Effects of external magnetic fields on atoms/subatomic particles - II

 

> Larmor precession - Spin excitation (Spin flipping)

 

References:

http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/larmor.html

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/larmor.html#c1

It may be helpful to refer to the section "Motion in Microcosm - Quantum Mechanics".

 

 

When a magnetic dipole moment is placed in a magnetic field it will experience a torque. 

 

We know how to calculate the torque exerted by the magnetic field on a coil with current I, surface A and n loops. It is described at the link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c2. We will use the same approach.

 

A magnetic source in general is inherently a dipole source that can be visualized as a coil, or let us say a loop with current I and area A as shown at Figure 1 (Ref. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1).

 

 

Figure 1: Representation of a magnetic dipole source in form of a loop with current I and area A. 

 

 

The magnetic (dipole) moment can be considered to be a vector μ with a value equal to I * A and a direction perpendicular to the current loop in the right-hand-rule direction. The torque is given by τ = μ * B.

 

IIf we consider a static magnetic moment or a classical current loop we will have the case presented in Figure 2 where the exerted torque induces the alignment of μ to B (Ref. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1).

 

 

 

 

Figure 2 : Magnetic field B exerts torque which aligns μ to B.

 

 

With the alignment of the magnetic moment to the magnetic field the system adopts the lowest energy configuration or state. In other words, in such as case the potential energy will be lowest. If the magnetic moment is anti-aligned, as shown in Figure 3 (Ref. http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magmom.html#c1), the magnetic moment will be at a high energy state. The difference of magnetic potential energy between aligned and anti-aligned states is: 

 

ΔU = 2μB

 

 

 

 

Figure 3 : The energy of the system depends on whether the magnetic dipole is aligned or anti-aligned with the magnetic field.

 

 

 

However, if we don't have a static magnetic moment but a case where the magnetic moment results from the motion of a particle e.g. an electron in orbit around the nucleus, then the magnetic moment is proportional to the angular momentum of the electron. The torque exerted will produce a change in the angular momentum that is perpendicular to it and the magnetic moment will precess around the direction of the magnetic field as shown at the image of the link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/larmor.html#c1. This is called Larmor precession.

 

 

The precession angular velocity is:

 

ω = ( ge/2mp ) * B

 

where g is the g-factor of the system.

 

The above relationship is also written as 

 

ω = γ * B

 

where γ is the gyromagnetic ratio or magnetogyric ratio (it is a ratio that associates a gyric or rotational property to a magnetic property).

 

The frequency linked to the precession angular velocity i.e. the precession angular frequency is termed Larmor frequency

ω = 2π/Τ <=> ω= 2πf <=> f= ω/2π

 

The Larmor frequency is associated with the "spin flip" or spin transition, i.e. the transition of the spin from the aligned state to the anti-aligned state, which involves an energy change of 2μB. In what way is it associated?

 

Let us consider the examples for the electron and the proton for 1 Tesla which are mentioned at http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/larmor.html

The Larmor frequency for the electron for 1 Tesla is 28.025 GHz.

The Larmor frequency for the proton for 1 Tesla is 42.5781 MHz.

 

Practically, that means that when in magnetic field of 1 Tesla, the electron spin will be doing approximatively 28 billion (1 Giga is 1 billion or 109) precession cycles per second.

 

What happens if we provide to the electron an electromagnetic wave with a frequency of 28 billion cycles per second or a frequency of 28 GHz? (To be exact 28.025 GHz).

 

The answer is that it will be able to take it up, to absorb it as it is compatible to its state!

Only this frequency can be taken up by the electron in these conditions!

 

And what will be the result? 

The energetic state of the electron spin will be increased (by ΔE=2μB) and as a result it will transition from the aligned state to the anti-aligned state (or from a parallel to an anti-parallel state).

 

This is termed spin excitation or spin flipping.

 

Given the above it is possible to draw the conclusion that the gyromagnetic ratio is a constant that enables us to know for a given intensity of magnetic field which will be the precession frequency and the frequency of electromagnetic radiation that can be absorbed.

 

There are tables that provide the gyromagnetic ratio and we can find it in two forms.

 

It can be given in radians (let us say the "equivalent" of degrees) per second in one period or in Hertz per period.

Note that "one cycle" or one Hertz is 2π radians per second.

Therefore by dividing the first form by 2π we obtain the gyromagnetic ratio in Hertz per period.

 

 

 

In conclusion as mentioned at the link (discussing NMR) http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/nmr.html#c1:

Excerpt from above link:"The Larmor frequency can be visualized classically in terms of the precession of the magnetic moment around the magnetic field, analogous to the precession of a spinning top around the gravity field. It can also be visualized quantum mechanically in terms of the quantum energy of transition between the two possible spin states for spin 1/2. This can be expressed as a photon energy according to the Planck relationship."

 

(A relevant link on ESR: http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/esr.html)

 

 

 

Some additional notes from Wikipedia https://en.wikipedia.org/wiki/Gyromagnetic_ratio:

"In physics, the gyromagnetic ratio (also sometimes known as the magnetogyric ratio in other disciplines) of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma".

 

Gyromagnetic ratio and Larmor precession | Main article: Larmor precession

"Any free system with a constant gyromagnetic ratio, such as a rigid system of charges, a nucleus, or an electron, when placed in an external magnetic field B (measured in teslas) that is not aligned with its magnetic moment, will precess at a frequency f (measured in hertz), that is proportional to the external field:

 

 

 

 

Gyromagnetic ratio for an isolated electron 

(...)

 

Gyromagnetic ratio for a nucleus

The gyromagnetic ratio of a nucleus plays a role in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). These procedures rely on the fact that bulk magnetization due to nuclear spins precession in a magnetic field at a rate called the Larmor frequency, which is simply the product of the gyromagnetic ratio with the magnetic field strength. With this phenomenon, the sign of γ determines the sense (clockwise vs counterclockwise) of precession.

 

Most common nuclei such as 1H and 13C have positive gyromagnetic ratios.[7][8] Approximate values for some common nuclei are given in the table below.[9][10]

 

Nucleus {\displaystyle \gamma _{n}} (106 rad s−1 T −1) {\displaystyle \gamma _{n}/(2\pi )} (MHz T −1)
1H 267.513 42.577 478 92(29)[13]
2H 41.065 6.536
3He −203.789 −32.434
7Li 103.962 16.546
13C 67.2828 10.7084
14N 19.331 3.077
15N −27.116 −4.316
17O −36.264 −5.772
19F 251.662 40.052
23Na 70.761 11.262
27Al 69.763 11.103
29Si −53.190 −8.465
31P 108.291 17.235
57Fe 8.681 1.382
63Cu 71.118 11.319
67Zn 16.767 2.669
129Xe −73.997 −11.777

 

 

Another table can be found at this link:

http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/nmr.html#c1