AC Measurement of Magnetic Susceptibility Episode 2 – Loss and Thermal Effects
Prof. Jeff Filippini Physics 401 Spring 2020 Outline Combine the tools and techniques we’ve learned to characterize magnetic properties of materials
• Magnetic Materials: Dia- and paramagnetism (Bonus: superconductors!) • Week 1 Refresher: AC Measurement of Susceptibility • Magnetic Losses: Complex permeability • Temperature Dependence: Curie-Weiss Law • Finishing the Final Lab: What you’ll get, what you need to do • Closing out the Semester
This is the second week of the final lab Report counts as your final exam
Physics 401 2 Reminder: Magnetic Response of Materials
Two things are often called the “magnetic field”: B and H
B H Magnetic induction Magnetic field intensity Magnetic flux density � = �! � + � Magnetizing field
Determines forces on Field created only by moving free charges moving free charges. via Lorentz force law: M Magnetization In vacuum, � = � �. Magnetic polarization ! �⃗ = � � + �⃗×� Field created only by moving bound charges, i.e. magnetic response of the medium
Physics 401 3 Reminder: Magnetic Response of Materials
� = �! � + � = �! 1 + � � = �!�"�
We classify materials into three major categories:
Diamagnetic � < 0 � < 1 Weakly repelled Paramagnetic � > 0 � > 1 Weakly attracted Ferromagnetic � ≫ 0 � ≫ 1 Strongly attracted
Physics 401 4 Magnetic Materials: Diamagnetism Many materials are diamagnetic (� < �), i.e. they magnetize in a
−5 way that reduces the applied field and are repelled from magnets. Material χv (10 ) Bismuth −16.6 Effect of paired electrons in all materials, generally very weak Carbon (diamond) −2.1 Carbon (graphite) −1.6 Copper −1.0 Lead −1.8 Mercury −2.9 Pyrolytic carbon −40.0 Silver −2.6 Superconductor −105 Water −0.91 Wikipedia Frog levitated in 16T field (Wikipedia) Levitating sample of pyrolytic carbon (Wikipedia)
Physics 401 5 Superconductors: the Meissner Effect
Normal Superconducting Superconductors are perfect diamagnets (� = −�)
At T Persistent image currents expel incident fields to maintain B=0. Above some critical field Hc, superconductivity is destroyed. Important for designing superconducting solenoid magnets Deep analogy with the Higgs mechanism in particle physics! Wikipedia Physics 401 6 Magnetic Materials: Paramagnetism In paramagnetic (� > �) materials, atomic/molecular Magnetic dipoles from unpaired electron spins align with Material susceptibility, external fields. χv [10−5] Tungsten 6.8 Caesium 5.1 Aluminium 2.2 Lithium 1.4 Magnesium 1.2 Sodium 0.72 Wikipedia Liquid oxygen trapped between poles of a strong magnet (Wikipedia) Physics 401 7 Magnetic Materials: Ferromagnetism In ferromagnetic (� ≫ �) materials, neighboring atomic/molecular dipoles from unpaired electron spins Material µ Brem (T) r spontaneously align with one another, forming domains. Fe, 99.8% pure 5000 1.3 Permalloy 100,000 0.7 Superpermalloy 1,000,000 0.7 Co, 99% pure 250 0.5 M~Ms Ni, 99% pure 600 0.4 Wikipedia M=0 E.V. Colla Physics 401 8 Measuring Permeability: DC Field H=H0 +Hs1 inωt B0 BfH= () DC part of the setup H0 � � � = � �� � ��� E.V. Colla Here Np is the number of turns in the DC primary coil Physics 401 9 Measuring Permeability: AC Modulation H =H0 +Hs1 inωt Bw B0 BfH= () Wavetek reference H = H01 + H sinωt ����� �� = ��� Rac H0 �� �� � ~ = =µ=µ µ � �� �� 0 r E.V. Colla Here Np is the number of turns in the AC primary coil Physics 401 10 Measuring Permeability: AC Toroid Flux Primary coil is a toroid of Np turns carrying a current Ip creates a magnetic field H: � � � = 2�� E.V. Colla … and this magnetizing field adds a flux dΦ to each turn of the pickup coil � � � � " �� � � � � � �Φ = � � ��⃗ = = ln 2� � 2� � ! Physics 401 11 Measuring Permeability: Pickup Coil Faraday’s Law �Φ � = −� Φ = � ��⃗ = � � ��⃗ �� Only the AC flux contributes to this time derivative: � � � � � Φ = Φ! + Φ" sin �� Φ = � � ��⃗ = ln 2� � #! $%& '( This AC flux is sourced by the current in coil L2: �!"= )"# � � � �2 ��!" � � � �2 �*+",-./ = −�0.",10 ln = −�0.",10 ln � cos �� 2� �3 �� 2� �3 � �4�� �2 � = −µ � � cos �� where � ≡ � ln � 4 0.",10 2� � 3 E.V. Colla Physics 401 12 Measuring Permeability: Pickup Coil Relative permeability Time derivative of source current Geometry of of core the toroid � �4�� �2 � = −��� � cos �� �4 ≡ �0.",10 ln � 2� �3 Geometric mutual inductance in the absence of the core In general: �� = � − ��′′ • The real component �′ appears at the Y (quadrature phase) channel of the lock- in, due to its �/2 phase shift with respect to the driving current (cos vs. sin!) • The imaginary component �′′ characterizes losses, and appears in-phase Physics 401 13 Hysteresis, Coercivity, and Work E.V. Colla Hysteresis necessarily involves energy losses from re-magnetization. If the domains are “sticky”, we need to do work to overcome that. � = � � �� � = � � �� = � ∗ � For uniform fields over volume V (analogous to dW = F dx) Physics 401 14 Aside: Loss from Complex Permeability � = � � + � = � − ��′′ � � Why is a material with complex permeability (�55 ≠ 0) lossy? Real μ : Zero r integral In analogy with �� = � ��, we have �� = � �� Complex μr: Nonzero integral! Physics 401 15 Calculating the Magnetic Induction, B � = � 1 + � � = � � � = �� �� Magnetics ZP44715-TC � = � � = ⟹ � = � � � �� 0.4 �� 0.2 Integrate using OriginPro to find B(H) contour 0.0 Work for any motion through this plane: (T) B -0.2 � = � � �� For uniform fields over volume V -0.4 (analogous to dW = F dx) -200 0 200 H (A/m) E.V. Colla Physics 401 16 Hysteresis and Remagnetization Losses �#$$% = � � �� = � ∗ �#$$% E.V. Colla Physics 401 17 Hysteresis and Remagnetization Losses �#$$% = � � �� = � ∗ �#$$% ! E.V. Colla Physics 401 18 Hysteresis and Remagnetization Losses �#$$% = � � �� = � ∗ �#$$% Magnetics ZP44715-TC Integrate from -0.0364912 to -0.0250877 0.4 80 60 0.2 40 0.0 B (T) B 20 Integrated Y3 Integrated -0.2 0 -0.4 -20 -200 0 200 -200 0 200 H (A/m) H (A/M) E.V. Colla Physics 401 19 Hysteresis and Remagnetization Losses �*++0 = � � �� = � ∗ �*++0 Units : V ()volume® m3 H()field® A• m-1 B(.magn induction )® kg•• s--21 A [V ••BH] ®m22•• kg s- º J(joule) This is the energy lost around a circuit of the loop. To make this a loss power, we need to account for the frequency: 7 � = = ��, where T is the period and f is the frequency *+66 8 E.V. Colla Physics 401 20 Temperature Dependence of Permeability In this experiment we measure �(�): magnetic permeability as a function of temperature We measure at fixed H0, and so fixed IDC. The default is to measure at IDC=0 (zero DC field). Our thermometer will be a T-type thermocouple. We’ll use the DMM to measure the EMF (voltage) across it, and from this infer the temperature. E.V. Colla Physics 401 21 Thermocouples Type Names of Materials T Range Platinum30% Rhodium (+) 2500 -3100F In a thermocouple, a pair of dissimilar metal wires B Platinum 6% Rhodium (-) 1370-1700C generates a temperature-dependent EMF W5Re Tungsten 5% Rhenium (+) 3000-4200F C W26Re Tungsten 26% Rhenium (-) 1650-2315C Seebeck effect: Charge carriers diffuse away from Chromel (+) 200-1650F E hot areas, setting up a (material-dependent) EMF Constantan (-) 95-900C that is a function of the temperature gradient. Iron (+) 200-1400F J Constantan (-) 95-760C Chromel (+) 200-2300F K Alumel (-) 95-1260C Nicrosil (+) 1200-2300F N Nisil (-) 650-1260C Platinum 13% Rhodium (+) 1600-2640F R Platinum (-) 870-1450C Platinum 10% Rhodium (+) 1800-2640F S Platinum (-) 980-1450C Copper (+) -330-660F T Constantan (-) -200-350C Type T at 0°C: dV/dT=41.5 μV/°C Wikipedia E.V. Colla Physics 401 22 Temperature Ramp Option 1: Manually change the voltage applied to the heater Option 2: Automate using the Omega PID* temperature controller * Proportional, Integral, Differential: Three standard linear operations to perform on error signal to determine control (feedback) signal to stabilize input at target E.V. Colla Physics 401 23 Temperature Measurement AC frequency (1-10kHz) Series resistor in AC loop (fixed) DMM resolution: Optimal=100nV E.V. Colla Physics 401 24 Temperature Dependence of Permeability Ferroxcube 3e8 250 cooling 200 150 100 ' (a.u.) ' c 50 0 300 350 400 450 E.V. Colla T(K) Physics 401 25 Temperature Dependence of Permeability heating cooling Ferroxcube 4A20 DT 100 (a.u.) heating ' c 300 cooling ? 200 0 (a.u.) ' c 376 378 380 382 384 386 388 390 T (K) 100 0 Hysteresis in Tc and χ 300 320 340 360 380 400 420 T (K) What is the source? E.V. Colla Physics 401 26 Temperature Dependence of Permeability dB µµµ()TTT==0 r () ( ) heating dH 300 cooling ? 200 (a.u.) ' 2 c 1.0 100 1 #2 0 0 B(a.u.) 300 320 340 360 380 400 420 0.5 (a.u.) B #1 T (K) -1 Slope#1 ~0.728 -2 0.0 -2 -1 0 1 2 Slope#2 ~0.546 0.00 0.05 0.10 0.15 H(a.u.) E.V. Colla Physics 401 27 Curie-Weiss Law Pierre Curie Pierre Weiss 1859 – 1906 1865 – 1940 � � = � − � E.V. Colla Physics 401 28 Temperature Dependence of Permeability Ferroxcube 3e8 0.4 20C 57C 80C 100C ~130C (T>Tc) 0.0 B (a.u.) B -0.4 -0.4 -0.2 0.0 0.2 0.4 I (A) E.V. Colla H ”Week 3”: Full curves at varying H0 Physics 401 29 Our (Socially Distanced!) Plan for the Final Lab • Apparatus is largely prepared, data collection is largely automated • We usually divide lab tasks up into three weeks: 1. B(H) for four samples at room temperature (χ, μ’’, hysteresis) 2. χ(H=0) for one sample at several temperatures (Curie-Weiss law ,TCurie) 3. B(H,T) for one sample at several temperatures (saturation, remanence, …) • We will post videos and data on the following schedule: • Week 1: Wednesday 4/22 • Week 2/3: Sunday 4/26 (combined, same sample) • You will perform the data analysis and write a report as usual, following the lecture and write-up • Contact me and your TAs with questions! Don’t wait! Physics 401 30 References • Information about magnetic materials can be found in: \\engr-file-03\phyinst\APL Courses\PHYCS401\Experiments\AC_Magnetization\Magnetic Materials • SR830 (Lock-in Amplifier) manual \\engr-file-03\phyinst\APL Courses\PHYCS401\Common\EquipmentManuals\SR830m.pdf E.V. Colla Physics 401 31 End of Semester • No lecture 4/27 or 5/4 – this is our last meeting! • Please turn in any late lab reports! You can revise and resubmit one report for a replacement grade. Coordinate with your TA! • Sunday, May 10th, 11:59 pm is the deadline for the final lab, and for all other reports/resubmissions. No extensions and no vouchers! • Grade thresholds may be adjusted if needed on a section-by-section basis if substantial disparities arise Physics 401 32 Thank You for your hard work Classical Physics Laboratory and especially for bearing with us during this most unusual and challenging semester Professor Jeff Filippini Best of luck in yourPhysics future 401 endeavors, in and out of theSpring laboratory! 2020 XKCD #683