"Solid Cold" (continued)
E. The theories and the facts
The figures illustrating Einstein's result give us a clue as to why 19th-century theory worked as well as it did, even though it took no account of the yet-to-be-discovered energy quanta. For all but the lower temperatures, the quantum hypothesis leads to practically the same results as the classical theory. And how low the temperature of a solid has to be to get very different results depends on how small the energy quanta are that the solid can absorb. The smaller the energy quanta (and the smaller the atoms' vibrational frequency), the more quanta it would take to raise the solid from a low to a high temperature-and the more closely the flow of energy quanta would resemble the continuous energy stream expected from 19th-century thermodynamic theory, instead of a "grainy" energy stream made of quanta.
We noted earlier that atoms' apparent inability to vibrate with arbitrary energies is odd, given the apparent ability of larger objects to vibrate at any energy whatsoever. If matter is made of such atoms, why don't we see larger objects also limited to certain vibrational motions? It turns out that we do, but it's not obvious. Again, the reason is that the energy quanta for ordinary vibrational frequencies are very small. A large weight on a spring will have almost the same motion if its energy increases or decreases by one quantum. If the motion becomes very different, a large number of the small energy quanta have to be gained or lost, and that would look to our unaided senses like a continuous energy change instead of a "grainy" one. So here, too, 19th-century mechanics' lack of a quantum hypothesis doesn't make it noticeably inaccurate.
It turned out that 19th-century theory was accurate in another respect as well. We noted earlier that certain features of atoms, which explain why they cannot absorb just any amount of energy, had yet to be discovered in 1906. Once we take account of these features, we find that the average energy per atom should be half a quantum higher than it would if Einstein's working assumptions were correct. So, instead of real atoms having less average energy than the atoms of classical theory-barely less at small temperatures, approaching a half-quantum less at high temperatures-real atoms should have more average energy-a half-quantum more at absolute zero temperature, but very little more at high temperatures. The higher the temperature, the more closely a real atom's average energy should equal that of a "classical" atom.
One other thing Einstein didn't take full account of is the way each atom's motion affects that of its neighbors. When one atom moves, it pushes against the atoms it approaches, and pulls at the atoms from which it moves away. This tends to get those other atoms moving in the same direction, or at least slow their motion if they were already moving in the opposite direction. At the same time, the neighboring atoms resist the motion of the first atom, tending to slow it down, stop it, and reverse its original motion. Thus when some atoms move back and forth, their neighbors begin to follow that motion, lagging behind it a little. And these atoms in their turn start the same process in their neighbors, spreading vibrations through the solid in waves-sound waves.
If an atom barely moved when its neighbors vibrated, the solid's energy would increase with temperature in much the same way as for Einstein's hypothetical solid. But in sound waves, each atom more or less keeps pace with its neighbors. The more closely an atom follows its neighbors' motion, the less it resists their motion, and the more time it will take all the atoms to complete a vibration. In other words, atoms' moving together lowers their vibrational frequency. The longer the sound waves are, the more closely neighboring atoms pace each other and thus the more their frequency lowers. So in a solid, the atoms have many possible vibrational frequencies, not just one. Just this additional fact implies a slightly different (and more accurate) relation between a solid's temperature and the average energy of its atoms. Still, this more accurate relation shows the same general trend as Figures 1 and 2-a high, nearly steady increase of average energy with temperature at high temperatures, a small but variable increase at low temperatures. (.....continued)