Apparatus and methods for cooling high temperature superconducting materials (HTSC) to superconductive temperatures within the range of 27.degree. K. to 77.degree. K. using a mixed refrigerant consisting of liquefied neon and nitrogen containing up to about ten mole percent neon by contacting and surrounding the HTSC material with the mixed refrigerant so that free convection or forced flow convection heat transfer can be effected.