Incoming links: Magnetic field
Physical Review Volume 53, page 318.
Not paywalled as of 2024! A miracle! It is barely one page long.
This is the paper that contains the first successful report of experimental nuclear magnetic moment observation.
They promise more at the end:and this promise was fulfilled on the later The Molecular Beam Resonance Method for Measuring Nuclear Magnetic Moments.
We have tried this experiment with LiC1 and observed the resonance peaks of Li and Cl. The effects are very striking and the resonances sharp (Fig. 1). A full account of this experiment, together with the values of the nuclear moments, will be published when the homogeneous field is recalibrated.
This figure sums it all up: they were measuring the intensity of one side of a molecular beam after a Stern-Gerlach experiment.
Then, they vary the constant magnetic field before the splitting, and at the same time apply a fixed radio frequency to the beam.
When the constant magnetic field makes the energy gap match the radio frequency input, nuclear spin of many atoms goes to the anti-aligned direction, the beam gets diverted, so the previously detected beam gets weaker.
Electromagnets allow us to create controllable magnetic fields, i.e.: they act as magnets that we can turn on and off as we please but controlling an input voltage.
Compare them to permanent magnet: on a magnet, you always have a fixed generated magnetic field. But with an electromagnet you can control the field, and even turn it off entirely.
This type of "useful looking thing that can be controlled by a voltage" tends to be of huge importance in electrical engineering, the transistor being another example.
A different and more elegant way to express Maxwell's equations by using the:instead of the:
The voltage changes perpendicular to the current when magnetic field is applied.
Hall effect experimental diagram
. Source. The Hall effect refers to the produced voltage , AKA on this setup.An intuitive video is:
The key formula for it is:where:
- : current on x direction, which we can control by changing the voltage
- : strength of transversal magnetic field applied
- : charge carrier density, a property of the material used
- : height of the plate
- : electron charge
Applications:
- the direction of the effect proves that electric currents in common electrical conductors are made up of negative charged particles
- measure magnetic fields, TODO vs other methods
Other more precise non-classical versions:
In some contexts, we want to observe what happens for a given fixed magnetic field strength on a specific plate (thus and are also fixed).
In those cases, it can be useful to talk about the "Hall resistance" defined as:So note that it is not a "regular resistance", it just has the same dimensions, and is more usefully understood as a proportionality constant for the voltage given an input current:
This notion can be useful because everything else being equal, if we increase the current , then also increases proportionally, making this a way to talk about the voltage in a current independent manner.
And this is particularly the case for the quantum Hall effect, where is constant for wide ranges of applied magnetic field and TODO presumably the height can be made to a single molecular layer with chemical vapor deposition of the like, and if therefore fixed.
A tiny idealized magnet! It is a very good model if you have a small strong magnet interacting with objects that are far away, notably other magnetic dipoles or a constant magnetic field.
The cool thing about this model is that we have simple explicit formulas for the magnetic field it produces, and for how this little magnet is affected by a magnetic field or by another magnetic dipole.
This is the perfect model for electron spin, but it can also be representative of macroscopic systems in the right circumstances.
The intuition for the name is likely that "dipole" means "both poles are on the same spot".
Different macroscopic magnets can be approximated by a magnetic dipole when shrunk seen from far away
. Source. As of 2023 the most important ones economicaly were:The main application is Magnetic resonance imaging. Both of these are have to be Liquid helium, i.e. they are not "high-temperature superconductor" which is a pain. One big strength they have is that they are metallic, and therefore can made into wires, which is crucial to be able to make electromagnetic coils out of them.
- Nb-Ti: the most widely used one. Used e.g. to create the magnetic fields of the Large Hadron Collider Up to 15 T.
- Nb-Sn: more expensive than Nb-Ti, but can reach up to 30 T.
Ciro Santilli once visited the chemistry department of a world leading university, and the chemists there were obsessed with NMR. They had small benchtop NMR machines. They had larger machines. They had a room full of huge machines. They had them in corridors and on desk tops. Chemists really love that stuff. More precisely, these are used for NMR spectroscopy, which helps identify what a sample is made of.
Basically measures the concentration of certain isotopes in a region of space.
Introduction to NMR by Allery Chemistry
. Source. - only works with an odd number of nucleons
- apply strong magnetic field, this separates the energy of up and down spins. Most spins align with field.
- send radio waves into sample to make nucleons go to upper energy level. We can see that the energy difference is small since we are talking about radio waves, low frequency.
- when nucleon goes back down, it re-emits radio waves, and we detect that. TODO: how do we not get that confused with the input wave, which is presumably at the same frequency? It appears to send pulses, and then wait for the response.
How to Prepare and Run a NMR Sample by University of Bath (2017)
Source. This is a more direct howto, cool to see. Uses a Bruker Corporation 300. They have a robotic arm add-on. Shows spectrum on computer screen at the end. Shame no molecule identification after that!Proton Nuclear Magnetic Resonance by Royal Society Of Chemistry (2008)
Source. This video has the merit of showing real equipment usage, including sample preparation.
Says clearly that NMR is the most important way to identify organic compounds.
- youtu.be/uNM801B9Y84?t=41 lists some of the most common targets, including hydrogen and carbon-13
- youtu.be/uNM801B9Y84?t=124 ethanol example
- youtu.be/uNM801B9Y84?t=251 they use solvents where all protium is replaced by deuterium to not affect results. Genius.
- youtu.be/uNM801B9Y84?t=354 usually they do 16 radio wave pulses
As seen from explicit scalar form of the Maxwell's equations, this expands to 8 equations, so the question arises if the system is over-determined because it only has 6 functions to be determined.
As explained on the Wikipedia page however, this is not the case, because if the first two equations hold for the initial condition, then the othe six equations imply that they also hold for all time, so they can be essentially omitted.
It is also worth noting that the first two equations don't involve time derivatives. Therefore, they can be seen as spacial constraints.
TODO: the electric field and magnetic field can be expressed in terms of the electric potential and magnetic vector potential. So then we only need 4 variables?
Quantum version of the Hall effect.
As you increase the magnetic field, you can see the Hall resistance increase, but it does so in discrete steps.
Gotta understand this because the name sounds cool. Maybe also because it is used to define the fucking ampere in the 2019 redefinition of the SI base units.
At least the experiment description itself is easy to understand. The hard part is the physical theory behind.
TODO experiment video.
The effect can be separated into two modes:
- Integer quantum Hall effect: easier to explain from first principles
- Fractional quantum Hall effect: harder to explain from first principles
- Fractional quantum Hall effect for : 1998 Nobel Prize in Physics
- Fractional quantum Hall effect for : one of the most important unsolved physics problems as of 2023
Solenoid means "tubular" in Greek.
Solenoids are simpler to build as they don't require insulated wire as in modern electrical cable because as the electromagnetic coils don't touch one another.
As such it is perhaps the reason why some early electromagnetism experiments were carried out with solenoids, which André-Marie Ampère named in 1823.
But the downside of this is that the magnetic field they can generate is less strong.
Magnetic field lines around a solenoid cross-section
. TODO accurate simulation or not?Applications: produce high magnetic fields forAs of the early 2020s, superconducting magnets predominantly use low temperature superconductors Nb-Ti and Nb-Sn, see also most important superconductor materials, but there were efforts underway to create practical high-temperature superconductor-based magnets as well: Section "High temperature superconductor superconducting magnet".
- Magnetic resonance imaging, the most important commercial application as of the early 2020s
- more researchy applications as of the early 2020s:
Wikipedia has done well for once:
The current to the coil windings is provided by a high current, very low voltage DC power supply, since in steady state the only voltage across the magnet is due to the resistance of the feeder wires. Any change to the current through the magnet must be done very slowly, first because electrically the magnet is a large inductor and an abrupt current change will result in a large voltage spike across the windings, and more importantly because fast changes in current can cause eddy currents and mechanical stresses in the windings that can precipitate a quench (see below). So the power supply is usually microprocessor-controlled, programmed to accomplish current changes gradually, in gentle ramps. It usually takes several minutes to energize or de-energize a laboratory-sized magnet.
Superconductivity: magnetic separation by University of Cambridge
. Source. The Stern-Gerlach experiment needs an inhomogenous magnetic field Updated 2024-12-23 +Created 2024-09-19
Needs an inhomogenous magnetic field to move the atoms up or down: magnetic dipole in an inhomogenous magnetic field. TODO how it is generated?
TODO understand.
Trapping Ions for Quantum Computing by Diana Craik (2019)
Source. A basic introduction, but very concrete, with only a bit of math it might be amazing:Sounds complicated, several technologies need to work together for that to work! Videos of ions moving are from www.physics.ox.ac.uk/research/group/ion-trap-quantum-computing.
- youtu.be/j1SKprQIkyE?t=217 you need ultra-high vacuum
- youtu.be/j1SKprQIkyE?t=257 you put the Calcium on a "calcium oven", heat it up, and make it evaporates a little bit
- youtu.be/j1SKprQIkyE?t=289 you need lasers. You shine the laser on the calcium atom to eject one of the two valence electrons from it. Though e.g. Universal Quantum is trying to do away with them, because alignment for thousands or millions of particles would be difficult.
- youtu.be/j1SKprQIkyE?t=518 keeping all surrounding electrodes positive would be unstable. So they instead alternate electrode quickly between plus and minus
- youtu.be/j1SKprQIkyE?t=643 talks about the alternative, of doing it just with electrodes on a chip, which is easier to manufacture. They fly at about 100 microns above the trap. And you can have multiple ions per chip.
- youtu.be/j1SKprQIkyE?t=1165 using microwaves you can flip the spin of the electron, or put it into a superposition. From more reading, we understand that she is talking about a hyperfine transition, which often happen in the microwave area.
- youtu.be/j1SKprQIkyE?t=1210 talks about making quantum gates. You have to put the ions into a magnetic field at one of the two resonance frequencies of the system. Presumably what is meant is an inhomogenous magnetic field as in the Stern-Gerlach experiment.This is the hard and interesting part. It is not clear why the atoms become coupled in any way. Is it due to electric repulsion?She is presumably describing the Cirac–Zoller CNOT gate.
A major flaw of this presentation is not explaining the readout process.
How To Trap Particles in a Particle Accelerator by the Royal Institution (2016)
Source. Demonstrates trapping pollen particles in an alternating field.Ion trapping and quantum gates by Wolfgang Ketterle (2013)
Source. - youtu.be/lJOuPmI--5c?t=1601 Cirac–Zoller CNOT gate was the first 2 qubit gate. Explains it more or less.
Introduction to quantum optics by Peter Zoller (2018)
Source. THE Zoller from Cirac–Zoller CNOT gate talks about his gate.- www.youtube.com/watch?v=W3l0QPEnaq0&t=427s shows that the state is split between two options: center of mass mode (ions move in same direction), and strechmode (atoms move in opposite directions)
- youtu.be/W3l0QPEnaq0?t=658 shows a schematic of the experiment
Split in the spectral line when a magnetic field is applied.
Non-anomalous: number of splits matches predictions of the Schrödinger equation about the number of possible states with a given angular momentum. TODO does it make numerical predictions?
www.pas.rochester.edu/~blackman/ast104/zeeman-split.html contains the hello world that everyone should know: 2p splits into 3 energy levels, so you see 3 spectral lines from 1s to 2p rather than just one.
p splits into 3, d into 5, f into 7 and so on, i.e. one for each possible azimuthal quantum number.
It also mentions that polarization effects become visible from this: each line is polarized in a different way. TODO more details as in an experiment to observe this.
Well explained at: Video "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)".
Experimental physics - IV: 22 - Zeeman effect by Lehrportal Uni Gottingen (2020)
Source. This one is decent. Uses a cadmium lamp and an etalon on an optical table. They see a more or less clear 3-split in a circular interference pattern,
They filter out all but the transition of interest.
- youtu.be/ZmObNFAqkBE?t=165 passes the lines through a polarizer, which shows how orbital angular momentum is carried by photon polarization
- youtu.be/ZmObNFAqkBE?t=370 says they are looking at 1D2 to 1P1 changes.
Zeeman Effect - Control light with magnetic fields by Applied Science (2018)
Source. Does not appear to achieve a crystal clear split unfortunately.