Dynamic Orientation of Nuclei with Spin &gt;1

S. K. Gupta; M. L. Narchal

doi:10.1364/AO.8.001677

Applied Optics
Vol. 8,
Issue 8,
pp. 1677-1686
(1969)
•https://doi.org/10.1364/AO.8.001677

Dynamic Orientation of Nuclei with Spin >1

S. K. Gupta and M. L. Narchal

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S. K. Gupta and M. L. Narchal, "Dynamic Orientation of Nuclei with Spin >1," Appl. Opt. 8, 1677-1686 (1969)

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Abstract

The dynamic orientation of nuclei with spin >1 is theoretically investigated in experimental samples where the nucleus is in well resolved hyperfine interaction with a paramagnetic ion of spin $\frac{1}{2}$ . It is shown that the contribution from off-diagonal elements of the relevant density matrix vanishes in the steady state, and the rate equation approach is justified. The nuclear orientation parameters are computed using the simplified method of partial distributions, both at low and high temperatures, and numerical results are presented for Pa⁴⁺ in Cs₂ZrCl₆.

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Dynamic Orientation of Spin 1 Nuclei. III

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	Relative populations

No		n_l	n₂_n₋_l₊₁
j	n₁ = n₂_n	l = 2 to n	l = 2 to n	Weight factor
1	[1,(n − 2)]	[0,(l − 2)]	[1,(n − l − 1)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻²
2	[1,(n − 3)]	[1,(n + l − 4)δ]	[2,(2n − l −3)]	Ww₂ⁿ⁻¹w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
3	[2,(2n − 7)]	[2,(2n + l − 8)]	[3,(3n − l −7)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³

	Relative populations					Weight factor

No			n_l		n₂_n₋_l₊₁
j	n₁	n₂ =n₂_n₋₁	l = 3 to n	n₂_n	l = 3 to n
1	[1,(n − 2)]	[1,(n − 1]	[0,l]	[2,(2n − 3)]	[1,(n − l + 1]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
2	[1,(n − 2)]	[1,(n − 1]	[1,(n + l − 3)]	[2,(2n − 3)]	[2,(2n − l − 2)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
3	[1,(n − 4)]	[2,(2n − 6)]	[1,(n + l − 5)]	[2,(2n − 5)]	[2,(2n − l − 4)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
4	[0,0]	[1,(n −2)]	[1,(n + l − 4)]	[1,(n − 1)]	[2,(2n − l − 3)]	Ww₂ⁿ⁻¹w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³

	Relative populations							Weight factor

No				n_l			n₂_n₋_l₊₁
j	n₁	n₂	n₃ = n₂_n₋₂	l = 4 to n	n_2n	n_{2n − 1}	l = 4 to n
1	[1,(n − 2)]	[1,(n − 1)]	[1,n]	[0,(l + 2)]	[2,(2n − 3)]	[2,(2n − 4)]	[1,(n − l + 3)]	Ww₁³w₂ⁿ⁻⁴w_pⁿ⁻¹[1+(w₁/w₁)]ⁿ⁻⁴
2	[1,(n − 2)]	[1,(n − 1)]	[1,n]	[1,(n + l −3)]	[2,(2n − 3)]	[2,(2n − 4)]	[2,(2n − l − 2)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
3	[1,(n − 2)]	[1,(n − 1)]	[2,(2n − 5)]	[1,(n + l − 3)]	[2,(2n − 3)]	[2,(2n − 4)]	[2,(2n − l − 2)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
4	[1,(n − 2)]	[1,(n − 1)]	[2,(2n − 5)]	[2,(2n + l − 8)]	[2,(2n − 3)]	[2,(2n − 4)]	[3,(3n − l − 7)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
5	[1,(n −3)]	[1,(n − 2)]	[2,(2n − 6)]	[2,(2n + l − 9)]	[2,(2n − 4)	[2,(2n − 5)]	[3,(3n − l − 8)]	Ww₂ⁿ⁻¹w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
6	[1,(n − 3)]	[1,(n − 2)]	[2,(2n − 6)]	[1,(n + l − 4)]	[2,(2n − 4)]	[2,(2n − 5)]	[2,(2n − l − 3)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
7	[2,(2n − 7)]	[2,(2n − 6)]	[2,(2n − 5)]	[2,(2n + l − 8)]	[3,(3n − 8)]	[3,(3n − 9)]	[3,(3n − l − 7)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
8	[2,(2n − 7)]	[2,(2n − 6)]	[2,(2n − 5)]	[1,(n + l − 3)]	[3,(3n − 8)]	[3,(3n − 9)]	[2,(2n − l − 2)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴

	Relative populations					Weight factor

No	n₁		n_l′	n₂_n₋_l₊₁	n₂_n₋_l′₊₁
j	l = 1 to k − 1	n_k = n_2n₋_k₊₁	l′ = k + 1 to n	l = 1 to k − 1	l′ = k + 1 to n
1	[1,(n + l − 3)]	[1,(n + k − 3)]	[0,(2k +l′ − 4]	[2,(2n − l − 2)]	[1,(n − l′ + 2k − 3)]	Ww₁³w₂ⁿ⁻⁴w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
2	[1,(n + l − 3)]	[1,(n + k − 3)]	[1,(n + l′ − 3)	[2,(2n − l − 2)]	[2,(2n − l′ − 2)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
3	[1,(n + l − 3)]	[2,(2n − k − 2)]	[2,(2n + l′ − 2k − 2)]	[2,(2n − l − 2)]	[3,(3n − l′ −2k − 1)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
4	[1,(n + l − 3)]	[2,(2n − k − 2)]	[1,(2n + l′ − 3)]	[2,(2n − l − 2)]	[2,(2n − l′ − 2)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
5	[1,(n +l − 4)]	[1,(n + k − 4)]	[0,(l′ + 2k − 5)]	[2,(2n − l − 3)]	[1,(n − l′ + 2n − 4)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
6	[1,(n +l − 4)]	[1,(n + k − 4)]	[1,(n +l′ − 4)]	[2,(2n − l− 3)]	[2,(2n − l′ − 3)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
7	[1,(n + l − 4)]	[2,(2n − k − 3)]	[2,(2n +l′ − 2k − 3)]	[2,(2n − l − 3)]	[3,(3n − l′ − 2k − 2)]	Ww₂ⁿ⁻¹w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
8	[1,(n + l − 4)]	[2,(2n − k − 3)]	[1,(n +l′ − 4)]	[2,(2n − l− 3)]	[2,(2n − l′ − 3)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
9	[2,(2n + l − 8)]	[2,(2n + k − 8)]	[1,(n +l′ + 2k − 9)]	[3,(3n − l − 7)]	[2,(2n − l′ + 2k − 8)]	Ww₁³w₂ⁿ⁻⁴w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
10	[2,(2n + l − 8)]	[2,(2n + k − 8)]	[2,(2n +l′ − 8)]	[3,(3n − l −7)]	[3,(3n − l′ − 7)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
11	[2,(2n + l − 8)1	[3,(3n − k − 7)]	[3,(3n +l′ − 2k − 7)]	[3,(3n − l − 7)]	[4,(4n − l′ −2k − 6)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵
12	[2,(2n + l − 8)]	[3,(3n − k − 7)]	[2,(2n +l′ − 8)]	[3,(3n − l − 7)]	[3,(3n − l′ − 7)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁵

	Relative populations			Weight factor

No	n_l		n₂_n₋_l₊₁
j	l = 1 to n − 1	n_n=n_n₊₁	l = 1 to n − 1
1	[1,(n + l − 3)]	[1,(2n − 3)]	[2,(2n − l − 2)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
2	[1,(n + l − 3)]	[2,(n − 2)]	[2,(2n − l − 2)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻³
3	[1,(n + l − 4)]	[1,(2n − 4)]	[2,(2n − l − 3)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
4	[1,(n + l − 4)]	[2,(n − 3)]	[2,(2n − l − 3)]	Ww₂ⁿ⁻¹w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
5	[2,(2n + l − 8)]	[2,(3n − 8)]	[3,(3n − l − 7)]	Ww₁²w₂ⁿ⁻³w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴
6	[2,(2n + l − 8)]	[3,(2n − 7)]	[3,(3n − l − 7)]	Ww₁w₂ⁿ⁻²w_pⁿ⁻¹[1+(w₁/w₂)]ⁿ⁻⁴

	Relative populations		Weight factor

No	N_l	N₂_n₋_l₊₁
j	l = 1 to n	l = 1 to n
1	[1,(n + l − 3)]	[2,(2n − l − 2)]	w₁w₂ⁿ⁻²w_pⁿ[1+(w₁/w₂)ⁿ⁻²
2	[1,(n + l − 4)]	[2,(2n − l − 3)]	w₂ⁿ⁻²w_pⁿ[1+(w₁/w₂)ⁿ⁻³
3	[2,(2n + l − 8)]	[3,(3n − l − 7)]	w₁w₂ⁿ⁻²w_pⁿ[1+(w₁/w₂)ⁿ⁻³

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Equations (64)

Applied Optics