On nuclear quadrupole moments

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Title:
On nuclear quadrupole moments
Physical Description:
4 p : ; 27 cm.
Language:
English
Creator:
Sternheimer, R
U.S. Atomic Energy Commission
Los Alamos Scientific Laboratory
Publisher:
U.S. Atomic Energy Commission, Technical Information Division
Place of Publication:
Oak Ridge, Tenn
Publication Date:

Subjects

Subjects / Keywords:
Nuclear quadrupole resonance   ( lcsh )
Quadrupole moments   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Bibliography:
Includes bibliographical references.
Statement of Responsibility:
by R. Sternheimer.
General Note:
Cover title.
General Note:
"AECU-923."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 702220266
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AA00012256:00001


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UNITED STATES ATOMIC ENERGY COMMISSION

AECU-923

ON NUCLEAR QUADRUPOLE MOMEN-S-..

By
R. Sternheimer _


Los Alamos Scientific Laboratory


IL Technical Information Division, ORE, Oak Ridge, Tennessee


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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


p Ze X e Q(3cos 1) (1)
+ (1)
2m 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 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


AECU-923










AECU-923


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 Thomas-Fermi

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/l-R times the value previously obtained without the induced effect.


3/Z Z-r Q(3cos z 1) (4)


0


(5a)








AECU-923


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 are also listed, together with the quadrupole moments1
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 [(Z-1) X-lJ 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
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AECU-923


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