## 3 A Model for the Measured Spectral Shape
### The Memory Function Formalism
The \(t=0\) values of the \(i\)-th order memory function \(K_{i}(t)\) can be determined by superimposing the sum rule fulfillment, which for the few lowest orders yields:  
\[K_{1}(0)=\left\langle\omega^{2}\right\rangle \tag{7a}\] \[K_{2}(0)=-\frac{\ddot{K}_{1}(0)}{2K_{1}(0)}=\frac{\left\langle\omega^{4} \right\rangle}{\left\langle\omega^{2}\right\rangle}-\frac{\left\langle\omega^ {2}\right\rangle}{\left\langle\omega^{0}\right\rangle}\] (7b) \[K_{3}(0)=-\frac{\ddot{K}_{2}(0)}{2K_{2}(0)}=\frac{1}{\Delta_{2}}\left[\frac{ \left\langle\omega^{6}\right\rangle}{\left\langle\omega^{2}\right\rangle}- \left(\frac{\left\langle\omega^{4}\right\rangle}{\left\langle\omega^{2} \right\rangle}\right)^{2}\right] \tag{7c}\]  
Here \(\left\langle\omega^{n}\right\rangle=i^{n}[\frac{d^{n}C(t)}{dt^{n}}]_{t=0}\cdot [C(0)]^{-1}=\int_{-\infty}^{+\infty}\omega^{n}C(\omega)d\omega\cdot[C(0)]^{- 1}\) is the \(n\)-order spectral moment of the correlation function \(C(\omega)\). Among all possible fluctuating variables, we are here interested in density fluctuations since the Fourier-Laplace transform of their correlation function, \(S(Q,\omega)\) is directly measured by both _IXS_ and _INS_. \(S(Q,\omega)\) can be derived as the Fourier anti-transform of the intermediate scattering function \(F(Q,t)=\int_{V}\left\langle\delta\rho(\overrightarrow{r},t)\delta\rho^{*}( \overrightarrow{r},0)\right\rangle exp\left(i\overrightarrow{Q}\cdot \overrightarrow{r}\right)d\overrightarrow{r}\). The latter can be expressed through the continued fractions expansion of Equation (4), whose 2-nd order truncation yields:  
\[\frac{F(Q,s)}{S(Q)}=\left[s+\frac{\left\langle\omega^{2}\right\rangle}{s+m_{ L}(Q,s)}\right]^{-1}\]  
Where \(m_{L}(Q,s)\) is the second order memory function for density fluctuations while \(S(Q)=F(Q,0)\) is the static structure factor. The \(S(Q,\omega)\) can be derived from Equation (7) as follows:  
\[\frac{S(Q,\omega)}{S(Q)}=\frac{1}{\pi}Re\left[\frac{F(Q,s=i\omega)}{S(Q)} \right]=\frac{1}{\pi}Re\left[i\omega+\frac{\left\langle\omega^{2}\right\rangle }{i\omega+m_{L}(Q,s=i\omega)}\right]^{-1}\]  
The last term of the above formula can be calculated explicitly to eventually obtain:  
\[\frac{S(Q,\omega)}{S(Q)}=\frac{1}{\pi}\frac{\left\langle\omega^{2}\right\rangle m _{L}{}^{\prime}(Q,\omega)}{\left[\omega^{2}-\left\langle\omega^{2}\right\rangle -\omega m_{L}{}^{\prime\prime}(Q,\omega)\right]^{2}+\omega^{2}[m_{L}{}^{ \prime}(Q,\omega)]^{2}} \tag{8}\]  
where \(m_{L}{}^{\prime}(Q,\omega)\) and \(m_{L}{}^{\prime\prime}(Q,\omega)\) are, respectively, real and imaginary parts of the Fourier transform of \(m_{L}(Q,t)\). At this stage, the problem of choosing the most appropriate model is shifted from \(S(Q,\omega)\) to \(m_{L}(Q,t)\), or \(m_{L}(Q,\omega)\). In many literature works on liquid systems attempting a memory function based modeling of the spectrum (see e.g. [13; 73]), it can be shown that a sensible _ansatz_ for the time dependent memory function is given by:  
\[\frac{dC(t)}{dt}=i\Omega\cdot C(t)-\int_{0}^{t}K(\tau)\cdot C(t-\tau)d\tau \tag{5}\]0  
The three terms in the right hand side of Equation (9) are (from left):  
1. **The thermal contribution. ** The first term describes diffusive thermal motions triggered by spontaneous temperature gradients. Its timescale and amplitude are \(\tau_{T}=\left(\gamma D_{T}Q^{2}\right)^{-\,1}\) and \(\Delta_{T}^{2}=(\gamma-1)(c_{T}Q)^{2}\) respectively, with \(D_{T}\), \(c_{T}\), \(\gamma\) and being the generalized diffusivity, the isothermal velocity and the constant pressure to constant volume specific heats ratio. 2. **The viscous relaxation contribution**. The second term accounts for relaxation processes affecting the viscosity and having a timescale \(\tau_{\alpha}\) and amplitude \(\Delta_{\alpha}^{2}=(c_{\infty}^{2}-\gamma c_{T}^{2})Q^{2}\), where \(c_{\infty}\) is the generalized infinite frequency or \"elastic\" sound velocity of the sample. 3. **The instantaneous contribution**.