ps1_soln

# ps1_soln - EE/BIOE 597e Problem Set 1 Solutions March 2007...

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Unformatted text preview: EE/BIOE 597e Problem Set 1 Solutions March 2007 Problem 1: (6 points) Why did the application papers distributed in lecture 1 focus exclusively on solid state QR spectroscopy? Is it possible to obtain a pure QR spectra from a liquid? The existence of a QR spectra requires a nonzero electrostatic interaction energy between the nuclear quadrupole moment eQ and the electric field gradient q zz . From equation B6 in the paper by Dehmelt, the classical electrostatic interaction energy is W Q = 1 4 eQq zz 3 2 cos 2 θ − 1 2 , where θ is the angle between the z axis of the coordinate system ( x, y, z ) used to represent the electric field gradient tensor resulting from electronic charges surrounding the nucleus and the z axis of the coordinate system ( x , y , z ) used to represent the electric quadrupole moment tensor of the nucleus. Following Dehmelt equation D1, represent the electrostatic interaction energy as W Q = 1 4 eQ ( q zz ) eff , where ( q zz ) eff is an effective electric field gradient at the center of the nucleus ( q zz ) eff = q zz 3 2 cos 2 θ − 1 2 . Observe that ( q zz ) eff varies between q zz and − q zz / 2 as θ varies between 0 and π/ 2, and vanishes when θ is 54 . 74 ◦ . And so any rotation of the electric field gradient from θ = 0 decreases the value ( q zz ) eff from q zz , and correspondingly, decreases the electrostatic interaction energy W Q . An additional rotation about some other axis will further decrease the value of ( q zz ) eff . In a liquid, where the axis of rotation continuously changes changes due to collisions between molecules, the value of ( q zz ) eff averages to zero. In short, because of the continual tumbling and collisions of the molecules, the electronic charge distribution surrounding the nucleus effectively has spherical symmetry, and the average value of electric field gradient tensor vanishes. As a result, QR experiments are not performed either in liquids or in solids near the melting point, where the molecules start tumbling as in liquids. Problem 2: (6 points) Discuss three reasons why QR spectra are frequently acquired with the sample cooled to 77 K. Reducing the sample temperature increases the signal-to-noise ratio (SNR) of QR measurements through several mechanisms: 1. Increased Population Difference Between Energy Levels: From Dehmelt equation C8, decreasing the sample temperature T increases the population difference n n ≈ N o hν kT (2 I + 1) between two consecutive energy levels, and as a result, increases the signal strength in both CW and pulsed spectroscopy experiments. To understand how increasing the population difference increases the signal strength in a CW spectroscopy experiment, we need to consider several relationships....
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ps1_soln - EE/BIOE 597e Problem Set 1 Solutions March 2007...

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