What Geiger and Marsden saw, however, was quite different. Most α particles indeed went through the foil with only small deflections. But a few were **scattered by large angles**, in a way which could only be explained by assuming that the positive charge of the gold atom and most of its mass formed **a tiny point-like concentration**. Only then was the bombarding particle able to approach it so close that its electric repulsion could deflect it by a big angle or even turn it back. Rutherford named this concentration **the atomic nucleus.** The electrons stayed far outside, like planets orbiting far from the Sun, a view indeed suggested by Hantaro Nagaoka in Japan.
It looked like a miniature planetary system, with the attraction between positive nucleus and negative electrons filling the place of gravity--an attraction which, like gravity, decreased like the inverse square of the distance R, like 1/R^{2}. An electron held in this manner should obey Kepler's laws, like a planet attracted by gravity, and move in a circle or ellipse. Its motion can then be viewed as the sum-total of two perpendicular oscillations, each reaching its greatest extent at the moment the other one passes the middle point (just consider the "shadows" of the motion on the x and y axes, shown in the drawing
!).

Each of these two (independent) motions is equivalent to an electric charge oscillating back and forth around an equilibrium point. By the laws of Maxwell and Hertz, however, such an electron is expected to emit electromagnetic radiation, which would reduce its energy. Because of this, unlike planets of the solar system, the electrons would quickly radiate away their energy and spiral inwards, until they reached the nucleus

Atoms, however, are (fortunately!) stable. How come?

The results of Balmer and Ritz seemed to tell that atoms had stable energy levels, and Einstein's formula suggested that energy at the atomic level seemed to be related to the Planck's constant **h**, measured in joule-seconds.
One quantity measured in joule-seconds was **angular momentum**. That is a mechanical property (just as energy, say) of any rotating object or collection of objects--even of the Earth, Sun or the solar system. It measures the total "amount of rotation"--adjusted for the **mass** involved, its **average distance** from the axis of rotation, and of course, its **rate** of rotation. It is a vector quantity, its direction following the rotation axis.

The rotations of Earth and Sun, and orbital motions of planets and moons, do involve angular momentum. A young Danish physicist named Niels Bohr, who arrived in England in 1912 and ended up working with Rutherford at the University of Cambridge, looked into the angular momentum of an electron orbiting a nucleus. It was guessed by then that as dimensions of a motion shrank to the atomic scale, Newton's laws were gradually modified, allowing electrons to have orbits where (for some yet-unknown reason) no energy was lost by broadcasting electromagnetic waves. Bohr went one step further: **perhaps orbits were stable when the angular momentum of the motion equaled h times some whole number**. In a large system, this might be millions of times larger than h, so the law allowed angular momentum (and energy) to change almost continuously. On the scale of a hydrogen atom, however, the possibilities were few, stable orbits were quite distinct, and so were their energy levels.

For circular orbits in a hydrogen atom this worked beautifully, and also gave a consistent value of the Rydberg constant. Allowing the angular momentum to take values of **nh/2π** with **n=1,2,3...** gave orbits with energies which exactly matched Balmer's hydrogen spectrum (the factor 2π can be justified). A promising first step--but how to continue from here?

Physical laws--the basic ones and those derived from them--usually make **precise, quantitative** statements, like **F=ma, E=mc**^{2}, E= hν. Very rarely does one encounter a law by which something is **approximately true** but not exactly (even though the approximation may be very good), e.g. valid for a long time but not to all eternity. Adiabatic invariance is of this kind.

Imagine a **pendulum** formed by a weight tied to a string, swinging back and forth under the influence of gravity. In each oscillation, as it descends to the bottom of its swing, it gains kinetic energy, then after having passed bottom, it slows down again, to stop briefly at the top of each swing. In the absence of friction and air resistance, one expects its total energy E (kinetic + potential) to stay unchanged, and the frequency ν and period T=1/ν of the oscillation will stay constant, too.

If the string starts at a pulley (see drawing) and is slowly pulled up, the length of the pendulum will gradually decrease, and since a shorter pendulum has a higher frequency ν and a shorter period T, these quantities are also expected to change. But the energy E **grows too**, because extra work is being done against the force stretching the string, so the swing becomes more vigorous. Suppose the string is drawn up until its length is **reduced by a certain fraction**--say, by half. It can be shown that the change in the ratio E/ν can be made **as small as we wish** by making the drawing-up process slow enough.

The ratio is not a real constant of the motion or "invariant" (like the energy in the undisturbed pendulum), but an approximate one, an "**adiabatic invariant**". The lack of constancy may be attributed to the fact that the tension in the string in any swing does not vary symmetrically--a small asymmetry is added by the fact the string is constantly being shortened, but slowing down the pull greatly reduces the asymmetry and its accumulated effects.

Notice here that the adiabatic invariant E/ν has the **same dimensions** of joule-sec (i.e. same combination of measurable quantities--also called "action") as the elusive Planck's constant **h**.

Adiabatic invariants occur in all sorts of periodic motions, including Kepler motion, which was taken as the model of the motion of an electron around the nucleus. Some physicists, notably Paul Ehrenfest, guessed that **perhaps electron orbits were stable when the adiabatic invariant I of their motion equaled h times some whole number**. For circular orbits in an atom, only one adiabatic invariant I existed, and allowing it to only take the values **I=nh** with n=1,2,3... gave orbits with energies which exactly matched Balmer's hydrogen spectrum.

In the end it turned out that the quantization of angular momentum, rather than of I, gave a better qualitative understanding of atoms of more than one electron. That is discussed in the section that follows.