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Figure 1. Non-interacting infinite fermion system. The ETF functions ff(x) up to n = 10 are displayed (broken curves) as functions of the momentum x = g/kg in units of the Fermi momentum kp. The exact Lindhard function f(x) is the full curve. The chain curve corresponds to the approximation n = 2 with fo = 1/4.

Figure 1 Non-interacting infinite fermion system. The ETF functions ff(x) up to n = 10 are displayed (broken curves) as functions of the momentum x = g/kg in units of the Fermi momentum kp. The exact Lindhard function f(x) is the full curve. The chain curve corresponds to the approximation n = 2 with fo = 1/4.

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Figure 2. Non-interacting infinite fermion system. The relative differences 1 — fere(x) /f (x) between the ETF functions /#,(x) (up to n = 10) and the exact Lindhard function f(x) are displayed as functions of the momentum x = g/kg in units of the Fermi momentum kp. The chain curve corresponds to the approximation n = 2 with £2 = 1/4.
Figure 2. Non-interacting infinite fermion system. The relative differences 1 — fere(x) /f (x) between the ETF functions /#,(x) (up to n = 10) and the exact Lindhard function f(x) are displayed as functions of the momentum x = g/kg in units of the Fermi momentum kp. The chain curve corresponds to the approximation n = 2 with £2 = 1/4.
The differential equation for 6 is more cumbersome in this case, but it can also be easily solved using Fourier transform techniques. For g 4 0, the M_; sum rule up to the nth order of the ETF approximation can be written as To determine the coefficients 6, y2; and 59; we expand the function 1/f5*(x) in kes of x in the region x < 2, and identify term by term with the above expression for 1/ FE@ (x) As they should, the obtained values for 62; coincide with those previously determined vor the free case. The following values are obtained for the other coefficients: where the additional gradient terms related to the effective mass contain new coefficient: v2; and 52; to be determined. It may be seen that for an interaction producing an effective mass m* = m independent of the density, the gradient expansion of the kinetic energy i: identical to that of the free case (10).
The differential equation for 6 is more cumbersome in this case, but it can also be easily solved using Fourier transform techniques. For g 4 0, the M_; sum rule up to the nth order of the ETF approximation can be written as To determine the coefficients 6, y2; and 59; we expand the function 1/f5*(x) in kes of x in the region x < 2, and identify term by term with the above expression for 1/ FE@ (x) As they should, the obtained values for 62; coincide with those previously determined vor the free case. The following values are obtained for the other coefficients: where the additional gradient terms related to the effective mass contain new coefficient: v2; and 52; to be determined. It may be seen that for an interaction producing an effective mass m* = m independent of the density, the gradient expansion of the kinetic energy i: identical to that of the free case (10).
Figure 3. Symmetric nuclear matter with Skyrme interaction. The relative differences 1 = ft @/ i¢ (x) between the ETF functions fer” (x) (up to n = 10) and the exact Lindhard function f Sk(x) are displayed as functions of the omentum x = q/kp in units of the Fermi momentum kp.
Figure 3. Symmetric nuclear matter with Skyrme interaction. The relative differences 1 = ft @/ i¢ (x) between the ETF functions fer” (x) (up to n = 10) and the exact Lindhard function f Sk(x) are displayed as functions of the omentum x = q/kp in units of the Fermi momentum kp.
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PhysicsNuclear matter
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