# Question about fast neutron leakage calculations

#### Warpspeed13

Ok so the equation for the probability that a fast neutron will not leek out of a non infinite mass is
Pfnl= exp(-(Bg)^2 Tth) so Bg is the geometric buckling and for a sphere that value is (pie/r)^2. My question is what is exp? And what is Tth / how do I get the value of it?

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#### mathman

Science Advisor
exp refers to the exponential function: exp(x) means ex. I don't know what Tth means.

#### Warpspeed13

Cool we'll one piece down. We've got 2/3 of the equation. So were in good shape for doing government work maybe not jazz but definitely government work.

#### Astronuc

Staff Emeritus
Science Advisor
Ok so the equation for the probability that a fast neutron will not leek out of a non infinite mass is
Pfnl= exp(-(Bg)^2 Tth) so Bg is the geometric buckling and for a sphere that value is (pie/r)^2. My question is what is exp? And what is Tth / how do I get the value of it?
It would help to have the source of the equation.

I believe Tth is the 'neutron age' in units of (length)2 where length would usually be in cm or m, depending on one's preference.

T would normally be the Greek letter 'tau' and th indicates thermal (it depends on the time it takes to slow from fast to thermal energies).

For the theory, see the following starting with equation 17.
http://mragheb.com/NPRE 402 ME 405 Nuclear Power Engineering/Fermi Age Theory.pdf

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#### Warpspeed13

The source was http://en.m.wikipedia.org/wiki/Six_factor_formula equation five. I was confused because it gave two answeres as to the value of Tth. Also what units is this formulas answer in? Percent or something else?

#### Astronuc

Staff Emeritus
Science Advisor
The source was http://en.m.wikipedia.org/wiki/Six_factor_formula equation five. I was confused because it gave two answeres as to the value of Tth. Also what units is this formulas answer in? Percent or something else?
The 6 parameters are dimensionless.

The values are well described.

η = The number of fission neutrons produced per absorption in the fuel > 1

ε = The fast fission factor = (total number of fission neutrons) / (number of fission neutrons from thermally induced fissions) > 1

the other four parameters are fractions or probabilities less than 1.

Generally, k > 1, but a reactor is 'critical' when k = 1, and k = k * nonleakage/nonloss probability.

The buckling has units of inverse length squared, e.g., cm-2, and age (tau) has units of cm2.

The exponential function gives dimensionless number.

#### Warpspeed13

Sorry if your having to spell this out but which Tth is used in the Pfnl equation? I'm Farley new to this.

Also off topic question: When a D-D fusion occurs is the resulting neutron always at 2.5Mev?

#### Astronuc

Staff Emeritus
Science Advisor
Sorry if your having to spell this out but which Tth is used in the Pfnl equation? I'm Farley new to this.
What does one mean by "which Tth"? The neutron age (to thermal energy) is a characteristic of a particular reactor core design, i.e. its a function of the materials and their cross-section. It is a simple approximation of a rather complex system. A given Tth value would likely be based on the most probable or mean fast neutron energy to the thermal cutoff value, or mean thermal energy. In two-group diffusion theory, Tth = D1/∑1, where the 1 signifies the fast group over which D and ∑ are evaluated.

Farley? Farley is the name of a nuclear plant.

Also off topic question: When a D-D fusion occurs is the resulting neutron always at 2.5Mev?
The d+d reaction has two possible outcomes, either 3He + n or t + p. In the case of the former, the total energy is at least 3.27 MeV, and of that, neutron receives about 2.45 MeV. The energy of the products would be a combination of the initial kinetic energy of the reactants and the binding energy for the products, which must be balanced, i.e., conservation of energy and momentum applies.

#### Warpspeed13

Cool thank you for the help.

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