§7 · fluorescence and its conditions · Light quanta: from entropy to an energy scale

Light quanta: from entropy to an energy scale: §7 · fluorescence and its conditions

Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt

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§7 · fluorescence and its conditions

An energy budget has conditions

When is emitted fluorescent light restricted to a lower frequency?

If an event receives only $h\nu_\text{in}$ , emits $h\nu_\text{out}$ , and retains a nonnegative remainder, energy conservation gives a bound on the outgoing frequency. It is a consequence of the assumed event budget, not evidence that all fluorescent events have that budget.

h\nu_{\mathrm{out}}\le h\nu_{\mathrm{in}}

The emitted quantum energy is at most the absorbed quantum energy in the restricted event budget.

Supply extra thermal energy or allow more than one absorbed quantum and the bound changes. The fluorescence laboratory's controls expose those extra-energy and channel assumptions rather than quietly creating energy.

Proportional fluorescence intensity requires further assumptions about absorption and conversion yield. Holding those assumptions fixed can make a weak-illumination rate proportional to the incident rate; the energy bound alone does not determine the yield.

Return to which energy sources are included in the fluorescent event. (included in this file)

Letters: the paper calls the frequency of the exciting light $\nu_1$ and that of the light produced $\nu_2$ , and takes both to consist of energy quanta of size $\frac{R}{N}\beta\nu$ . This explanation writes $\nu_{\mathrm{in}}$ and $\nu_{\mathrm{out}}$ , and hν for the quantum, with h a later name for Rβ/N.

The event: one exciting quantum of frequency $\nu_1$ is absorbed and, on its own, gives rise to one quantum of frequency $\nu_2$ . Quanta of other frequencies and energy of other kinds, heat for example, may be produced at the same time. The paper holds that the intermediate processes do not matter.
The energy supply: if the fluorescent substance is not a lasting source of energy, the absorbed quantum is all the energy the event has.
Write the budget: the energy of the produced quantum plus all other energy produced equals the energy of the absorbed quantum.
The other energy cannot be negative, so the produced quantum carries at most the absorbed energy: $\frac{R}{N}\beta\nu_2\le\frac{R}{N}\beta\nu_1$ .
Divide both sides by the positive number $\frac{R}{N}\beta$ : $\nu_2\le\nu_1$ . This is Stokes's rule, that fluorescent light does not have a higher frequency than the light that excites it. In the explanatory letters the budget reads as the formula below.

h\nu_{\mathrm{out}}\le h\nu_{\mathrm{in}}

The emitted quantum energy is at most the absorbed quantum energy in the restricted event budget.

Worked case: exciting light at $7.5\times10^{14}$ per second, violet. Under these conditions the budget allows fluorescent light at $5.5\times10^{14}$ , green, and rules out $8.5\times10^{14}$ , in the ultraviolet. In wavelengths: the fluorescent light is at the same or a longer wavelength.
The conditions are part of the result. The paper names where the rule may fail: when so many quanta are being transformed at once in each unit of volume that one produced quantum can take its energy from several exciting quanta, and when the exciting or produced light is not like black radiation within the range of Wien's law.
The same picture makes a prediction about weak light. Each exciting quantum causes its own elementary process, independently of the others, so with everything else unchanged the light produced is proportional to the exciting light, and there is no lower intensity below which the light would fail to excite at all.
The budget sets a bound on each event; it says nothing about how many absorbed quanta produce fluorescence in a given material. That yield is a separate question about the material.

Return to which energy sources are included in the fluorescent event. (included in this file)

No quantum comes out larger than one went in

Modern teaching equation, not the equation as printed.

h\,\nu_2 \le h\,\nu_1

h times the emitted frequency is at most h times the absorbed frequency.

If each emitted quantum is made from one absorbed quantum, it cannot carry more energy than that quantum brought, so the emitted light is no higher in frequency. This is Stokes's rule, read as an energy budget.

Assumptions

An event absorbs one light quantum and produces at most one emitted quantum in this idealized budget.
No additional energy reservoir contributes in the restricted case.
The light-quantum hypothesis is being applied to a transformation process.

Limits

The limit on frequency holds under these conditions; it does not forbid higher-frequency emission in general.
The budget does not predict actual material rates or spectra.
A dense field need not satisfy the independent-quantum assumptions used in the analogy.

Foundations included in this file

Energy of motion and inertia

If a body moves at the same speed but carries less energy of motion, what has changed?

A force that pushes a body through a distance does work on it, and the work becomes the body's energy of motion, its kinetic energy. That is separate from energy stored inside the body, as heat or in its chemistry.

For one body at everyday speeds, doubling the speed multiplies the energy of motion by four. At a fixed speed, doubling the mass doubles it. The mass-energy paper's title asks about a body's Trägheit, its inertia: its resistance to a change in its motion, which is measured by pushing it, not by weighing it.

K=\frac{1}{2}mv^2

In Newtonian mechanics, kinetic energy is one half times inertial mass times speed squared.

This formula holds at speeds small compared with the speed of light, and it is not exact close to that speed. So the mass-energy argument works with the exact expression and reads the mass from its low-speed limit, where it takes this form.

Worked example: Two bodies at the same speed, different masses

Two bodies move at 2 metres per second.
The 3 kg body has ½ × 3 × 2² = 6 joules of energy of motion; the 2 kg body has ½ × 2 × 2² = 4 joules.
The speed is the same, and the energies differ by 2 joules because the masses differ by 1 kg.

K_0-K_1=\frac{1}{2}(m_0-m_1)v^2

At the same low speed, the kinetic-energy drop equals one half times the mass decrease times the squared speed.

Here the two masses are given, so the example proves nothing about mass and energy. The 1905 argument runs the other way: it computes the energy difference from the light the body gives off, finds that it has this form, and reads the mass difference from the coefficient of ½v².

Where this lesson stops

At the same low speed, less energy of motion means less mass. The comparison gives only a difference in mass, never the total energy stored inside the body.

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Static laboratory examples

lq-07

This laboratory is not included in this file. The chapter's authored foundation examples remain available above. Open lq-07 online

Source references

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).
A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.