Fermi surfaces and kinetic densities

For INM arbitrary asymmetry, the Fermi energy of each particle species is different. Finite spin densities s_i break the isotropy of INM, creating the possibility that the Fermi surface will deform[72]. We are mainly interested in INM with small polarization, so we use the approximation that all Fermi surfaces are spherical.

In the mean-field approximation, $\tilde\rho_{q \sigma} (\vec{k})$, the density of particles in momentum space with the isospin projection q and spin projection $\sigma$, is

$$\tilde\rho_{q \sigma} (\vec{k}) = \left\{ \begin{array}{l} ... ...d \mbox{for $k > k_{{\rm F}, q \sigma}$}. \end{array} \right.$$ (36)

In asymmetric polarized INM, the relation between the Fermi momenta and the isoscalar scalar density reads

$$\rho_0 = \frac{2}{3 \pi^2} \; k_{\rm F}^3 = \frac{1}{6 \pi^2... ...igma = \uparrow, \downarrow} k_{{\rm F}, q \sigma}^3 \quad ,$$ (37)

with $k_{{\rm F}, q \sigma} = (6 \pi^2)^{1/3} \rho_{q \sigma}^{1/3}$. Here, $k_{\rm F}$ is the ``average'' Fermi momentum of the whole system. The kinetic density in momentum space for each particle species is given by

$$\tilde\tau_{q \sigma} (k) = k^2 \, \tilde\rho_{q \sigma} (k) ,$$ (38)

and the kinetic density in coordinate space is

$$\tau_{q \sigma} = {\textstyle\frac{{V}}{{10 \pi}}} k_{\rm F,... ...yle\frac{{3}}{{20}}} \, \beta \, \rho_{q \sigma}^{5/3} \quad ,$$ (39)

where $\beta = ( 3 \pi^2 / 2 )^{2/3}$. Various kinetic densities in the spin-isospin space are given by

$$\tau_0 \tau_{{\rm n}\uparrow}+ \tau_{{\rm n}\downarrow}+ \tau_{{\rm p}\u... ...downarrow} = {\textstyle\frac{{3}}{{5}}} \, \beta \, \rho_0^{5/3} F_{5/3}^{(0)}$$ =

$$\tau_1 \tau_{{\rm n}\uparrow}+ \tau_{{\rm n}\downarrow}- \tau_{{\rm p}\u... ...narrow} = {\textstyle\frac{{3}}{{5}}} \, \beta \, \rho_0^{5/3} F_{5/3}^{(\tau)}$$ =

$$\tau_{{\rm n}\uparrow}- \tau_{{\rm n}\downarrow}+ \tau_{{\rm p}\u... ...rrow} = {\textstyle\frac{{3}}{{5}}} \, \beta \, \rho_0^{5/3} F_{5/3}^{(\sigma)}$$ T₀ =

$$\tau_{{\rm n}\uparrow}- \tau_{{\rm n}\downarrow}- \tau_{{\rm p}\u... ... {\textstyle\frac{{3}}{{5}}} \, \beta \, \rho_0^{5/3} F_{5/3}^{(\sigma\tau)} ~,$$ T₁ = (40)

where F_m⁽⁰⁾, $F_m^{(\tau)}$, $F_m^{(\sigma)}$, and $F_m^{(\sigma\tau)}$ are functions of the relative excesses:

$${\textstyle\frac{{1}}{{4}}} \big[ \big( 1 + I_\tau + I_\sigma + I_{\sigma \tau} \big)^m + \big( 1 + I_\tau - I_\sigma - I_{\sigma \tau} \big)^m$$ F_m⁽⁰⁾ =

$$+ \big( 1 - I_\tau + I_\sigma - I_{\sigma \tau} \big)^m + \big( 1 - I_\tau - I_\sigma + I_{\sigma \tau} \big)^m \big]$$

This is a straightforward generalization of the corresponding definition for asymmetric unpolarized nuclear matter given in [23]. Similarly one defines

$$F_m^{(\tau)} {\textstyle\frac{{1}}{{4}}} \big[ \big( 1 + I_\tau + I_\sigma + I_{\sigma \tau} \big)^m + \big( 1 + I_\tau - I_\sigma - I_{\sigma \tau} \big)^m$$ =

$$- \big( 1 - I_\tau + I_\sigma - I_{\sigma \tau} \big)^m - \big( 1 - I_\tau - I_\sigma + I_{\sigma \tau} \big)^m \big] ,$$

$$F_m^{(\sigma)} {\textstyle\frac{{1}}{{4}}} \big[ \big( 1 + I_\tau + I_\sigma + I_{\sigma \tau} \big)^m - \big( 1 + I_\tau - I_\sigma - I_{\sigma \tau} \big)^m$$ =

$$+ \big( 1 - I_\tau + I_\sigma - I_{\sigma \tau} \big)^m - \big( 1 - I_\tau - I_\sigma + I_{\sigma \tau} \big)^m \big] ,$$

$$F_m^{(\sigma\tau)} {\textstyle\frac{{1}}{{4}}} \big[ \big( 1 + I_\tau + I_\sigma + I_{\sigma \tau} \big)^m - \big( 1 + I_\tau - I_\sigma - I_{\sigma \tau} \big)^m$$ =

$$- \big( 1 - I_\tau + I_\sigma - I_{\sigma \tau} \big)^m + \big( 1 - I_\tau - I_\sigma + I_{\sigma \tau} \big)^m \big] .$$

For calculations of INM properties, we also need derivatives of these functions. The first derivatives are given by

$$\frac{\partial F_m^{(\tau)}}{\partial I_\tau} \frac{\partial F_m^{(\sigma)}}{\partial I_\sigma} = \frac{\partial F_m^{(\sigma\tau)}}{\partial I_{\sigma \tau}} = m \, F_{m-1}^{(0)} ,$$ =

$$\frac{\partial F_m^{(0)}}{\partial I_\tau} \frac{\partial F_m^{(\sigma)}}{\partial I_{\sigma \tau}} = \frac{\partial F_m^{(\sigma\tau)}}{\partial I_\sigma} = m \, F_{m-1}^{(\tau)} ,$$ =

$$\frac{\partial F_m^{(0)}}{\partial I_\sigma} \frac{\partial F_m^{(\tau)}}{\partial I_{\sigma \tau}} = \frac{\partial F_m^{(\sigma\tau)}}{\partial I_\tau} = m \, F_{m-1}^{(\sigma)} ,$$ =

$$\frac{\partial F_m^{(0)}}{\partial I_{\sigma \tau}} \frac{\partial F_m^{(\tau)}}{\partial I_\sigma} = \frac{\partial F_m^{(\sigma)}}{\partial I_\tau} = m \, F_{m-1}^{(\sigma\tau)} ,$$ = (41)

while the second derivatives are

$$\frac{\partial^2 F_m^{(0)}}{\partial I_i^2} m (m-1) \, F_{m-2}^{(0)} ,$$ =

$$\frac{\partial^2 F_m^{(j)}}{\partial I_i^2} m (m-1) \, F_{m-2}^{(j)} ,$$ = (42)

for any $i,j=\tau, \sigma, \sigma\tau$. Functions of the order m=0 and m=1 are rather simple:

$$1 \quad , \quad F_0^{(i)} = 0 \quad ,$$ F₀⁽⁰⁾ =

$$1 \quad , \quad F_1^{(i)} = I_i \quad ,$$ F₁⁽⁰⁾ = (43)

for any $i=\tau, \sigma, \sigma\tau$. Some special values $F_m^{(i)}(I_\tau,I_\sigma,I_{\sigma\tau})$ appearing in limiting cases of INM are

$$1 \quad , \quad F_m^{(i)}(0,0,0) = 0$$ F_m⁽⁰⁾(0,0,0) =

F_m⁽⁰⁾(1,0,0) = F_m⁽⁰⁾(0,1,0) = F_m⁽⁰⁾(0,0,1) = 2^m-1

$$F_m^{(\tau)}(1,0,0) F_m^{(\sigma)}(0,1,0) = F_m^{(\sigma\tau)}(0,0,1) = 2^{m-1}$$ =

$$F_m^{(1)}(1,1,1) = 4^{m-1} \quad .$$ F_m⁽⁰⁾(1,1,1) = (44)

while F_m⁽ⁱ⁾ = 0 if I_i = 0 and one of the other I_j's is equal to 1, with the last equal to zero. These functions are useful when writing down the equation of state and its derivatives.