# Max temperature that can be obtained from radiation source when nonlinear interaction with matter is important?

Using a black body radiation as furnace it is usually said that the max achievable $T$ is limited to the T of the source. So a source of 3000K can not be used to heat a body above 3000K.

A long wavelength source let say $3000\ \mathrm{K}$ or nearly $1$ micron is considered. Having enough power and using a lens to focus, the intensity at the focus could exceed the binding potential of any core electron in any atom or some at least. The nonlinear interactions are important. So this 3000K source is ionizing radiation in the volume where the intensity is high enough to create "high" temperature plasma. (A small scale plasma experiment similar to the national ignition facility.)

1. What can we say about the entropy and the temperature in that volume? Is the temperature in the system much higher than $3000\ \mathrm{K}$? Let say intensity is high enough to doubly ionize He gas and create "hot" plasma.

2. Is there limit for the max temperature?

3. Once the matter is completely ionized would increase in intensity lead to higher T?

My question is not strictly related to lasers but here are few numbers for a laser source showing that the energies are "relatively small" for high nonlinear interactions: For a laser with $1\ \mathrm{mJ}$ at $1000\ \mathrm{nm}$ (3000K) beam diameter $20\ \mathrm{mm}$, $40\ \mathrm{fs}$ pulse duration, using $F=15\ \mathrm{cm}$ at the focus the peak intensity and field are $Ip=6.5 10^{16}\ \frac{\mathrm{W}}{\mathrm{cm}}, E= 7*10^9\ \frac{\mathrm{V}}{\mathrm{cm}}$. This is roughly on the scale what will completely remove the electrons from He.

• – dmckee Oct 14 '13 at 19:22
• Aside: the limit to heating this way is very fundamental; breaking it is essentially the same as breaking thermodynamics. Nor is it hard to understand: you just need to get it that radiative transfer is a two way process. – dmckee Oct 14 '13 at 19:25
• Here the re-radiation will come from electrons returning to their ions and radiation is at 24eV and 54eV lets say for simplicity. It looks hotter somehow to me. – Anonymous Oct 14 '13 at 20:56
• "Having enough power and using a lens to focus, the intensity at the focus could exceed [...]" This is also a fundamental misunderstanding there is a optical limit on the intensity that can be achieved. – dmckee Oct 15 '13 at 4:29

Well a Red photon at about 650 nm wavelength is about 2 electron Volts ( 2eV), so a one micron wavelength photon is about 1.3 eV (you can google up the exact number yourself).

So good luck on doing much ionizing with such low energy photons.

If your source is about like a 3,000 kelvin thermal radiation source, then 98% of that radiant energy lies between 0.5 microns and 8.0 microns wavelength, with a 1% tail at each end, so you will get less than 1% of your energy as 2.6 eV or higher photons.

• Not related to the question. Any wavelength is ionizing radiation provided it can have the peak intensity to overcome the binding potential of the electron. – Anonymous Oct 14 '13 at 20:50
• "Any wavelength is ionizing radiation provided [...]" This comment is more wrong than right. Though multi-photon ionization processes are an experimental fact the cross-section drop is phenomenal. – dmckee Oct 15 '13 at 4:27
• There are many issues, but the question is about the temperature limit and the entropy. – Anonymous Oct 15 '13 at 5:42
• Well anonymous, your two comments are quite beyond the scope of my limited education, so I am not really qualified to comment intelligently on them. – user26165 Oct 15 '13 at 7:20
• I seem to recall that when radiation from high Voltage (250kV) power lines reaches the ground, the energy density is such that the energy crammed into the space of a typical tissue organic molecule, is some 27 orders of magnitude two low to break any of the molecular bonds. – user26165 Oct 15 '13 at 7:25

Some points to consider:

1. There is no helium at the surface of the sun, all of it is in the core where it formed and/or sunk to, where the temperature is on the order of megakelvin--everything is ionised already. This I believe should answer the first question--if He is ionised at equilibrium, situation is way over 3000K.
2. Suppose He did exist at the surface. 3000K blackbody radiation may ionise He to He++, but my rough guess is 1/1,000,000 collisions are ionisation events, since some photons of 3000K emission (at the extreme end) may possess 79 eV. Use Planck's Law for exact probability.
3. This small likelihood drops to zero if you use a $1\mu m$ laser, since now 100% of your photons are less than 79eV (way less even for first ionisation requiring only 25eV). Doesn't matter how intense the laser is--this would increase the rate of ionisations assuming the individual photons already meet the threshold energy which they don't. You are looking at x-ray lasers.
• Sun was an example, the heating is through radiation away from the source. – Anonymous Oct 14 '13 at 20:51