Physics
Magnetic Moment
Magnetic moment refers to the property of a magnet or current-carrying loop that gives rise to a magnetic field. It is a measure of the strength and orientation of the magnetism in a material. The magnetic moment is a vector quantity, with both magnitude and direction, and is fundamental to understanding the behavior of magnetic materials.
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11 Key excerpts on "Magnetic Moment"
eBook - PDF- Duncan W. Bruce, Dermot O'Hare, Richard I. Walton(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The energy is minimised External magnet's North pole External magnet's South pole Magnetic field (H ) Magnetic dipole with moment μ F F Figure 1.2 A Magnetic Moment, , in an externally applied magnetic field, H. The energy of the Magnetic Moment is determined by its angle, , to the magnetic field. The orientation force, F, acts on the Magnetic Moment to minimise its energy. 6 MEASUREMENT OF BULK MAGNETIC PROPERTIES in Equation 1.3 if the dipole moves towards areas that contain larger magnitudes of H, where the magnetic flux lines are most dense. When confined to a particular direction, this force is proportional to the mag- netic field gradient, as described by Equation 1.4. In this equation, z is the component of the magnetic dipole in the z direction and dH∕dz is the field gradient. This effect is an important one, since the measurement of this translational force is the basis of some methods of determining the Magnetic Moment, as described in Section 1.2. F z = z dH dz (1.4) 1.1.2.2 The Quantum View On a sub-atomic scale, the origin of Magnetic Moments can be visu- alised as arising from the movement of charge, although in this case it is associated with the motion of electrons in orbitals (which is linked to the spin and orbital momentum of the electrons). However, this way of thinking should only be used to help conceptualise the effect. Electron spin is commonly imagined as the electron particle spinning about an axis, rather like the Earth's rotation, but in reality it is actually a more nebulous quantity, with the momentum being generated internally. Bulk materials evidently contain many atoms that have associated electrons, each having spin and orbital momentum. In cases where all electrons are paired, the sum of the momenta will cancel and no permanent Magnetic Moment is generated. However, external magnetic fields can still affect these materials (see the discussion on diamagnetism in Section 1.1.4.1).
eBook - PDFElectrons in Solids
An Introductory Survey
- Richard Bube(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
11 Magnetic Properties Literary references to the ancient lodestone, often invested with magical properties, testify to the length of time that magnetic materials have been known. Yet it is only in relatively recent years that magnetic materials have been applied to technological developments. A world of technical sophisti-cation separates the simple bar magnet made of steel from the 1-micrometer diameter magnetic domains in modern magnetic bubble memories made of garnet. In previous chapters we have treated the effects of a magnetic field on free electrons in a solid (Hall effect, magnetoresistance, cyclotron reso-nance), but except in the case of free-electron paramagnetism in Chapter 6, we have not treated the interaction between a magnetic field and the Magnetic Moments associated with electrons. It is these kinds of interaction that form the basis for our discussion of magnetic properties. There are two sources of an electronic Magnetic Moment: (1) the orbital electronic motion of electrons in atoms (crudely like the revolution of the earth about the sun), and (2) the intrinsic electron angular momentum (crudely like the rotation of the earth about its axis). ELECTRIC AND Magnetic MomentS The close parallel between electric and magnetic quantities and properties is demonstrated in the discussion of Maxwell's equations in Appendix C. 181 182 11. Magnetic Properties An understanding of both types of quantity is aided by a comparison of electric and magnetic dipoles and moments. If an electric field is applied to a polarizable material, a polarization P, the electric dipole moment per unit volume, is set up that opposes the applied electric field S. Relevant equations are given in Eqs. (C.2)-(C4). Most com-monly the polarization results from displacement of ionic or electronic charges to set up charge dipoles where the positive and negative charges are displaced by the distance d, and the electric dipole moment p = qa.
- A. O. Caldeira(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
2.1 Macroscopic Maxwell equations: The Magnetic Moment 7 D = E + 4π P, H = B − 4π M, (2.3) where P ≡ P(r, t ) and M ≡ M(r, t ) are, respectively, the polarization and mag- netization fields of the material being considered. As we are interested in magnetic phenomena we shall mostly be discussing the role played by H and, particularly, M. From the above equations (White, 2007) we conclude that the magnetization is actually due to the existence of the microscopic current density J mol (r, t ), which ultimately results from the stationary atomic motion of the electrons. Attributing a local current density J (i ) mol (r, t ) to the electronic motion about a given molecular or ionic position r i , we can associate a Magnetic Moment μ i (t ) ≡ μ(r i , t ) = 1 2c d r (r − r i ) × J (i ) mol (r , t ) (2.4) with that position. From this expression and a general representation of J (i ) mol (r , t ) in terms of μ i (t ) itself it can be shown (White, 2007) that the magnetization is written M(r, t ) = i (r − r i )μ i (t ) r , (2.5) where (r − r i ) is a function normalized to unity and strongly peaked about r i . Integrating the latter expression over the whole volume of the sample, we easily see that M(r, t ) is the total Magnetic Moment per unit volume. If we consider the presence of N e point electrons per ion (or molecule) at positions r (i ) k relative to r i with velocities v (i ) k , the local current density reads J (i ) mol (r) = N e k =1 ev (i ) k δ(r − r i − r (i ) k ), (2.6) which we can use in (2.4) to show that μ i = e 2mc N e k =1 r (i ) k × p (i ) k = e 2mc L i , (2.7) where p (i ) k = mv (i ) k is, in the absence of an external field, the canonical momentum of the k th electron at r i and L i ≡ N e ∑ k =1 r (i ) k × p (i ) k is the total electronic angular momentum at the same site.
- Christof M. Aegerter(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
As we have seen, this current leads to a Magnetic Moment of p m = Iπ r 2 = eπ ν r 2 = evr/2, where we have used the expression for the frequency in the final step. The product of the rotational speed v and the distance of rotation r is closely connected to the angular momentum of this rotation, L = m e v × r, 405 Properties of Magnetic Fields where m e is the mass of the electron; see Section 10.4.1. Therefore the magnetic dipole moment of a spinning charge is given by the following: p m = e 2m e L. Angular momenta are very closely related to quantum mechanics and hence magnetism is also rooted in there. Reconsidering the quantization rule of the Bohr model, we can see that angular momenta can only appear in chunks corresponding to a magnitude of | L| = ¯ h, where ¯ h is Planck’s constant. So the smallest chunk of magnetic dipole moment the electron can have is given by the following: p m = e ¯ h 2m e . An exact derivation of this is more complicated and also shows that the angular momentum of the electron (its spin) actually has the value of ¯ h/2. Therefore, then, every single electron has a Magnetic Moment, but in normal materials their orientations are arbitrarily arranged so that the average magnetization is zero. If one places such a material inside a magnetic field, the dipoles are arranged along the field, as we have seen for the electric dipoles in E fields. The corresponding material property is the magnetic permeability. Just as with the electric dipoles, the material property of the permeability can determine the size of the field in a material. The individual dipoles can also interact with one another, if a part of the aligned moments creates a field, which then aligns further moments. The interaction between the dipoles, which leads to the original alignment, is again given by quantum mechanics.
eBook - PDF- Edward Purcell(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
:j:Many people use the term gyromagnetic ratio for this quantity. Some call it the magnetogyric ratio. Whatever the name, it is understood that the Magnetic Moment is the numerator. MAGNETIC FIELDS IN MATTER Why don't we notice the magnetic fields of all the electrons orbiting in all the atoms of every substance? The answer must be that there is a mutual cancellation. In an ordinary lump of matter there must be as many electrons going one way as the other. This is to be expected, for there is nothing to make one sense of rotation intrinsi- cally easier than another, or otherwise to distinguish any unique axial direction. There would have to be something in the structure of the material to single out not merely an axis, but a sense of rotation around that axis! We may picture a piece of matter, in the absence of any external magnetic field, as containing revolving electrons with their various orbital angular momentum vectors and associated orbital Magnetic Moments distributed evenly over all directions in space. Consider those orbits which happen to have their planes approximately parallel to the xy plane, of which there will be about equal numbers with m up and m down. Let's find out what happens to one of these orbits when we switch on an external magnetic field in the z direction. We'll analyze first an electromechanical system that doesn't look much like an atom. In Fig. 11.12 there is an object of mass M and electric charge q, tethered to a fixed point by a cord of fixed length r. This cord provides the centripetal force that holds the object in its circular orbit. The magnitude of that force Fo is given, as we know, by Fo =-- r (24) In the initial state, Fig. 11.12a, there is no external magnetic field. Now, by means of some suitable large solenoid, we begin creating a field B in the negative z direction, uniform over the whole region at any given time.
No longer available |Learn moreAtomic Clusters
Introduction to the Nano World
- Michel Broyer, Patrice Mélinon(Authors)
- 2024(Publication Date)
- EDP Sciences(Publisher)
On the right, the resultant moment under the field is non-zero in the case of ferromagnetism (spins are oriented in the same direction as the field), ferrimagnetism, intermediate case between para- and ferromagnetism and diamagnetism (spins oriented in the opposite direction of the field). Fig. 7.2 – Periodic table of magnetic atoms (red circles). There are 79 magnetic atoms on the 103 elements. (The periodic table has for source https://commons. wikimedia.org/wiki/File:Periodic_table_large.svg). In an atom, the electron’s motion on their orbits generates a kinetic moment that we call orbital angular momentum noted L. Classically, we consider that the orbital motion of electrons creates electric currents, responsible for the Magnetic Moments of the atoms. To this orbi- tal moment one must add the proper kinetic moment of the electrons called Magnetism 275 Fig. 7.3 – Periodic table of magnetic atoms in the solid phase. By conven- tion, we consider only the magnetic order, i.e. the ferromagnetic (red circles), the antiferromagnetic (blue circles), and the ferrimagnetic (yellow circles). The other elements are said to be in non-ordered phases (paramagnetic and diamagne- tic). (The periodic table has for source https://commons.wikimedia.org/wiki/File: Periodic_table_large.svg). electronic spin noted S. The orbital Magnetic Moment is written −→ µ L = g L µ B L/ (7.1) µ B = 9.3×10 −24 J.T. is the Bohr magneton, (T Tesla, J Joule). In this notation L = [L(L + 1)] where L is an integer, L 2 = L(L + 1) 2 . µ B is sometimes given in GigaHz per Tesla via the relation µ B B = hν , as a first approximation, µ B is 14 GHz per Tesla (1 cm −1 = 30 GHz and 1 eV = 8065.5 cm −1 , see the units given at the beginning of the book) g L is the orbital Landé factor, in the hydrogen atom which has a single electron g L = 1. In a more complex atom, g L depends on the coupling, but in a pure LS coupling, g L = 1.
- David Jiles(Author)
- 2015(Publication Date)
- CRC Press(Publisher)
The Magnetic Moment m s due to the spin is then m p s e s = - eh m h 2 2 π π (10.18) Replacing 2 π p s /h by the spin quantum number s m s e = - 2 4 eh m s π (10.19) The quantity eh /4 π m e is known as the Bohr magneton , usually designated μ B , which has a value of 9.27 × 10 − 24 A m 2 . Replacing ( eh/ 4 π m e ) by μ B m s B = -2 μ s (10.20) where now because of the quantization of angular momentum 2 s = 4 π p s / h. Since s must be + -( ) ( ) 1 2 1 2 or , this means that the spin Magnetic Moment of an electron is 1 Bohr magneton. As discussed in Section 10.1.2, it is found that the spin gives twice the Magnetic Moment for a given angular momentum than does the orbit. As described above, there is no fundamental reason why these contributions should be equal; however, a precise treatment of this goes beyond the scope of this book. The situation is discussed by Born [4] and the difference in the relations between the spin and orbital Magnetic Moments and their respective angular momenta is attributed to relativistic effects. The total Magnetic Moment of an electron can be expressed in terms of multiples of the total angular momentum ( h/ 2 π ) j of the electron. m p p tot e tot B tot B = - = - = -g eh m h g h g j 4 2 2 π π μ π μ (10.21) where j is the total angular momentum quantum number, and for a particular electron m m m tot o s B B = + = + ( ) -μ μ l s 2 (10.22) The projection m j of the total angular momentum along the direction of a magnetic field is also quantized as shown in Figure 10.5a. 267 Electronic Magnetic Moments However, the assumption that the angular momentum is an integral multiple of the orbital quantum number l and the spin quantum number s as shown in Figure 10.5a is not quite valid. More precise values according to wave mechanics are p p o s and 2 2 2 2 2 1 2 1 = + = + ( ) ( ) ( ) ( ) h l l h s s π π .
eBook - PDFMagnetism in Medicine
A Handbook
- Wilfried Andrä, Hannes Nowak, Wilfried Andrä, Hannes Nowak(Authors)
- 2007(Publication Date)
- Wiley-VCH(Publisher)
We realize (as do most scientists) that such a confusing notation is very annoying, but now it is too late for it to be changed, as very large numbers of books and papers would need to be rewritten, making the whole operation absolutely out of question. Fortu- nately, we almost never simultaneously encounter h and H (or h and H) in the same problem. The vector field M formally introduced above has a very important physical meaning. To determine this, we recall the definition (Eq. 1.33) of the total mag- netic moment of a body and rewrite it using M as 1.2.3 Magnetic Field in Condensed Matter: General Concepts 39 m ¼ 1 2c ð V ½r hji d V ¼ 1 2 ð V ½r rot M dV ð1:44Þ Here, the integration volume can be expanded to contain the body inside it because outside a body hji ¼ 0. Rewriting the last integral in Eq. (1.44) as ð V ½r rot M dV ¼ þ S ½r ½dS M d V ð V ½½M ‘ r dV ð1:45Þ (the proof of Eq. (1.45) is a very nice exercise in vector analysis), we note that due to the mentioned expansion of the integration volume its bounding surface S is now outside the body where M ¼ 0 and hence the first integral in Eq. (1.45) van- ishes. Finally, rewriting a double vector product in the second integral as ½½M ‘ r ¼ M div r þ M ¼ 2M we obtain the desired result m ¼ 1 2c ð V ½r hji d V ¼ ð V M dV ð1:46Þ which shows that M is simply the density of the Magnetic Moment of the body (Magnetic Moment per unit volume). For this reason, M is called the magnetization vector. 1.2.3.2 Classification of Materials According to their Magnetic Properties The system of Eqs. (1.39), (1.42) and (1.43) div B ¼ 0 rot H ¼ 0 ð1:47Þ H ¼ B 4pM which describes the magnetic field in a condensed matter is clearly incomplete, be- cause we still do not know the relationship between M and H (or between B and H) inside a body. This relationship depends heavily on the material from which the body under study is made.
eBook - PDF- Frank Bovey(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
The maximum value of the angular momentum is 7, which is called the spin quantum number or more commonly simply the spin. Each nuclear ground state is characterized by just one value of 7. If 7 = 0, the nucleus has no angular momentum and no Magnetic Moment. If 7 is not zero, the nucleus will possess a Magnetic Moment , which is taken as parallel to the angular momentum vector. The permitted values of the vector moment along any chosen axis are described by means of a set of magnetic quantum numbers m given by the series: m = I, ( 7 -1 ) , ( 7 -2 ) , -7 (1-1) Thus, if 7 is ^, the possible magnetic quantum numbers are and . If 7 is 1, m may take on the values 1, 0, and 1 and so on. In general, then, there are 2 7 + 1 possible orientations or states of the nucleus. In the absence of a magnetic field, these states all have the same energy. In the presence of a uniform magnetic field H 0 , they correspond to states of different potential energy. For nuclei for which 7 is ^, the two possible values of m, and J, describe states in which the nuclear moment is aligned with and against the field 77 0 , respectively, the latter state being of higher energy. The detection of transitions of magnetic nuclei (often themselves referred to as spins) between these states is made possible by the nuclear magnetic resonance phenomenon. The magnitudes of nuclear Magnetic Moments are often specified in terms of the ratio of the Magnetic Moment and angular momentum, or magnetogyric ratio y, defined as r = % a * A spinning spherical particle with mass and charge e uniformly spread over its surface can be shown to give rise to a Magnetic Moment ehjAnMc, where c is the velocity of light. For a particle with the charge, mass and spin of the proton, the moment should be 5.0493 χ 10~ 24 erg/G on this model. Actually, this approximation is not a good one even for the proton, which is observed to have a Magnetic Moment about 2.79 times as great as the over simplified model predicts.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
Any non-uniform magnetic field whether caused by permanent magnets or by electric currents will exert a force on a small magnet in this way. Mathematically, the force on a small magnet having a Magnetic Moment m due to a magnetic field B is: ______________________________ WORLD TECHNOLOGIES ______________________________ where the gradient ∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B . To understand this equation, note that the dot product m · B = mB cos( θ ), where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions. Torque on a magnet due to a B-field Magnetic torque on a magnet due to an external magnetic field can be observed by placing two magnets near each other while allowing one to rotate. Magnetic torque is used to drive simple electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft. Magnetic torque τ tends to align a magnet's poles with the B-field lines (since m is in the direction of the poles this is equivalent to saying that it tends to align m in the same direction as B ). This is why the magnetic needle of a compass points toward earth's north pole.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Orange Apple(Publisher)
This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field whether caused by permanent magnets or by electric currents will exert a force on a small magnet in this way. ________________________ WORLD TECHNOLOGIES ________________________ Mathematically, the force on a small magnet having a Magnetic Moment m due to a magnetic field B is: where the gradient ∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B . To understand this equation, note that the dot product m · B = mB cos( θ ), where m and B represent the magnitude of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points 'uphill' pulling the magnet into regions of higher B-field (more strictly larger m · B ). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions having their own m then summing up the forces on each of these regions. Torque on a magnet due to a B-field Magnetic torque on a magnet due to an external magnetic field can be observed by placing two magnets near each other while allowing one to rotate. Magnetic torque is used to drive simple electric motors. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of electromagnets. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.
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