NOVEL M EASUR E MENT METHODS FOR ASSESSING MAGNETIC PROPERTIES OF FERROMAGNETIC MATER IALS by TODD WILLIAM FULTON A.S., Charles Stewart Mott Community College , 2006 A.A., Charles Stewart Mott Community College , 2006 A.G.S., Charles Stewart Mott Community College , 2006 B.S., University of Colorado Denver, 2017 A thesis submitted to the Faculty of the Gradu a te School of the University of Colorado in partial fulfillment of the degree of Master of Science Electrical Engineering Program 2018
ii This thesis for the Master of Science degree by Todd William Fulton has been approved f or the Electrical Engin e ering Program by Stephen D. Gedney, Chair Mark Golkowski Vijay Harid Date: Dec 12 , 2018
iii Fulton, Todd Willia m (M.S., Electrical Engineering Program) Novel Measurement Methods For Assessing Magnetic Properties O f Ferr omagnetic Materials. Thesis directed by Professor Stephen D. Gedney ABSTRACT The focus of this thesi s is the development of precisi on techniques for the measurement of non linear hysteretic susceptibility properties of ferromagnetic materials. A n ovel ap paratus i s developed and tested to prove its efficacy in obtaining desirable results, for the modeling of various types of ferromagnetic material sa mples. These new apparatus are supplemental to a standard vibrating sample magnetometer (VS M), which , histor ically has been primarily used for coercive and saturate magn etic field measurements, is now extended to susceptibility measurements to help further define the magnetization process. V ibrating samp le magnetometer measurements can be h am pered by rate depend ent effects , sample geometry , dependent non linear demagnetizing fields , as well a s volume fraction effects , and proposed methods addressing each of these obstacles have been developed. . New measurement methods are evaluated and then compared to other mea surement techniques, proposed numerical models, and numer ical approximations of non linear demagnetizing effects. The final goal of this effort is to develop a physical parameter based susceptibility model for non linear hysteresis. The form and conte nt of this abstract are approved. I recommend its publication Approved: Stephen D. Gedney
iv ACKNOWLEDGEMENTS I want to express my gratitude to my mentor Dr. Stephen D. Gedney. Your guidance, patience and wisdom are beyond reproach. Thank you for believin g in me and welcoming me as a member of your research team while helping me to further myself academically, personally , and professionally. I would also like to express a deep gratitude to my academic colleagues Dr. Carl Schneider, Mr. Sean Joyce, Mr. Kyl e Townsend, Mr. Mark Travers and Mr. David Gedney . It took all of us to help develop the techniques and theories presented in this thesis and each one of you help ed to solidify my understanding and confidence in this work. Beyond my peers, I would like to thank Dr. Mark Golkowski and Dr. Vijay Harid for serving as the committee members for my thesis panel. Each of you have inspired me on my journey here at the University of Colorado Denver and I am thankful for your participation and advice in the culminat ion of this chapter of my life. Last, thank you to the Colorado Nanofabrication Lab at the University of Colorado Boulder and Alexander Denton for helping to facilitate the annealing experiment.
v Table of Contents CHAPTER I. INTRODUCTION ................................ ................................ ................................ .................. 1 Overview ................................ ................................ ................................ ................................ ................... 1 Historical Work  ................................ ................................ ................................ ................................ ... 1 Quantum Theory ................................ ................................ ................................ ................................ ... 1 CHAPTER II: COOPERATIVE HYSTERESIS MODEL ................................ ................................ ........... 5 CHAPTER III: BASIC VSM PROCEDURES ................................ ................................ ............................. 9 Summary ................................ ................................ ................................ ................................ ................... 9 Density Measurement ................................ ................................ ................................ ............................... 9 Magnetic Moment Measurement ................................ ................................ ................................ ............ 10 CHAPTER IV: DEMAGNETIZATION FIELD ................................ ................................ ........................ 13 CHAPTER V: FIELD UNIFORMITY ................................ ................................ ................................ ....... 32 CHAPTER VI: DOUBL E DISK EXPERIMENT ................................ ................................ ...................... 35 Motivation ................................ ................................ ................................ ................................ ............... 35 Internal Field, Demagnetizing Factor ................................ ................................ ................................ ..... 39 Double Disk Sample Holder Prototype ................................ ................................ ................................ ... 41 Amplifier Circuit for Hall Chip Excitation ................................ ................................ ............................. 43 Filtering ................................ ................................ ................................ ................................ ................... 44 Double Disk, Allegro Hall Chip Measurements ................................ ................................ ..................... 46 Flexible Hall Probe ................................ ................................ ................................ ................................ . 47 Initial Design ................................ ................................ ................................ ................................ ....... 47 Robust Flexible Hall probe Design ................................ ................................ ................................ ..... 50 Double Disk Sample holder Results ................................ ................................ ................................ ... 51
vi CHAPTER VII: ANNEALED FERROMAGNETIC MATERIALS ................................ ....................... 54 Background ................................ ................................ ................................ ................................ ............. 54 Experiment Overview ................................ ................................ ................................ ............................. 55 Nickel Results ................................ ................................ ................................ ................................ ......... 59 Virgin Curve, Ni ................................ ................................ ................................ ................................ . 59 Hysteresis Curve, Ni ................................ ................................ ................................ ........................... 60 Differential Susceptibility, Ni ................................ ................................ ................................ ............. 61 CHAPTER VIII: TEMPERATURE DEPENDENCE ................................ ................................ ................ 62 Temperature Effects on the Coercive Field ................................ ................................ ............................ 62 Temperature Effect on Saturation Magnetization ................................ ................................ ................... 66 CHAPTER IX: SHAKE COIL ANYHYSTERETIC CURVE ................................ ................................ ... 71 Shake Coils Motivation ................................ ................................ ................................ ........................... 71 Description ................................ ................................ ................................ ................................ .............. 71 Results ................................ ................................ ................................ ................................ ..................... 73 CHAPTER X: CONCLUSION AND FUTURE WORK ................................ ................................ ........... 76 LIST OF REFERENCES ................................ ................................ ................................ ............................ 78
1 C HAPTER I. INTRODUCTION Overview A Vibrating Sample Magnetometer (VSM)  measures magnetic moment of a sample due to variations of an applied magnetic field. An advan tage of the VSM over a Faraday method is that the VSM directly measures the absolute magnetization of the sample under test, including permanent magnetization  . The objective of this effort is the measurement of the non line ar hysteretic susceptibility as a function of the applied field, including field rate dependence, temperature and geometry using the VSM. This thesis describes the use of and modifications of a standard VSM measurement facility that will enable such measu rements for thin disk ferromagnetic samples. This ultimate goal of this measurement laboratory is to facilitate the validation of an empirical Cooperative hysteresis model of magnetic susceptibility of ferromagnetic materials  . The University of Colorado Denver Magnetic Measurement Laboratory is equipped with a Microsense EZ7 VSM, and is funded by ONR Grant N00014 17 1 2328 awarded through the Office of Naval Research (ONR) DURIP program. Historical Work Quantum Theory A wel l known model of ferromagnetic materials is described by quantum theory . This model introduces the unit of the Bohr magneton which is the natural unit of the magnetic moment, and equivalent to the magnetic moment of single electr on spin. This Bohr magneton is calculated as: , ( 1 . 1 ) where is the charge of an electron, is the mass of an electron, is the speed of light. The magnetic moment of one gram atom of an element contain ing one Bohr magneton per atom is:
2 . ( 1 . 2 ) The number of Bohr magnetons per atom is found using the saturation of a single moment at , and the atomic weight as: . ( 1 . 3 ) The magnetic moment of an atom is found by the resultant of the electron spin and the orbital motion of its constituents as : , ( 1 . 4 ) where is the quantum number and ma y only be represented as having half integer values. The Land Ã© factor is calculated: . ( 1 . 5 ) If the moment of an atom is complet ely the result of electron spin then . Conversely, if the moment of the atom is due entirely by the orbital motion then , which happens when all of the spins are complimentary, that is, half up and half down spins. In ferromagnetic materials almost all of the moment is due to electron spins and it is a good approximation to use . Iron, for example h as a value of . The number of orientation s of the magnetic moment with respect to the magnetic field is and allows the description of the quantum mechanical analog of the Langevin function, known as the Brillouin function : , ( 1 . 6 ) where is the in tensity of magnetization is, is the intensity of saturation magnetization at absolute zero, and is : . ( 1 . 7 )
3 Typically, the contribution of the applied field is ignored compared to the molecular field term . Ferromagnetic mat erials such as Fe, Ni, Co, as well as various ferromagnetic alloys experience dramatic changes in magnetic properties when experiencing high temperatures at or above their respective Curie temperature. The Curie temperature is given as: , ( 1 . 8 ) and if the assumption is made that and , then: . ( 1 . 9 ) The quantum model has good agreement with empiri cal data below the Curie temperature. Above the Curie temperature, all ferromagnetic materials exhibit paramagnetic properties including a continuously decreasing susceptibility with increases in temperature. Fe, Ni, an d Co show unexpected variations in su sceptibility . Fe will experience changes in its crystal structure above the Curie temperature  . This thesis explores variations of saturation magnetization, which increases with decreasing temperatures, coercivity and suscep tibility at low and high temperatures, as well as permanent changes in releases of residual stresses through high temperature annealing processes above the Curie temperature. As described before, the main contribution of the moment in ferromagnetic mater ials is due to unpaired electron spins and very little moment is a consequence of the orbital motions. The electron configuration for Fe is shown in Figure 1 energy arra ngement of electrons in a subshell is obtained by putting electrons into separate orbitals of the subshell with the same spin before pairing electrons. The atoms with an excess of one type of spin can exhibit a net magnetism  . If an atom is free then orbital motion could contribute to the magnetic moment, but in solid state structures such as the disk samples presented in this thesis the atoms are fixed in place and do not have significant moment contributions due to orbital motion  . ( 1 . 10 )
4 Figure 1 Electron configuration for iron, a common ferromagnetic material 
5 C HA PTER II : COOPERATIVE HYSTERESIS MODEL The cooperative hysteresis model, first proposed by Dr. Carl Schneider, professor emeritus of the United States Naval Academy, presents an empirical model with parameters that represent physical properties of the material. The m agnetic susceptibility is typically expressed as : , ( 2 . 1 ) w her e is the induced magnetization ( ) and is the magnetic field inside the sample. is expressed as : . ( 2 . 2 ) Equation ( 2 . 2 ) has parameters that are each based on physical phenomena described briefly as: ( 2 . 3 ) The initial susceptibility is the change in magnetization with respect to a change in applied field coming from a demagnetized state in which each o f the magnetic domains are fully saturated in randomly distributed orientations giving a net zero aggregate magnetization as seen coming off from the origin on
6 the blue curve in Figure 2 . The saturation magnetization is defined as the strength of magnetization, a volume independent quantity, which can be obtained in a material if the applied field is increased indefinitely as shown in the top left subplot in Figure 2 . The coerciv e field is defined as a field that is applied which yields a net zero magnetization in a material on a major hysteresis loop. In other words , this is the point on hysteresis curve where the path crosses the zero magnetization boundary and the sign of magne tization in the sample switches. The coercive field points are shown in light blue in Figure 2 . Figure 2 Hysteresis plot, Magnetization (A/m) versus applied field H (A/m), displaying coercivity, reman ence, saturation magnetization, virgin curve and major loop The anhysteretic field is a quantity which is identified as the part of the magnetization process which occurs that is a completely reversible process. This quantity is easily observed by looking at the reversible susceptibility, shown in blue in Figure 3 . Nucleation magnetization is the region of magnetization where the domain walls have become completely saturated an d the domain walls have started to rotate. When the state of magnetization is less than nucleation magnetization, the magnetization is irreversible, such that if the intensity of the applied field is decreased the magnetization s tate would follow a new curve rather than the original curve traced during the increase in field. Beyond nucleation magnetization, the magnetization process is reversible meaning if the intensity of the applied field is
7 increased and decreased a magnetizat ion versus applied field plot would approximately continue to trace the same line. ,  refer to To better understand the meaning of the nucleation magnetization region, refer to Figure 3 , where the plot of susceptibility in red and the exponential model in green join the curve traced by the reversible susceptibility shown i n blue . The plot of a curve given by ( 2 . 2 ) usually will begin where the change of intensity of applied field switched from increasing to decreasing called a reversal. The magnetization state at that point is then referred to as , the magnetization at the last reversal . For a major loop this would correspond to the tips o f the curve where the maximum magnetic field is applied. Another important quantity to consider is the susceptibility as a function of the polycrystalline structure where: , ( 2 . 4 ) describes the properties of a material ffect on the magnetization process. ( 2 . 4 ) is a gene ral d efinition which extends to different crystal lattice structures. The product of the initial susceptibility and the polycrystalline texture term is then given as , the reversibl e susceptibility shown in blue in Figure 3 : . ( 2 . 5 )
8 Figure 3 Plot of the susceptibility versus magnetization coming off a negative field reversal for material A (red), the reversible susceptibility (blue), and the exponential model (green)
9 CHAPTER III : BASIC VSM PROCEDU RES Summary A standard vibrating sample magnetometer (VSM) setup suggested by the manufacturer for transverse measurements  utilizes an 8mm disk mounted to the end of a hollow quartz tube. The sample mounts between the poles of the VSM electromagnet using the quart z sample holder to suspend it is illustrated in Figure 4 . A un iform field is applied from the VSM electromagnet while simultaneously vibrating the sample vertically in the axial direction. This vibration causes a sinusoidal change in the magnetic flux, which passes between the poles. Faraday pickup coils located on the poles of the VSM translate this changing flux to an probe located at the poles is used to measure the appli ed field. The VSM can then record the magnetic moment versus the applied magnetic field. The magnetic moment is: , ( 3 . 1 ) where is the magnetization (permanent or induced) in the sample, and is the sample volume. Density Measurement To find the magnetization of the sample from the magnetic moment measurement, the volume of the sample is required. With appropriate calibration procedures, the magnetic moment can be measured to at least 4 significant figures. To maintain similar precision for estimating the magnetization, the volume of the sample should be known to at least as many signif icant figures. Due to defects in machining, it is diffic ult to measure the volume of a thin disk samples to this precision. To mitigate this, cylinder standards were accurately machined from which the density of each material is measured to better than 4 significant figures. Then, using a calibrated micro gram scale, a sample volume can be estimated to better than 4 significant figures via the mass and density. For each sample the mass is measured many
10 times and the average of the measurements is then ta ken. For iron, nickel, cobalt samples the density is a well known value which can be found through the National Institute of Standards and Technolog y website and shown in Table 1  . Material Density Iron (Fe) Nickel (Ni) Cobalt (Co) Table 1 National Institute of Standards and Technology (NIST) reported densities. The puri ty of these types of samples were 99.85 99.98% pure and thus good agreement was found when using these types of material for measurement. For other samples, including steel alloys, the density of the material is unknown and must therefore be foun d experimentally. Magnetic Moment Measurement Figure 4 A simplified diagram of the Vibrating Sample Magnetometer
11 As illustrated in Figure 4 the detection coils are located at the VSM p oles. The sample is typically centered between the poles. The detection coils measures the magnetic moment via a differential : , ( 3 . 2 ) where is the electromotive force ( ) about one turn of the coil shown in Figure 5 and is the net magnetic flux cutting through the turn. If the theory is then extended to multiple turns, then : . ( 3 . 3 ) The voltage induced across th e lead of the coil is equal to turns mult iplied by the time rate of decreasing flux pa ssing through each turn. By having two coils on each side of the electromagnet, this allows for a differential measurement of the using the process described in ( 3 . 3 ) and shown in Figure 6 . Figure 5 Differential measurement geometry for a vibrating sample magnetometer , curved lines represent the applied magnetic field Using the configuration shown in Figure 5 , and motion of the sample in the vertical direction, a flux change in all four coils shown in Figure 6 will occur. There are many configurations, which could be used for coil placement to obtain the di fferent components of the magnetization vector  . Most importantly,
12 groupings of two coils allows for switching which facilitates significant cancelation of noise due to field fluctuations, background fields and coil vibratio ns. Figure 6 Detection Faraday coils which are mounted on the poles of the VSM In order to properly measure the magnetic moment of a new sample, the calibration to a known nickel standard is first made. This calibrates the response to be recorded. The specific moment ( ) is a well known, documented quantity for nickel at various temperatures  . When calibrating the VSM, using the measured mass, th e moment is then calculated as : . ( 3 . 4 ) Using the calculated emu value, a field which will bring the nickel standard to a saturated magnetization state is then applied and a reading of the detection coils is obtained as a voltage. The calculated emu value from ( 3 . 4 ) is divided by this voltage and the proper is then obtained and is used to measure new materials of interest.
13 CHAPTER IV : DEMAGNETIZATION FIELD Figure 7 Schematic of magnetized sample as composite of microscopic dipoles  A very important part of this resea rch relates directly to understanding the non linear demagnetization effects which contribute to error in calculation of the correct susceptibility for a given material. To understand what demagnetization is, a simplified example geometry is shown in Figure 7 where a larger solid slab of magnetized ferromagnetic mater ial is modeled as a composite of microscopic dipoles  . The ends of the dipoles which meet their magnetic counterpart such that an equiv alent north pole touches ( 4 . 1 ) Which in short requires that the lines of magnetic field to close and thus not allow any real physical magnetic poles on the other side of the magnetized material. Depending on the geomet ry and the aspect ratio of a given magnetized sample, the easiest path for the field lines to follow is through the material  . These
14 magnetic field lines are nonlinear, but generally speaking, antiparallel to the direction o f the magnetization which created them. Realistically, the strength and direction of th e direction of the demagnetization field in non uniform for a given example, but often are approximated as constant, which is actually only true when considering an ell ipsoid geometry. In this research, thin disks used in the VSM are approximately ellipsoids, and although the demagnetizing factor is modeled as a function of susceptibility, the assumption that the demagnetizing field is constant throughout the material is assumed at a given instant in time. In contemporary VSM data analysis tools, such as the Microsense EasyVSM program, the option of a constant demagnetizing factor, unchanging with susceptibility, is given in the post processing options. This is really say i ng two things. The demagnetization effect is in fact large enough that it affects the error in the susceptibility, and that the use of a dem agnetization factor programmatically is still relatively inaccurate in colloquial systems. Non uniform Magnetizat ion in thin d isks The magnetization simulated by MagstrÃ¶m induced in a thin disk with a thickness to diameter ratio ( ) of for several susceptibilities is illustrated in Fi gure 8 . It is noted that if the susceptibility increases, the induced magnetization becomes non uniform., behaving as though North South magnetic poles are pushed more towards the center of the disk.
15 Mag s trÃ¶m Prediction of Demagnetizing Factor Suscepti bility Magnetization Fi gure 8 MagstrÃ¶m prediction of a thin disk sample, t/d ratio of 0.004, at different susceptibilities 
16 In Figure 9 , the demagnetizing factor as a function of the susceptibility as calculated by MagstrÃ¶m simulation is shown for a thin disk with a diameter of 8.01mm and thickness of 0.41mm or . Figure 9 Demag factor for thin disk sample, (8mm diameter, 0.41m m thickness), as a function of susceptibility Using instead a thin disk with diameter of 26mm and thickness of 0.13mm yields , the resulting demagnetizing factor is shown in Figure 10 . Figure 10 Demag factor for a thin disk sample, (26mm diameter, 0.13mm thickness), as a function of susceptibility 0.045 0.05 0.055 0.06 0.065 0.01 0.1 1 10 100 1000 10000 100000 Demag Factor Susceptibility Demag Factor VS Susceptibility 0 0.002 0.004 0.006 0.008 0.01 0.012 0.01 0.1 1 10 100 1000 Demag Factor Susceptibility Demag Factor VS Susceptibility Series1
17 Different geometries will have different aspect ratios, and thus different demagnetizing factors. This is one of the motivations whi ch fueled the desire to explore different geometries which were not standard practice for a VSM. The advantage of having a smaller demagnetization factor makes the total field to be closer to the expected applied field. With a smaller demagnetizing factor, there is a smaller magnetic field opposing the applied filed. If this is true, magnetization and applied field inside the material is approximated better allowing a better representation of the susceptibility. Having the demagnetization factor as a funct ion of the susceptibility gives closer to the accurate represen tation of the susceptibility. If a linear demag netization factor is used, then the demag netization factor is only valid for a small region of the magnetization process. If a variable demag netiz ation factor is calculated correctly , the entire susceptibility curve can be corrected to have the correct magnetic field internal to the sample at all times and allowing proper calculation of the material susceptibility. Correcting the Demagnetization Fa ctor from Xapp To correct the demagnetization factor from the apparent susceptibility , the axial magnetic field within a sample geometry due to the applied field can be expressed as: , ( 4 . 2 ) w here is the demagnetization factor, is the induced magnetization, and is the total field at the center of the sample (inside the disk). In general, is an operator. From, ( 4 . 2 ) once can solve for the magnetization as: . ( 4 . 3 ) The susceptibility is computed as: . ( 4 . 4 ) From ( 4 . 3 ) : , ( 4 . 5 )
18 where: ( 4 . 6 ) relative to the applied field measured by the VSM. One can solve from ( 4 . 5 ) : ( 4 . 7 ) Note that is expressed in this manner, since t he demagnetization factor is a function of the applied susceptibility, as will be detailed in a later section. Using MagstrÃ¶m to Compute the Demagnetization Factor MagstrÃ¶m can be used to compute the demagnetization factor with the field probe at the pole. To this end, the disk sample is exposed to a uniform magnetic field. Here, the field is assumed to be in the plane of the thin disk. MagstrÃ¶m can then compute the magnetization induced in the disk for a specific thickness to diameter ratio (t/d), and u niform susceptibility . Once the magnetization is computed, MagstrÃ¶m can compute the demagnetizing field inside or outside of the disk. MagstrÃ¶m computes the magnetization via a discretization of the Volume Integral Equation (V IE) de scribed as  : ( 4 . 8 ) where is the applied vector magnetic field, V is the volume of the magnetic material, is the magnetic susceptibility of the material, is the induced magnetization in V , and is the distance between the source point and the field point . The demagnetizing field at the field point is computed as  : ( 4 . 9 )
19 The magneto metric demagnetiz ing factor is computed as  : , ( 4 . 10 ) where is the magneti c moment of the disk, is the magnetic field moment of the disk, is the demagnetizing field in the disk, V is the volume of the disk, and the applied field is assumed to be applied along the x direction. For VSM measurements, it is most appropriate to work with a magnetometric demagnetizing factor, because the VSM actually measures the magnetic moment. This is computed from the MagstrÃ¶m simulation as a post processing step. For the linear problem, the s usceptibility is assumed to be constant, homogeneous, and isotropic. From ( 4 . 9 ) : . ( 4 . 11 ) Consequently, within the material volume, the demagnetizing field can conveniently be computed as: . ( 4 . 12 ) From ( 4 . 10 ) , the demagnetization factor is computed as: . ( 4 . 13 ) This can be i llustrated by the following example. Consider a thin disk that has a diameter d and thickness t , with a t/d ratio of 0.005. Figure 11 shows a top view example quadratic hexahedral mesh of a disk. For improved accuracy of the d emagnetization factor, the mesh has at least three layers of hexes through the thickness as shown in Figure 12 . As an example, consider a disk with d = 26mm, and thickness t = 0.13 mm. This disk has, a t/d ratio of 0.005. Table 2 illustrates the demagnetization factor of this disk as a function of the susceptibility. Figure 13 present s a plot of the data in Table 2 on a log log scale.
20 t/d 0.005 0.001 0.067754 0.0010 0.005 0.005 0.021248 0.0050 0.005 0.01 0.015431 0.0100 0.005 0.05 0.01075 0.0500 0.005 0.1 0.01013 0.0999 0.005 0.5 0.009397 0.4977 0.005 1 0.0090 74 0.9910 0.005 2 0.008659 1.9660 0.005 5 0.007985 4.8080 0.005 20 0.006935 17.5639 0.005 50 0.006382 37.9042 0.005 100 0.006088 62.1575 0.005 200 0.005898 91.7632 Table 2 Magnetometric Demagnetization Factor and apparent s usceptibility of a thin disk with t/d = 0.005 as a function of susceptibility X Figure 11 Top view of the quadratic hexahedral mesh of a thin disk with t/d=0.05 centered on the z=0 plane Figure 12 side view of the thin disk showing the three layers of hexahedron through the thickness of the disk
21 Figure 13 Magneto metric demagnetization factor ( ) as a function of the susceptibility of a thin dis k with t/d=0.005 From the MagstrÃ¶m data, the apparent susceptibility as a function of the material susceptibility X and the demagnetization factor can be computed from ( 4 . 7 ) : , ( 4 . 14 ) where is the appare nt susceptibility ( ) which is directed measured from VSM data when the probe is located at the pole. The values of for the thin disk with t/d are presented in the last column Table 2 . From this data, one can extract that can be used to correct the susceptibility. That is, if the susceptibility is computed directly from the VSM data, then one can use the MagstrÃ¶m predicted in order to estimate the true susceptibility using ( 4 . 7 ) . It is noted that by solving ( 4 . 14 ) for that the susceptibility is limited by the magnitude of the demagnetizing factor as:
22 ( 4 . 15 ) Probe under the disk It is observed from ( 4 . 7 ) that the maximum apparent susceptibility must be < , since is a positive real number. For a thin disk, the demagnetization factor can be on the order of . In fact, for the very thin disk used for Table 2 , is approximately . If is on the order of , and the true susceptib ility is on the order of , then 1 % error in will lead to 5 0 % errors in . If is on the order of , 1 % error in will leads to 10% percent error in . If is on the order of , then 1 % error in leads to 2% error in . Thi s motivates the need to reduce as much as possible. Unfortunately, there are physical limits on how small one can make t/d Whereas, the minimu m thickness of a disk sample is > 0.1 mm. Another alternative to reducing the effective demagnetization factor is to position the hall probe in the VSM system directly under the center of the disk. The motivation for doing so is that the Hall probe will be measuring a total field that is closer to the internal field than that at the pole. In this position, assume that the field measured by the probe is : . ( 4 . 16 ) where is the probe field, is the actual applied field, and is the effective demagne tization factor at the probe. In this scenario, the probe field, is the actual field driven by the VSM control system. From ( 4 . 16 ) one can solve for the app lied magnetization, which is the field at the VSM pole, from the probe field as : . ( 4 . 17 )
23 From ( 4 . 2 ) , the field inside the disk sample is then expressed as: . ( 4 . 18 ) Solving for the magnetization: ( 4 . 19 ) Then, computing the susceptibility from ( 4 . 19 ) , ( 4 . 20 ) where: ( 4 . 21 ) field measurement provided by the VSM. One can solve from ( 4 . 20 ) : ( 4 . 22 ) If the hall probe is close enough to the disk, then this leads to a direct reduction of the demagnetization factor of the sample. MagstrÃ¶m again is used to estimate the demagnetization factor at the probe, . Let be the c oordinate of the center of the H all probe, where the disk is assumed to be centered at (0,0,0). Next, let be the demagnetizing field at the probe coordinate, computed by MagstrÃ¶m. Note that is the field actually measured by the H all probe. The magneto metric demagnetization factor of the disk is computed using ( 4 . 10 ) . Si nce the problem is assumed to be linear, then the demagnetization factor at the probe can be assumed to be a liner scaling of To this end, it is assumed that :
24 ( 4 . 23 ) where is the demagnetizing field computed at the very center of the disk. One complication for the VSM, is that the sample is vibrating in the vertical direction. However, the probe is stationary. The sample has a sweep o f approximately +/ 3 mm. The H all probe is approximately 4 mm thick alon g the vertical direction. The H all probe must be placed benea th the sample approximately another 1 mm so that it does not touch the sample. As a result, the center of the hall probe is approximately 6 mm below the center of the sample. Table 3 Table 6 illustrate values of , , and as a function of for the thin disk as the distance of the hall probe varies from 5 mm to 6.5 mm, in 0.5 mm steps. For these calcula tions, was calculated from ( 4 . 22 ) as: ( 4 . 24 ) Graphs of the effective demagnetizing factors and are illustrated in Figure 14 Figure 17 . A fitted curve approximating the demagnetization factor as overlays the set of data points. It is found that this curve well approximates the demagnetization factor for susceptibilities > 0. 05. Below this, the curve deviates. Ho wever, as observed in Table 3 Table 6 , for susceptibilities this small, this has little impact on the estimated susceptibility from the apparent susceptibility.
25 0.01 0.009789 0.008258 0.001531 0.01 0.05 0.009753 0.008225 0.001528 0.049996 0.1 0.00971 0.008186 0.001524 0.099 985 0.5 0.009418 0.00792 0.001498 0.499626 1 0.009142 0.007665 0.001477 0.998525 2 0.008752 0.007295 0.001457 1.99419 5 0.008098 0.00664 0.001459 4.963799 20 0.007035 0.005396 0.00164 19.36495 50 0.006462 0.004581 0.001881 45.70165 100 0.006157 0.00 411 0.002047 83.0105 200 0.005959 0.0038 0.002158 139.6947 500 0.005818 0.003581 0.002236 236.0604 1000 0.005766 0.003502 0.002264 306.3909 Table 3 Magnetometric Demagnetization Factor, effective probe demagnetization factor a nd apparent susceptibility of a thin disk with t/d = 0.005 with probe 5 mm below the disk center 0.01 0.009789 0.007957 0.001832 0.01 0.05 0.009753 0.007925 0.001828 0.049995 0.1 0.00971 0.007887 0.001823 0.099982 0.5 0.009418 0.007628 0.001789 0.499553 1 0.009142 0.00738 0.001762 0.998241 2 0.008752 0.007019 0.001733 1.993094 5 0.008098 0.0 06377 0.001721 4.957347 20 0.007035 0.00515 0.001885 19.27349 50 0.006462 0.004345 0.002117 45.21352 100 0.006157 0.00388 0.002277 81.45446 200 0.005959 0.003575 0.002384 135.4323 500 0.005818 0.00336 0.002458 224.3371 1000 0.005766 0.003282 0.002484 287.0563 Table 4 Magnetometric Demagnetization Factor, effective probe demagnetization factor and apparent susceptibility of a thin disk with t/d = 0.005 with probe 5.5 mm below the disk center.
26 0.01 0.009789 0.007644 0.002145 0.01 0.05 0.009753 0.007613 0.00214 0.049995 0.1 0.00971 0.007577 0.002133 0.099979 0. 5 0.009418 0.007326 0.002092 0.499478 1 0.009142 0.007085 0.002057 0.997948 2 0.008752 0.006735 0.002017 1.991964 5 0.008098 0.006108 0.00199 4.950748 20 0.007035 0.004904 0.002131 19.18252 50 0.006462 0.004113 0.00235 44.74325 100 0.006157 0.003656 0.0025 79.99834 200 0.005959 0.003358 0.002601 131.57 500 0.005818 0.003148 0.002669 214.1723 1000 0.005766 0.003073 0.002693 270.7675 Table 5 Magnetometric Demagnetization Factor, effective probe demagnetization factor and ap parent susceptibility of a thin disk with t/d = 0.005 with probe 6.0 mm below the disk center. 0.01 0.009789 0.007325 0.002464 0.01 0.05 0.009753 0.007295 0.002458 0.049994 0.1 0.00971 0.00726 0.00245 0.099976 0.5 0.009418 0.007018 0.0024 0.499401 1 0.009142 0.006785 0.002357 0.997648 2 0.008752 0.006445 0.002307 1.990814 5 0.008098 0.005836 0.002262 4.944082 20 0.007035 0.00466 0.002375 19.09292 50 0.006462 0.003886 0.002577 44.29326 100 0.006157 0.003441 0.002716 78.64126 200 0.005959 0.003151 0.002808 128.073 500 0.005818 0.002947 0.00287 205.3187 1000 0.005766 0.002874 0.002892 256. 9237 Table 6 Magnetometric Demagnetization Factor, effective probe demagnetization factor and apparent susceptibility of a thin disk with t/d = 0.005 with probe 6.5 mm below the disk center.
27 Figure 14 Demagnetization factor as a function of apparent susceptibility computed using , with the probe located at the pole, and , with the probe located 5 mm beneath the center of th e disk. Fitting curves overlay each data set. Figure 15 Demagnetization factor as a function of apparent susceptibility computed using , with the probe located at the pole, and , with the probe located 5.5 mm beneath the center of the disk. Fitting curves overlay each data set.
28 Figure 16 Demagnetization factor as a function of apparent suscept ibility computed using , with the probe located at the pole, and , with the probe located 6 mm beneath the center of the disk. Fitting curves overlay each data set. Figure 17 Demagnetization factor as a function of apparent susceptibility computed using , with the probe located at the pole, and , with the probe located 6.5 mm beneath the center of t he disk. Fitting curves overlay each data set.
29 Figure 18 Virgin curve measured using the VSM for a thin disk sample with d = 26.08 mm, and t = 0.13 mm. Next, consider a thin disk sample of mild stee l with d = 26.08 mm, and t = 0.13 mm. The virgin M(H) curve was measured using the VSM with the probe placed directly under the sample. The magnetization is estimated by the VSM software from the measured magnetic moment. The magnetic field, H , is the pr obe field under the sample. This was measured using a sweep measurement with 0.2 Oe/s sweep rate. The curve is illustrated in Figure 18 . The apparent susceptibility, , can be computed using the slope o f the curve via a finite difference approximation from the data. The susceptibility can then be estimated using the demagnetization factor as a function of and the approximate formulation given in Figure 14 Figure 17 . As a first example, assuming the probe is placed 6 mm below the sample, the susceptibility is approximated using the formula in Figure 16 , . The susceptibility is then computed from and using ( 4 . 24 ) . The result of this is illu strated i n Figure 19 , which plots
30 X as a function of for small values of , where is the change in the magnetization from the demagnetized sate. Overlaying the graph is an exponential fit of the data. This in turn is fitted to the Cooperative exponential model, which near the virgin state is approximated as: ( 4 . 25 ) where is the initial susceptibility and is the cooperative field. For the data in Figure 19 , this leads to = 68.9, and = 663 A/m. This effort is reproduced in Figure 20 , assuming the probe is 6.5 mm below the sample. In this case, . A gain, fitting this to an exponential curve leads to = 69.6, and = 635 A/m. Figure 19 Susceptibility of the virgin curve computed using measured data of the VSM with the ha ll probe assumed to be 6 mm below the sample and using the demagnetization factor correction in ( 4 . 22 ) assuming for a thin disk sam ple with d = 26.08 mm, and t = 0.13 mm.
31 Figure 20 Susceptibility of the virgin curve computed using measured data of the VSM with the hall probe assumed to be 6.5 mm below the sample and using the demagnetization factor correct ion in ( 4 . 22 ) assuming for a thin disk sample with d = 26.08 mm, and t = 0.13 mm
32 . CHAPTER V : FIELD UNIFORMITY In order to obtain accurate results with larger disk sizes, the field uniformity needs to be consistent over a larger area of space. In the initia l factory installation, the pol es of the electromagnet were 2 inch pole faces, which were then replaced with 4 inch pole faces as shown in Figure 21 . Because this study mainly focuses on soft ferromagnetic materials, the high appli ed magnetic field achieved by concentrating the field s (up to 18,000 Oe) at the pole is unnecessarily large in order to saturate the samples. The prediction was made that by using a larger pole space there could be a better field uniformity in the area loc al to the sample by choosing to use flat faced 4 inch poles as shown on the right side of Figure 21 to replace the tapered 2 inch face poles. Figure 21 2 inch poles faces, tapered (LEFT) and 4 inch pole faces, flat (RIGHT) An experiment was performed along the horizontal or plane cross section of the applied field at various points between the electromagnet poles for both types of pole face configurations. Figure 22 shows a top down view of the magnetic field uniformity where the center is defined as the axial measurements. Additional measurements are made in front of and behind the axial measurements in the place at mm or slightly larger than the radius of the 25 mm diameter samples . In Figure 22 the
33 measurements around the center of the pole spacing are of the greatest significance. These measurements represent the loc al field which the sample will be exposed to during an experiment. The experiment setup for one of the these measurements consisted of one hall probe placed into a small hole at the face of the VSM pole, while a second probe is at the given locations and c onnected to a secondary Gauss meter , which is not controlled by the VSM software and is only partially calibrated to the probe. This means that the applied magnetic field is a well calibrated field controlled by the VSM software, but that the magnetic fiel d measurements between the pole spacing by the secondary probe should be assumed as fair approximations as relative measurements but not the exact field measurement at a given location. Figure 22 Field uniformity measurement f or 2 inch beveled pole (TOP), and 4 inch pole (BOTTOM) Several observations are made by comparing the measurements shown in Figure 22 . One common effect can be seen in both configurations where the magnetic field which is measure d at the pole deceases in amplitude the further from the poles where the applied field is measured and controlled by the VSM .
34 The furthest point from both poles is directly centered between the two poles. Looking at Figure 22 , thi s is where the lowest measurements were taken for both configurations. In the inch beveled pole, comparing the average of center measurements average of the pole measurements there is a decrease at the center of the magnet gap to a pproximately of the applied field at the poles. In the inch flat faced pole configuration the center magnetic field decreas es to 92% of the applied field. For the purposes of measuring the susceptibility, having this much of a discrepancy between the measured applied magnetic field and the actual magnitude of the incident field at the center will certainly cause large contributions to error. Having field uniformity is more important however when looking th e uniformity local to the sample. Assuming the proper magnitude can be controlled at the very center of the magnet gap, there needs to be close to the same applied field in a region on the order of the geometry size under test, otherwise nonlinear applied fields will contribute to error in the magnetization of the material. Comparing the average measured field near the center , the inch pole configurations see a variation from the center measurement at only mm from the center while the inch see only max error compared to the center measurement at the same distance .
35 CHAPTER VI : DOUBLE DISK EXPERIMENT Motivation The hall probe in the VSM electromagnet measures and controls the applied field. The system employs a feedback loop to minimize the error in th e applied field value using the Hall probe output. The difficulty with this setup is that the Hall probe only provides the applied field value. The field internal to the sample must be known if the magnetic properties of the sample material is to be full y understood. The demagnetizing effect of the sample prevents the full magnitude of the applied field from penetrating the disk. Using a linear approximation, the total internal field can be expressed as: , ( 6 . 1 ) such that: ( 6 . 2 ) where is the demagnetization factor and is the magnetization of the sample. The demagnetization factor, D , is a function of both the geometry of the sample, and the non linear susceptibility , and is hence a function of M and H . Consequently, extracting is non trivial. One of the objectives of this research is t o devise methods for measuring directly. In doing so, the ability to compute the material susceptibility as a function of the internal field where and: ( 6 . 3 ) On e approach to attempt to estimate the measurement of for the sin gle disk setup is to place the H all probe directly under the disk. A test fixture was designed to hold the Hall probe in position, as illustrated in Figure 23 . The Hall probe, placed orthogonally to the axial field of the VSM, measures the total field, as defined by ( 6 . 2 ) , approximately 4 mm, on average, below the test sample. Since the field is tangential
36 to the sample surface, the field will be continuous across the surface. Therefore, it provides an approximation to the internal field. Figure 23 Measuring the total field from below the single disk s ample using Gauss probe holder. In Figure 24 , the measured magnetization between three different test cases show the key differences between the incorporation of the demagnetizing field. In the blue graph, the h all probe placement in the pole without the use of a demagnetizing factor at all provides the least accurate des cription of the process. The green graph uses the same measurements from the hall probe in the pole, but a constant demagnetizing factor is applied and subtracted from the hall probe measurement to obtain an approximat ion of the field which the material A 25mm sample experiences. A constant demagnetizing fa ctor in this case chosen as 0.003 yields the new effective applied field where: , ( 6 . 4 ) and is the approximate linear demagnetizing factor, and is the m agnetiz ation. The third case being compared uses the theory that by measuring the effective applied field by centering the Hall probe
37 directly underneath the sample inherently measures the effect of demagnetizing field incorporated into the measurement of the app lied field. Comparing these results shows great promise in this technique. Figure 24 Magnetization (A/m) versus the Applied Field (A/m) The magnetization is approximated by assuming a uniform distribution of magnetization throu ghout the sample. That is, , where is the measured magnetic moment. By using m agnetization as a metric and converting the applied field to facilitates the calculation of the s usceptibility, allowing data to be compared to the contemporary susceptibility model, as well as data from other geometries and methodologies , i.e. flux metric pipes and toroids, being explored in parallel with the VSM work.
38 Figure 25 Magnetization (A/m) versus the Applied Field (A/m) (Zoomed) The magnetic susceptibility versus induced magnetization is computed using the three measurements in Figure 25 and illustrated by the plot in Figure 26 ole . The peak susceptibility is plateaued at approximately 1/ D . The manipulated data and the centered probe measurements provide a closer approximat ion to the susceptibility. However, both susceptibilities are too small to accurately represent the physical susceptibility of the material. There are several reasons for this. The first is an inadequate approximation of the total field. The centered t echnique does provide an improved approximation of the internal field by measuring underneath the sample. However, the proximity of the hall probe to the sample limits the accuracy of this technique. The sample is vibrating vertically approximately 3 mm with a static probe located just below the minimum height position of the vibrating sample. However the demagnetizing field will decrease in magnitude approximately by , with being the vertical distance between the disk and the Hall probe. Secondly, the magnetization in the disk is not uniform. This has be en clearly observed with MagstrÃ¶ m simulations. Thus, an effective volume fraction must be used to compensate when computi ng the magnetization from the mome nt. MagstrÃ¶ m has been used to compute this volume fraction which is a non linear function of the susceptibility. Figure 26 displays the susceptib ility plot with variable H all probe locations in a nd out of the pole. The probe in pole measurement is corrected using the original
39 linear demagnetizing factor while the probe under sample data has been corrected via a power law fit to the MagstrÃ¶m prediction of the magnetometri c demagnetizing factor as a function of the apparent susceptibility as: . ( 6 . 5 ) Figure 26 Differential Susceptibility of different pole placements, compared with the reversible susceptibility and the exponential susceptibility model T o fit the exponential model from ( 2 . 2 ) to the highest susceptibility curve shown in Figure 26 , the initial susceptibility , the saturation magnetization , coercive field , and anhysteretic field is used . Internal Field, Demagnetizing Factor A limitation of the single disk sample is that it becomes very difficult to measure the interior field at In this setup, the two disks are placed in parallel and again placed transverse to the field. A transverse Hall probe is then placed at the center of the disk pair to measure the axial fi eld of the VSM. Through MagstrÃ¶ m simulations, it can be shown that the field parallel to the disks is uniform to within a 1 %
40 variation between the disks. In this research, two different double disk sample holder a pparatus and probes are explored. The first double disk apparatus designed utilized an Allegro Hall chip sensor which mounted between the two thin disks. The ratio o f the disk separation to the disk diameters is small (< 0.2). Therefore, the probe measur ement provides a much better approximation to the internal field than the centered probe with the single disk measurement. The cost is that the demagnetizing factor of the double disk sample is approximately twice that of a single disk. A separate Hall p robe mounted between the two samples allows a measurement of the total field between the disks independent to the primary Hall probe mounted on the pole measuring the applied field. Thus, from ( 6 . 4 ) , the difference of the total field and the applied field measured by the Hall probe can also provide a measurement of the demagnetization factor . A second apparatus is tested in a later section with implements a FW Bell Hall flexible hall probe which communicates directly with the EasyVSM software allowing direct control of the VSM electromagnet via a control feedback of the total field measured between the disks. The advantages of this setup are the increased max imum field control and measurement, and inherent synchronization with the measured magnetic moment.
41 Double Disk Sample Holder Prototype Figure 27 Disassembled double disk sample holder The double disk sample holder is comprise d of five 3 D printed parts as shown in Figure 27 . The sample holder illustrated is designed for 25 mm diameter disks. Parts 1 and 2 are identical lids serving as a surface to mount the samples. A small applicat ion of vacuum grease to the bottom side of the lid allows for a temporary fixation of the sample to the lid. The vacuum grease also proves a dampening effect in the overall vibrating structure. Placing a Hall chip in the middle of part 3 provides the abili ty to measure a static or changing field in between the disks. Parts 4 and 5 are the top and bottom of the sample holder respectively.
42 Figure 28 Assembled double disk sample holder with Hall chip, samples mounted In Figure 28 , a fully glued and assembled double disk sample holder shows a Hall probe mounted with an insulating strip of electrical tape along with two samples mounted to the bottom of parts 1 and 2. The separation of the di sk pair is 4 mm, leading to a separation to diameter ratio of 0.16. The final step before a measurement involves placing the lids onto the assembly of parts 3, 4, 5 and wrapping the entire apparatus with PTFE Thread tape. The fully assembled double disk s ample holder screws into a custom rod, which suspends the sample between the poles of the electromagnet (shown in Figure 64 ). The same apparatus is constructed using a pair of Nickel standards of the same diameter and approximately the same thickness. This is used to calibrate the double disk measurement in a similar fashion to that described earlier for the single disk measurement . Using 2 25 mm Nickel standards, a moment measurement is taken and compared to the calculated value to obtain an calibration number for the double disk geometry.
43 Amplifier Circuit for Hall Chip Excitation The EasyVSM software relies on the use of a data acquisition device that hosts t wo separate Lock In Amplifiers (LIA). LIA1 reads the signal from the Faraday Coils to measure the magnetic moment. LIA2 degree axis from the axial coil. In this setup , the vector coil is removed and LIA2 is rather used for acquiring Hall chip signal. This inherently synchronizes the timing of the VSM sequence, the applied field, the Hall chip probe and LIA1 magnetic moment readings. In order to use LIA2 to measure t he hall chip signal, the Hall chip is biased with an amplitude modulated signal with the same frequency and phase as the VSM vibrator. This is done using the summing circuit shown in Figur e 29 . The 70 Hz AC sign al input to the circuit is the synchronous output of LIA1. A 5 V DC offset is supplied by an external power supply. The first stage is a summing circuit, feeding into the left side of .The second stage is a simple unity gain inve rting amplifier circuit returning the excitation signal: ( 6 . 6 ) The resi stor values from Table 7 are correctly chosen for an Allegro A1324 Hall effect sensor to operate in a linear region with a modulated 70 Hz bias preventing the Hall chip from becoming saturated over the range of +/ 600 Oe. Figur e 29 Summing circuit for AC modulated Hall chip excitation
44 Table 7 Resistor values for AC modulating circuit Filtering The Easy VSM provides the ability to apply filtering to the data. Depending on the type of sequence executed, different settings are used. The two main types of sequences when measuring the magnetization versus applied field are the step method and the sweep method. The step method changes the applied field in uniform increments and holds while a user defined number of measurements of the magnet ic m oment are measured. Next , an average is taken over all the all samples at that applied magnetic field and saved, before stepping the intensity to the next applied magnetic field value. Alternatively, using a sweep method, a user can choose a field rate, wh ich decides how fast the applied magnetic field sweeps from a start value to and end value, while continuous samples of the magnetic moment are being taken. Averages are also available in this setting, but do not affect the overall time of the process. Usi ng the sweep method there should be a smaller time constant of 10ms, filter order of 6db/Octave, as well as the optional use of a synchronous filter. When moving to the step method, a larger time step of 0.1 s and a filter order of 24dB/Octave are being use d. In Figure 31 , in the setting for a step method, the original Easy VSM software only provides the option to allow a single value for the EMU RANGE option. This became a problem when starting to use the second LIA to record the Hall chip readings. To mitigate this , a custom version of the EasyVSM was provid ed by Microsense, which provides the ability to choose the SIGNAL RANGE and SIGNAL RANGE2 options independently. Resistor R1 R2 R3 R4 R5 Value 100 K 4.7 K 4.7 K 8.2 K 8.2 K
45 Figure 30 Setting s for the Sweep Method Figure 31 Original option settings for Step Method
46 Figure 32 Updated with new independent SIGNAL RANGE, SIGNAL RANGE2 option for Step Method Double Disk , Allegro Hall Chip M easurements One issue with the Allegro Hall chip is that the sensor became completely saturated at Oe which is not sufficient to obtain saturation magnetization for ferromagnetic materials accurately. Additionally, although this concept would allow internal field measurement it did not allow field rate changes to be controlled directly. Using the double disk sample holder design combined with the filtering techniques for the obtained signal from the Allegro Hall c hip mounted betw een the samples , the remaining nois e was an issue when measuring small fields. For the exponential susceptibility model to be accurate and for veri fication and validation of the H all chip solution, minor loop data and virgin curve processes must be realiza ble for the instrument. Because the relative noise was so high in this cheaper hall chip sensor, combined with the inability for direct applied field control and the chips small operating range, the choice was made to investigate different hall probe solut ions, which could give better empirical data, allowing for more accurate conclusions to de drawn.
47 Flexible Hall Probe Initial Design The requirement of a quality applied field measurement was still necessary in order to deliver measurement techniques that were not only state of the art but also yield results with low noise and high accuracy. By using a FW Be ll flexible hall probe which coul d work with the Gauss meter and the EasyVSM software, it was possible to drive the applied field as a function of the t otal field measured as a superposition of the applied filed and the demagnetizing field from between two disks. The FW Bell 6000 series hall probe allowed measurable up to kG, which is beyond the requirements of the ferromagnetic measurement studied in this thesis and higher than the maximum field obtainable by the VSM electromagnet at the inch spacing using inch poles of approximately kG. Figure 33 FW Bell Ha ll probe size and specifications summary
48 Figure 34 Flexible Hall probe double disk sample holder pieces (left), assembled shown with probe inside (right) One of the first problems to overc ome for the flexible hall probe was creating an accurate way to calibrate the probe. For a standard VSM hall probe, a National Institute of Standards and Technology (NIST) standard reference magnet of 1005+/ 0.5% Oe is used in conjunction with a zero Gauss chamber are used to set the Hall P robe gain and offset respectively. These apparatus are shown in Figure 35 .
49 Figure 35 Zero Gauss chamber, Probe Reference Magnet 1005+/ 0.5% Gauss, Microsen s e Model F CM 10 field control module with standard reading from the probe reference magnet shown in display. The magnet is designed typically for the standard probes. Initial measurements using the reference magnet yielded unreliable and non repeatable results . Bec ause of gaps between the probe and the walls of the reference magnet, it was difficult to ensure that the probe was orthogonal to the reference field. Additionally centering the probe was an issue. Using a Solidworks2017, Prusa IK3 MK2 3 D printer and PLA plastic a new reference magnet form was produced to hold and position the flexible hall probe directly in the center of the reference magnet with acceptable repeatability as shown in Figure 36 . Figure 36 Initial design for reference magnet insert which allows repeatable positioning for the flexible hall probe calibration The reference holder design worked quite well and the next stage was to test the overall design. A calibration was performed and verif ied by re measuring the calibration standard. Next, material A was
50 going to be tested. During the first experiment, a large amount of vibration in the support holding the flexible probe was present. When switching to the different material, a decision was made to trying a softer packing foam as a support to hold the flexible hall probe. While inserting the probe through the foam the tip of the probe was caught and bent and broke the tip of the sensor off. This brought attention to two issue s . The first issu e was that there was a need to dampen the vibration rece ived at the flexible H all probe support apparatus. The second issue, which needed to be solved, was how to make the end of the probe more robust by creating a fixture to protect it at all times. Robu st Flexible Hall probe Design Figure 37 Protective cover for the flexible Hall prob e tip, allowing rigid co vering in the reference magnet, zero Gauss chamber and sample holder Learning from the previous mistake, a protect ive cover for the Hall probe tip was introduced as shown in Figure 37 to the double disk sample holder design to allow continuous protection in both the calibration process involving the reference magnet and zero Gauss chamber as shown in Figure 38 and the double disk sample holder itself as shown in Figure 39 . This design is also a 3 D printed PLA plastic using Solidworks2017 as a CAD tool. One of the issues raised here is that w hen utilizing 3 D printed designs, calibration of the printer, extruder temperature, and infill can all a ffect the outcome of the design and its accuracy to the CAD drawing itself. In this case, there are competing geometries with the orientation of the pr otective cover requiring the need for exactness is both extruder height as well as placement. Because the level of accuracy desired is smaller than the tolerances of the 3 D printer, the design for the cover was produced as close as possible and then sande d to an exact fit for the reference magnet and the double disk sample holder. It is also worth noting that the CAD design of the protective cover and the sample holder are produced in and parts and supe r glued together respectively. The reasoning
51 behind this is the part of the geometries which are not supported by the base layer would otherwise require a support material print to hold up the design. This causes additional roughness and inconsistency in t he print than is easily avoided using the proper dissection of the geometry and reassembling it. Figure 38 Calibration geometries allowing continuous use of the protective flexible Hall probe tip cover Figure 39 Double disk sample holder, adjusted to accommodate the protective cover for the flexible Hall probe tip Double Disk Sample holder Results With the latest iteration of the double disk sample holder initial results show the ability to m easure high susceptibility has increased dramatically. Using the saturate curve shown in Figure 40 and using the
52 approximation that the demag factor is close to on the major saturate loop the linear dema g factor approximation for material A is close to as shown and used in Figure 42 to create a relevant comparison to the exponential model from ( 2 . 2 ) shown in green with the initial susceptibility , the saturation magnetization , coercive field , and anhysteretic field . The results have given even better results than originally anticipated. Initial findings show a n order of magnitude higher , smooth measurements of apparent susceptibilities (before a demagnetizing factor correction) above compared to with the VSM standard 8mm disk with 1 inch pole spacing with the field driven from the pole of the electromagnet. Because the center of the disk separation is driving the applied field, the attenuation of t he max field strength at the pole over the distance to the sample is inherently incorporated i nto the measurement s control feedback loop . Figure 40 Virgin and Saturate Loop (Major Loop), using the double disk sample holde r wi th the applied field driven from between the Material A disks
53 Figure 41 Virgin and Saturate Loop (Major Loop) , zoomed using the double disk sample holder with the applied field driven from between the material A disks Figure 42 measured and linear demagnetizing factor corrected differential susceptibility overlaid with model fit, measurements using the double disk sample holder with the applied field driven from between the material A disk
54 CHAPTER VII : ANNEALED FERROMAGNETIC MATERIALS Background Ferromagnetic material such as Ni, Fe, and Co have magnetic properties which ar e heavily influenced by applied and residual stresses to the material and can change them from soft magnetic materials to semi hard when plastic deformation occu rs. Ni, for example, with high values of magne to strictive consta nts, coupled with a moderate ma gnetocrystalline anisotropy, puts effective constraints on the domain topology through long range stresses. Additionally, short range stress fields generated by dislocations can strongly hind er domain wal l displacements. Single crystal and polycrystalline samples that initially have low coercivities in an annealed state see orders of magnitude increases in coercivity after applications of tensile straining or heavy cold rolling. Because of this these effects , residually stressed materials demonstrate a lar ge decrease in the initial susceptibility, whose analysis can aid in the effort to explain contributions from domain wall displa cements and coherent rotations in polycrystalline Ni samples  . To generate samples which are u sed in a vibrating sample magnetometer, residual stresses can easily One option which is studied here utilized cold rolling the metallic samples into a sheet to obtain a uniform thickness. The sheet can then have a circular cut removed via a punch or stamp removal process. This is the type of material preparation which is described in this study. In this lab, alternative methods include cutting a wafer off a l athed cylinder using a band saw or using electric discharge machining (EDM) . The wafer can then be placed in a magnetically held position and slowly polished until the correct thickness and uniformity is obtained. In this process, the residual stress over the largest part of the material surface area which is introduced by the polishing is only a ffecting the sample at a depth on the order of the grit size of polishing product used.
55 Experiment Overview Beginning this experiment, a naÃ¯ve choice was made to try a PID controlled ceramic oven to see the effect of oxidation which would occur if a slab of metallic material was annealed at . The result of this is shown at full temp and after cooling is shown in Figure 43 . With oxidation seemingly destroying the exterior layer a depth which is larger than the thickness of the samples to be tested, this oven was clearly not going to suffice. Figure 43 Steel plate at 1000Â°C (Le ft), steel plate after cooling, heavily oxidized, brittle (Right) An alternative solution was found via the University of Colorado shared equipment network at the Colorado Nanofabrication Lab at the University of Colorado Boulder . This oven allows an inert gas such as argon to be pumped into a cylindrical ceramic oven to displace the oxygen and other air constituents interior to the oven. The samples themselves were placed on a quartz sample holder boat, which was flat on top and semicircular on the bottom. The boat was inserted into the argon bath inside the ceramic heater
56 and argon flow was continued for over 30 minutes to achieve a high percentage of argon interior to the cylinder as shown in Figure 44 . Figure 44 Geometry of annealing experiment, inert argon flooded ceramic cylinder with thermocouple reading A thermocouple was inserted into the center next to the sample holder boat through the hole where t he A rgon flowed out of the cylinder. This allo wed the observation of the approximate temperature that the samples were being exposed to. Control of the temperature was manually controlled via a variable resistor knob located on the front of the apparatus. Careful adjustments were made watching the the rmocouple readings and increasing the temperature over approximately 45 minutes to 1000Â°C. The samples then remained in the oven at a constant for minutes. The oven heat was then backed down from and turned off. The oven cooled back to room temperature overnight with the samples inside the oven which argon continuing to flow. A graph of the annealing temperature process in shown in Figure 45 .
57 Figure 45 Thermocouple reading tracking the temperature internal to the cylindrical ceramic insulated oven during the annealing experiment Figure 45 , Figure 46 , and Figure 47 show the samples before and after the annealing process respectively. In Figure 47 , you can see that the surface of the material has turned a slate gray color as a result of oxidation. With the ef fort that was made to use the Argon bath annealing ove n it was frustrating to see oxidation , but it is safe to assume that this is a less dramatic effect than w ould have been obtained than if n o inert gas was used to decrease oxygen present surrounding the material. Figure 46 Before annealing, top, left to right: Fe, Ni, Co 8mm diameter, 1mm thick samples; bottom: 2 25 mm diameter 1mm thick Ni samples, placed on a quartz sample holder plate. 0 200 400 600 800 1000 1200 0 50 100 150 200 250 Temperature ( C) Time (min) Thermocouple Reading
58 Figure 47 After annealing, top, left to right: Fe, Ni, Co 8mm diameter, 1mm thick samples; bottom: 2 25 mm diameter 1mm thick Ni samples placed on a quartz sample holder plate Because using the mass to determine the volume of the material and subsequently the oxi dation it was important to remove the oxidized material and re weigh the samples allowing only the mass contributing to the magnetization to be considered. First , a fine grit sandpaper was used to polish the samples. This worked to remove some to of the ma terial but exposed another issue. As the annealing process occurred, residual stresses which were in the material were released and cause small deformations in the samples themselves. This caused an inconsistent removal of the oxidized material when using sandpaper as shown in Figure 48 . To remove the rest of the oxidation, the samples were placed in muriatic acid to dissolve the ou ter layer of oxidized material. Figure 48 After annealing and polishing with fine grit sandpaper, the uneven nature of the surface is revealed, a result of release of residual stress , 25 mm diameter, 1mm thick Ni sample example .
59 Nickel Results By using the maximum susceptibility of the saturate loops where , a linear demagnetizing factor of and is applied to the virgin curves and major loop for the Ni sample before and after annealing respectively. Notice that after annealing the susceptibility ha s increased dramatically for the Ni sample as seen in the virgin, major loop and susceptibility plots Figure 49 , Figure 51 , and Figure 53 respectively. It is also observed that t he coercive field has been dramatically reduced by an order of magnitude from t o . Virgin Curve, Ni Comparison of the results for Ni in Figure 49 is made to a similarly conducte shown in Figure 50  , using the coversion from Tesla to is : . ( 7 . 1 ) Figure 49 Virgin curves, before and after annealing cold r olled and stamped 8 mm diameter , 1 mm thick Ni disk
60 Figure 50 "Magnetization process in thin Ni Sheets: Effect of cold rolling and recrystallization annealing", FIG. 2. (a) Initial magnetization curves in cold rolled and recrystallized polycrystalline Ni she ets , Full symbols correspond to a flux metric measurement technique, open symbols correspond to an alternating gradient force magnetometer measurement.  . Hysteresis Curve , Ni Figure 51 Hyster esis Plot, before and after annealing cold rolled and stamped 8 mm diameter, 1 mm thick Ni disk
61 Figure 52 Coercive field (Zoomed), before and after annealing cold rolled and stamped 8 mm diameter, 1 mm thick Ni disk Differentia l Susceptibility, Ni Figure 53 Differential susceptibility, before and after annealing cold rolled and stamped 8 mm diameter, 1 mm thick Ni disk
62 CHAPTER VIII : TEMPERATURE DEPENDENCE Temperature Effects on the Coercive Field One of the features of the vibrating sample magnetometer is the ability to control the ambient temperature conditions samples whic h are less than 8mm in diameter. A small quartz tube with a titanium coating provides an insulating tube which covers the sample. Then the air inside is replaced by an inert gas which flows from below the sample and out the top of the cryostat insulated tu be. Nitrogen gas is chosen for temperatures cooler than room temperature to near liquid nitrogen temperature. The coolest steady controllable experimental temperature produced in this lab was at . For high temperature experiments ranging from room temperature to Argon gas is chosen to fill the cryostat insulated tube. It is very important to purge the tube completely by allowing the gas to flow for approximately minutes at the be ginning of the experiment at a flow rate of standard cubic feet per hour (SCFH) ensuring all of the air has been purged. Any remaining air could contain humidity of small amounts and cause the tubes to form ice and constrict or c log airflow of the inert gas. Reduced or clogged airflow can corrupt the ability to control temperature , and potentially allow back f low of air into the cryostat cau sing m oisture inside which adversely a ffects the thermocouple control feedback loop which m onitors and c ontrols the temperature. In this experiment , a test of hysteresis loops is done for material A at three different temperatures. The first experiment done is at , using the argon gas and heating oven. The high temperat ure experiment is chosen as the first experiment since it will not cause any ambient moisture to condense and affect the thermocouple feedback as discussed before and shown on the outside of the cryostat tube in Figure 54 for a co ld experiment at . Subsequent experiments are done at room temperature and then at using the cryostat, nitrogen gas, and liquid nitrogen. In this process, gaseous nitrogen is released from a container a nd then flows through a heat exchange coil submerged in a Dewar of liquid
63 nitrogen. Higher flow rates up to 20 SCFH and above allow the coolest temperatures close to liquid nitrogen to be attainable. Figure 54 Temperature effe ct on hysteresis curve, coercive field and saturation magnetization, cryostat concealing 8mm material A under test , 150Â°C . Figure 55 shows the results for the three temperatures , , and their corresponding hysteresis curves. One noticeable difference is that increases in temperature are associated with a higher max susceptibility as shown in the steepest slopes of the hysteresis curve in Figure 55 and also plotted as the susceptibi lity in Figure 57 . Another observation which is expected is that the coercive field decreases with increasing temperature  . This is best illustrated using a zoomed window of the hysteresis plot in Figure 56 .
64 Figure 55 Temperature effect on hyster esis curve and the coercive fiel d Figure 56 Zoomed in look at the temperature effects on the co ercive field
65 Figure 57 Temperatu re effects on the differential susceptibility for three temperatures Figure 58 illustrates a comparison of the proposed model in ( 2 . 2 ) and the extracted susceptibility from the experiment. The susceptibility for the is calculated using a constant demagnetizing factor of to crudely correct the data. This data set comes from earlier stages of the ex perimental apparatus using the inch poles, single 8 mm diameter disk sample, and the applied field measured at the face of the p oles . The fit is better through the reversible region where the susceptibility is the lowest and the sample is near saturated past nucleation. The fi t near and at the coercive field is not as accurate, where small changes in the applied field lead to large changes in magnetization, and consequently noise in the applied field can lead to large errors in this region. Another observation is that the overall shape of the curve has been shifted especially though the cooperative region of the susceptibility plot shown in Figure 58 . To obtain a fit of this data which is heavily influence by temperature, the initial susceptibility , the saturation magnetization , coercive field , and anhysteretic field is used.
66 Figure 58 Material in 150Â°C, corrected measured data in dark blue, exponential susceptibility model shown in cyan Temperature Effect on Saturati on Magnetization Figure 59 Temperature effect on saturation magnetization, cryostat concealing 8mm nickel sample on quartz sample holder An o ther temperature study was conducted to measure the relationship between the saturatio n magnetization at low temperature s . I n this experiment, 4 inch poles and a single 8mm nickel sample is
67 used to find the saturation magnetization as tested from the same applied field of Oe and shown in Figure 59 . In theory , the only way to find a true saturation magnetization of a sample would be to increase the applied field indefinitely. At Oe in applied field, changes in the noise of the measurement of the magnetization are hi gher than the change in magnetization with respect to increasing applied field. Thus, it is a good approximation to say that the sample is near its saturate value and this applied field is held as a constant for comparison between temperatures. This is wel l known phenomena and it will eventually need a place in the final version of the susceptibility model once predictions at a single temperature are consistent, repeatable and predictable. Fe lix Bloch studied ferromagnetic materials made calculations of the variations of saturation magnetization at low temperatures using the Heisenberg model while considering exchange forces between atoms and making the assumption that all the atomic spins inside a given material are parallel. The an tiparallel spins are then assumed to be sparse and not interact with the other atoms. The problem can then be posed as a material which has one active electron spin per atom with each atoms taking positons at points in a cubic lattice yielding: , ( 8 . 1 ) and t he constant is specific to type of cubic la ttice which is being studied such as: . ( 8 . 2 ) Then , by sub stituting the relationship: , ( 8 . 3 )
68 which is based on the atomic model and only consid ering nearest neighbor interactions for a body centered lattice such as iron equation ( 8 . 1 ) becomes: . ( 8 . 4 ) Historically, this equation has given good agreement between theoretical predictions and empirical data for the power, but less consistent correlation with the constant  . A plot of the spontaneous magnetization normalized by the spontaneous magnetization at absolute zero versus the sample temperature normalized by t he Curie temperature is shown in Figure 60 . Figure 60 Variation of spontaneous magnetization with temperature for nickel and iron, and comparison with theory  . Notice the Weiss theory and Modified Weiss theory  have poor agreement with the empirical results for nickel and iron in Figure 60 . The nickel and iron both however have good agre ement with the law. Experimental data measured for saturation magnetization as a function of temperature is shown in Figure 61 . Using a polynomial fit to extrapolate the data, the intercept is used to approximate the saturation m agnetization at absolute zero for the nickel sample .
69 Figure 61 Polynomial fit to get the saturation magnetization at T = 0K The saturation magnet ization at absolute zero approximation is used to normalize the saturation magnetization at different temperatures normalized by the C urie temperature and plotted in Figure 62 . The measured data is fit using the law and varying the constant until the form: ( 8 . 5 ) is finally found as a reasonable fit shown as the orange plot in Figure 62 . This confirms that the relationship found for low temperature saturation magnetization has a reasonable correlation to the law and could serve as a starting point at how to incorporate a low temperature dependence for saturation magnetization into the predictive susceptibility model this research aims to ascertain. y = 0.3638x 2 + 18.8x + 516598 4.900E+05 4.950E+05 5.000E+05 5.050E+05 5.100E+05 5.150E+05 5.200E+05 0 50 100 150 200 250 300 Saturation Magnetization (A/m) Temperature (K) Fit to Find Msat at 0 K Measured Poly. (Measured)
70 Figure 62 Variation of the sat uration magnetization as a function of temperature, Texp(3/2) fit 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 0 0.1 0.2 0.3 0.4 0.5 MSat (T)/ Msat(0K) T/ Variation of Msat, Ni Measured Data T^(3/2) Law
71 CHAPTER IX : SHAKE COIL ANYHYSTERETIC CURVE Shake Coils Motiva tion For demagnetization of the samples, the EasyVSM has a built in function that uses the existing electromagnet to alternate the applied field from positive to negative. After each cycle from minimum to maximum, an execution of the loop repeats with a de creased amplitude. This process creates smaller and smaller minor loops while slowing approaching a demagnetized state. Using the VSM software and the electromagnet to complete this process is time consuming and impedes access to quick results and the prod uctivity of trial and error type processes that require demagnetization. To mitigate this, the CUD lab added a pair of secondary coils mounted in the VSM geometry allowing absolute control through an outside signal generator and amplifier. These secondary coils act as an alternative demagnetization apparatus. A second motivation for producing a secondary coil was creating the ability to apply a secondary field with absolute control during an otherwise standard VSM test procedure. For example, a saturate hy steresis loop executed by the VSM electromagnet superimposed by an alternating magnetic field shows a reduction in the coercive field. Data analysis concludes that the alternating field is thus reducing the magnetic hysteresis in a given sample, which aids in the goal of measuring the anhysteretic curve of the material. Description The secondary coil design closely follows that of a Helmholtz coil configuration. The key difference being that for a true Helmholtz configuration the radius of each coil is equa l to the separation of both coils. Ideally, this would provide a nearly uniform field in the gap between the coils. Our geometry does not allow for this exact ratio.
72 Each coil has an effective length of 0.01m, inner diameter of 0.098m, and outer diameter of 0.108m. Each coils itself has 11 layers of wire with a polyurethane coat between each layer yielding an aggregate turn density of 105turns/0.01m. Figure 63 SOLIDWORKS model of one secondary (shake) coil, front (left), ba ck (right) The holder itself has a few other design properties. The backside of the sample holder shown in Figure 63 (right) shows the need for a recessed face for a proper fit onto the VSM. This guarantees proper radial alignment. Another important function was preventing the coils from pulling toward each other. The Figure 64 shows the fully mounted shake coil system. Each coil is producing a field in the same direction at any given time. The spacers shown in Figure 64 near the top and bottom of the apparatus prevent the natural attrac tion of the coils from pulling them off their mount on the VSM.
73 Figure 64 Mounted Shake Coil Results In Figure 65 , a magnetization versus applied field plot completes two loops to near saturate values of magnetic field to the sample while the VSM electromagnet continues its same process. Note that in the loop shown in black there is an o bvious decrease in the coercive field. This tells us that the shake coils is helping to eliminate magnetic hysteresis. One hope is that with the proper calibration, there might be a
74 total elimination of hysteresis, which would make our black curve represen tative of the anhysteretic curve. This would allow the direct analysis of reversible susceptibility. Figure 65 Shake Coil usi ng 1 Amp, 50 Hz secondary field The hysteresis curve shown in Figure 66 shows a second example using the shake coil to apply a secondary alternating field. In this case, increasing the current to 2 Amps through the secondary coils produces a larger decrease in the coercive field.
75 Figure 66 Shake coil usi ng 2 Amp, 50 Hz secondary field
76 CHAPTER X : CONCLUSION AND FUTURE WORK The goal set out in this work was to develop state of the art techniques to measure the non linear hysteretic properties of ferromagnetic materials. Several dif ferent apparatus were built and tested as supplements to a vibrating sample magnetometer. The secondary shake coils show promise in ascertaining the measurement of the reversible susceptibility which is a curve which scales the proposed exponential suscept ibility model. Further work needs to be done to solve how to properly drive the secondary coils allowing perfect reversibility to be measured. It was shown by theory and measurement that the demagnetizing factor of the sample under test can greatly impact internal to the disk can be estimated by a parallel plate structure with the disk sam ples and placing the Hall probe between the samples. A proof of concept using a n Allegro hall chip driven by an AC modulated signal is presented . Although the signal integrity of the Allegro Hall chip did not meet the standards of precision desired, the experimental 25 mm double disk 3 D printed geometry suspended from a custom sampl e holder between a non standard 3 inch pole spacing did return motivating results. After a study of field uniformity and considering the desired tests and materia ls, a decision to switch from inch to inch pole faces for the electromagnet dramatically improved the measurements and uniformity . A new study was conducted to drive the electromagnet using a precision Hall probe between the mm samples that measured the superimposed total field. After a small setback with a less robust design failure a protected Hall probe design allowed better than satisfactory measurement results. Additional analysis and validation of this new apparatus is now an exciting new frontier for vibrating sampl e magnetometers. Studies were conducted as well to understand the effects of temperature on ferromagnetic materials showing changes in the saturation magnetization, coercive field and susceptibility. Once static temperature models are verified for differen t apparatus such as pipe and toroid experiments, the VSM could prove useful in studying these temperature effects and their applications to a more dynamic model. Production of the samples themselves also raises the issue of the need to minimize
77 residual st resses to the thin disk samples. These stresses can potentially increase coercivity and lower the susceptibility of a given sample. Although pure materials can be annealed as shown in this study to see how dramatic these stresses could be, annealing change s the overall chemistry of a given alloy sample and then subsequent measurements would not be a good representation of the original material and its designed preparation techniques.
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