Strong interaction between light and electrons (2) "Four states of interaction between photon and spins"
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- January 29, 2021 |
- Electron Spin Resonance (ESR) |
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A conventional ESR spectrometer uses a cavity for microwave irradiation and detection of ESR absorption. On the resonance state, it can be considered as a model that spins absorb energy of โ๐=๐๐๐ต๐ต and then release it to the lattice system one way, whereย h: Planck constant,ย ฮฝ: frequency,ย g:ย g-value, ฮผB: Bohr magneton, andย B: magnetic flux density.
However, the interaction between photon of microwave and spins of electrons is a little more complex in fact.
Figure 1 is a modelized drawing that expresses energy flow of microwave photon and spins. The cavity resonates with angular frequencyย ๐c, relaxes with velocity ofย ๐
๐=๐๐ย /ย ๐๐ขย , which is inversely proportional to unloaded Q value of the cavity. On the other hand, spins do precess with an angular frequency ofย ๐๐=ย ๐พ๐ย ๐ต๐ย under the static magnetic fieldย ๐ต๐.
When the resonant condition ofย ๐๐ย =ย ๐๐ย is satisfied, excited electron spins that absorbed microwave energy relax with the velocity ofย ๐พ๐ย (half width: half width at half maximum (HWHM)), which corresponds to spectral line width. At this time, photon and electron spins exchange energy with a coupling constantย ๐๐. The coupling constantย ๐๐ย is expressed as[1]
whereย ๐๐๐ ๐๐๐กย is the square root of the filling factor of the cavity,ย ๐พ๐ย is gyromagnetic ratio of the electron, โ is reduced Planck constant (h/2ฯ),ย ๐0ย is vacuum permeability,ย ๐๐ย is the volume of the cavity,ย Nย is number of magnetic ions, andย Sย is spin quantum number.
Fig.1 Energy flow in spinโcavity system.
Four states of interaction between photon and spins
Interaction between photon and spins can be categorized to four states according to the relation between the coupling constant (๐๐), relaxation velocity (๐
๐) of photon, and relaxation velocity (๐พ๐) of spins[1].
- ใweak couplingใ:
A state that corresponds to the condition ofย ๐๐<ย ๐
๐ย andย ๐๐ย <ย ๐พ๐. Normal ESR measurement is done in this state.
- ใPurcell effectใ:
A state that corresponds to the condition ofย ๐๐ย <ย ๐พ๐ย andย ๐๐ย >ย ๐
๐.
- ใstrong couplingใ:
A state that corresponds to the condition ofย ๐๐ย >ย ๐พ๐ย andย ๐๐ย >ย ๐
๐. It behaves as a "quasi particle" that is unified by photon and spins.
- ใmagnetically induced transparent (MIT)ใ:
A state that corresponds to the condition ofย ๐๐ย <ย ๐
๐ย andย ๐๐ย >ย ๐พ๐.
Special care should be taken to the state of "Purcell effect" and "strong coupling", in practical ESR measurements.
Extraordinary spectral line shape, as shown inย Application Note ER200006Eย ย , might be due to the interaction between microwave photon and spins. An excess sample amount would produce an unexpected effect according to this photon- spin interaction, because equation 1 indicates that coupling constant is proportional to the square root of spin numbers. As shown in Fig. 2, resonant frequency shift of the cavity (dotted lines corresponded to AFC balance) on measuring spectrum can help to check the abnormal interaction.
Fig. 2ใFrequency shift of the cavity on measuring ESR spectra (conf.ย Application Note ER200006Eย ).
(a) Normal (Set_B). AFC balance does not almost move. (b) Excess sample (Set_A). AFC balance moves hard.
Investigating the frequency shift of the cavity around the resonant region, we can estimate what kind of interaction the system can be categorized to. Figure 3 is a mapping graph that the frequency spectra of the cavity ("Set_A"ย configuration as shown in Application Note ER200006E) is arrayed on the respective magnetic fields. Based on the simulation using the equation of Q-dip (S11ย parameter) shown in references[2][3], coupling constant was estimated to ca 1.2 MHz. This situation is inferred to the state of "Purcell effect", because it is under the condition ofย ๐๐ย / 2๐ <ย ๐พ๐ย / 2๐ andย ๐๐ย / 2๐ >ย ๐
๐ย / 2๐ , where spectral half width (HWHM๏ผhalf width at half maximum) is ca 121 ฮผT (๐พ๐ย / 2๐=3.39 MHz), Quย value of the cavity is ca 18,000 (๐
๐ย / 2๐=0.52 MHz). Such a situation, normal ESR spectra can not be obtained. Especially, using high Q cavity that includes an excess sample might induce "Purcell effect", and it should be taken care of in the interaction. Furthermore, when the sample is not a paramagnet, but a ferromagnet, and its line width is narrow, we might encounter the state of "strong coupling"ย (continued toย Application Note ER200008Eย ).
![Fig.3ใField dependence of the frequency spectra (Q-dips) . (a) Observed. (b) Simulation results based on the equation in the literature[3].](https://www.jeol.com/solutions/applications/details/product_file/file/2014_04e.jpg)
Fig.3ใField dependence of the frequency spectra (Q-dips).
(a) Observed. (b) Simulation results based on the equation in the literature[3].
Reference: [1] X. Zhang, C-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett.ย 113, 156401 (2014).
[2] E. Abe, H. Wu, A. Ardavan, and J. J. L. Morton,ใAppl. Phys. Lett.ย 98, 251108 (2011).
[3] Patent, US10288707B2 "Relaxation time measuring method and magnetic resonance measuring apparatus".