I looked at the contexts of the "angular momentum" cluster around the spike: angular momentum of electrons; angular momentum about the nucleus; anti-neutron differs from the neutron in that its intrinsic magnetic moment is directed parallel to the spin angular momentum; the quantum number of angular momentum, the spin Hamiltonian, and the angular momentum Hamiltonian; quantum theory of scattering, etc. Orbital dynamics contributed some, but much less, it seems. I would not be surprised, however, by many spurious correlations. The turn of 1960-s was a "cultural moment" that lifted all boats, the romance with nuclear physics and space exploration were only two aspects of it. So were high hopes for thinking computers, loosening of the iron curtain, and the rise of youth counterculture and rock-n-roll.
The bulk seems to come from expositions of quantum theory, presumably prompted by its technologization that created a new audience of engineers (lasers, semiconductors, transistors, nuclear energy, etc.). The so-called magic-angle spinning in nuclear magnetic resonance was also discovered in 1958-59, see Timeline of quantum mechanics.
The second contributing factor seems to be the great search for the new subatomic particles in 1950-s after the large accelerators were built, followed by the efforts to make sense of the "particle zoo" in 1960-s. "Had I foreseen this, I would have gone into botany", Pauli quipped. Soviet movie Nine Days of One Year, that romanticized the endeavor, came out in 1962 and became a hit. The spin of course is the distinction between bosons and fermions so it came up a lot.