The $^3$He($alpha$,$gamma$)$^7$Be reaction is a widely studied nuclear reaction; however, it is still not understood with the required precision. It has a great importance both in Big Bang nucleosynthesis and in solar hydrogen burning. The low mass number of the reaction partners makes it also suitable for testing microscopic calculations. Despite the high number of experimental studies, none of them addresses the $^3$He($alpha$,$gamma$)$^7$Be reaction cross sections above 3.1-MeV center-of-mass energy. Recently, a previously unobserved resonance in the $^6$Li(p,$gamma$)$^7$Be reaction suggested a new level in $^7$Be, which would also have an impact on the $^3$He($alpha$,$gamma$)$^7$Be reaction in the energy range above 4.0 MeV. The aim of the present experiment is to measure the $^3$He($alpha$,$gamma$)$^7$Be reaction cross section in the energy range of the proposed level. For this investigation the activation technique was used. A thin window gas-cell target confining $^3$He gas was irradiated using an $alpha$ beam. The $^7$Be produced was implanted into the exit foil. The $^7$Be activity was determined by counting the $gamma$ rays following its decay by a well-shielded high-purity germanium detector. Reaction cross sections have been determined between $E_{cm} = 4.0 - 4.4$ MeV with 0.04-MeV steps covering the energy range of the proposed nuclear level. One lower-energy cross-section point was also determined to be able to compare the results with previous studies. A constant cross section of around 10.5 $mu$barn was observed around the $^7$Be proton separation energy. An upper limit of 45 neV for the strength of a $^3$He($alpha$,$gamma$)$^7$Be resonance is derived.
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