## 5 Numerical Results and Discussion  
We have displayed the \((q,\alpha)\)-distribution [15] in Equation (1) for various values of spectral index \(\alpha\) and \(q\) for numerical demonstration. The spectral indices \(q\) and \(\alpha\) represent the influence on the tail of super thermal particles and the shoulder of the velocity-disturbed graph, respectively. Figure 1a,b are plotted for condition \(-1<q\leq 1\). The effect of different values of the spectral index \(\alpha(0.1,0.15,0.2)\) for the fixed value of \(q\) (=0.7) is shown in Figure 1a. \(\alpha\) illustrates the high-energy plasma elements on the shoulder of the graph. The distribution function \(f_{CT}(V_{j})\) as a function of \(V_{j}\) is plotted for different values of \(\alpha(0.1,0.15,0.2)\) at a fixed value of \(q\) (=0.9), as shown in Figure 1b. This figure shows that \((q,\alpha)\)-distributed reduces to Maxellian at \(q=0.9\) and \(\alpha=0\). Figure 1c,d depict the \((q,\alpha)\)-distribution for different values of non-extensive spectral index \(q\geq 1\). It is observed that for increasing values of \(\alpha\), the prominent effects of distribution are shown on the top shoulders, while the magnitude of the velocity distribution curves are nearly the same on the initial and end regions. Figure 1c,d show the values of the non-extensive spectral index q. [PERSON] discovered that as the value of increases, the prominent effects of plasma species are observed on the top shoulders but have no effect on the tails of velocity distribution curves. In the framework of Cairn-Tsallis-distributed dusty plasmas, the \((10)\) and \((14)\) are numerically solved, considering the typical properties of laboratory plasmas. In recent work, we evaluated the DSP (\(\psi\)), versus dust-density parameter \(P_{d}\), for species \(Xe^{+}-F^{-}-SF_{6}^{-}\) and \(Ar^{+}-F^{-}-SF_{6}^{-}\). Furthermore, the mass of ions associated with \(Xe^{+}\) and \(Ar^{+}\) is composed of \(131m_{P}\) and \(40m_{P}\), respectively, where \(m_{p}\) is mass of a proton. The mass ratios for \(Xe^{+}-F^{-}-SF_{6}^{-}\) and \(Ar^{+}-F^{-}-SF_{6}^{-}\) are \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\), and \(\eta_{4}\) are \((131/19)^{1/2}\), \((131/146)^{1/2}\), \((40/19)^{1/2}\), and \((40/146)^{1/2}\), respectively. Previously, the IA waves in plasmas (multi-ions) were explored numerically and experimentally [18], with the two kinds of negative ions. Figure 2 illustrates the DSP versus dust-density parameter \(P_{d}\) for the various values of spectral index \(\alpha(0.0,0.01,0.02)\); the fixed values of spectral index \(q\) and the values of other parameters include \(Z_{Xe}=Z_{F}=Z_{Ar^{+}}=Z_{SF}=1\), \(U_{Xe^{+}}=U_{Ar^{+}}=U_{F^{-}}=U_{SF_{6}^{-}}=U_{X^{e+}}=0.5\), \(\Delta_{1}=\Delta_{2}=\Delta_{3}=\Delta_{4}=0.4\), \(\varepsilon_{1}=(131/0.00055)^{1/2},\varepsilon_{2}=(40/0.00055)^{1/2}\). It is shown in Figure 2a that the magnitude of DSP varies directly with the \(\alpha\), by taking into account the \(Xe^{+}-F^{-}-SF_{6}^{-}\) plasmas (using Equation (10)). The same effect spectral index \(\alpha\) is observed in \(Ar^{+}-F^{-}-SF_{6}^{-}\) plasmas (using Equation (14)). Furthermore, the dust surface potential in \(Xe^{+}-F^{-}-SF_{6}^{-}\) is frequently larger than in \(Ar^{+}-F^{-}-SF_{6}^{-}\), owing to the higher mass of \(Ar^{+}\). For spectral index \(\alpha\) with other parameters held constant, the dust \(DSP\) demonstrates almost monotonic behavior. Using Equations (10) and (14), we show the profiles of the \(DSP\) vs. dust parameters \(P_{d}\) for both types of plasmas with different values of spectral index \(q\) (=0.90.05,1.0) and a fixed value of alpha (=0.01) for both types of plasmas in Figure 3a,b, respectively. Both. Figure 3a,b show that as the value of \(q\) increases, the dust-surface potential drops. Figure 2: (Color online) \(\psi\) versus \(P_{d}\) for \(\alpha\)(0.0,0.01,0.02) with \(q=0.9\); (**a**) xenon–fluorine–sulfur hexafluoride as given in Equation (10) (**b**) for argon–fluorine–sulfur hexafluoride plasma as given in Equation (14).