Quantum Mechanical Properties of Nuclei: Lecture 4 - Nuclear Sizes, Shapes, and Moments - , Study notes of Chemistry

Download Quantum Mechanical Properties of Nuclei: Lecture 4 - Nuclear Sizes, Shapes, and Moments - and more Study notes Chemistry in PDF only on Docsity! Quantum Mechanical Properties of Nuclei Lecture 4 Sizes and Shapes of Nuclei • How big are nuclei? • 1-10 fm • sharp cutoff model, R=r0A1/3 fm r0=1.07 r0=1.2 r0=1.44 The “Halo Nuclei” 208 Ph jj Nuclear Spin and Parity • Nuclei have an intrinsic “spin” angular momentum, akin to the spin of the electron. • For odd A nuclei, the spin J is half integer. (1/2,3/2, etc.) For even A nuclei, the spin J is integer. (0,1,2,etc.). The magnitude of the spin is multiple of hbar. Parity (the symmetry properties of the nuclear wave function) ! "(r,s) = +"(#r,#s) $ = + "(r,s) = #"(#r,#s) $ = # For a central potential, V=V(r) ! " = (#1)l A=4 2=0 a2 a=3 Sphere Quadrupoles Octupoles Hexadecapoles OBLATE y ~ ‘ J - V9 Vv PROLATE 2 -@- Neutrons Octupoles K isomers spherical Shape coexistence, shape transitions spherical —> strongly deformed Protons Electric and magnetic moments • Electric moments are measures of the distribution of electric charge. Magnetic moments measure the distribution of electric currents. Classical Analogy ! magnetic dipole moment = iA µ = iA = ev 2"r # $ % & ' ( "r2( ) = evr 2 l = mvr µ = evr 2 • m m = el 2m gyromagnetic ratio = ) = µ l = e 2m Electric Quadrupole Moments ! charge at point = "d# = " r2drsin$d$d%( ) potential at P = d& d& = "d# ' = "d# D 2 + r2 ( 2Drcos$( ) 1/ 2 P2 cos$( ) = 3 2 cos2$ ( 1 2 P1 cos$( ) = cos$ d& = "d# D 1+ r D P1 cos$( ) + r D ) * + , - . 2 P2 cos$( ) +L / 0 1 2 3 4 V = 1 D "d# volume 5 / 0 1 2 3 4 + 1 D 2 "rcos$d# volume 5 / 0 1 2 3 4 + 1 D 3 "r2 3 2 cos2$ ( 1 2 ) * + , - . d# +L volume 5 / 0 1 2 3 4 ! Q = r 2"(r) 3 2 cos 2# $ 1 2 % & ' ( ) * r 2 drsin#d#d+,,, ! R 2 = 1 2 a 2 + c 2( ) = r0A1/ 3 Q = 2 5 Ze a 2 " c 2( ) oe \f J \ | a “—y | | 7 04} — 814 4050 82, 1 OS; = 06, -o71 4 ~0a} 09] 4 o 10 2 30 4 50 60 70 80 90 100 110 120 190 140 150 Zon