My Math Forum Working Some Physics, Stuck on Algebra

 Algebra Pre-Algebra and Basic Algebra Math Forum

 May 17th, 2015, 06:50 PM #1 Newbie   Joined: May 2015 From: USA Posts: 3 Thanks: 0 Working Some Physics, Stuck on Algebra I'm working on laser physics, specifically steady state inversion and I can't get past this one algebra part... Here are what is known: $N_1= R_1T_1 + R_2T_2A_{21}T_1$ $N_2= R_2T_2$ These two equations are inserted into $N_2/g_2 > N_1/g_1$ When the values for N_1 and N_2 are substituted in, the answer the book gives is $\frac{R_2T_2g_1}{R_1T_1g_2}[1-\frac{g_2}{g_1}A_{21}T_1] > 1$ I do not know how they got to that equation, I'm stuck. When I substitute and rearrange the numbers I get: $\frac{R_2T_2g_1}{g_2(T_1R_1+R_2T_2A_{21}T_1)} > 1$ Anyway, I really can't figure this one out. Any help would be appreciated. If you are wondering, this is suppose to be the equation for Steady-state inversion, the condition necessary for a laser to work. N_1 and N_2 are the numbers of atoms in high and low states, g_1 and g_2 are their respective degeneracies, T1 and T2 is the time period they spend in those states and A_21 is a constant based on the atomic structure... Thanks you much!
May 17th, 2015, 07:45 PM   #2
Math Team

Joined: May 2013
From: The Astral plane

Posts: 2,257
Thanks: 927

Math Focus: Wibbly wobbly timey-wimey stuff.
Quote:
 Originally Posted by JamesBrady I'm working on laser physics, specifically steady state inversion and I can't get past this one algebra part... Here are what is known: $N_1= R_1T_1 + R_2T_2A_{21}T_1$ $N_2= R_2T_2$ These two equations are inserted into $N_2/g_2 > N_1/g_1$ When the values for N_1 and N_2 are substituted in, the answer the book gives is $\frac{R_2T_2g_1}{R_1T_1g_2}[1-\frac{g_2}{g_1}A_{21}T_1] > 1$ I do not know how they got to that equation, I'm stuck. When I substitute and rearrange the numbers I get: $\frac{R_2T_2g_1}{g_2(T_1R_1+R_2T_2A_{21}T_1)} > 1$ Anyway, I really can't figure this one out. Any help would be appreciated. If you are wondering, this is suppose to be the equation for Steady-state inversion, the condition necessary for a laser to work. N_1 and N_2 are the numbers of atoms in high and low states, g_1 and g_2 are their respective degeneracies, T1 and T2 is the time period they spend in those states and A_21 is a constant based on the atomic structure... Thanks you much!
You are both right. Here's the book derivation:
$\displaystyle \frac{R_2T_2}{g_2} > \frac{R_1T_1 + R_2T_2 A_{21}T_1}{g_1}$

$\displaystyle \frac{R_2T_2}{g_2} > \frac{R_1T_1}{g_1} + \frac{R_2T_2 A_{21}T_1}{g_1}$

$\displaystyle \frac{R_2T_2}{g_2} - \frac{R_2T_2 A_{21}T_1}{g_1} >\frac{R_1T_1}{g_1}$

Multiply both sides by $\displaystyle \frac{g_1}{R_1T_1}$
$\displaystyle \frac{R_2T_2g_1}{R_1T_1g_2} - \frac{R_2T_2A_{21}}{R_1} > 1$

Here comes a trick: I'm going to multiply the second term one the LHS by $\displaystyle 1 = \frac{g_1}{g_2} \cdot \frac{g_2}{g_1} \cdot \frac{T_1}{T_1}$. (How do I know to do this? I solved the problem backward. )
$\displaystyle \frac{R_2T_2g_1}{R_1T_1g_2} - \frac{R_2T_2 A_{21}T_1g_1g_2}{R_1T_1g_1g_2} > 1$

Now factor:
$\displaystyle \frac{R_2T_2g_1}{R_1T_1g_2} \cdot \left ( 1 - \frac{A_{21}T_1g_2}{g_1} \right ) > 1$

-Dan

 May 18th, 2015, 06:17 AM #3 Newbie   Joined: May 2015 From: USA Posts: 3 Thanks: 0 Thanks a lot Dan! That helps a lot.
 May 18th, 2015, 08:41 AM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 James, there are 7 variables used: A21, g1, g2, R1, R2, T1, T2 Can you give me an actual/typical value for each? Thanks. I'm trying to understand the book's equation (the (1 - ?) portion): how the results can be > 1 after a multiplication by (1 - ?). Btw, quite an interesting equation... Thanks from topsquark
 May 18th, 2015, 01:33 PM #5 Newbie   Joined: May 2015 From: USA Posts: 3 Thanks: 0 So inside the square brakets: $[1 - \frac{g_2}{g_1}A_{21}T_1] > 1$ is called the "necessary but not sufficient condition" for achieving population inversion. Population inversion is when you have more atoms per unit volume in the high state rather than the low state. This condition can be rearranged to give: $A_{21} < \frac{g_1}{g_2T_1}$ $A_{21}$ is called the Einstein A coefficient, it describe the rate at which atoms decay to the lower level. A typical value for A_21 would be 5*10^7 s^-1. So together with N_1 (in atoms/unit volume), we get $\frac{atoms}{m^3s}$ a typical value for T (or tau actually) is 30ns, this is the "flourescence lifetime", it describes how long an atom lasts in the high or low states. The g values are the degeneracy values for each energy state. They are integers. In quantum physics there are many different ways to get to the same energy levels, if there are three different ways, the degeneracy value is 3. The R values are the pump rates. In order for a laser to work, you have to continuously pump atoms into the high state, there's lots of ways to do it, but they all involve energizing the atoms in some way, unfortunately, I do not have a good typical value since I'm still a novice at this. But if I find one I will message you. Last edited by JamesBrady; May 18th, 2015 at 01:39 PM. Reason: Forgot g values
May 18th, 2015, 04:13 PM   #6
Math Team

Joined: Oct 2011

Posts: 14,597
Thanks: 1038

Quote:
 Originally Posted by JamesBrady But if I find one I will message you.
No need to. You've given me enough to make relevant google searches.
I was kind of interested in reading up, not so much the calculations;
I'll leave that to Tricky Danny

 May 18th, 2015, 08:52 PM #7 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 Another way to get to the book's format. R1, T1 = a, m R2, T2 = b, n g1, g2 = u, v A21 = k So, right before Dan gets tricky, we're in this position: bnu / (amv) - bnk / a > 1 And we want to change above to this format: **** bnu / (amv)[1 - x] > 1 So bnux / (amv) = bnk / a Simplify: x = mvk / u **** bnu / (amv)[1 - mvk / u] > 1 ...same as Dan. Note to James: when working with longish equations with unwieldy variables as is the case here, I find that changing the variables as I did here saves a whole lot of writing (and paper!). Something to remember... Thanks from topsquark

 Tags algebra, physics, stuck, working

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post chrisjtumble Calculus 1 March 18th, 2015 10:16 AM Kinroh Algebra 7 October 23rd, 2014 03:12 PM Kinroh Physics 2 October 1st, 2013 05:16 PM mathkid Algebra 1 January 30th, 2013 09:32 AM kreshnikk1 Algebra 3 January 18th, 2012 10:34 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top