On the relationship between the Method of Least Squares and Gram-Schmidt orthogonalization

The method of Least Squares is due to Carl Friedrich Gauss. The Gram-Schmidt orthogonalization method is of much younger date. A method for solving Least Squares Problems is developed which automatically results in the appearance of the Gram-Schmidt orthogonalizers. Given these orthogonalizers an induction-proof is available for solving Least Squares Problems.

Quadrupole moments of radium isotopes from the 7p^2P_3/2 hyperfine structure in Ra II

The hyperfine structure and isotope shift of ^{221- 226}Ra and ^{212, 214}Ra have been measured in the ionic (Ra 11) transition 7s^2 S_{1/2} - 7p ^2 P_{3/2} (\lamda = 381.4 nm). The method of on-line collinear fast-beam laser spectroscopy has been applied using frequency-doubling of cw dye laser radiation in an external ring cavity. The magnetic hyperfine fields are compared with semi-empirical and ab initio calculations. The analysis of the quadrupole splitting by the same method yields the following, improved values of spectroscopic quadrupole moments: Q_s(^221 Ra)= 1.978(7)b, Q_s (^223 Ra)= 1.254(3)b and the reanalyzed values Q_s(^209 Ra) = 0.40(2)b, Q_s(^211 Ra) = 0.48(2)b, Q_s(^227 Ra)= 1.58(3)b, Q_s (^229 Ra) = 3.09(4)b with an additional scaling uncertainty of ±5%. Furthermore, the J-dependence of the isotope shift is analyzed in both Ra II transitions connecting the 7s^2 S_{1/2} ground state with the first excited doublet 7p^ P_{1/2} and 7p^ P_{3/2}.