## 3 Results and Discussion
### Effects of Elemental Substitution on NbN Structure  
While not explicitly defined within the models discussed, crystal structure has a significant impact on property, and the crystal structure is significantly impacted by chemistry. Substituting different elements into the NbN matrix can significantly impact its crystallinity and phase stability, as seen in Figure 6. The X-ray diffraction (XRD) patterns, ingested from the ICSD database [23; 24; 25], for NbN with various elemental substitutions demonstrate the impact on crystallinity, phase stability, and peak intensity, with Ti and Ta additions having minor impact as compared to Co. Even if the impact is embedded in the analysis,  
Figure 5: Actual versus predicted Tc by composition type. Each subplot represents a scatter plot for a specific type of superconducting material composition: (**a**) Oxide, (**b**) Stannide, (**c**) Telluride, (**d**) Nitride, (**e**) Fluoride, and (**f**) Sulfide. The actual Tc values (\(x\)-axis) are plotted against the predicted Tc values (\(y\)-axis), obtained from a machine learning model. Data points are represented as lime-colored squares with black edges, where each point corresponds to a material within the filtered dataset for the respective composition type. The black dashed line signifies the line of ideal prediction, illustrating where the actual and predicted Tc values would be equal.  
as demonstrated by the model accuracy, for prediction of new systems, this issue must still be considered.  
Transition from theoretical predictions to practical application requires a careful examination of whether the identified substitutions truly maintain or enhance the structural features necessary for optimal superconductivity. To address the structural issues, we follow the Hume-Rothery (HR) rules [26; 27; 28] to identify substitutional elements which have closer solubility with Nb and have higher likelihood to maintain stability when substituted for Nb (Figure 7).  
We explored the substitutional potential within the NbN superconductor system, aiming to enhance Tc by invoking the HR rules as a theoretical scaffold. Initially, we addressed the atomic size factor by comparing the atomic radii differences, ensuring that the discrepancy between the host and the substituting atoms did not exceed the critical 15% threshold that could lead to significant lattice distortion and impede solubility. Furthermore, we took into account the electrochemical factor by examining the electronegativity of the elements in question. In line with HK's guidance, we considered the propensity for elements with differing electronegativities to form compounds rather than solid solutions. The atomic size factor is essential for maintaining the integrity of the NbN crystal lattice. A substitute with a significantly different atomic radius could distort the lattice, affecting the electronic properties crucial for superconductivity. Additionally, we evaluated the relative valency factor, acknowledging that metals with lower valency have a higher likelihood of dissolving metals with higher valency, as per historical observations in alloys containing copper, silver, or gold with higher-valency metals. Through these HR parameters, we filtered and identified suitable candidates for substitution in the NbN matrix. Ta is seen to be most similar to Nb by these rules, and then Ti, Sc, Zr, Hf, and V are also similar. Numerous other elements including Co are found in the second layer of similarity, which  
Figure 6: X-ray diffraction (XRD) patterns of NbN and its various elemental substitutions: Nb\({}_{4}\)Co\({}_{2}\)N, NbTaN\({}_{2}\), NbTiN\({}_{2}\), and pure NbN. The patterns illustrate the impact on crystallinity, phase stability, and peak intensity. The reference pattern for pure NbN shows high crystallinity and structural order.  
would suggest structural change with some addition. These findings match the XRD patterns in Figure 6.