Length Changes of Unconfined Nb$_3$Sn Rutherford Cables During Reaction Heat Treatment

In order to predict volume changes of Nb$_3$Sn coils during reaction heat treatment (RHT), coefficients of thermal expansion (CTE) of the involved materials need to be known. While the CTEs of bulk materials are widely available, there is a lack of knowledge about the dimensional changes of the Nb$_3$Sn Rutherford cables. By extensometry we have measured the length changes of 11 T dipole Nb$_3$Sn Rutherford cables during the RHT. Similar to the length change behavior of multifilament wires, after an initial cable length increase the cable length decreases when the temperature exceeds about 150°C, due to relaxation of the Nb subelements when the Cu matrix is annealed. At higher temperatures the cable length increase is determined by the CTE of the unreacted Nb, before the onset of Nb$_3$Sn formation leads to an increased expansion rate. During cooling, the thermal contraction of fully reacted Nb$_3$Sn cable is similar to that of Nb$_3$Sn bulk.