The effects of neutron displacement damage on materials are sensitive to neutron energy spectra. In controlled neutron damage experiments, a well characterized neutron flux spectrum is critical in determining the equivalent dose for displacement damage. Two techniques were used to characterize the neutron flux spectra in the University of Texas at Austin TRIGA research nuclear reactor. The first technique uses a standard method of measuring the reaction rates of two identical metal foils, one of which was irradiated in a Cd cover, the other of which was irradiated bare. Assuming an analytic form of the neutron spectrum the reaction rates were used to determine an approximate spectrum. The second technique uses the reaction rates measured from a set of activated metal foils along with two spectral unfolding techniques to approximate and then refine the neutron spectrum. A Matlab code was developed which fits radiative capture reaction rates to an approximate spectrum using a least squares approach. The result was used as an initial guess in a second Matlab code which refines the epithermal and fast energy ranges of the spectrum using reaction rates from threshold reactions. Errors in the reaction rates calculated from the resulting spectrum to the measured reaction rates were used to assess the accuracy of the final neutron spectrum. / text