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Would it be possible through a clever arrangement of solar dishes, lenses and mirrors etc, to create fusion at the focal point?

Imagine you arrange your solar dishes, lenses and mirrors in such a way, that the final stage would be lenses fixed up in the air, surrounding the target to be fused from all sides, forming a sphere. The lenses would be cooled as much as possible and the light incoming would be at the maximum possible without destroying the lense.

What size would the lenses have to be? (any size is allowed - note that this is just a thought experiment, not concerned with if it's practical or not)

Are there any physical constrains which would prevent for any fusion to occur, no matter how we arrange this experiment?

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You question isn't specific enough; it needs a little work to clarify the fusion setup. For example, what fuel type are you talking about fusing? Is there confinement, so that this is a thermal fusion reaction, or would just one fusion reaction be sufficient?

For example, the temperature required to overcome the Coulomb barrier for deuterium-tritium fusion is $4.5 \times 10^7 \mathrm{K}$, which corresponds to an energy per particle of $E \approx\frac{3}{2} k_\mathrm{B} T$ $\approx 10^{-15} \mathrm{J}$. So the simplest way to answer your question, then, is to say that you just need to give this much energy to a deuteron, and direct it perfectly towards its target tritium. And you could do that in principle with only that much solar energy. But the area needed for that would depend on how long you apply the force to the deuteron, because the sun is giving you power (which is energy per unit time).

Now, that's probably cheating -- though I can't say for sure, because your question isn't specific enough. Let's say you actually want to just heat up some sample that is magically contained, but still lets all that solar energy in. Basically, you need to balance the heat output against the heat input. Modeling the magical ball of fuel as a blackbody of area $a$, you can use the Stefan-Boltzman law to say that the power output just from blackbody radiation is $$ I_\mathrm{out} = a\, \sigma\, T^4. $$ If you collect solar energy over an area $A$, and perfectly transfer all of that energy to your fuel, the energy input is in the neighborhood of $$ I_\mathrm{in} = A\, (1 \mathrm{kW}/\mathrm{m}^2). $$ You need to balance the output and input energies to get the temperature $T$, so $$ T = \sqrt[4]{\frac{A}{a\, \sigma}\, (1 \mathrm{kW}/\mathrm{m}^2)} \geq 4.5 \times 10^7 \mathrm{K}. $$ So you can see that the area $A$ over which you have to collect energy is inversely proportional to the area $a$ of your ball (or torus) of fuel, according to this equation. For example, if your fuel is a cube of side $1\mathrm{cm}$, it has area $8 \times 10^{-4}\mathrm{m}^2$ $$ A \gtrsim (4\times 10^{30}\mathrm{K}^4) (8 \times 10^{-4}\mathrm{m}^2) (6 \times 10^{-8}\, \mathrm{\mathrm{W}\, \mathrm{m}^{-2}\mathrm{K}^{-4}}) / (10^3 \mathrm{W}/\mathrm{m}^2) \approx 10^{17} \mathrm{m}^{2} $$ That's a good deal larger than the earth. In fact, it's larger than the surface area of the entire sphere surrounding the sun at the radius of Earth's orbit. So this says that the sun doesn't provide enough energy to ignite a blackbody in the shape of a cube of side $1\mathrm{cm}$. This suggests to me that (1) my math is wrong; (2) you'd have to have a smaller pellet of fuel; (3) you'd have to magically recycle that blackbody radiation somehow; or (4) you'd need a different star.

But of course, all of this is still ignoring what may be your motivation for asking this question. This presumably would be a very poor method for capturing energy, because the energy released by fusion would need to be captured. And any way I know of doing this would be easier to do just by directly converting the solar energy.

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  • $\begingroup$ Interesting, but there is a reason i kept the details on which fuel to use and what kind of setup to have in order to contain the fuel or how exactly to use the energy coming out of this process. The reason is that i wanted to leave the details to the experts which know better than me. I was interested to see if it's feasible at all, but according to your calculations, it is not, as the area size would have to be way too large for getting this setup on earth. Which is a bit surprising to me. $\endgroup$ – pZombie May 2 '15 at 20:40
  • $\begingroup$ Not sure why you came up with these numbers, but a practical fusion reactor with 1GW electric output will probably require a couple of minutes of electrical power inout in the range of 10-100MW to start up, after which it will be self-sufficient. $\endgroup$ – CuriousOne May 2 '15 at 22:17
  • $\begingroup$ @pZombie I didn't say it couldn't be done. In fact, I showed one way to achieve fusion: use the energy to accelerate a single particle, and it can (potentially) fuse with another. Even using the sun, my calculation shows that you just need a small enough target. $\endgroup$ – Mike May 3 '15 at 1:56
  • $\begingroup$ @CuriousOne Yes, but even my highly speculative readings of the question didn't stretch it as far as "use solar energy to start up ITER++"... The question seemed to me to be about direct energy transfer to cause fusion. I didn't assume a self-sustaining nuclear reaction or any confinement. $\endgroup$ – Mike May 3 '15 at 2:00
  • $\begingroup$ I made that comment to Mike. The answer to your question is "no" with optics alone and "absolutely" when done right. $\endgroup$ – CuriousOne May 3 '15 at 2:07

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