of cows. If we take the sqrt(3), we get a "real" number. However, if we take the sqrt(-4), its clear the answer is not another "real" number.  Imaginary numbers were first used by Renaissance mathematicians who also attached a great deal of mystical significance to them. Leibniz, one of the inventors of calculus, said, "The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world,  that amphibian between being and not being, which we call the imaginary root of negative unity." It was Gauss, around 1830, who really understood imaginary numbers and how to use them.

     Gauss understood that a new kind of complex number could be written, and this number would contain a real part and an imaginary part. Typical complex numbers might be 5+3i or a+bi, where 5 or "a" are the real parts and 3i or bi are the imaginary parts. Gauss showed that these numbers could be added, (a+bi )+ (c+di )= (a+c)+(b+d)i , or multiplied, (a+bi)* (c+di )= a*c+a*di+c*bi+d*b*i2 and since i 2= sqrt(-1)2=-1,  the whole expression equals (ac-bd)+(ad+cb)i. Once  mathematicians understood these concepts, a whole new area of mathematics was open to study. Even relatively simple  ideas, like prime numbers, changed. The prime number 2 can be factored as (1+i )*(1-i ), 5 as (2+i )*(2-i ), and 29 as (5+2i )*(5-2i ), but 7, 11, and 19 cannot be factored. Surprisingly, Gauss found that many ideas in number theory are easier to solve using complex, rather than real numbers, as originally posed.

       One interesting and suggestive use for complex numbers comes from repeatedly multiplying i. by itself.


i*i= -1     


 i *i*i =  -i


i *i *i *i = 1           


i *i *i *i*i = i             


i *i *i *i*i*i= -1            


     Note how the sequence repeats itself after four iterations. This, so to speak, means you're going in circles. We can think of a multiplication by i as a rotation of 90 degrees. By labeling the x axis by the real part of the complex number, and the y axis with the imaginary part, complex numbers can be represented by arrows in the plane. The number 3+2i can be placed on this coordinate system by going three steps along the (real) x-axis and two steps  parallel to the (imaginary) y-axis. Rotating this vector by 90 degrees means multiplying 3+2i by i =3i -2 (See drawing. It really works). This idea of arrows rotating around the  origin is very useful--recurring phenomena like alternating electrical currents, pendulums, the wave nature of light, are all often represented using complex numbers.




Two complex numbers can be added by placing the tail of  the arrow representing the second complex number at the head of the first arrow (you are not allowed to rotate the second arrow). The sum is the new arrow drawn from the origin of the coordinate system to the head of the second arrow. If we add 1+2i to 3+2i we first place the arrow for 3+2i as above.  Now, from the head of the arrow go one more step parallel to   the real -axis and two steps parallel to the imaginary y-axis. You (or your pencil) are now standing4+4i steps from the origin.



Complex numbers, since they can be represented by a that have both length and direction, are used to represent many physical quantities. (Note: Gamow has a great story  about using complex numbers to find buried treasure on a desert island in his book One Two Three... Infinity ( George Gamow, Bantam Books, New York 1967). There is more about complex numbers (and their applications) in the books Asimov on Numbers by Isaac Asimov (Pocket Books, New York, 1977), and Gauss by Ian Stewart in Scientific American, July 1977.)  

Myth of the Month:

     At the Spring Equinox a number of classes tried to balance an egg on its pointy end. Supposedly, the alignment of the earth's axis and the sun in some way help "balance" the egg. Although you can get an egg to balance at any time of year by sprinkling some salt on the table for it to rest on, or by using an egg with a rough surface texture, the stars have nothing to do with the egg's propensity to fall. As it turns out, the gravitational pull of your stomach on the egg is about one billion times more critical than the sun in determining if the egg will fall left or right (Hint: just calculate the difference in forces across the egg due to the sun or your body. It is the difference in forces across the egg which help orient top and bottom.)


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Contact Greg Blonder by email here - Modified Genuine Ideas, LLC.