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Special Reading Topics

by: Ericka Ryan I

Special Reading Topics PHYS 7910

Marketplace > University of Utah > Physics 2 > PHYS 7910 > Special Reading Topics
Ericka Ryan I
The U
GPA 3.5


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This 55 page Class Notes was uploaded by Ericka Ryan I on Monday October 26, 2015. The Class Notes belongs to PHYS 7910 at University of Utah taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/230023/phys-7910-university-of-utah in Physics 2 at University of Utah.


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Date Created: 10/26/15
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Vt 1 2 aha7 gar Mr www 40 7M Memu 75 ux 2 11 1 03 svwm 6 MD as W and Ag4 Asiawit Marvinx 9k ute Wynne 650 Chapter 31 Diamagnetism and Paramagnetism When a solid contains some ions with partially fi EiEectronic shel ls its magnetic behavior is very diiTerent Before we can apply the general result 3120 to this case we must review the basic facts about the lowlying states of such ions v cRoUND sTATE OF IONS WITH A P ARriALLY FILLED SHELL HUND S RULES 39 Suppose we have a free16 atom or ion in which all electronic shells are lled or empty except for one whose oneelectron levels are characterized by orbital an gular momentum I Since for given I there are 21 1 values I can have I I l I 2 l and two possible spin directions for each 2 such a shell will contain 221 l oneelectron levels Let n be the number of electrons in the shell with 0 lt it lt 221 1 If the electrons did not interactwith one another the ionic ground state Would be degenerate re ecting the large number of ways of putting it electrons into more than it levels Hov vevet this degeneracy is considerably though in general not completely lifted by electronelectron Coulomb interactions as well as by the electron spin orbit interaction Except for the very heaviest ions where spinorbit coupling is Very strong the lov yest lying levels after the degeneracy is lifted can be described by a simple set of rules justi ed both by complex calculations and by the analysis of atomic spectra Here we shall simply state the rules since we are more interested in their implications for the magnetic prOperties of solids than in their Underlying justi cation 7 l Russel Saunders Coupling To a good approximation13 the Hamiltonian of the atom or ion can be taken to commute with the total electronic spin and39orbital angular momenta S and L as well as with the total electronic angular momentum J L 8 Therefore the states of the ion can be described by quantum numbers L Li S S J and J2 indicating that they are eigenstates of the operators L2 L 2 32 J2 and J2 With eigenvalues LL 1 L1 SS 1 S2 JJ l and J2 respectively Since lled shells have zero orbital spin and total angular momentum these quantum numbers describe the electronic con guration of the partially lled shell as well as the ion as a whole 2 Hun d s First Rule Out of the many states one can form by placing n electrons into the 22l 1 levels of the partially lled shell those that lie lowest in energy have the largest total spin S that is consistent with the exclusion principle To see what that value is one notes that the largest Value S can have is equal to the largest magnitude that S can have if it lt21 1 all electrons can have parallel Spins without multiple occupation of any oneelectron leVel in the shell by assigning them levels with different values of 2 HenceS 41 when n S 21 1 When n 2l 4 1 15 We shall discuss how the behavior of the free atom or ion is modi ed by the crystalline environment on pages 656 659 17 The rules are discussed in most quantum mechanics texts See for example L D Landau and E M Lifshitz Quantum Mechanics Addison Wesley Reading Mass 1965 i 39 l 13 The total angular momentum J is always a good quantum number for an atom or ion but Land S are good quantum numbers only to the extent that spinorbit coupling is unimportant Hund s Rules 651 S has its maximum value 1 5 Since electrons after the Zl 39391th are required by 39 the exclusion principle to have their spins opposite to the spins of the rst 21 l S is reduced from its maximum value by hair a unit tor each electron atter the 21 1th 3 Huml s Second Rule The total orbital angular momentum Lof the lowest lying states has the largest value that is consistent with Hund s rst rule and with the exclusion principle To determine that value one notes that it is equal to the largest magnitude that szcan have Thus the rst electron in the shell will go into a level with lzl equal to its maximum Value I The second according to rule 2 must have the same spin as the rst and is therefore forbidden by the exclusion principle from having the same value of 2 The best it can do39is to have 1239 l 1 leading to a total L of l l 1 21 1 Continuing in this way if the shell is less than half lled we will have L l l l 39 i l n 1 When the shell is pre cisely half lled all Values of I must be assumed and therefore L 0 The second half of the shell is lled with electrons with spin opposite to those in the rst half 39 r and therefore the exclusion principle allows us again to go through the same series of values for L we traversed in lling the rst half 4 Hzmd s Third Rule The rst two rules determine the values of L and S assumed by the states of lowest energy This still leaves 2L lZS 1 possible states These can be further classi ed according to their total angular momentum J which according to the basic rules of angular momentum composition can take on all integral values between L S and L S The degeneracy of the set of 2L l2 1 states is lifted by the spinorbit coupling which within this set of states can be represented by a term in the Hamiltonian of the simple form ML 8 Spinorbit coupling will favor maximum J parallel orbital and spin angular momenta if A is negative and minimum J antiparallel orbital and spin angular momenta if xl is positive As it turns out A is positive for shells that are less than half lled and negative for shells that39are more than half lled As a result the value J assumes in 39 the states of lowest energy is 39 J IL SI nltt21 1 J L quotS It 2 21 l 3129 39 In magnetic problems one usually deals only with the set of 2L 1ZS 1 states determined by Hund s rst two rules all others lying so much higher in energy as to be of no interest Furthermore it is often enough to consider only the N 1 lowest lying of these speci ed by the third rule V v The rules are easier to apply than their description might suggest indeed in determining the lowest lying Jmultiplet known as a term for ions in a solid one really encounters only 22 cases of interest I to 9 electrons in a l shell l 2 or 1 toll3 electrons in an f shell l 319 For unfortunate historical reasons the groundstate multiplet in these cases is not describedby the simple triad of numbers 9 Partially llecl pshclls contain valence electrons and invariably broaden into bands in the solid Thus the con guration of the electrons they contain in the solid is in no sense a slight distortion of the Con guration in the free atom and the analysis of this chapter is inapplicable 39 iii if its a r39 if at 3511 652 Ch a pter 3 l Diam agnetismandllaramagnetism 39 v SLJ Instead the orbital angular momentum L is given39by a letter according to I the hallowed spectroscopic Code 0123456 XSPDFGHI The spin is speci ed by af xing the number ZS 1 known as the multiplicity to the letter as asuperpre x and only J is given as the number J af xed as a right subscript Thus the lowest lying J multiplet is described by the symbol 23 Xi The cases of major interest for the study of magnetism in solidsare given in Table 312 Table 312 I v f I GROUND STATES OF iONS WITH FAR l iALLY FILLEB 1 OR jLSH39EL39LS As CONsTR UCTED39 FROM HUNb s RULES 3130 when i 2 4 n I 39 39 I L 21 QwaNomwww l 2 3 4 5 6 7 8 9 0 f Shell z 3 b H T2 12 0 1 2e3 K1 lt lt lt m 1T 1 it it LT lt1lt gt gt lt lt Ommaammcmmmmmw SSS gt y gt gt lt lt Squ he quotT spin spin s EBA MAME 3Q V EC MM Pa CO M 22 34 H52 3amp1Lisz 30154 3M BdgllABd QS Wslgdi qg LJJq cWtsz H V K t l Mk amp OM30 Let L Mo gal Luz 9K Edatil Qm 32 0LVMQ 62 0 55 35 L zed j gaugo J 1 2 0 2 K 1 4 LT 392quot 2 97 2932 W 1 F 3 FILM Z 3F2 3 If T T 52 3 32 232 Lt IT T T F 2 2 0 EDD L 3 Ir 4 T 7 I 572 0 92 697 2 A N T T T 2 2 q SDLl 3r M M r T 3L 3 072 LIP Z 5 N N N F 1quot 4 3 LBS L 3 1 1 1 2 2052 to w M m M m o o 0 50 25 Yr 9re vceca39rc XI v X l Z EA f g L 5 WDNFQ H I Q 552 EwanCc Cem yumkow 1 462 231 ZIP 332 31 390 43 Ogt MM 5322 2 22 4 2e Lo H 2 m qu s Cw W WW mag MM m Cuz 9546 gt 3019 gtgt EE Tg 12 MN gm 73 m L20 gag nk Miro jugaw Ls 5 z Mme H NwMaJQg UuM onvygNeDr 76 52255 0LI Ckh vw Tum AM M 01L MW Mom 4 4 a IJZJ 9 3 WLHk LS we mesh t T 39vToz7 E g Waive amyuQM me 3 SakSthg TX 2H Sign TV 739 7 4 quota7141 723274j j25 392 L U 2 in A LJLL J45 fL J2JZJ39 C LeJ Jz 0 H 1 2 N Z 6 r 1 gm of 3m 1 I 75 z 744 72 71 W 3 rr a L50 91 A0ZLIgt 4132 71 t3ww39bh Tl ljlMgt t15 3rk UVUMgt QJ l helm 1 2 I J GUNS W MPH 9 3 51 f AZ z u i2 oz Mz J PM JZ WJ 2sz how J U Zg MINML 1 L 1M7 IJZ IML iii sz 37 1 i 5 2quot 61 D j1gtizjzlgt imstl 17042 o waLS U1W1J7wgtgt J JW1gtJZquotMLgt gt 39 an J I v 4rd J JIW gtJLW2 WW1ML J1 undzmbg I 7 g Q5 ES c 2 g s C KN E g 39 u 2 V 4 II H Q 2 3 N I h h K Mam MW WW aggth Hms aWVOQJA 3 m N03 WA Q km 2 Mm Mzmm 0 31 LUchQ MW mam 9 t j J I HK WMMZ w 7 1 Ws WC auaxsmb W WWJMW AZ h 8 WA add2 quot75 LDMW I 1J1Jld2 d7 gt 5 d A A M Q3 etcR Ou M475 95k w39x39HA Qt T3 V v aw a wt Woke E2Q3 34quot465 Cawa 9 S v M s 3 43967 Q all s4m 529m 3 Qddz M wT39A L W1 393j j 52jzgt 9 M SVyS L 39 9ng Ml j drjl ocGLur ml om A h A 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