Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism. NNSE508 / NENG452 Lecture #14

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1 1 Lecture contents agnetic properties Diaagnetis and paraagnetis Atoic paraagnetis NNSE58 / NENG45 Lecture #14

2 agnetic units H /r V s 1 Wb T 1 T Wb T 1H A A Fro Treolet de Lacheisserie, 5 NNSE58 / NENG45 Lecture #14

3 agnetic susceptibility 3 SI: H H R H R 1 CGS: H 4 H R R H 14 Fro Cullity, 9 NNSE58 / NENG45 Lecture #14

agnetic properties of aterials 4 agnetization or agnetic dipole density Diaagnetic ~ - 1-5

4 agnetic properties of aterials 4 agnetization or agnetic dipole density Diaagnetic ~ Paraagnetic ~ +1-5 Ferroagnetic spontaneous agnetization, large Fro Cusack, 1963 NNSE58 / NENG45 Lecture #14

Diaagnetis (classical) Arises fro Lentz s law: when agnetic flux changes in a circuit, a current is induced which opposes the change of flux 5 Orbiting electron creates agnetic dipole (circulating

5 Diaagnetis (classical) Arises fro Lentz s law: when agnetic flux changes in a circuit, a current is induced which opposes the change of flux 5 Orbiting electron creates agnetic dipole (circulating current) charge I q period In agnetic field, Lorentz s force is added to centrifugal force And corresponding change of rotational frequency If change in orbital otion is sall ( ) The energy associated with this frequency is IA The change in frequency can be associated with induced agnetic dipole oent: 1 R F qrh qr v qh q H q H E q J T H 4 ohr agneton A qr 4 H = H Lentz NNSE58 / NENG45 Lecture #14

6 Diaagnetis (classical) contd. 6 Sall agnetic field-induced agnetic dipole oent: Now we can apply the result to spherical closed-shell ato Averaging over 3D gives ean square radial distance Su over all Z electrons in the ato Su over all atos in a unit volue, density N, to obtain agnetization Finally susceptibility All atos and ions display diaagnetic response Alost independent of teperature olar susceptibility is often used to describe agnetis of atos (should be ultiplied by olar volue to obtain diensionless susceptibility) H q ZN 6 R R 3 R qr 4 H Laror or Langevin diaagnetic susceptibility x y R 1 x y z R 3 q ZN 6 H R olar susceptibilities of soe atos and ions (x1-6 c 3 /ole) Fro urns, 199 NNSE58 / NENG45 Lecture #14

7 Paraagnetis 7 Contrary to diaagnetis, paraagnetis arises fro non-zero agnetic oents: Free electron (Pauli) spin paraagnetis Langevin atoic paraagnetis An electron has an intrinsic agnetic dipole oent associated with its spin S, equal to ohr agneton: We can expect that the agnetic dipoles will rotate towards low-energy state ( U fro to ) The fraction of electrons with agnetic oents parallel to agnetic field exceeds the anti-parallel fraction by H For n free electrons, the agnetization ut we need to take band structure into account! g s =.3 q S q J A T n H 4 For =1 T (H = 8x1 5 A/ ) U 58 ev.67 K Field alignent is weak! NNSE58 / NENG45 Lecture #14

8 Paraagnetis of free spins H n agnetization is ~1 ties higher than observed in real aterials 8 In a band only a theral fraction of electrons contributes to paraagnetis (copare to transport) E F Energy vs. density of states In a agnetic field before the spins reorient In equilibriu agnetization is n H F Siilar to transport, ore accurate averaging over the distribution function gives susceptibility 3 n F For exaple, for Na Fro urns, 199 NNSE58 / NENG45 Lecture #14

9 Langevin atoic paraagnetis 9 Siilar to free spins, if an ato has a agnetic oent, it can align along the agnetic field eff agnetization of a aterial with atoic density N is (averaging included) 1 eff H Neff 3 And susceptibility Ato with orbital, spin and total angular oenta, L,S, and J = L+S, will have agnetic oent With Lindé g-factor Coplications 3 g J N eff 1 Quantu echanical averaging of J Ions eff gj J L S J S L J Quenching of orbital oentu in the crystal field (Stark splitting of L+1 degeneracy ) C T Curie law for paraagnetics With Curie constant C N eff =.3 NNSE58 / NENG45 Lecture #14 3k Often spin-only oent is used with convention: g J =, L=, J=S and axiu oent H =J Need to be careful with scientific texts!

10 Quantu echanical averaging over (J+1) projections With rillouin function: Atoic paraagnetis - Quantu theory J gj J gj Je J N Ng g J J J JJ y Neff J y J J e J 1 J 1 1 y J ( y) coth y coth J J J J sat agnetic oent vs. H/T 1 with y eff gjj k T k T If agnetic energy is sall copared to theral energy, y << 1, rillouin function gives ( y 1) y J J 1 3J This results in classic susceptibility with quantu averaged 3 N eff g eff J J( J 1) NNSE58 / NENG45 Lecture #14

11 Ground states of ions predicted by Hund s rules agnetic oents of ions S 1 L J Values of agnetic oents of 4f and 3d ions in insulating copounds 11 Fro urns, 199 NNSE58 / NENG45 Lecture #14

12 In KCr(SO 4 ). 1 H O copound, the only agnetic ato is Cr 3+ : 3 d-electrons: S=3/, L=3, J=3/ g Atoic paraagnetis - Exaple J J S L 1 J 5 Fro experient: g J =, L=, J = S =3/, oent is deterined by spin, orbital coponent is quenched agnetic oent of KCr(SO 4 ). 1 H O, at fields up to 5, Oe and at 4. K 1 g J ( J 1) eff J axiu (asyptotic) value: H = J = 3 (copare to the classic value) Fro Cullity, 9 NNSE58 / NENG45 Lecture #14

13 13 Lecture recap Diaagnetis (susceptibility is negative ) All atos Classical, due to addition of agnetic oent to the electron orbital current and (Pauli) paraagnetis Due to alignent of spins of free electrons Spins of the electrons at the Feri surface can be affected Atoic paraagnetis Due to alignent of existing agnetic oents of atoic electrons Needs quantu assessent of agnetic oents of electrons (S, L, J) and statistics NNSE58 / NENG45 Lecture #14

Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism Ferromagnetism. Molecular field theory Exchange interaction

Lecture contents. Magnetic properties Diamagnetism Band paramagnetism Atomic paramagnetism Ferromagnetism. Molecular field theory Exchange interaction 1 Lecture contents Magnetic properties Diamagnetism and paramagnetism Atomic paramagnetism Ferromagnetism Molecular field theory Exchange interaction NNSE 58 EM Lecture #1 [SI] M magnetization or magnetic