This dissertation presents the results of a search in the exclusive photon plus missing transverse energy ([gamma] + E T) finnal state in proton antiproton collisions at a center of mass energy of 1.96 TeV using the Collider Detector at Fermilab experiment. The strategy used here is to search for delayed photons coming from gauge mediated supersymmetric events with the exclusive production of X~01 2![gamma]G~. In these models the ~ 01 is the lightest neutralino and has nanosecond lifetime before decaying to a photon ([gamma]) and gravitino (~G) which exits the detector unrecorded. In order to search for this process we select collisions that have a single photon plus missing transverse energy and little other activity in the detector and examine the arrival time of the photon. This arrival time is then compared against expectations from a data driven background of the standard model sources. In the data collected from the Fermilab Tevatron collider from December 2004 to June 2010, representing 6.3 fb-1 of data, we observe 322 events in the photon arrival timing region from 2 nanoseconds to 7 nanoseconds with a data driven background prediction of 257 ± 35. An excess of 65 events is observed, equivalent to a standard deviation (N [sigma]) of 1.65 from the null hypothesis.