Talk:Bateman equation

There is both a summation and a product involving index j, which is rather confusing. Would someone please modify this to use a third index variable?

Bateman Equation
The equation seems to have an error in the product term, an error inherited from reference 3. The two methods of noticing the inconsistency in the old formula:

- the decay constant ($$\lambda$$) has units of 1/time, and so the final equation must have equal numbers of $$\lambda$$ terms in the numerator and denominator

- considering the N=2 case, the Bateman equation should reduce to:
 * $$N_2(t) = \lambda_1 N_1 (0) \left ( \frac{e^{-\lambda_1 t}}{\lambda_2 - \lambda_1} + \frac{e^{-\lambda_2 t}}{\lambda_1 - \lambda_2} \right ) + N_2 (0) e^{-\lambda_2 t}$$

instead the previous version of the equation would simplify to:
 * $$N_2(t) = \lambda_1 N_1 (0) \left ( \frac{e^{-\lambda_1 t}}{\lambda_2 - \lambda_1} + \frac{e^{-\lambda_2 t}}{\lambda_1 - \lambda_2} \right ) + \lambda_1 N_2 (0) e^{-\lambda_2 t}$$

where the extra $$\lambda_1$$ in the final term presents both a unit problem and gives a illogical answer at t=0:
 * $$N_2 (t=0) = \lambda_1 N_2 (0)$$

I do not have a source to confirm this math, but it checks out thoroughly in test cases and never will result in a unit problem. Further I will note that that both the new and old are valid solutions to the differential equations, because multiplication by constants will yield the same derivative; so no clarity can be obtained from there.

Since this equation is such a mess, I've added a lot of parentheses and brackets to clarify order of operation and extend of the sums/products. This may violate some style guide and I would welcome improvement in presentation. The rearrangement of the summation over i and product over j is important, as the product should depend on the range of i.

These edits do not explicitly address comment 1 above, but the additional parentheses may clarify the use of the j notation to render the concern accounted for. — Preceding unsigned comment added by 130.132.173.207 (talk) 20:12, 7 August 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just modified 1 one external link on Bateman Equation. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
 * Added archive https://web.archive.org/web/20130927064244/http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf to http://chemistry.sfu.ca/assets/uploads/file/Course%20Materials%2012-1/NUSC%20342/L9.pdf

When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).

Cheers.— InternetArchiveBot  (Report bug) 07:43, 28 October 2016 (UTC)

insert if feel good
E.g. for the simple case of a chain of three isotopes with only the first species initially present, the Bateman equation gives:

$$   A \xrightarrow{\lambda_A} B \xrightarrow{\lambda_B} C $$


 * $$ N_{A}= N_{A_0} e^{- \lambda_{A}t} $$


 * $$ N_{B}= N_{A_0}\frac{ \lambda_{A}}{\lambda_{B}-\lambda_{A}}( e^{- \lambda_{A}t} - e^{- \lambda_{B}t} ) $$


 * $$ N_{C}=N_{A_0} \lambda_{B}\lambda_{A} \left[\frac{ e^{- \lambda_{A}t}} {(\lambda_{A}-\lambda_{B})\lambda_{A}} + \frac{e^{- \lambda_{B}t}} {(\lambda_{B}-\lambda_{A})\lambda_{B}}

+ \frac{1} {\lambda_{A}\lambda_{B}}\right] = N_{A_0} \left[\frac{\lambda_{B} e^{- \lambda_{A}t}} {(\lambda_{A}-\lambda_{B})} + \frac{ \lambda_{A}e^{- \lambda_{B}t}} {(\lambda_{B}-\lambda_{A})} + 1 \right] $$

If two decays have equal rates, the Bateman equation fails because the first introduces a resonant driving term in the differential equation for the second; in this example


 * $$ N_{A}= N_{A_0} e^{- \lambda t} $$


 * $$ N_{B}= N_{A_0}\lambda t e^{- \lambda t}  $$


 * $$ N_{C}=N_{A_0} ( 1 - e^{- \lambda t} - \lambda t e^{- \lambda t}) $$

that may be easily obtained either by direct solution or taking the $$ \lambda_{A}\rightarrow\lambda_{B} $$ limit of the above equations. pietro151.29.249.152 (talk) 11:41, 1 March 2017 (UTC)

I suggest in addition

state that lambda for the last species must set to zero to employ the equation for it

While this can be solved explicitly for i=2, -> While this can be solved easily for i=2,

finally, I have a suspect on the cause of the accuracy loss: the true root is the experimental uncertainty in the rate constants, in particular when the relative order of two of them is not certain (I have started to read the original reference, but I stopped feeling that refer to computing powers that belong to the past)

Implicit assumption NB(0) = 0 in the solution for the simple case of a chain of three isotopes.
In the solution for the case with a chain of three isotopes, the solution implicitly assumes that NB(0) = 0. Since it is an example, I think making this assumption is okay, but it should be explicitly stated in the text. Do others agree? If so, could someone change it? I am myself not familiar with editing wikipedia. 193.110.36.16 (talk) 06:27, 1 February 2019 (UTC)