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Lh2 UNITED STATES ATOMIC ENERGY COMMISSION AECU923 ON NUCLEAR QUADRUPOLE MOMENS.. By R. Sternheimer _ Los Alamos Scientific Laboratory IL Technical Information Division, ORE, Oak Ridge, Tennessee  .b I~P Reproduced direct from copy as submitted to this office. PRINTED IN USA PRICE 5 CENTS ftl AEC, Oak Ridge, Tenn., 101350675A26167 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 ThomasFermi model. For the electrons of maximum energy E = O, the momentum p is given by p Ze X e Q(3cos 1) (1) + (1) 2m r 3 4r where X is the ThomasFermi function at a point in the electron cloud, r is the length of the vector from the nucleus to this point and 0 is the angle in cluded by this vector and the axis of the nuclear quadrupole moment Q. The density of electrons p is 8ITp3/3h Let A p be the density due to the second term of (1). Thus, 2 Ap = 8PoP (2) 3 h where Ap is the change of momentum associated with the term containing Q, and Po would be the maximum momentum p for Q = O. We have, 2 2 p Ap e Q(3cos e 1) m 3 4r AECU923 AECU923 From (1), (2), (3) we obtain 2 2 me Ap = m 2 2 hr The potei.tial due to Ap is that of a quadrupole amount AQ: S212 2mez 3/2 AQ = 2 r2(3cos2 )Apr sin OdOdr = 6 me 0o o0 h Upon substituting r = (. 88534 aH/Z1/3 ) X, where X is the ThomasFermi variable (aH = Bohr radius), we obtain A Z(. 7707)3/2 AQ = 51T 5ir 0 Q dx . o 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: 2 E =AQ dr (6) o r where A is a constant. For the interaction E with the induced moment, the penetration of the electron inside the core lead to: the penetration of the electron inside the core leads to: S(1. 7707)3/ AQ / 2 1 x AQ 5r 3 o r o 00 Xx dx' + r 1 dx dr. (7) 1 r xr The difference in sign of E 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 = EAQ/E , then Q is 1/lR times the value previously obtained without the induced effect. 3/Z Zr Q(3cos z 1) (4) 0 (5a) AECU923 We write, R = .2998 (8) l/r 3> where O V> = v 3X dx' + r2 f dx' dr, (9) rL r J o o x > 3 dr, (10) 0 with: / v2dr = 1. o Table I gives the values of R for eight elements. The values of 0V> and 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 [(Z1) XlJ 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: B0, B l, Ga Ga Eu , 107 113 69 71 151 f) 8) h) i) 113 j) 115 153' 175' 176' In In AECU923 U 0 o u C) /1 E u N I 0 *a i o Ln 0O o o mSNN o N L N ffl U 'r' r r v4 5 o f LA o *' M OCN N P ~rj fn N '~ 0 U< s0 '0 LA N 0 N 0' P LA r oo t oo ca Jn LN N 0 a N SN \O lN L N o) N Ln 00 Lo LA f Ni N LA) LA 0 N N ^ 0 '  ni ri N & Tl .J LA 'M '0 0 N LA (4* ^ i 0' r^ , a N n 'r 0 o V) S . < u U >S U END OF DOCUMENT I Jr A UNIVERSITY OF FLORIDA 3 1262 08917 1093 1 
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