7; 
 
 UNITED STATES ATOMIC ENERGY COMMISSION 
 
 AECU-923 
 
 ON NUCLEAR QUADRUPOLE MOMEN 
 
 By 
 
 R. Sternheimer 
 
 
 
 Los Alamos Scientific Laboratory 
 
 Technical Information Division, ORE, Oak Ridgo, Tennessee 
 
1 
 
 Reproduced direct from copy 
 as submitted to this office. 
 
 PRINTED IN USA 
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 AEC, Oak Ridge, Tenn., 10-ia-SO— 675-A26167 
 
On Nuclear Quadrupole Moments 
 
 R. Sternheimer 
 Los Alamos Scientific Laboratory, Los Alamos, N. M. 
 
 It was pointed out by Professor Rabi that the hyperfine splitting 
 due to the nuclear quadrupole moment includes the effect of an electric 
 quadrupole moment induced in the electron shells. In order to obtain a 
 crude estimate of the moment induced in a core of closed shells we consi- 
 der the Thomas-Fermi model. For the electrons of maximum energy E = O, 
 the momentum p is given by 
 
 2 2 2,2, 
 
 p Ze X e Q(3cos 9 - l) . . 
 
 2 m r ,3 
 
 4r 
 
 where X is the Thomas -Fermi function at a point in the electron cloud, r is 
 the length of the vector from the nucleus to this point and 9 is the angle in- 
 
 cluded by this vector and the axis of the nuclear quadrupole moment Q. The 
 density of electron! 
 term of (1). Thus, 
 
 3 3 
 density of electrons p is 8irp /3h . Let A p be the density due to the second 
 
 Ap = 8 ^Po^P , (2) 
 
 h 
 
 where Ap is the change of momentum associated with the term containing Q, 
 and p would be the maximum momentum p for Q = O. We have, 
 
 P Q AP e Q(3cos 9-1) (3) 
 
 4r 
 
 AECU-923 1 
 
AECU-923 
 
 From (1), (2), (3) we obtain 
 
 Ap = it 
 
 2 3/2 , 
 2 me ' \{ ZX 
 
 H 2 2 
 h r 
 
 in 
 
 Q(3cos 9 - 1) . (4) 
 
 The potential due to Ap is that of a quadrupole amount AQ; 
 
 } r°° Z Z ? iA 2 -> 2 V^ 
 
 AQ = 27T J / r (3cos - l)Apr 2 sin 6d0dr = — 2me 
 
 o o 
 
 
 1 VXrdr. (5) 
 o 
 
 1/3 
 
 Upon substituting r = (. 88534 a H /Z ' ) X, where X is the Thomas-Fermi 
 variable (a^ = Bohr radius), we obtain 
 
 Ar> 2(l.7707) 3/2 
 
 AQ z — i — 1 q 
 
 5tt 
 
 
 dx . 
 
 (5a) 
 
 We shall consider the case of a single valence electron; its radial wave func 
 tion times r will be called v. The energy of interaction E with the nuclear 
 moment can be written: 
 
 E Q = -AQ 
 
 o r 
 
 J + 
 
 dr , 
 
 (6) 
 
 where A is a constant. For the interaction E with the induced moment, 
 
 the penetration of the electron inside the core leads to: 
 
 2(1. 7707) 3/2 AQ 
 
 AQ 
 
 5tt 
 
 o [r Jo x r J 
 
 dr. (7) 
 
 The difference in sign of E q and E reflects the fact that the electrons 
 
 concentrate in the region where the potential due to the nuclear Q is positive, 
 
 thus tending to compensate the effect of the nucleus. If we let R = -E /E 
 
 AQ' Q" 
 then Q is 1/1-R times the value previously obtained without the induced effect. 
 
AECU-923 
 
 We write, 
 
 <V> 
 
 R = .2998 ; , (8) 
 
 <i/, 3 > 
 
 °° 2 
 
 v 
 
 / r 3 
 o 
 
 dr, (10) 
 
 with: / v dr = 1. 
 
 Table 1 gives the values of R for eight elements. The values of 
 \V/ and \l/r / are also listed, together with the quadrupole moments 
 as determined at present and the corrected values, in the cases where 
 data are available. The valence electron functions were obtained by means 
 of the Thomas -Fermi potential L(Z-l) X-l_j e/r. 
 
 A more detailed discussion will be given in a forthcoming paper. 
 
 It is a great pleasure to thank Professor Edward Teller, who sug- 
 gested this problem, for many helpful discussions. I am also indebted 
 to Drs. H. M. Foley and H. Snyder for stimulating discussions. 
 
 References: 
 
 1. J. E. Mack, Rev. Mod. Phys. 22, 64 (1950) 
 
 2. In the last two columns of this table the subscripts refer as follows 
 
 a) b) c) d) e) 
 
 to different isotopes: ^ > B , Ga, , ^a , Eu , 
 
 0- g) T h ) T T H3 j). 115 
 
 EU 153' LU 175' LU 176* In ' In • 
 
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 END OF DOCUMENT 
 
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