Linear Polarisability interesting problem

[DanSandberg]

Linear Polarisability $$\alpha_{ij}$$ is the amplitude of the electric field induced in the molecule in the i direction given a unit amplitude field in the j direction, hence

$$E_i^{induced} = \alpha_{ij} E_j^{incident}$$
(observing summation convention)

Extending this physical interpretation, $$\beta_{ijk}$$ (X₂) is the amplitude of the electric field induced in the i direction given an unit incident field in the j direction applied after unit incident field in the k direction has already been applied, so

$$E_i^{induced} = \beta_{ijk} E_j^{incident2} E_k^{indicent1}$$

But there is a third-order term (and fourth-order and fifth-order, etc.) so what do the indices represent in these cases?

Also, there is only one applied (indicent) field so what does it mean to say "the incident field in the j direction" compared to "the incident field in the k direction"?

 

[weejee]

Also, there is only one applied (indicent) field so what does it mean to say "the incident field in the j direction" compared to "the incident field in the k direction"?

You just take components in j and k directions from the same incident electric field. I'm not sure about the right "physical picture" concerning $$\beta_{ijk}$$.

 

[DanSandberg]

You just take components in j and k directions from the same incident electric field. I'm not sure about the right "physical picture" concerning $$\beta_{ijk}$$.

I think you are correct about me being wrong. For example, the third-order term is a fourth-rank tensor. So... in that case its _{$$\gamma$$ijkl}. If my picture were right then the i^th direction induced electric field would be the result of the incident field in j^th, k^th and l^th directions but now we've got too many dimensions, unless two letters are degenerative in some way.

Can anyone help me sort this out?

 

[DrDu]

I just want to remark that the polarisability determines not the induced field as a function of the external field but the polarisation as a function of the total field. The total field being the sum of the external and the induced field and the polarization in general not being equal to the induced field but (up to maybe some constants depending on the system of Maxwell equations chosen) $$P(x,t)=-\int_{-\infty}^t dt' j_{ind}(x,t')$$.

 

[weejee]

Indices in $$\beta_{ijk}$$ or $$\gamma_{ijkl}$$ are not necessarily all distinct. We can of course think of something like $$\gamma_{xxxx}$$.

 

[DanSandberg]

Indices in $$\beta_{ijk}$$ or $$\gamma_{ijkl}$$ are not necessarily all distinct. We can of course think of something like $$\gamma_{xxxx}$$.

OKAY! okay. But my question is, what is the SIGNIFICANCE of $$\gamma_{xxxx}$$

So by my earlier (and likely incorrect) attempt to explain it, this would be the constant of proportionality between the incident electric field and the polarizability of the molecule for the third-order interaction, where

P(t)_x1 = gamma_x1x2x3x4} (E_x2(t)+E_x3(t)+E_x4(t))

where the polarizability along direction x₁ is a function of the incident field in directions x₂, x₃, and x₄

where x₁=x₂=x₃=x₄

is this correct?

 

[DrDu]

No, it's not the sum but the product of the fields. beta and gamma describe a non-linear dependence of polarization on the field. If the field is linearly polarized, let's say in x direction,
then
$$P_i=\alpha_{ix}E_x+\beta_{ixx}E_x^2.$$

 

[DanSandberg]

No, it's not the sum but the product of the fields. beta and gamma describe a non-linear dependence of polarization on the field. If the field is linearly polarized, let's say in x direction,
then
$$P_i=\alpha_{ix}E_x+\beta_{ixx}E_x^2.$$

Okay - I understand now. Thank you very much for your patience with me.