Overhauser and Jeffries-Abragam Effect in a Four-Level System

Mohan Lal Narchal; W. A. Barker

doi:10.1364/AO.2.000787

Applied Optics
Vol. 2,
Issue 8,
pp. 787-792
(1963)
•https://doi.org/10.1364/AO.2.000787

Overhauser and Jeffries-Abragam Effect in a Four-Level System

Mohan Lal Narchal and W. A. Barker

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Mohan Lal Narchal and W. A. Barker, "Overhauser and Jeffries-Abragam Effect in a Four-Level System," Appl. Opt. 2, 787-792 (1963)

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Abstract

The method of partial distributions is used to compute the normalized population distribution in an electron-nuclear coupled spin system in which I = S = 1/2. The expressions obtained are more general than those reported previously in that they include the effects of competitive relaxation and competitive radiative induced transition probabilities. It is clear that positive and negative enhancement may be obtained either by the Overhauser or the Jeffries-Abragam effect irrespective of the relative magnitudes of the nuclear Zeeman and hyperfine interaction energies. A small enhancement is not necessarily due to incomplete saturation, but may well be the result of compensating emission and absorption.

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k	n₁	n₂	n₃	n₄	Number of times the distribution occurs
1	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	4
2	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{2} (1 - \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	1
3	$\frac{N}{4} (1 - \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	1
4	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{3 Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	3
5	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} - \frac{δ}{2})$	3
6	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 + Δ - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - Δ - \frac{δ}{2})$	1
7	$\frac{N}{4} (1 + \frac{3 Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{2})$	3
8	$\frac{N}{4} (1 + Δ + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 - Δ + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	1
9	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	3
10	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	16

k	n₁	n₂	n₃	n₄	Number of times the distribution occurs
1	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{δ}{2})$	2
2	$\frac{N}{4} (1 - \frac{Δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2})$	1
3	$\frac{N}{4} (1 + \frac{Δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2})$	1
4	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{4})$	$\frac{N}{4} (1 + \frac{3 Δ}{4} - \frac{δ}{4})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{3 δ}{4})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{4})$	3
5	$\frac{N}{4}$	$\frac{N}{4} (1 - δ)$	$\frac{N}{4} (1 + δ)$	$\frac{N}{4}$	1
6	$\frac{N}{4}$	$\frac{N}{4} (1 + Δ)$	$\frac{N}{4} (1 - Δ)$	$\frac{N}{4}$	1
7	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{4})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{3 δ}{4})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} + \frac{δ}{4})$	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{4})$	3
8	$\frac{N}{4} (1 + \frac{3 Δ}{4} + \frac{δ}{4})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{4})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{4})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{3 δ}{4})$	3
9	$\frac{N}{4} (1 + δ)$	$\frac{N}{4}$	$\frac{N}{4}$	$\frac{N}{4} (1 - δ)$	1
10	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{3 δ}{4})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{4})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{4})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} - \frac{δ}{4})$	3
11	$\frac{N}{4} (1 + Δ)$	$\frac{N}{4}$	$\frac{N}{4}$	$\frac{N}{4} (1 - Δ)$	1
12	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	16

Case	Enhancement ɛ
A.1	$1 + \frac{w_{41} - w_{32}}{w_{41} + w_{32} + 2 w_{1}} \frac{Δ}{δ}$
A.2	1
B.1	≈0
B.2	$\frac{2 w_{1}}{w_{42} + w_{31} + 2 w_{1}} (≪ 1)$
B.3	$\frac{w_{1} + w_{32}}{w_{1} + w_{2} + 2 w_{32}} - \frac{w_{2} + w_{32}}{w_{1} + w_{2} + 2 w_{32}} \frac{Δ}{δ}$
B.4	$\frac{1}{2} [\frac{(w_{1} + w_{2}) (2 w_{1} + w_{31} + w_{42})}{(w_{1} + w_{2}) (w_{1} + w_{31} + w_{42}) + w_{1} w_{2} + w_{31} w_{42}} - \frac{2 w_{1} w_{2} + 2 w_{31} w_{42} + (w_{1} + w_{2}) (w_{31} + w_{42})}{(w_{1} + w_{2}) (w_{1} + w_{31} + w_{42}) + w_{1} w_{2} + w_{31} w_{42}} \frac{Δ}{δ}]$
B.5	$\frac{w_{1} + w_{41}}{w_{1} + w_{2} + 2 w_{41}} + \frac{w_{2} + w_{41}}{w_{1} + w_{2} + 2 w_{41}} \frac{Δ}{δ}$
B.6	$\frac{1}{2} [\frac{(w_{1} + w_{2}) (2 w_{1} + w_{31} + w_{42})}{(w_{1} + w_{2}) (w_{1} + w_{31} + w_{42}) + w_{1} w_{2} + w_{31} w_{42}} + \frac{2 w_{1} w_{2} + 2 w_{31} w_{42} + (w_{1} + w_{2}) (w_{31} + w_{42})}{(w_{1} + w_{2}) (w_{1} + w_{31} + w_{42}) + w_{1} w_{2} + w_{31} w_{42}} \frac{Δ}{δ}]$

Case	Nuclear polarization P_N
A.1	$- \frac{1}{2} \frac{γ_{n}}{∣ γ_{n} ∣} [\frac{w_{32} - w_{41}}{w_{32} + w_{41} + 2 w_{1}} Δ - δ]$
A.2	$\begin{array}{l} - \frac{1}{2} \frac{a}{∣ a ∣} [\frac{w_{2} (w_{1} - w_{2}) + (w_{1} w_{42} - W_{2} w_{31})}{2 W_{1} W_{2}} Δ] (W_{1}, W_{2}) ≫ w_{i j} \\ - \frac{1}{2} \frac{a}{∣ a ∣} [\frac{w_{2} + w_{42}}{w_{1} + w_{2} + 2 w_{42}}] Δ (W_{2} ≪ w_{i j} ≪ W_{1}) \\ \frac{1}{2} \frac{a}{∣ a ∣} [\frac{w_{2} + w_{31}}{w_{1} + w_{2} + 2 w_{31}}] Δ (W_{1} ≪ w_{i j} ≪ W_{2}) \end{array}$
B.1	$- \frac{1}{4} \frac{γ_{n}}{∣ γ_{n} ∣} [\frac{w_{2} (W_{1} - W_{2}) + (W_{1} w_{32} - W_{2} w_{41})}{W_{1} W_{2}} Δ - (\frac{w_{1} (W_{1} + W_{2}) + (W_{1} w_{32} + W_{2} w_{41})}{W_{1} W_{2}}) δ]$
B.2	$\frac{1}{2} \frac{a}{∣ a ∣} [{- \frac{w_{42} - w_{31}}{w_{42} + w_{31} + 2 w_{1}} - \frac{(w_{1} + w_{2}) (w_{42} - w_{31})}{W (w_{42} + w_{31} + 2 w_{1})}} Δ + \frac{w_{1} + w_{2}}{W} δ]$
B.3	$- \frac{1}{2} \frac{γ_{n}}{∣ γ_{n} ∣} [\frac{w_{1} + w_{32}}{w_{1} + w_{2} + 2 w_{32}} Δ - \frac{w_{1} + w_{32}}{w_{1} + w_{2} + 2 w_{32}} δ]$
B.4	$\frac{1}{4} \frac{a}{∣ a ∣} [\frac{(w_{1} + w_{2}) (w_{31} - w_{42})}{(w_{1} + w_{2}) (w_{1} + w_{31} + w_{42}) + w_{1} w_{2} + w_{31} w_{42}} (Δ + δ)]$
B.5	$- \frac{1}{2} \frac{γ_{n}}{∣ γ_{n} ∣} [- \frac{w_{2} + w_{41}}{w_{1} + w_{2} + 2 w_{41}} Δ - \frac{w_{1} + w_{41}}{w_{1} + w_{2} + 2 w_{41}} δ]$
B.6	$\frac{1}{4} \frac{a}{∣ a ∣} [\frac{(w_{1} + w_{2}) (w_{31} - w_{42})}{w_{1} w_{2} + (w_{1} + w_{31}) (w_{1} + w_{2} + w_{42}) + w_{2} w_{42}} (Δ - δ)]$

k	n₁	n₂	n₃	n₄	Number of times the distribution occurs
1	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	4
2	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{2} (1 - \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	1
3	$\frac{N}{4} (1 - \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	1
4	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{3 Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	3
5	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} - \frac{δ}{2})$	3
6	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 + Δ - \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{δ}{2})$	$\frac{N}{4} (1 - Δ - \frac{δ}{2})$	1
7	$\frac{N}{4} (1 + \frac{3 Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} - \frac{δ}{2})$	3
8	$\frac{N}{4} (1 + Δ + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	$\frac{N}{4} (1 - Δ + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{δ}{2})$	1
9	$\frac{N}{4} (1 + \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{3 Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{4} - \frac{δ}{2})$	3
10	$\frac{N}{4} (1 + \frac{Δ}{2} + \frac{δ}{2})$	$\frac{N}{4} (1 + \frac{Δ}{2} - \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{4} + \frac{δ}{2})$	$\frac{N}{4} (1 - \frac{Δ}{2} - \frac{δ}{2})$	16

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Applied Optics