Keywords: 
• Slichter modes /
• Chebyshev collocation method

Abstract: The translational free oscillations of the Earth's innen core, which are often called Slichter modes, are studied. The periods of them are computed by solving the governing ordinary differential equations singular at the Earth center using the Chebyshev collocation method. The results are in accordance with that of many other authors^[7,8].The main difficulties of the problem are risen by the singularity of the equations at the Earth center. If some usual numarical methods are used, the equations must be treatedspecially at the Earth center. As this problem contains an infinite number of equations which are coupled one another, truncation must be made in practical solution. The necessity of special treatment at the Earth center hinders to truncate at higher order and hence limits the accuracy of the result. The Chebyshev Collocation method proposed in this paper is able to overcome the difficuties of the singularity without further special development. Hence one can truncate at as higher order as one want as far as his computing device permits. In this paper, the author has chosen to solve a system of 16 ordinary differential equation. This corresponds to a truncation of order 4 in the inner core and mantle and of order 8 in the outer core. Within the knowledge of the author, this level of truncation has never been achieved before.