Empirical scaling formulas for critical current and critical field for commercial NbTi

Abstract

Short sample 4.2 K experimental facilities are plentiful, but equipment for measurements of current as functions of temperature and field is scarce. An analysis has been made of published data comprising at least six manufacturers and spanning a range of critical current density at 4.2 K, 8 T of 50 to 108 kA/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , and linear equations have been found to fit the data over a wide range of field B and temperature T. For a constant temperature of 4.2 K, the following expression holds for B in the range of 3 to 10 T: j <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (B, T = 4.2 K) = j <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> [1 - 0.096B], where [B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</inf> (4.2 K)] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> = 0.096 with a standard deviation of 3% for ten samples. The constant j <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> can be determined for any sample from a single point measurement at a convenient field. For a constant field of 8 T, the following expression holds for T in the range of 2 to 5.5 K: j <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (B = 8 T, T) = j' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</inf> [1 - 0.177T], where [T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (8 T)] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> = 0.177 with a standard deviation of less than 1%. Linear equations have also been obtained for higher fields and lower temperatures. The critical field vs temperature is B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</inf> (T) = B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</inf> (0) [1 - (T/T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (0)) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ], where B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</inf> (0) = 14.5 T, T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (0) = 9.2 K, and n = 1.7 (not 2, which is used in theoretical derivations). For more accurate critical temperature calculations above 10 T, this equation can be used with the modification B <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c2</inf> (0) = 14.8 T. No one simple power law for the upper critical field holds over the whole temperature range.