The neutron total cross sections for He, Be, C, A1, Fe, Cu, Cd, W, Pb and U have been measured to an accuracy of 2 % at 5.7 ± 0.6 GeV/c. The total cross section versus atomic weight data was fit to an absorbing uniform grey sphere optical model. The best fit parameters are R = aoA1/3 where a o = 1.27 =e0.01 fm, and a nucleon mean free path in nuclear matter of 3.0 ±0.2 fm. The energy dependence of the nucleonnucleus total cross sections is discussed in terms of the nucleon-nucleon total cross sections and a screening term.