The classical model scaled inversely with the square of the magnetic field.
If the classical model is correct, small increases in the field lead to much longer confinement times.
If the Bohm model is correct, magnetically confined fusion would not be practical.
Early fusion energy machines appeared to behave according to Bohm's model, and by the 1960s there was a significant stagnation within the field.
The introduction of the tokamak in 1968 was the first evidence that the Bohm model did not hold for all machines.
where B is the magnetic field strength, T is the electron gas temperature, e is the elementary charge, kB is the Boltzmann constant.
It was first observed in 1949 by David Bohm, E. H. S. Burhop, and Harrie Massey while studying magnetic arcs for use in isotope separation.
Fortunately there are exceptions where the diffusion rate is lower, otherwise there would be no hope of achieving practical fusion energy.
Lyman Spitzer considered this fraction as a factor related to plasma instability.
[2] Generally diffusion can be modeled as a random walk of steps of length
In a magnetized plasma, the collision frequency is usually small compared to the gyrofrequency, so that the step size is the gyroradius
On the other hand, if the collision frequency is larger than the gyrofrequency, then the particles can be considered to move freely with the thermal velocity vth between collisions, and the diffusion coefficient takes the form
In this regime, the diffusion is maximum when the collision frequency is equal to the gyrofrequency, in which case
Considering the approximate nature of this derivation, the missing 1/16 in front is no cause for concern.
Regions of higher or lower electric potential result in eddies because the plasma moves around them with the E-cross-B drift velocity equal to E/B.
These eddies play a similar role to the gyro-orbits in classical diffusion, except that the physics of the turbulence can be such that the decorrelation time is approximately equal to the turn-over time, resulting in Bohm scaling.
Another way of looking at it is that the turbulent electric field is approximately equal to the potential perturbation divided by the scale length
is then independent of the scale length and is approximately equal to the Bohm value.
The theoretical understanding of plasma diffusion especially the Bohm diffusion remained elusive until the 1970s when Taylor and McNamara[3] put forward a 2d guiding center plasma model.
The concepts of negative temperature state,[4] and of the convective cells[5] contributed much to the understanding of the diffusion.
The process can be a transport driven by the thermal fluctuations, corresponding to the lowest possible random electric fields.
Due to the long range nature of Coulomb interaction, the wave coherence time is long enough to allow virtually free streaming of particles across the field lines.
The 2D plasma model becomes invalid when the parallel decoherence is significant.
Once ions move across the magnetic field by the gyro-center shift, this movement generates spontaneous electric unbalance between in and out of the plasma.
However this electric unbalance is immediately compensated by the electron flow through the parallel path and conducting end wall, when the plasma is contained in the cylindrical structure as in Bohm's and Simon's experiments.
Simon recognized this electron flow and named it as 'short circuit' effect in 1955.
[8] With the help of short circuit effect the ion flow induced by the diamagnetic drift now becomes whole plasma flux which is proportional to the density gradient since the diamagnetic drift includes pressure gradient.
, (here n is density) for approximately constant temperature over the diffusion region.
The other front coefficient of this diffusion is a function of the ratio between the charge exchange reaction rate and the gyro frequency.
A careful analysis tells this front coefficient for Bohm's experiment was in the range of 1/13 ~ 1/40.