How to raise the height of a hat when it is over just an empty space?

Question

I would like to write a hat over just an empty space, like \hat{ }. My problem is that I need it to be at a certain height, which happens to be higher that the standard height I get with the previous command. I tried to raise the hat using \vspace{} like this: \hat{\vspace{.1cm}} (changing the .1cm to different values), but got no result; the hat remains at the same height. So I would like to know of an alternative way of doing what I need.

Answer 1

Here I put \hat over a hard space, otherwise it takes up no width. Then, I can use \raisebox to lift it higher.

\documentclass{article}
\begin{document}
$a \hat{~} b \raisebox{2pt}{$\hat{~}$} c$
\end{document} 

To obey the smaller mathstyles, I make the macro \highhat[] that can take a real number as an optional argument, specifying the number of "local-mathstyle points" to raise the hat (default 2\LMpt).

\documentclass{article}
\usepackage{scalerel}
\newcommand\highhat[1][2]{\ThisStyle{\raisebox{#1\LMpt}{$\SavedStyle\hat{~}$}}}
\begin{document}
$a \hat{~} b \highhat c \highhat[4] d$

$\scriptstyle a \hat{~} b \highhat c \highhat[4] d$

$\scriptscriptstyle a \hat{~} b \highhat c \highhat[4] d$
\end{document} 

Answer 2

One option that might help you is to place the hat over a phantom character. The phantom character will not be shown, but still takes up space.

\documentclass{article}
\begin{document}
$\hat{~} \hat{b} \hat{\phantom{b}}$
\end{document}