The Sun, Hydrogen Bombs, and the physics of fusion

(copyright 2002, Richard A. Muller)


Opening anecdote:


When Charles Darwin published his Origin of Species, he had a lot to say that went beyond evolution.  He had found fossil sea shells high up in the Andes mountain ranges, and had concluded that these regions had once been under the sea.  He estimated that the Earth must be at least 300 million years old, based on his calculation that it would have taken that long for erosion to create some of the great valleys in England.  That was adequate time, he concluded, for natural selection to account for changes in species. 


But when his book was first published, the great physicist William Thompson, later knighted as Lord Kelvin (after whom the Kelvin temperature scale is named) objected.  He told Darwin that the Earth could not possibly be that old, or the sun would have completely burned out.  Even if the Sun were made completely out of fuel, such as coal, it would have used up all its energy long ago.  A more energic source, Kelvin suggested, was meteors!  (Even Kelvin knew that meteors carried much more energy than coal!)  Kelvin estimated that meteoric heat could account for a solar lifetime as long as 30 million years.

Darwin had no answer.  In the second and later editions of his book, he removed his thesis that the Earth was hundreds of millions of years old. 

Of course, we now know that Darwin's original conclusion was correct.  The Earth is about 4,500 million years old.  The Sun has been burning for longer than that.  It isn't fission: the Sun contains very little uranium or plutonium.  So what is the answer?  What did Kelvin get wrong?

Kelvin, and everybody else at that time,  didn't know there was a way of burning hydrogen that yielded a million times more energy per gram than coal.  It gives more energy per gram of fuel than nuclear fission.  It is called nuclear fusion.


Amusing digression.  There is no universal agreement in professional publications on how to refer to "The Earth."  Some people point out that we donŐt refer to "The Mars" or "The Jupiter", and that "Earth" is a proper name, just like Mars and Jupiter.  So some people always say "Earth" without the "the": "The age of Earth is about 4.5 billion years."


Power from the Sun

                  "Fusion" refers to the coming together of particles, in contrast to "fission" which is the separation of particles.  It may seem strange that you can get energy by bringing particles together, but it is true, if you pick the particles correctly.  The basic source of fusion in the Sun comes from bringing together  4 hydrogen nuclei to make helium.  In the process, other things are also created.  These are 5 gamma rays (for which we use the symbol g), two neutrinos (symbol n), and two positrons (symbol e+).  (A positron is just like an electron, but while the electron has a negative electric charge, the has a positive charge).  In symbols, we write this as:


4 H --> He + 2 e+ + 5 g + 2 n


Isn't that fission?  Notice that there are more particles after the reaction than before!  So why is this called fusion, rather than fission? It is really based on the convention that a new element has been created, helium, that is heavier than any of the nuclei that we started with (which were all hydrogen). 

Energy from fusion.  It is all those other particles (e+, g, n) that carry off a great deal of energy.  The neutrino (n) escapes out of the Sun, but the other particles collide with other atoms (mostly hydrogen), sharing their energy, heating the sun.  It is the heat of the sun that makes it shine.  The amount of energy released in the one fusion reaction given above is 25 MeV. 


Did you forget what an "MeV" is?  If so, read on.  The unit called the "electron volt", abbreviated "eV" (small e, capital V) is a tiny unit of energy.  It is useful when talking about individual atoms, since chemical reactions typically have an energy between 0.1 and 10 eV.   An MeV is a million eV. 


Digression.  The V is often capitalized, presumably because it arises from a person's name, Volta.  (Yet the word "volt" is often not capitalized, which goes to show that there are no absolute rules followed.) 


More numbers than you need to remember (unless you are a Chemistry major):  1 eV = 1.6 x10-19 joules = 3.8 x10-23 Calories.  So if you have one mole of material (6 x1023 atoms), and each atom releases 1 eV of energy, then the energy released is 3.8 x10-23  x 6 x1023  = 23 Cal per mole.


To compare this with previous work: our table in Chapter 1 shows that methane, burned in air, releases about 13 Cal of heat per gram.  One mole of methane is 16 grams, so methane releases 13x16 = 208 Cal per mole.  That's about 9 eV per molecule.


Why the world is interesting.  The Sun is a star, and the fusion that takes place in the Sun is very similar to the fusion that takes place in stars.  If the only fusion that took place was the combining of 4 hydrogens into one helium, then the world would be a very very dull place.  The reason is that complex life, as we know it, requires heavier atoms such as carbon and oxygen.  Not much of interest (life, intelligence) can happen when all you have is hydrogen and helium.  The only molecules you can get is H2.


We believe that the heavier atoms such as carbon and oxygen are created inside stars.  Carbon is formed from the fusion of three helium atoms.  All the carbon in you body, all the oxygen in the atmosphere, was once buried deep in a star, where it had been created.  Fortunately, for those of us who like an interesting world, that star eventually exploded, spewing its debris into space.  Eventually the material clumped up to form a new star (which we call the Sun) and a bunch of planets (Earth, Venus, Mars, etc.).  With all the carbon and oxygen on the Earth, life (as we know it) was possible.  We are literally made out of the ashes of an exploded star.  The Sun is a secondary star, created from such debris of an earlier star.


Unrequired details of fusion.  You don't have to know the following, unless you are interested: The fusion reaction that we gave above, with 4 hydrogens turning into one helium and a bunch of other things, does not usually take place in just one step.  If the star is a primary star, made of only hydrogen and helium, then the first step is that two hydrogens will combine to form a deuteron (nucleus contains one proton and one neutron), a positron, and a neutrino.  In symbols, the reaction looks like this:

                                    H  +  H -->  d  +  e+ + n

Then the deuteron combines with a Hydrogen to create an isotope of helium called "Helium-3" and written as 3He:

                                    d + H -->  3He + g

Finally, two Helium-3 atoms combine to form ordinary helium:

                                    3He + 3He --> He + 2 H + g

Note that in the end, the original hydrogens have been transformed into helium plus a few other things.


In "secondary" stars, such as the Sun, the reactions are different because there is carbon available.  In the sun, the first reaction is a fusion of hydrogen with carbon!

                  H  +  C --> 13N + g

This is followed by 5 other reactions.  In the end, the carbon is replaced, so there is no net carbon gain.  The only atoms used up are four hydrogens, and the net things produced are helium, two positrons, five gamma rays, and two neutrinos.  This is the reaction described in the first fusion equation I gave you.  This series of reactions is called "the Carbon cycle".


In case any of you really want to know the complete carbon cycle, here it is.  Preceding superscripts refer to the atomic weight of the nucleus.

12C + 1H --> 13N + g                                            106 years

13N --> 13C + e+ + n + g                                  10 minutes

13C + 1H --> 14N + g                                            2 x105 years

14N + 1H --> 15O + g                                            3 x107 years

15O --> 15N + e+  + n + g                                2 minutes

15N + 1H --> 12C + 4He                                      104 years


On the right side of each reaction, I indicated how long it takes to happen, on average.   It is surprisingly long.  That's why the sun hasn't used up all its fuel yet.  It is burning very slowly.  Despite that, it's pretty bright.



Why isn't the Earth a star?  Why doesn't fusion take place on the surface of the Earth?   Or inside a tank of hydrogen gas?   The simple reason is that they aren't hot enough.   Why does that matter?


The nuclei of all the elements have protons and neutrons in them.  The protons give them a positive charge.  Positive charges repel.  In ordinary matter, two nuclei will never get very close, because of this repulsion. 


To overcome the repulsion, you have to give the nuclei enough energy that they are not stopped by the electric force.  Energetic atoms are hot atoms.  So if you heat the atoms enough, they will have enough kinetic energy to allow their nuclei to touch.  Once the hydrogen is hot enough, they will have sufficient energy (and velocity) to overcome this repulsion.  The temperature that you need depends on the specific fusion reaction, and how often you need to have the fusion take place.  The temperature in the center of the sun is believed to be about 15 million K.  That's hot, but not too hot.  Even at 15 million C what takes place is a relatively slow fusion; most of the hydrogen fuel has not yet burned.  (Good thing.  If it burned faster, the Earth would have been too hot for life.)


If we are going to use fusion to generate electricity, then we need hotter temperatures.  The plans for controlled thermonuclear reactors (CTR, in the jargon of the energy business) have temperatures of 100 million C. 


If you want the reactions to take place in a millionth of a second (to make a hydrogen bomb), then the temperature must be even greater.


How did the Sun get so hot?  Of course, the Sun is very hot now, because of the fusion that is taking place.  But how did the Sun get hot in the first place?  We believe it was simply the gravitational attraction of the matter that made up the Sun.  When all that matter got pulled together, the mutual gravity gave it kinetic energy; when it all settled, that energy was converted into heat.  This will happen only if the object is big enough so that the temperature rises over a million C.  If the mass isn't that large, then the fusion isn't ignited, and the object never turns into a star.  In fact, the definition of a star, used by most astronomers, is an object that is large enough that the heat at the core ignites fusion.


The outer surface of the sun is not that hot.  It's temperature is about 6000K.  That is not hot enough for fusion.  All of the fusion takes place at the core of the sun.  The heat works its way out, and the surface then glows at a much lower temperature.


Jupiter is not quite big enough to become a star.  It's mass is about 0.1% of the Sun's.   It is believed that to become a star (i.e. to ignite fusion), an object must be substantially larger, about 8% of the mass of the sun (almost 100 times more massive than Jupiter).


The Hydrogen Bomb


There is another fusion reaction that takes place at relatively low temperature.  It is

d + t --> He + n

In this equation, the d stands for deuterium (hydrogen with an extra neutron) and the t stands for tritium (hydrogen with two extra neutrons).  This is the reaction that is used in the hydrogen bomb.  The neutron comes out with high energy of 14.1 MeV, and the He flies off with an energy of 3.5 MeV; the total energy released is 17.6 MeV..


That may not sound like much energy.  But we started with light objects.  The d has atomic mass 2, and t has atomic mass 3.  So we get 17.6/5 = 3.5 MeV per atomic mass unit.  In fission, we start with U-235, and that has 235 mass units.  The energy released in a fission is about 200 MeV.  That comes to 200/235 = 0.85 MeV per atomic mass unit.  This means that with an equal weight of fuel, fusion produces more energy, by a factor of 3.5/0.85 = 4.1.


In the hydrogen bomb, the fusion is ignited by a fission bomb.  The fission part is called the primary, and the fusion part is called the secondary.  Thus, there are two bombs in the hydrogen bomb.  The temperature of the exploding fission bomb is so high (reaching a billion C) that the reactions occur very rapidly, much faster than on the Sun. 


The "Super".

The initial idea of a hydrogen bomb was that a stick of hydrogen would be ignited, at one end, by a fission bomb.  If you wanted a bigger explosion, then use a longer stick (more hydrogen).  That scheme was called the "Super" and it never worked.  Calculations showed that the end cooled off too rapidly, and the burn would not continue.


Edward Teller and Stanislaw Ulam solved this problem by using a different geometry.  The radiation from the fission bomb would be trapped in a shell (made, typically, of U-238).  This radiation would compress and heat the hydrogen.  In this geometry the fusion would be sufficient to make most of the hydrogen fuse.


As I mentioned in the chapter on nuclear weapons, there are additional tricks used in actual bombs.  Instead of tritium, the fuel can use the stable isotope Lithium-6 (6Li).  The neutrons from the fission can break this up (i.e. make it fission) to produce tritium.  In addition, the fast neutrons can make the U-238 shell fission, and this produces even more energy.


The name.  The fusion bomb is called the hydrogen bomb because it uses hydrogen as its fuel.  Even tritium and deuterium are considered to be isotopes of hydrogen.  The bomb is sometimes called the H-bomb.  It is also called a Thermonuclear bomb, since it uses heat (the "thermo" part) to ignite the nuclear reactions. 


Terrorist fusion bomb?  This isn't likely.  It is hard enough to make a fission bomb, but then making the geometry just right so that the  fission bomb will ignite the hydrogen (rather than just blow it apart) is extremely difficult.  It would be even harder if you couldn't test the fission.  Most experts agree that a terrorist group could not make a working fusion bomb.  Even a country would have difficulty, if they couldn't test their fission weapons.