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NUCLEAR PROPERTIES

The soft tissues of organisms are full of hydrogen, because all organic molecules are mainly hydrogen and carbon, and hydrogen in molecules is basically a naked proton, with spin 1/2, so putting any organism in a magnetic field, and inducing the splitting between the two magnetic substates, offers a unique radio-frequency transition that allows the detailed imaging of soft tissues! [The jargon is “nuclear magnetic resonance.”]

Internal excitations of nuclei consist of two main categories... first are single-nucleon states, as described by the so-called “independent particle model” and shell model. Then there are collective excitations, in which many or all nucleons take part, the quantum equivalent of vibrational excitations (in spherical nuclei), and the quantum equivalent of rotations, in deformed nuclei. Because of pairing of nucleons, all nuclei with an even number of both protons and neutrons have a J = 0 ground state. The vibrational and rotational excitations are built on that state. The single nucleon states are easiest to see when a single nucleon is in an otherwise empty state, with all other low-lying states completely filled and having J = 0.

The same single-nucleon states are independently available for single protons and for single neutrons. The “magic numbers” of Z and N occur when a shell of states is filled and there's a large gap before the next state is reached. The closed shells have J = 0, of course, because there is a spin-spin interaction that pairs two identical particles to S = 0. As a result, all nuclei with even numbers of protons and even numbers of neutrons have a J = 0 ground state.

Collective vibrational states in spherical nuclei. The vibrations are of course quantum vibrations, with the vibrational field quantized.

Rotational states in deformed nuclei. The “rotations” are of course quantum excitations and have nothing to do with classical rotations.

The mathematics of transition from one state to another is the same for all quantum processes. The probability of transition, λ, (aka decay constant) is independent of time. This probability is per unit time and can also be related to the average lifetime τ by τ = 1/λ. The solution to the differential equation is N(t)= N₀ e^-λt = N₀e^-t/τ. It is also convenient to introduce the half-life, t_1/2 = 0.693 τ, which is the time-independent time required for half the excited states to de-excite.

Quantum Transitions

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