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# boltz - E 1 The population diﬀerence divided by the sum...

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Physics 225/315 Feb. 15, 2008 State Populations in Magnetic Field The energy difference between the two energy levels of a magnetic moment of a spin 1/2 particle in a magnetic field B 0 is Δ E = ¯ 0 ω 0 = - γB 0 Assume B 0 = 1 Tesla Δ E = ¯ hc c ω 0 2 π ω 0 42 MHz Δ E 200 eV - nm 3 10 8 10 9 2 π 42 10 6 Hz 1 . 8 10 - 7 eV At room temperature kT 1 / 40 eV Δ E kT 40 × 1 . 8 10 - 7 7 10 - 6 The probability of a two state system being in a state with energy E 0 is 1 Z e - E 0 /kT where Z = e - E 0 /kT + e - E 1 /kT see Moore, Volume T, Example T6.2 The lower energy state (with the larger population) has energy E 0 and the higher,
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Unformatted text preview: E 1 . The population diﬀerence divided by the sum of the populations is P-P 1 P + P 1 = e-E /kT-e-E 1 /kT e-E /kT + e-E 1 /kT = 1-e-( E 1-E ) /kT 1 + e-( E 1-E ) /kT ≈ 1-e Δ E/kT 2 ≈ Δ E 2 kT ≈ 3 . 5 10-6 Thus only about one in a million molecules contribute to the next magnetization of the sample. However there are of order 10 23 molecules present so the signal is due to about 10 17 magnetic moments....
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