Geomagnetism Part 1: Basic Principles and Material Properties

1. GeomagnetismPart 1: Basic Principles and Material Properties

2. Magnetism Like a lot of phenomena in Physics, understanding magnetism requires an understanding of quantum theory, but perhaps more than most. We’ll need to get into this a bit, but there are some useful ideas we can discuss without going too deeply. Scientists first investigating magnetism noticed a lot of similarities between magnetic fields and electrical fields, and so presumed they were due to the same physical mechanism. In fact, Gauss proposed that Coulomb’s law for the forces between electrical charges could be modified for magnetic force, except that the property eo, the electrical permittivity, is replaced by something called the magnetic permeability - mo. Carl Friedrich Gauss

3. Thus, the electric force: becomes the magnetic force Carl Marks? Where P1 and P2 are called magnetic poles. The main difference is that the magnetic field always looks like there are two poles of opposite sign in some proximity to each other, but as far as we know the concept of a magnetic pole is a pure fiction.

4. Nevertheless, it turns out to be a useful one. Like the E field, we can define the B field as the field produced by a single (fictional) pole: and define a magnetic potential W as the work done to bring an second pole in from infinity: Since poles always occur in pairs (+ and -) it is useful to compute the potential field of a dipole; it is simply the sum of the potentials of individual poles. In the following we will consider an analogy to Electrical Dipoles, where we use charges q of opposite sign.

5. Dipole Moment It will be very useful to define a quantity called the Dipole Moment, defined as the magnitude of the poles times the distance between them and the defined direction is toward the positive pole. For magnetic dipoles the dipole moment m = pd, where p is the pole strength. At points far from the dipole (r >> d), we have where q is the angle between d and r Electric Dipole. For Magnetic, substitute p for q and m for P

6. Potential V for an Electric Dipole. For the Magnetic equivalent, substitute W for V, p for q, m for k, and m for p

7. Torque on a Magnetic Dipole Suppose a magnetic field B is oriented at an angle q with respect to the dipole. The Magnetic force is Bp on the positive pole and –Bp on the negative pole. Thus there are equal and opposite forces a distance dsinq/2 from the center. The total torque is then t = 2Bpdsinq/2 = Bpdsinq = m x B We will come back to the magnetic torque concept in a different context in a bit.

8. If Poles are Fictitious, What’s Really Going On? Well, a very realistic way to think about magnetism is that it is always produced by moving charges (currents). These can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits, or even the individual spinning of electrons or an atomic nucleus.

9. Then, the magnetic field B is defined in terms of force on moving charge in the Lorentz force law. Both the electric field and magnetic field can be defined from this law: The electric force is straightforward, being in the direction of the electric field if the charge q is positive, but the direction of the magnetic part of the force is given by the right hand rule. Hendrick Antoon Lorentz

10. The SI unit for magnetic field is the Tesla, which can be seen from the magnetic part of the Lorentz force law Fmagnetic = qvB to be composed of (Newton x second)/(Coulomb x meter). A smaller magnetic field unit is the Gauss (1 Tesla = 10,000 Gauss), and an even smaller one often used in Geophysics is the gamma (g), which is 10-5 Gauss or 1 nanoTesla. Nikola Tesla

11. Magnetic Dipole Moment Revisited Let’s use this way of thinking to revisit the idea of a magnetic dipole moment. Consider a charge q moving through a magnetic field B with velocity v. Lorentz says: If we consider the current as being the total charge of N charges that passes through a volume of area A and length dl in a unit time, then the total force dF on this element is In the case of a wire of length L perpendicular to the magnetic field:

12. The torque exerted by the magnetic force on a wire (see figure below) is given by • = BILWsinq • The magnetic moment of a loop is m = IA where A is the area of the loop, or m = NIA for N loops. • Since A = LW, the torque is then written as • = mBsinq

13. The direction of the magnetic moment is perpendicular to the current loop in the right-hand-rule direction. • Considering torque as a vector quantity, this can be written as the vector product • = m X B • Since this torque acts perpendicular to the magnetic moment, then it can cause the magnetic moment to precess around the magnetic field at a characteristic frequency called the Larmor frequency. • If you exerted the necessary torque to overcome the magnetic torque and rotate the loop from angle zero to 180 degrees, you would do an amount of rotational work given by the integral

14. As seen in the geometry of a current loop, this torque tends to line up the magnetic moment with the magnetic field B, so this represents its lowest energy configuration. The potential energy associated with the magnetic moment is so that the difference in energy between aligned and anti-aligned is DU = 2mB These relationships for a finite current loop extend to the magnetic dipoles of electron orbits and to the intrinsic magnetic moment associated with electron spin. Also important are nuclear magnetic moments.

15. Magnetic Domains It will be essential for Geomagnetism to understand the magnetic properties of solids, and in particular how secondary fields are induced and how permanent magnets are made. We begin with the idea of Magnetic Domains. The “long range order” which creates magnetic domains in ferromagnetic materials arises from a quantum mechanical interaction at the atomic level. This interaction is remarkable in that it locks the magnetic moments of neighboring atoms into a rigid parallel order over a large number of atoms in spite of the thermal agitation which tends to randomize any atomic-level order. Sizes of domains range from a 0.1 mm to a few mm. When an external magnetic field is applied, the domains already aligned in the direction of this field grow at the expense of their neighbors.

16. If all the spins were aligned in a piece of iron, the field would be about 2.1 Tesla. A magnetic field of about 1 T can be produced in annealed iron with an external field of about 0.0002 T, a multiplication of the external field by a factor of 5000! For a given ferromagnetic material the long range order abruptly disappears at a certain temperature which is called the Curie temperature for the material. The Curie temperature of iron is about 1043 K. The microscopic ordering of electron spins characteristic of ferromagnetic materials leads to the formation of regions of magnetic alignment called domains.

17. The main implication of the domains is that there is already a high degree of magnetization in ferromagnetic materials within individual domains, but that in the absence of external magnetic fields those domains are randomly oriented. A modest applied magnetic field can cause a larger degree of alignment of the magnetic moments with the external field, giving a large multiplication of the applied field.

18. These illustrations of domains are conceptual only and not meant to give an accurate scale of the size or shape of domains. The microscopic evidence about magnetization indicates that the net magnetization of ferromagnetic materials in response to an external magnetic field may actually occur more by the growth of the domains parallel to the applied field at the expense of other domains rather than the reorientation of the domains themselves as implied in the sketch below.

19. Some of the more direct evidence we have about domains comes from imaging of domains in single crystals of ferromagnetic materials. They suggest that the effect of external magnetic fields is to cause the domain boundaries to shift in favor of those domains which are parallel to the applied field. It is not clear how this applies to bulk magnetic materials which are polycrystalline. Keep in mind the fact that the internal magnetic fields which come from the long range ordering of the electron spins are much stronger, sometimes hundreds of times stronger, than the external magnetic fields required to produce these changes in domain alignment. The effective multiplication of the external field which can be achieved by the alignment of the domains is often expressed in terms of the relative permeability.

20. Domains may be made visible with the use of magnetic colloidal suspensions which concentrate along the domain boundaries. The domain boundaries can be imaged by polarized light, and also with the use of electron diffraction. Observation of domain boundary movement under the influence of applied magnetic fields has aided in the development of theoretical treatments. It has been demonstrated that the formation of domains minimizes the magnetic contribution to the free energy.

21. Magnetic Field Strength H The magnetic fields generated by currents and calculated from Ampere's Law or the Biot-Savart Law are characterized by the magnetic field B measured in Tesla. Illustration of the Biot-Savart Law But when the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise about what part of the field comes from the external currents and what comes from the material itself.

22. It has been common practice to define another magnetic field quantity, usually called the "magnetic field strength" designated by H. It can be defined by the relationship H = B0/m0 = B/m0 - M and has the value of unambiguously designating the driving magnetic influence from external currents in a material, independent of the material's magnetic response. The quantity M is called the magnetization of the material.

23. The relationship for B can be written in the equivalent form B = m0(H + M) H and M will have the same units, amperes/meter. To further distinguish B from H, B is sometimes called the magnetic flux density or the magnetic induction. You can think of H as the ambient field, M as the field induced in a material by the presence of H (for example by realigning the magnetic domains) and B as the sum of the two.

24. Another commonly used form for the relationship between B and H is B = mmH Where mm = Kmm0 m0 being the magnetic permeability of space and Km the relative permeability of the material. If the material does not respond to the external magnetic field by producing any magnetization, then Km = 1.

25. Another commonly used magnetic quantity is the magnetic susceptibility which specifies how much the relative permeability differs from one. Magnetic susceptibility km = Km – 1 For paramagnetic and diamagnetic materials the relative permeability is very close to 1 and the magnetic susceptibility very close to zero. For ferromagnetic materials, these quantities may be very large. Note that B = m0(H + M) = mmH = Kmm0H So M = (Km – 1)H = kmH

26. Relative Permeability The magnetic constant m0 = 4p x 10-7 T m/A is called the permeability of space. The permeabilities of most materials are very close to m0 since most materials will be classified as either paramagnetic or diamagnetic. But in ferromagnetic materials the permeability may be very large and it is convenient to characterize the materials by a relative permeability.

27. When ferromagnetic materials are used in applications like an iron-core solenoid, the relative permeability gives you an idea of the kind of multiplication of the applied magnetic field that can be achieved by having the ferromagnetic core present. So for an ordinary iron core you might expect a magnification of about 200 compared to the magnetic field produced by the solenoid current with just an air core. This statement has exceptions and limits, since you do reach a saturation magnetization of the iron core quickly, as illustrated in the discussion of hysteresis.

28. Hysteresis When a ferromagnetic material is magnetized in one direction, it will not relax back to zero magnetization when the imposed magnetizing field is removed. It must be driven back to zero by a field in the opposite direction. If an alternating magnetic field is applied to the material, its magnetization will trace out a loop called a hysteresis loop. The lack of retraceability of the magnetization curve is the property called hysteresis and it is related to the existence of magnetic domains in the material. Once the magnetic domains are reoriented, it takes some energy to turn them back again.

29. This property of ferrromagnetic materials is useful as a magnetic "memory". Some compositions of ferromagnetic materials will retain an imposed magnetization indefinitely and are useful as "permanent magnets". The magnetic memory aspects of iron and chromium oxides make them useful in audio tape recording and for the magnetic storage of data on computer disks.

30. It is customary to plot the magnetization M of the sample as a function of the magnetic field strength H, since H is a measure of the externally applied field which drives the magnetization . Variations in Hysteresis Curves There is considerable variation in the hysteresis of different magnetic materials.

31. Hysteresis in Magnetic Recording Because of hysteresis, an input signal at the level indicated by the dashed line could give a magnetization anywhere between C and D, depending upon the immediate previous history of the tape (i.e., the signal which preceded it). This clearly unacceptable situation is remedied by the bias signal which cycles the oxide grains around their hysteresis loops so quickly that the magnetization averages to zero when no signal is applied. The result of the bias signal is like a magnetic eddy which settles down to zero if there is no signal superimposed upon it. If there is a signal, it offsets the bias signal so that it leaves a remnant magnetization proportional to the signal offset.

32. Coercivity and Remanence in Permanent Magnets A good permanent magnet should produce a high magnetic field with a low mass, and should be stable against the influences which would demagnetize it. The desirable properties of such magnets are typically stated in terms of the remanence and coercivity of the magnet materials.

33. To Review: When a ferromagnetic material is magnetized in one direction, it will not relax back to zero magnetization when the imposed magnetizing field is removed. The amount of magnetization it retains at zero driving field is called its remanence. It must be driven back to zero by a field in the opposite direction; the amount of reverse driving field required to demagnetize it is called its coercivity. If an alternating magnetic field is applied to the material, its magnetization will trace out a loop called a hysteresis loop. The lack of retraceability of the magnetization curve is the property called hysteresis and it is related to the existence of magnetic domains in the material. Once the magnetic domains are reoriented, it takes some energy to turn them back again. This property of ferrromagnetic materials is useful as a magnetic "memory". Some compositions of ferromagnetic materials will retain an imposed magnetization indefinitely and are useful as "permanent magnets".

34. The table below contains some data about materials used as permanent magnets. Both the coercivity and remanence are quoted in Tesla, the basic unit for magnetic field B. Besides coercivity and remanence, a quality factor for permanent magnets is the quantity (BB0/m0)max. A high value for this quantity implies that the required magnetic flux can be obtained with a smaller volume of the material, making the device lighter and more compact.

35. The alloys from which permanent magnets are made are often very difficult to handle metallurgically. They are mechanically hard and brittle. They may be cast and then ground into shape, or even ground to a powder and formed. From powders, they may be mixed with resin binders and then compressed and heat treated. Maximum anisotropy of the material is desirable, so to that end the materials are often heat treated in the presence of a strong magnetic field. The materials with high remanence and high coercivity from which permanent magnets are made are sometimes said to be "magnetically hard" to contrast them with the "magnetically soft" materials from which transformer cores and coils for electronics are made.

36. Rare Earth Magnets The permanent magnets which have produced the largest magnetic flux with the smallest mass are the rare earth magnets based on samarium and neodynium. Their high magnetic fields and light weight make them useful for demonstrating magnetic levitation over superconducting materials.

37. The samarium-cobalt combinations have been around longer, and the SmCo5 magnets are produced for applications where their strength and small size offset the disadvantage of their high cost. The more recent neodynium materials like Nd2Fe14B produce comparable performance, and the raw alloy materials cost about 1/10 as much. They have begun to find application in microphones and other applications which exploit the high field and light weight. The production is still quite costly since the raw allow must be ground to powder, pressed into the desired shape and then sintered to make a durable solid.

38. Ferromagnetism Iron, nickel, cobalt and some of the rare earths (gadolinium, dysprosium) exhibit a unique magnetic behavior which is called ferromagnetism because iron (ferric) is the most common and most dramatic example. Samarium and neodynium in alloys with cobalt have been used to fabricate very strong rare-earth magnets. Ferromagnetic materials exhibit a long-range ordering phenomenon at the atomic level which causes the unpaired electron spins to line up parallel with each other in a region called a domain. Within the domain, the magnetic field is intense, but in a bulk sample the material will usually be unmagnetized because the many domains will themselves be randomly oriented with respect to one another.

39. Ferromagnetism manifests itself in the fact that a small externally imposed magnetic field, say from a solenoid, can cause the magnetic domains to line up with each other and the material is said to be magnetized. The driving magnetic field will then be increased by a large factor which is usually expressed as a relative permeability for the material. There are many practical applications of ferromagnetic materials, such as the electromagnet.

40. Ferromagnets will tend to stay magnetized to some extent after being subjected to an external magnetic field. This tendency to "remember their magnetic history" is called hysteresis. The fraction of the saturation magnetization which is retained when the driving field is removed is called the remanence of the material, and is an important factor in permanent magnets. All ferromagnets have a maximum temperature where the ferromagnetic property disappears as a result of thermal agitation. This temperature is called the Curie temperature. Ferromagntic materials will respond mechanically to an impressed magnetic field, changing length slightly in the direction of the applied field. This property, called magnetostriction, leads to the familiar hum of transformers as they respond mechanically to 60 Hz AC voltages.

41. Magnetic Properties of Ferromagnetic Materials

42. Diamagnetism The orbital motion of electrons creates tiny atomic current loops, which produce magnetic fields. When an external field is applied, it produces a torque which induces a precision of the magnetic moment at the Larmor frequency. The direction of the precession is in the opposite direction of the electron orbit, and produces a field that opposes the applied field (Lenz’s Law). The result is a material with a negative susceptibility.

43. All materials are inherently diamagnetic, but if the atoms have some net magnetic moment as in paramagnetic materials, or if there is long-range ordering of atomic magnetic moments as in ferromagnetic materials, these stronger effects are always dominant. Diamagnetism is the residual magnetic behavior when materials are neither paramagnetic nor ferromagnetic. Any conductor will show a strong diamagnetic effect in the presence of changing magnetic fields because circulating currents will be generated in the conductor to oppose the magnetic field changes. A superconductor will be a perfect diamagnet since there is no resistance to the forming of the current loops.

44. Paramagnetism Some materials exhibit a magnetization which is proportional to the applied magnetic field in which the material is placed. These materials are said to be paramagnetic and follow Curie's law: All atoms have inherent sources of magnetism because electron spin contributes a magnetic moment and electron orbits act as current loops which produce a magnetic field. In most materials the magnetic moments of the electrons cancel, but in materials which are classified as paramagnetic, the cancelation is incomplete.

45. Magnetic Susceptibilities of Paramagnetic and Diamagnetic Materials at 20°C

46. For ordinary solids and liquids at room temperature, the relative permeability Km is typically in the range 1.00001 to 1.003. We recognize this weak magnetic character of common materials by the saying "they are not magnetic", which recognizes their great contrast to the magnetic response of ferromagnetic materials. More precisely, they are either paramagnetic or diamagnetic, but that represents a very small magnetic response compared to ferromagnets.

47. Magnetostriction Why does a transformer hum? You may have noticed the humming sound associated with a transformer or a fluorescent light ballast. For U.S. circuits, that hum will be at 120 Hz since the iron material associated with the transformer core responds mechanically to the magnetic field which is impressed upon it. The effect is called magnetostriction, and it is one of the magnetic properties which accompanies ferromagnetism. For 60 Hz applied magnetic fields in AC electrical devices such as transformers, the maximum length change happens twice per cycle, producing the familiar and sometimes annoying 120 Hz hum.

48. In formal treatments, a magnetostrictive coefficient L is defined as the fractional change in length as the magnetization increases from zero to its saturation value. The coefficient L may be positive or negative, and is usually on the order of 10-5. There is an elastic strain energy associated with the deformation, leading to some dissipation of energy in transformer cores. If the magnetostriction acts to contract a specimen, then this will act against any tensile stress on the material and leads to a larger value for the Young's modulus for the material.

49. Two examples of measurements of this phenomenon are included in the table below.

50. It is also observed that applied mechanical strain produces some magnetic anisotropy. If an iron crystal is placed under tensile stress, then the direction of the stress becomes the preferred magnetic direction and the domains will tend to line up in that direction. Ordinarily the direction of magnetization in iron is easily changed by rotating the applied magnetic field, but if there is tensile stress in the iron sample, there is some resistance to that rotation of direction. Bulk solid samples may have internal strains which influence the domain boundary movement.