[Tex/LaTex] How to get an exponent hat

Question

I would like to use a hat in exponent for a notation, and I have found no satisfactory way of doing this. If I go for $f^{\hat{\null}}$, the hat is far too high, while $f^\wedge$ gives to big and too acute a hat. I would like to have the regular hat, but placed in exponent just as if it where usually drawn in the center of the line, not on the top. Any idea?

Precision: I seek a symbol that I could use quite like \wedge, in the sense that I should be able to write $f^\solution$ and more importantly $f_x^\solution$ with the hat and the index aligned correctly. A solution that would need to rewrite all my $f^\wedge$ in a slightly different way but would give good alignment without too much twiddling would already be good.

Additional precision: I also have to deal with double anitderivatives, so I need to be able to have the equivalent of $f^{\wedge\wedge}$, with both hats on level. The really best solution would be to extract the symbol used by \hat, without the surrounding command (i.e., the command that puts this symbol on top of the next letter).

Answer 1

You can get a decent result with

$f\hat{\mkern6mu}$

or

$f{\mathchar"5E}$

In the second case the spacing is perhaps too much; here they are side by side with a symbol after them to show the spacing