Question about absorption line spectra

[songoku]

Homework Statement

A beam of light consists of a continuous range of wavelengths. The light passes through a cloud of cool gas. The spectrum of the light emerging from the cloud of cool gas is viewed using a diffraction grating. Explain why this spectrum contains a number of dark lines

Relevant Equations

Absorption line spectra

The answer from answer key is electrons in gas absorb photon energy and causes electron to move to higher energy level. When electrons de-excite, photons emitted in all directions so dark lines occur.

My question is if photons are emitted in all directions when electrons move from higher to lower energy level, is there possibility that some photons are actually emitted in the direction of the screen so the wavelengths that should be missing turn out to be seen on the screen?

Thanks

 

[kuruman]

My question is if photons are emitted in all directions when electrons move from higher to lower energy level, is there possibility that some photons are actually emitted in the direction of the screen so the wavelengths that should be missing turn out to be seen on the screen?

Yes, and you can estimate the fraction of photons of a particular wavlength that reach an area element $dA$ on the screen for any particular absorption line. It's a simple calculation for which you need to know the distance from the source to the screen.

 

[songoku]

Yes, and you can estimate the fraction of photons of a particular wavlength that reach an area element $dA$ on the screen for any particular absorption line. It's a simple calculation for which you need to know the distance from the source to the screen.

Is the formula I = P/A where I is the intensity of the wave?

Thanks

 

[kuruman]

Is the formula I = P/A where I is the intensity of the wave?

Thanks

What constitutes “the wave”?

 

[songoku]

What constitutes “the wave”?

Photon emitted during electron transition

 

[kuruman]

Right, and that photon has a wavelength that is characteristic of that particular transition. Consider the point on the screen where the absorption dark line appears. If there is no cloud of gas, that dark line will not be there. Instead, there will be a bright line of the characteristic wavelength that will have intensity, say, $I_0$.

Now, make some reasonable assumptions about the geometry of the dark line on the screen and the gas cloud-to-screen distance. Some photons will, as you say, be emitted and reach the part of the screen where the dark line appears. Can you estimate the intensity of light at that part of the screen as a fraction of $I_0$? This should be an order of magnitude calculation.

 

[songoku]

Right, and that photon has a wavelength that is characteristic of that particular transition. Consider the point on the screen where the absorption dark line appears. If there is no cloud of gas, that dark line will not be there. Instead, there will be a bright line of the characteristic wavelength that will have intensity, say, $I_0$.

Now, make some reasonable assumptions about the geometry of the dark line on the screen and the gas cloud-to-screen distance. Some photons will, as you say, be emitted and reach the part of the screen where the dark line appears. Can you estimate the intensity of light at that part of the screen as a fraction of $I_0$? This should be an order of magnitude calculation.

The geometry of the dark line on the screen maybe can be assumed to be like small part of surface area of sphere since the photon is emitted to all directions. The distance from gas cloud to screen maybe assumed to be 3 meters.

I am not sure what formula I should use. The one I can think of is I = P/A

Thanks

 

[kuruman]

I am not sure what formula I should use. The one I can think of is I = P/A

Stop thinking in terms of a formula to substitute in. Think of the physical situation instead.

Start by considering that the source of monochromatic scattered light emits photons uniformly over a solid angle $~4\pi$. What fraction of the total number of emitted photons reaches the area on the screen that you have described?

 

[songoku]

Stop thinking in terms of a formula to substitute in. Think of the physical situation instead.

Start by considering that the source of monochromatic scattered light emits photons uniformly over a solid angle $~4\pi$. What fraction of the total number of emitted photons reaches the area on the screen that you have described?

In my scenario, is the intensity $\frac{1}{9}I_{0}$?

Thanks

 

[kuruman]

In my scenario, is the intensity $\frac{1}{9}I_{0}$?

Thanks

I don’t know because you have not explained your scenario in any detail that I can understand.

 

[haruspex]

The distance from gas cloud to screen maybe assumed to be 3 meters.

That's rather close for an interstellar gas cloud, no?
But say the distance is R. What area is the reemitted light spread over when it reaches the spectrometer? Over what area does the spectrometer collect?

 

[songoku]

That's rather close for an interstellar gas cloud, no?
But say the distance is R. What area is the reemitted light spread over when it reaches the spectrometer? Over what area does the spectrometer collect?

Oh sorry, I thought everything was inside the lab, including the gas cloud.

Let say the distance is $R$.

The area the reemitted light spread over when it reaches the spectrometer is $4\pi R^{2}$

I think The spectrometer collects over area that is equal to the area of slit used. Is this correct?

Thanks

 

[kuruman]

I think The spectrometer collects over area that is equal to the area of slit used. Is this correct?

Collects what "over area that is equal to the area of slit used"? Only the scattered light or some other kind of light as well?

 

[songoku]

Collects what "over area that is equal to the area of slit used"? Only the scattered light or some other kind of light as well?

I think there will be more than one monochromatic light being remitted in all directions and there will also be another light that is not absorbed by the gas passing through the slit so there will be multiple light going through the slit.

 

[kuruman]

I think there will be more than one monochromatic light being remitted in all directions and there will also be another light that is not absorbed by the gas passing through the slit so there will be multiple light going through the slit.

You seem to be unsure so let's consider an "experiment" to have something specific to talk about. Imagine being in a room with a window facing the Sun. The window is painted black except for a slit that runs from top to bottom that allows a sliver of light to enter the room. This light is incident on a screen inside the room at distance, say, 2 m from the window. I will ask three questions and give you the first two answers to get you started. You have to answer the third question.

Question 1: What do you see on the screen?
Answer 1: A bright sliver of "white" sunlight with fuzzy edges running from the top of the screen to the bottom.

Now suppose you put a diffraction grating at 1 m from the window so that it reflects the incoming sunlight onto the screen.

Question 2: What do you see on the screen now?
Answer 2: A resolved spectrum that looks like what's on the figure below.

Now imagine that a huge hydrogen cloud appears somewhere between the Sun and your room so that all the sunlight that enters your room has passed through the cloud.

Question 3: How will the above spectrum change and why?

Bonus question: The statement of the problem says that the gas is cool. Why do you think that is?

 

[songoku]

Now imagine that a huge hydrogen cloud appears somewhere between the Sun and your room so that all the sunlight that enters your room has passed through the cloud.

Question 3: How will the above spectrum change and why?

There will be dark lines superimposed on the continuous spectrum because electrons in the cloud absorb lights with energy equal to the difference in certain energy level inside the cloud. The lights being absorbed are the ones with dark lines on the screen.

Bonus question: The statement of the problem says that the gas is cool. Why do you think that is?

So there will be more electrons on the ground state for absorbing the photons

 

[kuruman]

Good. Do you consider your original question answered or not?

 

[songoku]

Good. Do you consider your original question answered or not?

Yes

I am just still curious how to calculate the intensity of reemitted photons.

Thanks

 

[kuruman]

Yes

I am just still curious how to calculate the intensity of reemitted photons.

Thanks

You can only calculate the ratio of intensities with and without the cloud in place. Suppose the intensity of the photons of the resonant frequency is $I_0$ at the slit on the window when there is no cloud. What fraction of that intensity is scattered out of the beam when the cloud is put in place? Think solid angle.

 

[songoku]

You can only calculate the ratio of intensities with and without the cloud in place. Suppose the intensity of the photons of the resonant frequency is $I_0$ at the slit on the window when there is no cloud. What fraction of that intensity is scattered out of the beam when the cloud is put in place? Think solid angle.

I think it would be $\frac{I_{0}}{4\pi}$. Is that correct?

Thanks

 

[kuruman]

I think it would be $\frac{I_{0}}{4\pi}$. Is that correct?

Thanks

Yes.

 

[songoku]

Thank you very much for all the help and explanation kuruman and haruspex