{'fraction_non_alphanumeric': 0.024469820554649267, 'fraction_numerical': 0.0065252854812398045, 'mean_word_length': 4.482142857142857, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Spin Echoes after Arbitrary N Pulses

Based on the Bloch equations, the general response of a nuclear spin-½ system to multiple RF pulses with arbitrary phases and flip angles is presented. The general solution is expressed in a form including the pathway vectors specifying the magnetization paths generated by a string of pulses. The pathway vectors are useful for predicting the exact positions, magnitudes, and number of echoes. Also, these vectors make it easy to trace the physical origin of an echo formation. It is shown that the maximum number of echoes after N pulses is (3N-1 - 1)/2 using this concept.