Hyperfine Interaction Energy in Electrodynamics Lectures

phy 712 electrodynamics 10 10 50 am mwf olin n.w
1 / 27
Embed
Share

Explore the concepts of hyperfine interaction energy in electrodynamics, covering magnetic dipoles interactions, sources of magnetism in atoms, evaluation of magnetic fields, and electron current densities. Delve into the complexities of evaluating magnetic fields near nuclei in atomic systems.

  • Electrodynamics
  • Hyperfine Interaction
  • Magnetic Fields
  • Atom Magnetism
  • Electron Current
rayaanacharya
rizz_ana Follow

Uploaded on | 0 Views

Download Presentation

Please find below an Image/Link to download the presentation.

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.

You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.

Download Presentation

The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.

E N D

Presentation Transcript

  1. PHY 712 Electrodynamics 10-10:50 AM MWF Olin 107 (Make up lecture 2/17/2014 at 9 AM) Plan for Lecture 13: Continue reading Chapter 5 1. Hyperfine interaction 2. Macroscopic magnetization density M 3. H field and its relation to B 4. Magnetic boundary values 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 1

  2. 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 2

  3. Hyperfine interaction energy: Interactions between magnetic dipoles Sources of magnetic dipoles and other sources of magnetism in an atom: Intrinsic magnetic moment of a nucleus Intrinsic magnetic moment of an electron Magnetic field due to electron current Interaction energy between a magnetic dipole m and a magnetic field B: N e ( ) eJ r e = m B E r int N = N ) r 8 3 B B In this case: (0) E r e J int e N N 3 ( = + N 3 B ( ) r 0 ( ) r N 3 4 r N 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 3

  4. Hyperfine interaction energy: -- continued = B B (0) E e J int e N N Evaluation of the magnetic field at the nucleus due to the electron current density: The vector potential associated with an electron in a bound state of an atom as described by a quantum mechanical wavefunction can be written: nlm ( ) r l 2 ( ) r z r r r e m nlm = 3 A ( ) r 0 m d r l J l 2 2 4 | | sin B r e e = A We want to evaluate the magnetic field in the vicinity of the nucleus r ( 0). 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 4

  5. Hyperfine interaction energy: -- continued 2 ( ) r r z r r r ( ) l nlm e m nlm = = 3 B A 0 (0) m d r l J J l 2 2 4 | | sin r r 0 e e e 2 r 0 ) ( r r 3 r r z r | ( ) e m = 3 B r 0 ( ) m d r o l 2 2 4 | sin r e r 0 2 ( ) r r z r ( r ) e m nlm = 3 B 0 0 ( ) m d r l o l 3 2 2 4 sin r e = 2 r z r z 1 x y ( ) ( cos ) cos sin cos cos sin sin ). 2 2 ( ) r ( ) r 2 2 z sin r e m r e m nlm nlm = = 3 3 B 0 z 0 0 ( ) m d r m d r l l o l l 3 2 2 3 4 sin 4 r r e e 1 e m z 0 = m l 3 4 ' r e 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 5

  6. Hyperfine interaction energy: -- continued = B B (0) E H e J int HF e N N Putting all of the terms together: r r 3( )( ) 8 e L = + e + 3 N e N e N 0 ( ) r . H N HF 3 3 4 3 r m r e In this expression the brackets indicate evaluating the expectation value relative to the electronic state. 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 6

  7. Macroscopic dipolar effects -- Magnetic dipole moment m = 2 1 ( ) r 3r r J d Note that the intrinsic spin of elementary particles is associated with a magnetic dipole moment, but we often do not have a detailed knowledge of J(r). Vector potential for magnetic dipole moment m r ( ) r = A 0 3 4 r 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 7

  8. Macroscopic magnetization ( ) ( ) = 3 M r m r r i i i Vector potential due to free current Jfree(r) and macroscopic magnetization M(r). Note: the designation Jfree(r) implies that this current does not also contribute to the magnetization density. ( ) ' r ( ) ( r ) J r ' 3 M r r r ' ' ( ) r free = + 3 A 0 ' d r r 4 r ' 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 8

  9. Vector potential contributions from macroscopic magnetization -- continued ( ) ' r ( ) ( r ) J r ' 3 M r r r ' ' ( ) r free = + 3 A 0 ' d r r 4 r ' Note M that ' : ( ) ( r ) 3 r r r ' 1 ( ) r = M ' ' r r ' r ' ( ) r ( ) ' r ' r M r M r ' ' = ' + r ' ( ) r ( ) r + r ' J M ' ' ' ( ) r free = 3 A 0 ' d r r 4 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 9

  10. Vector potential contributions from macroscopic magnetization -- continued ( ) r ( ) r + J M ' ' ' ( ) r free = 3 A 0 ' d r r r 4 ' = A Note that for the = case that 0 : ( ) r ( ( ) r ) ( ( ) r = 2 B A A ( ) ( ) ) ) ( ) r ( ) r = + 3 3 r r J M 0 ' 4 ' ' d r ' ' free 4 ( r ( ) M 0 ( ) J 0 = + J r M = r 0 free ( ( ) ( ) r ) ( ) r B free 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 10

  11. Magnetic field contributions ( ( ) r the ( ) r ) ( ) r = B M J 0 0 free density Define magnetic flux : ( ) r = ) ) = ( ) r H r B M ( ) 0 0 ( ) r ( ) r that H r J ( B ( ( ) r the free ( ) r flux M J 0 0 free ( ) r ) B ( Note the magnetic density H ( r ) Define magnetic field ( ) r = ) ( ) r H r B M 0 0 ( ) r H r J ( free 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 11

  12. Summary H of equations J total of magnetosta tics : ( ) ( = r ( ) r = B r 0 ( ) ) + B r = r J r ) free ( ) B ( 0 ( ) r ) H r M ( 0 ( ) = ( ) r 1 = J For the case that 0: n free = = H r B r ( ) 0 2 ( ) 0 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 12

  13. ( ) r Jfree For the B case = r that : H ( ) = 0 ( ) r 0 boundary At : 1 n = n n H H 1 2 2 = n n B B 1 2 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 13

  14. Example magnetostatic boundary value problem 0z M r a = M r ( ) M0 0 r a = = H r H r r ( ) 0 ( ) ( ) H ( ) B ( 0 ( ) H ) r = + B r H r M r ( ) 0 ( ) r ( ( ) r ) = = + M ( ) 0 ( ) r = 2 r M ( ) H 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 14

  15. Example magnetostatic boundary value problem -- continued 0z M r a = M r ( ) M0 0 r a ( ) r = 2 r M ( ) H ( ) ' r M r 1 ' r ' = 3 r ( ) ' d r H 4 ( ) r M r 1 ' 1 ( ) r = 3 M ' ' ' ' d r r r r 4 ' ' ( ) r M r 1 ' = 3 ' d r r 4 ' 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 15

  16. Example magnetostatic boundary value problem -- continued ( ) r 0z M r a M r 1 ' = M r ( ) = 3 r ( ) ' d r H M0 0 r a r 4 ' For this example: a 1 r M = 2 ( ) r 0 4 ' ' r dr H 4 z 0 2 2 M z a r = = ( ) r 0 For : a r M 0 H 2 6 3 z 3 3 M a z r a = = ( ) r 0 For : a r M 0 H 3 3 3 z r 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 16

  17. Example magnetostatic boundary value problem -- continued 0z M r a = M r ( ) M0 0 M r a H ) z M = = = r r r z 0 0 For : ( ( ) ( ) r a H H 3 3 3 3 z r 3 M a z M a z = = = r H r r 0 0 3 For : ( ) ( ) ( ) r a H H 3 ) M 3 5 3 ( r ) r r r ( = + B r H r M ( ) ( ) 0 z z 2 M = = H r B r 0 0 For : ( ) ( ) r a 0 3 3 3 z r 3 M a z = H r M 0 3 For : ( ) r a 3 5 r r 3 z r 3 a z = B r 0 3 ( ) 0 3 5 r r 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 17

  18. Check boundary values: z M M = = ) a r r r H r H z 0 0 For : ( ) ( r a 3 3 3 z r 3 M a z = H r 0 3 For : ( ) r a 3 5 r r 3 r z M a = ) a r r H 0 3 ( 3 a 2 z 2 M M = = ) a r r r B r B z 0 0 For : ( ) ( r a 0 0 3 3 3 z r 3 M a z = B r 0 For : ( ) r a 0 3 5 3 r r 3 2 1 a 3 M a a = ) a r r r B z 0 ( 0 3 5 3 a 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 18

  19. Variation; magnetic sphere plus external field B0 M r a M0 B0 0 = M r ( ) 0 r a superposit By r ion : For : a 2 ( ) r = + B B M 0 0 0 3 1 1 ( ) r = H B M 0 0 3 = 0 ( ) r ( ) r + B H B 2 3 0 0 ( ) r ( ) r = B H isotropic an For " paramagnet material, ic" 3 = M B 0 0 0 + 2 PHY 712 Spring 2014 -- Lecture 13 02/17/2014 19 0 0

  20. Summary H of equations J total of magnetosta tics : ( ) ( = r ( ) r = B r 0 ( ) ) + B r = r J r ) free ( ) B ( 0 ( ) r ) H r M ( 0 ( ) = ( ) r Jfree For the B case = r that : H ( ) = 0 ( ) r 0 At boundary = : 1 n n n H H 1 2 2 = n n B B 1 2 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 20

  21. Magnetism in materials ( ) r ( ( ) linear ) = + B H r M r ( ) 0 materials For H B = with magnetism : paramagnet material ic 0 diamagneti material c 0 ferromagne For B f = antiferrom tic, agnetic materials ( ) H (with hysteresis ) 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 21

  22. Example: permalloy, mumetal 0 ~ 104 B0 Spherical shell a < r < b : a b 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 22

  23. Example: permalloy, mumetal 0 ~ 104 -- continued B0 For this H case r = : a ( ) = 0 b ( ) r = B 0 ( ) r B H r ( ) Continuity B boundaries at : = n H continuous = n continuous 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 23

  24. Example: permalloy, mumetal 0 ~ 104 -- continued B0 a b = H r r Let : r B ( ) ( ) r H ( ) 0 = = 2 r 0 r ( l ) 0 H ( ) = l For a ( ) cos r P H l l ( ) l = + l r B For b ( ) cos l a r r P H l l + 1 l r ( ) l = + r 0 For b ( ) cos cos l r r P H l + 1 l r 0 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 24

  25. Example: permalloy, mumetal 0 ~ 104 -- continued Applying boundary l = conditions B0 a (only terms 1 contribute : ) b = = At a 2 1 r 1 1 3 a 0 = + 1 a a 1 1 2 a B = = 0 At b 2 2 1 1 r 1 3 3 b b 0 0 B + = + 0 1 1 b b 1 2 2 b b 0 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 25

  26. Example: permalloy, mumetal 0 ~ 104 -- continued B0 a b When the dust clears : 9 / B = 0 0 ( )( ) ( ) ( ) 1 3 2 + + 2 / 1 / 2 2 / / 1 a b 0 0 0 0 1 9 a / 2 b B ( 1 ) 0 ( ) 3 / / 0 0 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 26

  27. Energy associated with magnetic fields Note previously We : used without proof - - m B magnetic a on force the dipole external an in field is : ( ) = implies F m B that This energy associated with aligning a m B magnetic U dipole external an in field is given by : = m B energies c Macroscopi - - 1 ( ) r ( ) r = 3 B H shown that be can It : W d r B 2 1 ( ) r ( ) r = 3 E D analogy to In : W d r E 2 02/17/2014 PHY 712 Spring 2014 -- Lecture 13 27

Related


No related presentations.

More Related Content