# What is the time interval for photon emission of the electron ?

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#### Ron Hughes

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##### What is the time interval for photon emission of the electron ?
« on: 04/11/2010 14:54:06 »
A photon of a 122nm is absorbed by the hydrogen atom, how long does it take for the electron to re-emit that photon?
From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. Sherlock Holmes.

#### Soul Surfer

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##### What is the time interval for photon emission of the electron ?
« Reply #1 on: 04/11/2010 16:53:33 »
An interesting and deceptively simple question with a complex answer

122nm corresponds to the Lyman alpha excitation of the hydrogen atom that is lifting the electron from the first to the second level.

How long it stays there depends a great deal on the conditions that the atom is in the higher the pressure the more the atom is jostled by others and the more likely it is to emit its quantum and fall back to the ground state.

In a situation where there are many atoms absorbing and emitting this length of time can be measured as the line width of the spectrum line.

The reason is that the shorter the period between absorption and emission the greater is the uncertainty in the energy level.

It is interesting to note that this Lyman alpha spectrum line in the ultra violet is a very important tool for looking at the very early universe because it is emitted well by young active stars and quasars this broad high density spectrum is also absorbed by cool material in between creating the Lyman alpha forest of fine lines which are red shifted into the visible or infra red regions for distant galaxies.   This means that a single spectrum line profile of a distant quasar can tell us about the conditions and locations of many galaxies between the quasar and the earth.
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#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #2 on: 04/11/2010 18:28:20 »
It depends on the intrinsic width of the line: from Heisenberg relation you have

ΔE*Δt ~ h/4π     E = photon energy

so

hΔν*Δt ~ h/4π    ν = photon frequency

Δν*Δt ~ 1/4π

Δt ~ 1/(4π*Δν)

and so from the intrinsic width Δν you can evaluate Δt.

Example: if Δν = 10MHz = 107Hz, Δt ~ 1/(4π*107) ~ 8 ns.

Of course you can find Δν if you have Δλ, remembering that ν = c/λ:

dν = -c/λ2 dλ  →   Δν = ν*Δλ/λ.

« Last Edit: 04/11/2010 18:29:52 by lightarrow »

#### Bored chemist

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##### What is the time interval for photon emission of the electron ?
« Reply #3 on: 04/11/2010 19:06:43 »
The  line width and half life are related to one another but that doesn't really answer the question.
The rate of spontaneous emission of photons is given here.
http://en.wikipedia.org/wiki/Spontaneous_emission
As Souls Surfer says, it's complicated.

#### Ron Hughes

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##### What is the time interval for photon emission of the electron ?
« Reply #4 on: 04/11/2010 19:43:06 »
Thanks for the help.
From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. Sherlock Holmes.

#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #5 on: 05/11/2010 18:01:00 »
The  line width and half life are related to one another but that doesn't really answer the question.
I believed it does.

Quote
The rate of spontaneous emission of photons is given here.
http://en.wikipedia.org/wiki/Spontaneous_emission
where it says that:

<<...where N(0) is the initial number of light sources in the excited state, t  is the time and Γrad  is the radiative decay rate of the transition. The number of excited states N thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value (\frac{1}{e}-time). The radiative decay rate Γrad  is inversely proportional to the lifetime τ12...>>

So I don't understand why it doesn't answer the question (it's possible I didn't grasp well the very question).
« Last Edit: 05/11/2010 18:03:33 by lightarrow »

#### Soul Surfer

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##### What is the time interval for photon emission of the electron ?
« Reply #6 on: 05/11/2010 18:44:39 »
I agree that there is an intrinsic line width for every spectrum line but this depends strongly on the particular transition in the atom involved some decay quickly and are broad band but some last a long time producing very narrow line-widths.  These are the ones that are used in time and frequency standards.  I had a quick search but could not find the intrinsic line width of the Lyman alpha line so could not quote it.
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#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #7 on: 05/11/2010 18:57:18 »
« Last Edit: 05/11/2010 19:21:11 by lightarrow »

#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #8 on: 05/11/2010 19:29:48 »

#### Soul Surfer

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##### What is the time interval for photon emission of the electron ?
« Reply #9 on: 06/11/2010 17:38:07 »
Useful pages explaining the process lightarrow but it does not explain why one transition is long lived and another is very quick.  I am familiar with the very narrow line widths of "forbidden"  transitions and also those associated with locked nuclei in Mossbauer spectra but I am not familiar with anything (other than measurement ) that will tell us what the natural line-width (or decay time) is likely to be
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#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #10 on: 06/11/2010 19:06:48 »
You're right. If I find something I write it here.

Ok, found something. The natural width of a line, in the absence of particular situations (example: metastabile levels) is proportional to the square  of the excited level's dipole moment and to the cube of the energy difference between the two levels at which the transition happens.

Edit: ...and so it doesn't depend on the frequency emitted *only* as the OP question seemed to suggest.
« Last Edit: 07/11/2010 14:30:19 by lightarrow »

#### granpa

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##### What is the time interval for photon emission of the electron ?
« Reply #11 on: 06/11/2010 20:39:45 »
The natural width of a line, in the absence of particular situations (example: metastabile levels) is proportional to the square  of the excited level's dipole moment and to the cube of the energy difference between the two levels at which the transition happens.
i.e. the cube of the frequency of emitted light

for reference:

http://en.wikipedia.org/wiki/Larmor_formula

http://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force

why dont 'indent' tags work
« Last Edit: 07/11/2010 01:30:46 by granpa »

#### Bored chemist

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##### What is the time interval for photon emission of the electron ?
« Reply #12 on: 06/11/2010 21:08:22 »
If someone asks me what the circumference of the Earth is and I tell them it's roughly 3.14 times the diameter I haven't really told them the answer.
If someone asks me the lifetime of an excited state of a hydrogen atom and I tell them they can calculate it from
Δt ~ 1/(4π*Δν)
I am not sure I have helped them much.

#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #13 on: 07/11/2010 14:17:24 »
The question was:
What is the time interval for photon emission of the electron ?
I have answered that question because that was the title, so I thought this was the main question, that is, how one can compute it *in general*.
He also asked, specifically, the time interval for the Lyman-α. I didn't ask that in my first post but in my third.
« Last Edit: 07/11/2010 14:19:20 by lightarrow »

#### Ron Hughes

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##### What is the time interval for photon emission of the electron ?
« Reply #14 on: 07/11/2010 15:29:25 »
Is it possible that the Lyman transitions are due to some type of resonance between the electron and proton?
From a drop of water a logician could infer the possibility of an Atlantic or a Niagara without having seen or heard of one or the other. Sherlock Holmes.

#### lightarrow

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##### What is the time interval for photon emission of the electron ?
« Reply #15 on: 08/11/2010 15:55:33 »
Try to be more specific with the term "resonance between the electron and proton".
« Last Edit: 08/11/2010 15:58:59 by lightarrow »