Intrinsic and Spectroscopic Electric Quadrupole Moments

[James_1978]

Hi...I am trying to calculate the Spectroscopic Electric Quadrupole Moment. I have found an explanation of this in a book by E. Segre called "Nuclei and Particles". However, I am unable to get the correct answer. I hope someone can help me fill in the blanks.

The energy for the quadrupole term is as follows$$W_{Q} = \frac{1}{2} \psi_{zz} \int \rho(r) \left[\frac{3}{2} z^{2} - \frac{r^{2}}{2}\right] d \tau$$

Here Segre introduces nuclear coordinates

$$eQ = \int \rho_{n} (3 \zeta^{2} - r^{2}) d\tau = 3Q_{\zeta \zeta} - Q_{r r}$$

The nuclear charge distribution is symmetric around $\zeta$. This gives

$$\int \rho_{n} \xi^{2} d \tau = e Q_{\xi \xi} = Q_{\eta \eta}$$

and

$$\int \rho_{n} \xi \eta d \tau = e Q_{\xi \eta} = Q_{\eta \zeta} = Q_{\xi \zeta} = 0$$

The relationship between the x,y,z and \xi, \eta, \zeta give

$$\xi^{2} + \eta^{2} + \zeta^{2} = x^{2} + y^{2} + z^{2} = r^{2}$$

Therefore

$$Q_{r r} = Q_{x x} + Q_{y y} + Q_{z z} = Q_{\xi \xi} + Q_{\eta \eta} + Q_{\zeta \zeta}$$

Moreover

$$z = \xi cos \xi z + \eta cos \eta z + \zeta cos \zeta z$$

With

$$cos^{2} \xi z + cos^{2} \eta z + cos^2 \zeta z = 1$$

Calculating $z^{2}$ and $Q_{z z}$

$$Q_{z z} = Q_{\xi \xi} ( 1 - cos^{2}) + Q_{\zeta \zeta}$$

Then on the next page Segre shows how using the equations above we can get

$$3Q_{z z} - Q_{r r} = Q\left(\frac{3}{2} cos^{2} - \frac{1}{2}\right) = Q P_{2} (cos \theta)$$

I have struggled with this and would appreaciate the help. Also, this is my second post. I am familiar with latex but unsure if it shows up.

 

[AndyW]

Hello there,

Thank you for reaching out for help with your calculation of the Spectroscopic Electric Quadrupole Moment. I am happy to assist you.

Firstly, it is important to understand the basic concept of the electric quadrupole moment. It is a measure of the distribution of electric charge in a system. It is defined as the second moment of the charge distribution around an axis, and is related to the shape of the system.

Now, let's take a look at the equations provided by Segre. The first equation, W_{Q}, is the energy for the quadrupole term. This is a standard equation used to calculate the quadrupole moment. The terms \psi_{zz} and \rho(r) are the electric potential and the charge distribution, respectively.

Next, Segre introduces nuclear coordinates, which are used to simplify the calculation. The first equation, eQ, is the quadrupole moment in terms of the nuclear coordinates \zeta and r. The second equation is the quadrupole moment in terms of the coordinates \xi and \eta, which are related to the x and y coordinates. This is done to take advantage of the symmetry of the nuclear charge distribution around \zeta.

The next set of equations show the relationship between the x,y,z and \xi, \eta, \zeta coordinates. This is important because it allows us to express the quadrupole moment in terms of the x,y,z coordinates, which are easier to work with.

Finally, the last equation shows how the quadrupole moment can be expressed in terms of the Legendre polynomial P_{2}(cos \theta). This is done by using the equations above to substitute for Q_{r r} and Q_{z z}. This equation is important because it allows us to calculate the quadrupole moment using the angle \theta, which is the angle between the nuclear axis and the z-axis.

I hope this explanation has helped you understand the equations better. If you are still struggling, I suggest seeking help from a colleague or a mentor who has experience with these types of calculations. Good luck!