An <i>ab Initio</i> Answer to Long-Debated Questions about Superconducting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>Nb</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mi>Sn</mml:mi> </mml:math>

<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>Nb</a:mi> <a:mn>3</a:mn> </a:msub> <a:mi>Sn</a:mi> </a:math> is the workhorse superconductor for high-field applications; however, several fundamental aspects remain open 70 years after its discovery. Using anharmonic Migdal-Eliashberg theory, we resolve three key questions: (1) the cubic-to-tetragonal martensitic transition is weakly first order; (2) the superconducting gap is strongly anisotropic but continuous over the Fermi surface; and (3) the drop of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>H</c:mi> <c:mrow> <c:mi>c</c:mi> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:math> across the transition arises from both reduced electron-phonon coupling and a redistribution of Fermi velocities. The orbital origin of the gap anisotropy suggests targeted doping strategies: <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>Sn</e:mi> </e:math> -site substitution can increase both <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>T</g:mi> <g:mi>c</g:mi> </g:msub> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>H</i:mi> <i:mrow> <i:mi>c</i:mi> <i:mn>2</i:mn> </i:mrow> </i:msub> </i:math> , while <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>Nb</k:mi> </k:math> -site substitution raises <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>H</m:mi> <m:mrow> <m:mi>c</m:mi> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> at the cost of lowering <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>T</o:mi> <o:mi>c</o:mi> </o:msub> </o:math> .