Electron Spin Resonance

Abstract: In this post (and in subsequent ones) we will describe an apparatus for demonstrating the phenomenon of electron spin resonance (also called paramagnetic electronic resonance). The phenomenon arises from the interaction of the magnetic moment of unpaired electrons with radio frequency electromagnetic radiation in the presence of external magnetic fields. This phenomenon is exploited in the homonymous spectroscopic technique used in the analysis of chemical compounds. The basic concepts of the EPR technique are similar to those of nuclear magnetic resonance, but in this case it is the electron spins that are excited instead of the spins of the atomic nuclei. This project refers to the great work done by Jabolatorium (http://www.jabolatorium.com/Projects/Esr/esr.html) to which our thanks go for making the project available.

Introduction

In 1925, two young researchers, Goudsmit and Uhlenbeck proposed the notion of electron spin, the angular momentum of spin, S, applying the same quantization rules that govern the orbital angular momentum of atomic electrons. In particular, in a given direction, say the z direction, the component S_z is:

S_z = m_s h/2π

Where m_s = ±1/2. To make a classical analogy we can say that the spin vector, once a direction has been established, can only point upwards or downwards.
Since the electron is a charged particle, it is natural that its spin angular momentum is associated with a spin magnetic moment. Electron spin resonance (ESR) describes the interaction of the magnetic spin moment of electrons with the magnetic component of electromagnetic radiation, usually in the radio frequency range (MHz-GHz). Each electron therefore has a magnetic dipole moment deriving both from the angular momentum of spin and from the orbital moment, however the only electrons that contribute to generate a non-zero magnetic dipole are the unpaired electrons, characteristic of paramagnetic substances.
Only the compounds that possess unpaired electrons, for example free radicals, therefore have a non-zero magnetic dipole moment that can interact with electromagnetic radiation.

Theory

As we wrote in the previous paragraph, the electron has a spin magnetic moment μ_s given by:

μ_s = -[e/2m_e] g S = ±1/2 [eh/4πm_e] g  [1]

Where S is the spin moment and g is the Landè factor (related to the gyromagnetic ratio, g = 1 for a pure orbital moment, g = 2 for a free electron). When the electron is immersed in a magnetic field B, the magnetic moment will be characterized by an energy given by the scalar product of the two quantities:

U = -μ_s x B = ±1/2 [eh/4πm_e] g B  [2]

Therefore, ignoring the spin-orbit interactions, a given energy level for an atomic electron will be split, due to the spin, into two distinct levels each corresponding to a distinct spin value, which differ in energy by an amount equal to:

ΔE = U₊ – U_– = [eh/4πm_e] g B  [3]

This levels splitting is known as the Zeeman effect and can be observed with optical spectroscopy techniques. Spin electron resonance (ESR) therefore refers to the situation in which photons of a frequency f are absorbed or emitted during the transitions between these two levels U₊ and U_–. Using the Planck relation ΔE = hf we obtain for the frequency f (also called Larmor frequency) the following relation:

f = geB/4πm_e [4]

By measuring f as a function of B and knowing the charge and mass values ​​of the electron e and m_e, the Landé factor g can be determined. The theory described above is exemplified in the following image which describes the splitting of the energy level in the presence of a magnetic field B and the transition induced by a radiofrequency.

We can try to make some quantitative estimates to understand the orders of magnitude of the quantities involved. Our goal is to create a demonstration apparatus that can be made without too many technical difficulties. For this reason we can establish a range of about 40-50MHz for the excitation radio frequencies, since working at higher frequencies (GHz) is technically more difficult and the components are more expensive. From relation [4] we can obtain the value of field B which turns out to be of the order of 2mT. A magnetic field of this magnitude can easily be generated by a pair of Helmholtz coils with dimensions of 2-3cm, with a number of turns of 200-300 and a current of less than 1A.

Basic Scheme

In the following image we have described the basic scheme of our apparatus for electronic spin resonance.

Helmholtz Coils and RF Coil

The external magnetic field that split the energy levels is produced by a pair of Helmholtz coils. These coils, spaced by a value equal to the radius of the solenoid, have the characteristic of producing an almost constant field in the central area between the two solenoids. In the central area is placed the RF coil which houses the sample being analyzed inside. The magnetic field generated by the coils is not constant but is made to vary slowly with continuity from a minimum of zero to a maximum of about 4mT (sweep signal). The magnetic field is measured by a Hall sensor whose output is sent to the microcontroller. A modulating signal of lower intensity at a frequency of 1KHz is superimposed on this slowly varying field. This modulating signal has the purpose of optimizing the detection of the signal using a Lock-In amplifier.

Robinson Oscillator

The RF coil is powered by a Robinson oscillator tuned to approximately 42MHz. In correspondence with the resonance condition, the RF energy absorbed by the sample increases and this causes the Q factor of the equivalent RLC circuit and the amplitude of the detected signal to vary in equal measure. The oscillator driver signal is sent to a prescaler which divides its frequency and converts it into a TTL signal before being sent to the microcontroller, on which the frequency reading is carried out. The prescaler is used because usually the microcontroller does not have a clock so high as to easily read frequencies of 40-50MHz. The ESR signal is sent to a bandpass filter centered at 1KHz in order to eliminate low and high frequency noise.

Lock-In Detector

The lock-in detector receives the 1KHz synchronism signal and the filtered ESR signal. This technique allows to detect and amplify very weak signals, significantly increasing the S/N ratio. The output of the Lock-In detector is a continuous or slowly varying signal that is further amplified by a signal amplifier before being sent to the microcontroller ADC converter.

Modulation and Lock-In Detection

The signal we take from the Robinson oscillator is a sinusoidal signal at about 42MHz whose amplitude is proportional to how close we are to the resonance condition of the sample. Quantifying a signal like this and distinguishing it from noise is not easy. The modulation technique comes to our “rescue” alongside the lock-in detection, described in the post Lock-in Amplifier.
A 1KHz modulating signal is superimposed on the polarization magnetic field which causes a similar oscillation on the amplitude of the output signal. The signal taken from the Robinson oscillator is initially filtered by a band-pass filter centered at 1KHz and then sent to the lock-in amplifier which is locked to the modulating signal. In the following diagram we show the lock-in detection scheme which uses modulation of the magnetic field. The first graph shows the trend of the intensity of the RF absorption by the sample under analysis, with the maximum corresponding to the resonance condition. A modulating signal is superimposed on the magnetic field which gives rise to a modulated signal d∆I/dB₀ which varies with the same modulation frequency. The second graph shows the amplitude of the detected ESR signal which turns out to be the first derivative of the absorption function.

ESR Apparatus

Following the principles described in the previous paragraphs, we have created a simple demonstration apparatus for electronic spin resonance. The “heart” of the system are the Helmholtz coils with the RF coil and the Robinson oscillator. There is also the PSoC 5lp series microcontroller, with its evaluation PCB including LCD display. Measurement data are recorded by the microcontroller and sent via USB connection to a Linux PC which takes care of recording on file and displaying on graph. The image below shows the complete apparatus.

Experimental Results

The graph below shows the data recording for a sample of the TTF-TCNQ salt. It is an organic compound used in some types of electrolytic capacitors. The compound is characterized by unpaired electrons for which it clearly presents the phenomenon of electronic resonance. The upper graph shows the trend of the magnetic field measured by the Hall sensor, while the lower graph shows the trend of the ESR signal. As you can see, the magnetic field has a “sawtooth” trend with a maximum value that is determined by software. The ESR signal has the classic “butterfly” trend expected from the occurrence of resonance conditions in correspondence with a magnetic field of about 1.5mT.

Conclusions

In this article we have described a DIY apparatus for electronic spin resonance. It is a mainly demonstrative apparatus with which we will try to do some experiments on the physics of the phenomenon. In the next post we will describe in some greater detail all the hardware parts of the project: Electron Spin Resonance – HW Part.

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