Quick Reminder About Magnetism
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Atomic Orbitals
Visualization of Atomic Orbitals Source: Robert R. Gotwals, Jr., Department of Chemistry, North Carolina School of Science and Mathematics, Durham, NC, 27705. Email: [email protected]. Copyright ©2007, all rights reserved, permission to use for non-commercial educational purposes is granted. Introductory Readings Where are the electrons in an atom? There are a number of Models - a description of models that look to describe the location and behavior of some physical object or electrons in atoms. It is important to know this information, as behavior in nature. Models it helps chemists to make statements and predictions about are often mathematical, describing the behavior of how atoms will react with other atoms to form molecules. The some event. model one chooses will oftentimes influence the types of statements one can make about the behavior – that is, the reactivity – of an atom or group of atoms. Bohr model - states that no One model, the Bohr model, describes electrons as small two electrons in an atom can billiard balls, rotating around the positively charged nucleus, have the same set of four much in the way that planets quantum numbers. orbit around the Sun. The placement and motion of electrons are found in fixed and quantifiable levels called orbits, Orbits- where electrons are or, in older terminology, shells. believed to be located in the There are specific numbers of Bohr model. Orbits have a electrons that can be found in fixed amount of energy, and are identified with the each orbit – 2 for the first orbit, principle quantum number 8 for the second, and so forth. -
Basic Magnetic Measurement Methods
Basic magnetic measurement methods Magnetic measurements in nanoelectronics 1. Vibrating sample magnetometry and related methods 2. Magnetooptical methods 3. Other methods Introduction Magnetization is a quantity of interest in many measurements involving spintronic materials ● Biot-Savart law (1820) (Jean-Baptiste Biot (1774-1862), Félix Savart (1791-1841)) Magnetic field (the proper name is magnetic flux density [1]*) of a current carrying piece of conductor is given by: μ 0 I dl̂ ×⃗r − − ⃗ 7 1 - vacuum permeability d B= μ 0=4 π10 Hm 4 π ∣⃗r∣3 ● The unit of the magnetic flux density, Tesla (1 T=1 Wb/m2), as a derive unit of Si must be based on some measurement (force, magnetic resonance) *the alternative name is magnetic induction Introduction Magnetization is a quantity of interest in many measurements involving spintronic materials ● Biot-Savart law (1820) (Jean-Baptiste Biot (1774-1862), Félix Savart (1791-1841)) Magnetic field (the proper name is magnetic flux density [1]*) of a current carrying piece of conductor is given by: μ 0 I dl̂ ×⃗r − − ⃗ 7 1 - vacuum permeability d B= μ 0=4 π10 Hm 4 π ∣⃗r∣3 ● The Physikalisch-Technische Bundesanstalt (German national metrology institute) maintains a unit Tesla in form of coils with coil constant k (ratio of the magnetic flux density to the coil current) determined based on NMR measurements graphics from: http://www.ptb.de/cms/fileadmin/internet/fachabteilungen/abteilung_2/2.5_halbleiterphysik_und_magnetismus/2.51/realization.pdf *the alternative name is magnetic induction Introduction It -
Magnetic Fields Magnetism Magnetic Field
PHY2061 Enriched Physics 2 Lecture Notes Magnetic Fields Magnetic Fields Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor. Magnetism Is ubiquitous in every-day life! • Refrigerator magnets (who could live without them?) • Coils that deflect the electron beam in a CRT television or monitor • Cassette tape storage (audio or digital) • Computer disk drive storage • Electromagnet for Magnetic Resonant Imaging (MRI) Magnetic Field Magnets contain two poles: “north” and “south”. The force between like-poles repels (north-north, south-south), while opposite poles attract (north-south). This is reminiscent of the electric force between two charged objects (which can have positive or negative charge). Recall that the electric field was invoked to explain the “action at a distance” effect of the electric force, and was defined by: F E = qel where qel is electric charge of a positive test charge and F is the force acting on it. We might be tempted to define the same for the magnetic field, and write: F B = qmag where qmag is the “magnetic charge” of a positive test charge and F is the force acting on it. However, such a single magnetic charge, a “magnetic monopole,” has never been observed experimentally! You cannot break a bar magnet in half to get just a north pole or a south pole. As far as we know, no such single magnetic charges exist in the universe, D. -
The Magnetic Moment of a Bar Magnet and the Horizontal Component of the Earth’S Magnetic Field
260 16-1 EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH’S MAGNETIC FIELD I. THEORY The purpose of this experiment is to measure the magnetic moment μ of a bar magnet and the horizontal component BE of the earth's magnetic field. Since there are two unknown quantities, μ and BE, we need two independent equations containing the two unknowns. We will carry out two separate procedures: The first procedure will yield the ratio of the two unknowns; the second will yield the product. We will then solve the two equations simultaneously. The pole strength of a bar magnet may be determined by measuring the force F exerted on one pole of the magnet by an external magnetic field B0. The pole strength is then defined by p = F/B0 Note the similarity between this equation and q = F/E for electric charges. In Experiment 3 we learned that the magnitude of the magnetic field, B, due to a single magnetic pole varies as the inverse square of the distance from the pole. k′ p B = r 2 in which k' is defined to be 10-7 N/A2. Consider a bar magnet with poles a distance 2x apart. Consider also a point P, located a distance r from the center of the magnet, along a straight line which passes from the center of the magnet through the North pole. Assume that r is much larger than x. The resultant magnetic field Bm at P due to the magnet is the vector sum of a field BN directed away from the North pole, and a field BS directed toward the South pole. -
PH585: Magnetic Dipoles and So Forth
PH585: Magnetic dipoles and so forth 1 Magnetic Moments Magnetic moments ~µ are analogous to dipole moments ~p in electrostatics. There are two sorts of magnetic dipoles we will consider: a dipole consisting of two magnetic charges p separated by a distance d (a true dipole), and a current loop of area A (an approximate dipole). N p d -p I S Figure 1: (left) A magnetic dipole, consisting of two magnetic charges p separated by a distance d. The dipole moment is |~µ| = pd. (right) An approximate magnetic dipole, consisting of a loop with current I and area A. The dipole moment is |~µ| = IA. In the case of two separated magnetic charges, the dipole moment is trivially calculated by comparison with an electric dipole – substitute ~µ for ~p and p for q.i In the case of the current loop, a bit more work is required to find the moment. We will come to this shortly, for now we just quote the result ~µ=IA nˆ, where nˆ is a unit vector normal to the surface of the loop. 2 An Electrostatics Refresher In order to appreciate the magnetic dipole, we should remind ourselves first how one arrives at the field for an electric dipole. Recall Maxwell’s first equation (in the absence of polarization density): ρ ∇~ · E~ = (1) r0 If we assume that the fields are static, we can also write: ∂B~ ∇~ × E~ = − (2) ∂t This means, as you discovered in your last homework, that E~ can be written as the gradient of a scalar function ϕ, the electric potential: E~ = −∇~ ϕ (3) This lets us rewrite Eq. -
Ph501 Electrodynamics Problem Set 8 Kirk T
Princeton University Ph501 Electrodynamics Problem Set 8 Kirk T. McDonald (2001) [email protected] http://physics.princeton.edu/~mcdonald/examples/ Princeton University 2001 Ph501 Set 8, Problem 1 1 1. Wire with a Linearly Rising Current A neutral wire along the z-axis carries current I that varies with time t according to ⎧ ⎪ ⎨ 0(t ≤ 0), I t ( )=⎪ (1) ⎩ αt (t>0),αis a constant. Deduce the time-dependence of the electric and magnetic fields, E and B,observedat apoint(r, θ =0,z = 0) in a cylindrical coordinate system about the wire. Use your expressions to discuss the fields in the two limiting cases that ct r and ct = r + , where c is the speed of light and r. The related, but more intricate case of a solenoid with a linearly rising current is considered in http://physics.princeton.edu/~mcdonald/examples/solenoid.pdf Princeton University 2001 Ph501 Set 8, Problem 2 2 2. Harmonic Multipole Expansion A common alternative to the multipole expansion of electromagnetic radiation given in the Notes assumes from the beginning that the motion of the charges is oscillatory with angular frequency ω. However, we still use the essence of the Hertz method wherein the current density is related to the time derivative of a polarization:1 J = p˙ . (2) The radiation fields will be deduced from the retarded vector potential, 1 [J] d 1 [p˙ ] d , A = c r Vol = c r Vol (3) which is a solution of the (Lorenz gauge) wave equation 1 ∂2A 4π ∇2A − = − J. (4) c2 ∂t2 c Suppose that the Hertz polarization vector p has oscillatory time dependence, −iωt p(x,t)=pω(x)e . -
Magnetism, Angular Momentum, and Spin
Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v. -
Chapter 7 Magnetism and Electromagnetism
Chapter 7 Magnetism and Electromagnetism Objectives • Explain the principles of the magnetic field • Explain the principles of electromagnetism • Describe the principle of operation for several types of electromagnetic devices • Explain magnetic hysteresis • Discuss the principle of electromagnetic induction • Describe some applications of electromagnetic induction 1 The Magnetic Field • A permanent magnet has a magnetic field surrounding it • A magnetic field is envisioned to consist of lines of force that radiate from the north pole to the south pole and back to the north pole through the magnetic material Attraction and Repulsion • Unlike magnetic poles have an attractive force between them • Two like poles repel each other 2 Altering a Magnetic Field • When nonmagnetic materials such as paper, glass, wood or plastic are placed in a magnetic field, the lines of force are unaltered • When a magnetic material such as iron is placed in a magnetic field, the lines of force tend to be altered to pass through the magnetic material Magnetic Flux • The force lines going from the north pole to the south pole of a magnet are called magnetic flux (φ); units: weber (Wb) •The magnetic flux density (B) is the amount of flux per unit area perpendicular to the magnetic field; units: tesla (T) 3 Magnetizing Materials • Ferromagnetic materials such as iron, nickel and cobalt have randomly oriented magnetic domains, which become aligned when placed in a magnetic field, thus they effectively become magnets Electromagnetism • Electromagnetism is the production -
Chapter 22 Magnetism
Chapter 22 Magnetism 22.1 The Magnetic Field 22.2 The Magnetic Force on Moving Charges 22.3 The Motion of Charged particles in a Magnetic Field 22.4 The Magnetic Force Exerted on a Current- Carrying Wire 22.5 Loops of Current and Magnetic Torque 22.6 Electric Current, Magnetic Fields, and Ampere’s Law Magnetism – Is this a new force? Bar magnets (compass needle) align themselves in a north-south direction. Poles: Unlike poles attract, like poles repel Magnet has NO effect on an electroscope and is not influenced by gravity Magnets attract only some objects (iron, nickel etc) No magnets ever repel non magnets Magnets have no effect on things like copper or brass Cut a bar magnet-you get two smaller magnets (no magnetic monopoles) Earth is like a huge bar magnet Figure 22–1 The force between two bar magnets (a) Opposite poles attract each other. (b) The force between like poles is repulsive. Figure 22–2 Magnets always have two poles When a bar magnet is broken in half two new poles appear. Each half has both a north pole and a south pole, just like any other bar magnet. Figure 22–4 Magnetic field lines for a bar magnet The field lines are closely spaced near the poles, where the magnetic field B is most intense. In addition, the lines form closed loops that leave at the north pole of the magnet and enter at the south pole. Magnetic Field Lines If a compass is placed in a magnetic field the needle lines up with the field. -
New Measurements of the Electron Magnetic Moment and the Fine Structure Constant
S´eminaire Poincar´eXI (2007) 109 – 146 S´eminaire Poincar´e Probing a Single Isolated Electron: New Measurements of the Electron Magnetic Moment and the Fine Structure Constant Gerald Gabrielse Leverett Professor of Physics Harvard University Cambridge, MA 02138 1 Introduction For these measurements one electron is suspended for months at a time within a cylindrical Penning trap [1], a device that was invented long ago just for this purpose. The cylindrical Penning trap provides an electrostatic quadrupole potential for trapping and detecting a single electron [2]. At the same time, it provides a right, circular microwave cavity that controls the radiation field and density of states for the electron’s cyclotron motion [3]. Quantum jumps between Fock states of the one-electron cyclotron oscillator reveal the quantum limit of a cyclotron [4]. With a surrounding cavity inhibiting synchrotron radiation 140-fold, the jumps show as long as a 13 s Fock state lifetime, and a cyclotron in thermal equilibrium with 1.6 to 4.2 K blackbody photons. These disappear by 80 mK, a temperature 50 times lower than previously achieved with an isolated elementary particle. The cyclotron stays in its ground state until a resonant photon is injected. A quantum cyclotron offers a new route to measuring the electron magnetic moment and the fine structure constant. The use of electronic feedback is a key element in working with the one-electron quantum cyclotron. A one-electron oscillator is cooled from 5.2 K to 850 mK us- ing electronic feedback [5]. Novel quantum jump thermometry reveals a Boltzmann distribution of oscillator energies and directly measures the corresponding temper- ature. -
1 Introduction 2 Spins As Quantized Magnetic Moments
Spin 1 Introduction For the past few weeks you have been learning about the mathematical and algorithmic background of quantum computation. Over the course of the next couple of lectures, we'll discuss the physics of making measurements and performing qubit operations in an actual system. In particular we consider nuclear magnetic resonance (NMR). Before we get there, though, we'll discuss a very famous experiment by Stern and Gerlach. 2 Spins as quantized magnetic moments The quantum two level system is, in many ways, the simplest quantum system that displays interesting behavior (this is a very subjective statement!). Before diving into the physics of two-level systems, a little on the history of the canonical example: electron spin. In 1921, Stern proposed an experiment to distinguish between Larmor's classical theory of the atom and Sommerfeld's quantum theory. Each theory predicted that the atom should have a magnetic moment (i.e., it should act like a small bar magnet). However, Larmor predicted that this magnetic moment could be oriented along any direction in space, while Sommerfeld (with help from Bohr) predicted that the orientation could only be in one of two directions (in this case, aligned or anti-aligned with a magnetic field). Stern's idea was to use the fact that magnetic moments experience a linear force when placed in a magnetic field gradient. To see this, not that the potential energy of a magnetic dipole in a magnetic field is given by: U = −~µ · B~ Here, ~µ is the vector indicating the magnitude and direction of the magnetic moment. -
Spin the Evidence of Intrinsic Angular Momentum Or Spin and Its
Spin The evidence of intrinsic angular momentum or spin and its associated magnetic moment came through experiments by Stern and Gerlach and works of Goudsmit and Uhlenbeck. The spin is called intrinsic since, unlike orbital angular momentum which is extrinsic, it is carried by point particle in addition to its orbital angular momentum and has nothing to do with motion in space. The magnetic moment ~µ of silver atom was measured in 1922 experiment by Stern and Gerlach and its projection µz in the direction of magnetic fieldz ^ B (through which the silver beam was passed) was found to be just −|~µj and +j~µj instead of continuously varying between these two as limits. Classically, magnetic moment is proportional to the angular momentum (12) and, assuming this proportionality to survive in quantum mechanics, quan- tization of magnetic moment leads to quantization of corresponding angular momentum S, which we called spin, q q µ = g S ) µ = g ~ m (57) z 2m z z 2m s as is the case with orbital angular momentum, where the ratio q=2m is called Bohr mag- neton, µb and g is known as Lande-g factor or gyromagnetic factor. The g-factor is 1 for orbital angular momentum and hence corresponding magnetic moment is l µb (where l is integer orbital angular momentum quantum number). A surprising feature here is that spin magnetic moment is also µb instead of µb=2 as one would naively expect. So it turns out g = 2 for spin { an effect that can only be understood if linearization of Schr¨odinger equation is attempted i.e.