Talk:Lawson criterion

message: Moving original triple product discussion here, it's very poor and doesn't describe the origin of the triple product correctly. The actual origin is the substitition of \langle \sigma v \rangle = 1.1 \times 10 ^{-24} T^2, which is an approximation to the \langle\sigma v\rangle curve, into the Lawson criteria. I will fix this at a later date if it has not already been corrected.
The old version is here for reference:
==The "triple product" n_{\rm e}T\tau_{\rm E}==
A still more useful figure of merit is the "triple product" of density, temperature, and confinement time, n_{\rm e}T\tau_{\rm E}. For most confinement concepts, whether inertial, mirror, or toroidal confinement, the density and temperature can be varied over a fairly wide range, but the maximum pressure attainable is a constant. When that is the case, the fusion power density is proportional to p^2\langle\sigma v\rangle/T^2. Therefore the maximum fusion power available from a given machine is obtained at the temperature where \langle\sigma v\rangle/T^2 is a maximum. Following the derivation above, it is easy to show the inequality
:n_{\rm e} T \tau_{\rm E} \ge \frac{12k_{\rm B}}{E_{\rm ch}}\,\frac{T^2}{\langle\sigma v\rangle} 
For the special case of tokamaks there is an additional motivation for using the triple product. Empirically, the energy confinement time is found to be nearly proportional to n1/3/P2/3. In an ignited plasma near the optimum temperature, the heating power P is equal to the fusion power and therefore proportional to n2/T2. The triple product scales as
:nTτ \propto nT (n1/3/P2/3) \propto nT (n1/3/(n2/T2)2/3) \propto T -1/3
Thus the triple product is only a weak function of density and temperature and therefore a good measure of the efficiency of the confinement scheme.
The quantity \frac{T^2}{\langle\sigma v\rangle} is also a function of temperature with an absolute minimum at a slightly higher temperature than \frac{T}{\langle\sigma v\rangle}.
For the D-T reaction, the physical value is about
:n_{\rm e} T \tau_E \ge 10^{21} \mbox{keV s}/\mbox{m}^3
This number has not yet been achieved in any reactor, although the latest generations of machines have come close. For instance, the TFTR has achieved the densities and energy lifetimes needed to achieve Lawson at the temperatures it can create, but it cannot create those temperatures at the same time. ITER aims to do both.
CnlPepper (talk) 17:39, 12 December 2007 (UTC)
replies: text: I think you are wrong. Different portions of the Lawson criterion can be approximated by different power laws depending on the temperature around which the expansion is made. The sigma-v ~ T2 scaling is a consequence of the assumption of fixed pressure. Besides, it is pretty heavy-handed to delete a whole section just because you find parts of the description poor. Anyway, I won't have time to do much before next week. We'll see what you've got by then. --Art Carlson (talk) 18:23, 12 December 2007 (UTC)
replies: text:  Apologies for being too heavy handed, I brought my bad mood onto wikipedia. I've put it back for now. Though it has already confused some of the students here at Culham. The origin of the T^2 scaling, as far as we have ever enountered, is simply a mathematical approximation to the sig-v curve to within 10\% over the range 10-20 KeV. In Lawson criteria is is assumed that the reactions are occuring at the optimal temperature (~25KeV). In the Triple product this assumption is relaxed. See Tokamaks by John Wesson for example. CnlPepper (talk) 14:11, 13 December 2007 (UTC)
thread_title:  Triple Product 

message: The triple product is listed as being proportional to temperature to the -1/3.  However, the equation that leads to that seems to clearly give temperature to the 7/3, the error coming from two division operations.  The following sentence claims that the triple product is a weak function of temperature.  Could someone explain why they would call this weak or why this error would have lasted so long?  Am I missing something? PSimeon (talk) 05:17, 29 October 2008 (UTC)
replies: text: Good eye. The problem was not the result but the expression used for the power. It's been there forever, but I guess no one ever reads mathematical derivations. --Art Carlson (talk) 06:40, 29 October 2008 (UTC)
thread_title:  Temperature dependence of Triple Product 

message: Substituted at 21:44, 29 April 2016 (UTC)
thread_title: Assessment comment

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thread_title:  External links modified 

message: The lead contains a sentence that says if the Lawson criteria are achieved, "that leads to net energy output." Do we know as a fact that a triple-product result within the boundary defined by Lawson DOES or WILL lead to net energy output? Or would it be more accurate to say "that it MAY lead to net energy output" or "that is necessary for net energy output.<BR>
StevenBKrivit (talk) 21:42, 20 November 2020 (UTC)
thread_title: LEAD