l D =[kT eo / 4p n e e2 ] 1/2
Where k is Boltzmann’s constant and e is the electronic charge. Substituting the values for those two constants one can write a simpler numerical form:
l D =10 0.84 (ÖT/ Ön)
Technically, this applies only to electrons. The ions will have a Debye length peculiar to the species of ion, e.g. for helium ions one will use Z = 2 and Lithium ions Z = 3. For this ion-species based Debye length we may write:
l D,s =[kT s eo / 4p Z s2 n s e2 ] 1/2
Where the subscript ‘s’ refers to the ion species.
Example Problem: A helium plasma is at a temperature of 10 6 K, and exhibits a number density of 10 10 /m3. Find the Debye length.
For helium plasma we have Z s =2. T s= T He = 10 6 K
n s = n He = 10 10 /m3
l D,s =[(1.38 x 10-23 (10 6 K) (8.85 x 10 -12 F/m) / (4p) 22 (10 10/m3) e2 ] ½
l D,s = 0.30 m
We have: Ñ 2 F = e (n e - n i)/ e o
Adjusting the potential for the case for r >> l D,s we have:
j D = q / 4p eo r (exp –r/ l D)
L = n o l3 D
This definition implies:
L = n o l3 D >> 1
N = 4p n o l3 D / 3
1)For the helium plasma in the example problem, compute the plasma parameter, and the number of particles in the Debye. sphere.
2)Compare your values with the same ones for the solar atmosphere, for which n o = 10 20 /m3 and T = 10,000K.
3)Compare the values for a laser fusion plasma with a particle density n o = 10 29 /m3 and temperature T = 10 7 K) to the plasma of the tail of Earth’s magnetosphere with n o = 10 6 /m3 and T = 10 7 K
4)Can a plasma with n o = 10 6 /m3 be maintained at an electron temperature of 100K? (Hint: Calculate the density limit using the plasma parameter).
5) In the limit 1/N << 1 and 1/N ® 0 we say a plasma is “collisionless”. Do any of the plasmas cited in the previous problems qualify as collisionless?