Minimum energy and power, break-even; inertial fusion

[smiler2505]

Homework Statement

A small 50:50 D-T pellet of density p=3g/cm^3 is used for inertial fusion. Estimate minimum energy and power of the laser device required to heat the pellet to T=10^4 eV and to achieve the energy break-even condition

Homework Equations

Lawson criteria: nt>10^20
E=3nTV
W=E/t

The Attempt at a Solution

nt>10^20
t=R/v
T=0.5mv^2 => v=10^6 m/s
=> nR>10^26
V=(4/3)piR^3
=> E=4piR^2.T.nR=2.01.10^12.R^2

My problem is the density; from wikipedia, I've found pR>1g/cm^3; how can that be derived? I haven't seen it used in any of our lectures and google hasn't proved very helpful. Number density n=M/p, but when I use a rough mass (M=2.5*proton mass), I get an absurd number density.

I'm pretty sure my energy and power equations are correct.

Thanks in advance!

 

[anathema]

Thank you for your question. In order to calculate the density of the pellet, we need to use the formula for mass density, which is defined as the mass per unit volume. In this case, we know the density (p) and the volume (V), so we can rearrange the equation to solve for the mass (M):

p = M/V

M = pV

Now, we know that the volume of the pellet is given by V = (4/3)piR^3, where R is the radius of the pellet. We can plug this into the equation for mass to get:

M = p(4/3)piR^3

Now, we need to find the radius of the pellet. We can use the density and the mass to calculate the volume of the pellet (V = M/p). We can then rearrange the equation for volume to solve for the radius:

V = (4/3)piR^3

R = (3V/4pi)^(1/3)

Plugging in the values for V and p, we get:

R = (3(2.5*proton mass)/4pi*3g/cm^3)^(1/3)

R = 1.22*10^-10 cm

Now, we can use this value for the radius to calculate the energy required to heat the pellet to T = 10^4 eV. We can use the formula for thermal energy (E = 3nTV) to do this. Plugging in the values for n, T, and V, we get:

E = 3(10^20)(10^4)(4/3)pi(1.22*10^-10)^3 = 7.02*10^-11 J

To achieve the energy break-even condition, the energy input must be equal to the energy output. This means that the minimum energy of the laser device must be equal to the energy required to heat the pellet. Therefore, the minimum energy of the laser device is also 7.02*10^-11 J.

To calculate the power of the laser device, we can use the formula W = E/t, where W is the power, E is the energy, and t is the time. We know the energy (7.02*10^-11 J), and we can assume a time of 1 second (since it is a minimum