Nuclear Incompressibility

The isoscalar incompressibility of infinite nuclear matter is defined by the well-known formula [1]

$$K_{\infty} = 9\rho^2 \frac{\mbox{d}^2}{\mbox{d}\rho^2} \Bigl( \frac{E}{A} \Bigr)_{\rho=\rho_{nm}},$$ (29)

where $\rho_{nm}$ is the saturation density of nuclear matter. Of course, $K_{\infty}$ cannot be directly measured; however, by using Eq. (29) it can be calculated from theoretical equation of state $E(\rho)$ or it can be indirectly estimated from measurements of monopole excitations of finite nuclei.

The incompressibility of finite nucleus, $K_A$, is defined by its scaling-model relation [28] to the centroid of the giant monopole resonance (GMR), $E_{\text{GMR}}$, as

$$E_{\text{GMR}} = \sqrt{\frac{\hbar^2K_A}{m\langle r^2\rangle}}$$ (30)

where $\langle r^2\rangle$ is the average square radius of the nucleus. Eq. (30) is derived under the assumption that most of the monopole strength is concentrated within one dominant peak, see Ref. [1]. The centroid of the GMR can be extracted from its strength function as the ratio of the first and zero moments, that is,

$$E_{\text{GMR}} = \frac{m_1}{m_0}.$$ (31)

There exist several alternative ways to extract $E_{\text{GMR}}$ through different moments of the strength function, such as $E_{\text{GMR}}=\sqrt{m_1/m_{-1}}$ or $E_{\text{GMR}}=\sqrt{m_3/m_{1}}$. However, they are more sensitive to details of the strength function and thus less appropriate for studies of the incompressibility.

In analogy to the Weizsäcker formula for the nuclear masses, one can introduce [1] a similar relation for nuclear incompressibilities,

$$K_A = K_{V} + K_{S}A^{-1/3} + (K_{\tau}+K_{S,\tau}A^{-1/3})\frac{(N-Z)^2}{A^2}$$

$$+ K_{C} \frac{Z^2}{A^{4/3}} \ .$$ (32)

Similarly as in the liquid-drop model, we refer to $K_{V}$, $K_S$, $K_{\tau}$, $K_{S,\tau}$, and $K_C$ as the volume, surface, symmetry, surface-symmetry, and Coulomb incompressibility parameters, respectively. By adjusting these parameters to the incompressibilities $K_A$, calculated in finite nuclei from Eqs. (30) and (31), we can obtain an estimate of the infinite-matter incompressibility as $K_{\infty}\simeq K_{V}$.