Abstract

This lab seeks to find the g-factor of DPPH, which should be near the Lande g-factor for the electron (2.0023). This is done by placing a sample of DPPH into a magnetic field and exposing it to microwaves. It was found that the Lande g-factor of the sample was 2.00823 ± 0.00024.

Introduction

The g-factor was first described by Alfred Lande in 1921. This was part of his explanation of the Zeeman effect. It is a term in the energy level expressions of an atom in a weak magnetic field.

Theoretical Background Info

The electron has a quantum spin number, s = ½ and magnetic moment, ms = + ½ and ms = -½. When placed in the presence of an external magnetic field, B, the magnetic moments align themselves either parallel or antiparallel to the applied field. This is known as the Zeeman effect. The two alignments have a difference in energy given by, ΔE = gμ_bB, where g is the electron’s Lande g-factor and μ_b is the Bohr magneton. This equation implies a direct proportionality between the energy difference and the applied magnetic field.

This proportionality allows for a wide range of electromagnetic radiation to be used, given the corresponding field. A microwave was chosen for this experimental setup. A higher frequency leads to a larger the energy splitting which results in smaller error in detecting a spin flipping.

An unpaired electron can shift between the two energy levels by absorption or emission of a photon of frequency ν = gμ_bB/h, where h is Plank's constant. For this reason a chemical with one unpaired electron. DPPH fits this requirement and therefore is chosen for this experiment.

Due to the Maxwell-Boltzmann distribution there are more electrons in the lower energy state than the higher one initially. Therefore when bombarded with microwaves there is a net absorbance. In order to find the maximum of this plot the first derivative is taken. Below is a plot showing the absorbance curve as well as the first derivative.

Experimental Method

This experimental setup consists of a microwave system, a sample of DPPH, Helmholtz coils, oscilloscope, audio amplifier, gauss meter, lock-in amplifier, and computer data acquisition system.

The source of microwaves is a 100 mW Gunn diode. After the waves are emitted by the diode they pass through a switch, then into a ferrite attenuator. This attenuator serves to decouple the source from the rest of the experimental setup.

The frequency meters and tuning stubs are used to tune the wave frequency so that it will resonate in the waveguide.

The crystal detector attached to the magic tee is used to measure the relative power of the waves rather than the absolute power.

The attenuators are used to change the power of the passing wave without affecting the frequency. This effect can be used to amplify the signal or, as is the case with the ferrite attenuator, decouple parts of the setup.

The resonator cavity is used to amplify the ESR signal. This will only work if the microwave frequency is properly tuned to result in resonance inside of the cavity.

The DPPH sample is placed into the resonator cavity between the Helmholtz coils and is the subject of study in this experiment.

The Helmholtz coils are used to generate an approximately spatially constant magnetic field. There are three parts to the Helmholtz coils used in this experiment, constant, ramping coils, and alternating. The constant coils are used to set the magnetic field to within 30 G of the calculated field. The ramping coils are used to sweep the field which results in a more precise tuning of the magnetic field. The alternating coils are connected to the lock-in amplifier and set to alternate at around 200 Hz. This alternating field allows sweeping of the range in-between steps of the ramping coils. This technique results in a finer sweep of the magnetic field and smaller error.

The gauss meter is used to measure the magnetic field. The measurement given by the gauss meter is highly affected by the orientation of the probe. This results in a large amount of uncertainty.

The magic tee is used to divide the signal from the source between the reference arm and the sample. One would expect that the signal would be divided between B, C, and D, but due to polarization differences the signal from A does not enter D. The signal read at the crystal detector is the difference between the reference arm and the sample arm. Therefore once the signals are balanced a null output is read at D.

Lock-in amplifier is used to reduce the noise in the signal. This is achieved because the lock-in amplifier acts as a narrow-bandpass filter with the ability to change the filtering frequency. The filtering frequency was set to 200 Hz for this experiment.

Experimental results

B (kG)g-factor
13385.522.00787
23384.532.00846
33384.552.00845
43385.0952.0081
μ3384.9242.00823
σ0.412180.00024

Uncertainty Evaluation

The largest source of error in this experiment is the gauss meter and more specifically the Hall probe. The measured magnetic field is highly dependent on the orientation of the probe. If the probe is not exactly perpendicular to the magnetic field the reading is inaccurate. To counteract this problem another gauss meter was used to calibrate the field measurement.

Discussion and Conclusion

This experiment showed that the Landé g-factor for the DPPH sample is close to the g-factor for a free electron. The result confirms the theoretical prediction based on molecular structure. This lab also explained the importance and proper operation of a lock-in amplifier as well as a Gauss meter.