Matter Density versus Distance for the Neutrino Beam from Fermilab to Lead, South Dakota, and Comparison of Oscillations with a Variable and a Constant Density

This paper is divided into two parts. In the first part, the material densities passed through for neutrinos going from FNAL to Sanford Laboratory are calculated using two recent density tables, Crustal [G. Laske, G. Masters. Z. Ma, and M. Pasyanos, Update on CRUST1.0 -- A 1-degree global model of Earth's crust, Geophys. Res. Abstracts 15, EGU2013-2658 (2013)] and Shen-Ritzwoller [W. Shen and M. H. Ritzwoller. Crustal and uppermost mantle structure beneath the United States, J. Geophys. Res.: Solid Earth 121, 4306 (2016)], as well as the values from an older table PEMC [A. M. Dziewonski, A. L. Hales and E. R. Lapwood, Parametrically simple earth models consistent with geophysical data, Phys. Earth Plan. Int. 10 12 (1975)]. In the second part, neutrino oscillations at Sanford Laboratory are examined for the variable density table of Shen-Ritzwoller. These results are then compared with oscillation results using the mean density from the Shen-Ritzwoller tables and one other fixed density. For the tests made here, the mean density results are quite similar to those found using the variable density vs distance.