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123B_1_EE 123B W11 lecture 14 chapter 11 posting

123B_1_EE 123B W11 lecture 14 chapter 11 posting - Chapter...

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Unformatted text preview: Chapter 11. Diamagne0sm and Paramagne0sm Lecture 14 2/15/11 Today Lecture Overview magne0c materials Primer on Hydrogen atom Magne0c suscep0bility, magne0c moment Langevin diamagne0c equa0on (classical approach) •  Quantum mechanical approach •  •  •  •  Last Lecture •  Discussed take ­home exam Homework #5 Due Tuesday – 2/17/11 •  11.1 – given electronic wavefunc0on, calculate <r> and χ. •  11.2 – Apply Hund’s rules to find ground state configura0on of several ions. •  Describe use of diamagne0c materials in levita0on. •  Plot M, as in Figure 11.4 for two ions Cr2+ and Fe2+. Magne0c field, H and Flux, B B Magne0c Materials •  All atoms have several types of spin and magne0c moment associated with nuclei, electrons, molecules. •  Electron spin and magne0c moments have largest effect on material proper0es. •  Each electron has a magne0c moment and spin iden0fied through quantum numbers, which can be affected by an applied field. B Magne0c Materials •  The origin of magne0sm lies in the orbital and spin mo0ons of electrons and how the electrons interact with one another. •  The best way to introduce the different types of magne0sm is to describe how materials respond to magne0c fields. •  The main dis0nc0on is the collec0ve interac0on of atomic magne0c moments. Some materials have very strong interac0on between atomic moments. Categories of Magne0c Materials •  Magne0c suscep0bility M χ = (CGS) B •  Diamagne0sm •  Magne0c momentum is induced  ­ no collec0ve magne0c interac0ons. •  Paramagne0sm •  •  •  Magne0c momentum is spontaneous.  ­ exhibit long ­range magne0c order below a certain cri0cal temperature. Ferromagne0sm •  Ferromagne0c materials are usually An0ferromagne0sm considered magne0c (ie., behaving like iron). Diamagne0sm – in electromagne0sm •  Lenz’s law: In response to changing flux B, an induced (diamagne0c) current flows in coils to oppose change. N=5 turns I => Binduced => serves to stabilized B I = (charge)(revolutions per unit time) Area of loop = π r 2 1 eB I = ( − Ze)( ) 2π 2 m − Ze2 B 2 µ=( )r 4m magnetic moment Nµ − NZe2 χ= =( ) r2 B 6 mc 2 Diamagnetic Susceptibility Diamagne0c current establishes a diamagne0c field and moment. Diamagne0c Materials •  Diamagne0c substances are composed of atoms which have no net magne0c moments (ie., all the orbital shells are filled and there are no unpaired electrons). • The orbital mo0on of electrons creates 0ny atomic current loops, which produce magne0c fields. • When an external magne0c field is applied to a material, these current loops will tend to align in such a way as to oppose the applied field. • This may be viewed as an atomic version of Lenz's law: induced magne0c fields tend to oppose the change which created them. • Diamagne0sm is a fundamental property of all mader, although it is usually very weak. It is due to the non ­coopera0ve behavior of orbi0ng electrons when exposed to an applied magne0c field. Diamagne0sm – Nega0ve Suscep0bility • When exposed to a field, a nega0ve magne0za0on is produced and thus the suscep0bility is nega0ve. Diamagne0c Materials Some well known diamagne0c substances, in units of 10 ­8 m3/kg, include: •  Quartz (SiO2)  ­0.62 •  Calcite (CaCO3)  ­0.48 •  Water  ­0.90 Some well known diamagne0c elements in units of 10 ­6 cm3/mole, include: •  He  ­1.9, Ne  ­7.2 •  Ar  ­19.4, Kr  ­28.0 •  Xe  ­43.0 Uses  ­ Diamagne0c Materials MRI •  An important applica0on of diamagne0c materials is magne0c resonance imaging (MRI). •  How it works is that when carbon ­based atoms in the body are exposed to a strong magne0c field, they are slightly repelled by the field. This movement of the atoms can be detected and used for analysis. Levita+on •  Homework – explain use of diamagne0c materials in levita0on. Paramagne0c Materials •  Atoms or ions have a net magne0c moment due to unpaired electrons in par0ally filled orbitals. •  Individual magne0c moments do not interact magne0cally, and like diamagne0sm, the magne0za0on is zero when the field is removed. •  In the presence of a field, there is now a par0al alignment of the atomic magne0c moments in the direc0on of the field, resul0ng in a net posi0ve magne0za0on and posi0ve suscep0bility. One of the most important atoms with unpaired electrons is iron. Paramagne0c Materials •  The efficiency of the field in aligning the moments is opposed by the randomizing effects of temperature. •  This results in a temperature dependent suscep0bility, known as the Curie Law. •  At normal temperatures and in moderate fields, the paramagne0c suscep0bility is small (but larger than the diamagne0c contribu0on). Unless the temperature is very low (<<100 K) or the field is very high paramagne0c suscep0bility is independent of the applied field. Summary •  Magne0c response of a material involve orien0ng magne0c dipole of an electron. •  Magne0c dipole Magne0c moment Magne0c moment has three sources: •  Electron spin (paramagne0c) •  Electron orbital angular momentum (paramagne0c) •  Change in orbital moment induced by applied magne0c field (diamagne0c). Electron spin Electron orbital dipole moment Magne0za0on •  Magne0za0on M Flux density Applied field Vector sum of all dipoles Induced magne0za0on In linear regime (diamagne0c, paramagne0c) : B = µ H ⇒ permeability M = χ H ⇒ Susceptibility In diamagnetic and paramagnetic materials, 4π M << H , so µ 1, B~H M χ = (CGS); B µM χ = (SI) B Diamagne0c Suscep0bility •  Suscep0bility, magne0za0on are nega0ve and temperature insensi0ve. Diamagne0sm – Magne0c moment defini0on •  Next consider an electron with charge e, revolving in circle, A with angular velocity ωo under central force F (Charge passes any point every 2π/ω) = 1/v; 2πv=ω Induces current, I and magne0c moment I = eω o / 2π , IA eω o a µm = = c 2c 2 2 F = mω o a µm A a  ­e v •  Next, inves0gate effect of external magne0c field on rota0on change Magne0c moment arises from induced current by moving charge. Diamagne0sm – Larmor frequency •  New force is induced by applied H •  Charge rota0onal frequency changes to compensate for new F Find = ev × B  ­e v vind Ftotal = F − Find eω aH mω a = mω a − c 2 2 0 Change in orbital frequency is the diamagne0c response to external field. Larmor frequency Diamagne0sm – atom in field •  Energy associated with this new frequency ∆ω ΔE = Δω Change in orbital frequency is the diamagne0c response to external field. Change in orbital frequency changes magne0c moment of each electron. •  Consider an atom with closed shells under an applied field. •  There is a small change in orbital frequency thus a small induced magne0c moment opposite to the field. •  We can find magne7c moment due to H from Lamour frequency. ea 2 µ= Δω 2c No7ce a nega7ve number so as H increases, the induced dipole increases, opposing the external field. Diamagne0sm – contd. Applied to a closed shell atom: r is average radial distance between electron and nucleus NZe2 =− < r 2 > (CGS) N: number of atoms Z: number of electrons 6 mc 2 µo NZe2 =− < r 2 > (5) SI units 6m Classical Langevin result – Larmor diamagne7c suscep7bility or Langevin suscep7bility. No7ce a nega7ve number so as H increases, the induced dipole increases, opposing the external field. < r 2 > is known for closed-shell atomic configurations Wave Func0on – H atom •  Wave func0on – H atom, 1s Charge density ( ) He:  ­1.9 Ne:  ­7.2 Ar:  ­19.4 Xe:  ­43.0 What is it good for? Hydrogen atom  ­ primer 321 Principle quantum number l ≡ Orbital quantum number Magne0c quantum number 2l+1 values Spin quantum number 100 l=0 s shell 211 l =1 p shell l=2 d shell 311 l =1 3p shell Hydrogen atom •  Look at periodic table and consider how electrons fill orbitals or or or paramagne0c! allowed Not paramagne0c!  ­13.6eV Periodic Table Periodic Table Periodic Table Electron Orbit Magne0c Moment -introduce quantization to describe diamagnetic activity. + r  ­  ­e e µ = IA = − L 2 mc Quantum number of e orbit magne0c moment Bohr magneton Larmor Precession •  Angular frequency, precession Area of loop πρ 2 Orbital quantum number Precession induces a current: cylindrical calcula0on involves r for e ­ distribu0on Spin ­orbit coupling – vector defini0ons Orbital angular momentum oven wriden without ħ Spin angular momentum Total angular momentum Magne0c quantum number For half ­full shells: ∑l i i =0 For full shells: Paramagne0sm Electronic paramagne0sm (posi0ve contribu0on to χ) is found in: •  Atoms, molecules, lawce defects for odd number of electrons, where total spin ≠ 0. e.g. Free sodium atoms; gaseous nitric oxide (NO); organic free radicals; F centers in alkali halides. •  Free atoms and ions with a partly filled inner shell. e.g. transi0on elements; ions isoelectronic with transi0on elements; rare earth and ac0nide elements; Mn2+, Gd3+, U4+. •  A few compounds with an even number of electrons. e.g. molecular oxygen; organic biradicals. •  Metals. Here, χ>0, Paramagne0sm – contd. •  Consider free electrons i.e. ideal magne0c gas p electrons in an incomplete shell. ith electron has: Orbital quantum number: li Orbital angular momentum: ħli or L Spin quantum number: si Spin angular momentum: ħsi or S li=0,1,2,3,… designates shell. si=+⅟₂,  ­⅟₂. •  How do i momentums couple? Russell ­Saunders (LS coupling) coupling – small atoms (Z<30) ...
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