Resonance frequency of a free electron?

Question

Calculate the resonance frequency of a free electron in a magnetic field of 1 T. What is the associated wavelength?

Answer 1

Perhaps you mean the cyclotron resonance frequency. If so, download thee the NRL Plasma Formulary.

Answer 2

You do not specify what kind of effect you're studying, so I will assume that your question relates to magnetic resonance and the Zeeman effect. If you are referring to cyclotron resonance, that is a different problem.

The energy of a magnetic dipole moment in a magnetic field is E = -μ . B, where μ is the dipole moment vector and B is the magnetic field vector. For the sake of simplicity, say B is in the +z direction, such that E = -μ_z |B|, where μ_z is the z-component of μ. In quantum mechanics μ_z is a quantum mechanical operator, so you actually have to find E = -<μ_z> |B|, where <μ_z> is the average value (expectation value) of the operator representing μ_z.

Let's assume that the only source of angular momentum in this problem is the intrinsic spin angular momentum of the particle; this assumption is certainly valid for a free electron. The operator that represents μ_z is g_S μ_B S_z/h-bar, where g_S is the g-factor (a constant relating the angular momentum to the magnetic moment), μ_B is the Bohr magneton (a constant), S_z is the operator for the z-component of spin angular momentum, and h-bar is the reduced Planck constant (= h/(2 pi)). So the energy of the spin in the magnetic field is -g_S μ_B /h-bar |B|. A free electron has a g-factor approximately equal to -2 (using this sign convention), but in matter environmental effects cause slight deviations in the effective g-factor, which can be probed by electron paramagnetic resonance (EPR) spectroscopy.

So, to calculate the energy levels you need to be able to calculate , which is easy: it evaluates to M_s h-bar, where M_s is the spin quantum number for the electron (+1/2 or -1/2). So you can see that the free electron has two magnetic energy levels, one with energy -(g_S)(μ_B)(+h-bar/2)/h-bar |B| = -g_S μ_B |B|, and one with energy -(g_S)(μ_B)(-h-bar/2)/h-bar = +g_S μ_B |B|. The splitting (energy difference) between the two levels is

ΔE = |g_S| μ_B |B|.

Resonance occurs when the energy of the radiation equals the energy splitting between the two levels. The energy of the radiation can be calculated from E = h ν, where ν is the frequency of the radiation and h is the Planck constant. Therefore,

ν = |g_S| μ_B |B|/h.

If you want to be accurate, the free electron g-factor is about -2.002 (most people use a sign convention in their equations where g_S is positive, though). The Bohr magneton is μ_B = 9.274 × 10^-24 J/T. The Planck constant is h = 6.626 × 10^-34 J s. So v = |-2.002| × (9.274 × 10^-24 J/T) × |1 T|/(6.626 × 10^-34 J s) = 2.803 × 10^10 s^-1 = 28.03 GHz. A good rule of thumb for electron spin resonance is is that the resonance frequency is about 28 GHz/Tesla.

If by "associated wavelength" you mean the wavelength of the radiation, you can use the relation λ ν = c for electromagnetic radiation. Here λ is the wavelength of the radiation and c is the speed of light. λ = (2.998 × 10^8 m.s)/(2.803 × 10^10 s^-1) = 0.01070 m = 1.070 cm. This is in the microwave region of the electromagnetic spectrum.

Interestingly, the electron cyclotron frequency also works out to be about 28 GHz/Tesla. The fact that this is approximately equal to the magnetic resonance frequency is an "accident" of |g_S| being approximately equal to 2, since the expression for the cyclotron resonance frequency does not involve the g-factor.

Answer 3

In physics, resonance is the tendency of a device to oscillate at greater effective amplitude at some frequencies than at others. those are known by means of fact the device's resonant frequencies (or resonance frequencies). At those frequencies, even small periodic using forces can produce super amplitude vibrations, by means of fact the device shops vibrational capability. while damping is small, the resonant frequency is approximately equivalent to the organic frequency of the device, it is the frequency of unfastened vibrations. Resonance phenomena happen with all kinds of vibrations or waves: there is mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave purposes. Resonant structures would be used to generate vibrations of a particular frequency (e.g. musical contraptions), or %. out particular frequencies from a complicated vibration containing many frequencies Nuclear magnetic resonance (NMR) is the call given to a actual resonance phenomenon related to the commentary of particular quantum mechanical magnetic properties of an atomic nucleus interior the presence of an utilized, exterior magnetic field. Many scientific ideas make the main NMR phenomena to examine molecular physics, crystals and non-crystalline components by way of NMR spectroscopy.that's often called ATOMIC RESONANCE.

Answer 4

The electron has spin of h-bar / 2. The energy associated with that is the spin is proportional to the B-field dotted with the Bohr magneton. Then use E = h-bar omega to find your frequency.