I'm interested about whether or not a conventional neon sign such as this one could emit x-ray that could be harmful.
The sign that you can see in the picture is just below my office window so I'm basically sitting like 8 hours a day about 2 meters away from the sign which is functioning the whole day. Just curious if this can cause any harm of any kind because of the long exposure.
Neon signs do not produce X-rays. Although they operate at high voltages, the gas within the tubes prevent electrons from gaining enough energy to produce X-rays.
However, if one wants to split hairs, all emission of electromagnetic waves is probabilistic. Even an incandescent bulb will produce a negligible amount of X-rays. Maybe one photon per second, or maybe one photon per year. I don't know the rate but it is very small compared to the more or less astronomical number of photons produced by an X-ray machine. I would guess that the amount of X-ray radiation you would receive from a neon light might well be less than what you would receive sitting under a beach umbrella on a sunny day.
Edit: I just did some back-of-the-envelope calculations to estimate the power of the X-rays emitted by an incandescent bulb.
Assume an incandescent bulb emits black body radiation according to Planck's Law, and is at 2700 Kelvins. Assume further that X-rays consist of photons with frequency 3 x 10^16 Hz.
To find the relative intensity of X-rays to yellow light, I divided the Planck's Law result for X-rays by that of yellow light and get this approximation.
$$B_{rel} = e^{\left(\frac{h}{k_{B}T}\right)\left({f_{yellow}-f_{x-ray}}\right)}$$
where \$h\approx 6.6 \times 10^{-34} joules\$ is Planck's constant. \$k_{B}\approx 1.38 \times 10^{23} [complicated-units]\$ is Boltzmann's constant, \$T\$ is the temperature in Kelvins.
$$\frac{h}{k_{B}} \approx 4.8 \times 10^{-11} sK$$
$$\frac{h}{k_{B}\times 2700K} \approx 1.778 \times 10^{-14} s$$
$$e^{\left(1.778 \times 10^{-14}\right)\left(3 \times 10^{16}\right)} = e^{\left(5.333 \times 10^2\right)} \approx 4 \times 10^{231}$$
So, (with a bit of further approximation) there is somewhere around \$4 \times 10^{231}\$ times more yellow light emitted from an incandescent light bulb than x-ray radiation. AFAIK, that is more than the number of atoms in the visible universe.
This was just a fun calculation and not meant to be precise.
High energy electrons produce x-rays when they strike atoms, usually metal. That's how a scanning electron microscope can detect elements. Basically, you need electrons accelerated with a high voltage and a long mean free path to get x-rays. Cathode ray tubes (CRT) produce x-rays when the electron beam strikes the phosphor. As Math said the noble gas in neon lights makes the mean free path of the electrons too short to gain the energy for the electrons to produce x-rays.
As shown with the black body radiation equations given by Math, to produce x-rays based on the temperature of the object takes temperatures well above our normal capabilities. That is not how we produce x-rays, nor is it how light is produced by neon lights.
Neon lights produce light by electrons interacting with the electron shells of the noble gases atoms. When the electrons lose energy, the interaction with the electron shells produces a photon with the energy equal the amount of energy lost by the electron.
We produce x-rays in a similar fashion, but consider this.
An electron accelerated across a field of 1V has an energy of 1 electron volt (eV). Thus, the upper limit of the energy in eV produced is the voltage of the power source.
Electrons cannot lose more energy than they have. Thus, cannot produce photons with a higher energy than they have.
X-ray photons have energies in keV (>1keV).
Figure 1. Photon energies of the electromagnet spectrum (Taken from "Μελέτη της διακριτικής ικανότητας μηχανής ακτίνων-Χ με τεχνικές προσομοίωσης Monte-Carlo," p. 16)
Neon signs can be operating with 2V to 15kV, but for an electron to gain the energy of the power source, it needs to be accelerated from the negative electrode and strike the positive electrode without colliding with atoms, causing energy loss. In a neon sign electrons collide extensively with noble gas atoms. As the electrons lose energy in the atom's electron shells, they produce visible light photons (1.75 to 3.1eV). The unobstructed path between the electrodes to produce x-rays isn't there for neon signs.
