Why do fusion reactors only work for a few seconds?

[rogerk8]

Hi!

I wonder why it is so hard to make a fusion reaction last for more than a few seconds.

I have read a few elementary courses in Plasma Physics but do not remember so much.

What I do rememeber is that they said that whenever a fusion reaction starts within a Tokamak the plasma tends to approach the wall and when it does it of course cools off and that is when the fusion stops.

But do I remember correctly?

I mean, there are other ways to start a fusion reaction.

I am thinking of laser-beams. But the same time-problem seem to excist there.

I am very excited about the ITER-project which I think is about to start.

Reading some information about ITER says that they actually think the output energy will exceed the input energy to such an extent that they, if it wasn't a pure experimental site, could hook it up onto the elctrical grid.

Sounds amazing!

Hope they are right because there is quite much hydrogen here on the planet. Even if the type of hydrogen (Deuterium?) is very rare but yet available in our seas.

Further more, the only "exhaust" is harmless Helium.

Roger

 

[Bill_K]

Reading some information about ITER says that they actually think the output energy will exceed the input energy to such an extent that they, if it wasn't a pure experimental site, could hook it up onto the elctrical grid.

It will be more impressive when the dollar value of the output energy exceeds what it costs to produce it.

Further more, the only "exhaust" is harmless Helium.

Everything in the reactor's surroundings gets struck by the high-energy particles produced, becoming radioactive.

 

[rogerk8]

The radioactivity is at least kept within the reactor.

 

[Astronuc]

The radioactivity is at least kept within the reactor.

Until the reactor is repaired and the activated part removed. Then it must be stored onsite, or shipped to a radioactive waste repository.

 

[Drakkith]

What I do rememeber is that they said that whenever a fusion reaction starts within a Tokamak the plasma tends to approach the wall and when it does it of course cools off and that is when the fusion stops.

If I remember correctly it's something similar to the following:

The issue has to do with instabilities in the plasma. These instabilities make the reactor unable to contain part of the plasma, which is then able to touch the reactor walls. When the hot, ionized plasma touches the walls, it "scorches" the wall and heavy particles from the wall itself are thrown into the plasma. This introduction of heavy elements that cannot fuse into the plasma wrecks the reaction as a whole.

I am thinking of laser-beams. But the same time-problem seem to excist there.

This is by design. The laser pulse needs to deliver its energy in an extremely short time or there simply won't be enough pressure on the fuel pellet to cause fusion. After fusion another pellet is shot into the chamber and ignited. Well, it's supposed to be that way at least. They weren't able to get the fusion to occur correctly, so it never really went anywhere.

Hope they are right because there is quite much hydrogen here on the planet. Even if the type of hydrogen (Deuterium?) is very rare but yet available in our seas.

Too bad we need tritium for it to work, which only occurs in very very very small amounts naturally, decaying with a half life of about 15 years I believe.

 

[the_wolfman]

I wonder why it is so hard to make a fusion reaction last for more than a few seconds.

Lets break things up into magnetic (MFE) and inertial confinement fusion (ICF).

In ICF you use laser or particle beams to compress a fuel pellet to extremely large density. Once its compressed you quickly heat the pellet. Hopefully igniting fusion before the pellet expands and blows itself apart. The duration is the time between initial compression on the the deconstruction of the pellet. By their nature ICF experiments last only fractions of a second. An ICF power-plant will have to repeat this process at a rate of ~10 pellets a second.

MFE experiments are a different story. If fact there are a ton of different types MFE experiments Each one is a different story. In fact the Large Helical Devices, a stellarator in Japan) has run for hours. My understanding is that they could run even longer. But honestly there wasn't much a reason to do so.

Another type of MFE device is the tokamak. They're operation is usually limited to a few seconds or less. This because use induction to drive large (MA) toroidal currents. As such there operational life time is limited by the number of V-S stored in the capacitor banks to drive these currents. In fact, non-solenoidal (non-induction) start up and current drive is a hot area of active tokamak research.

Another thing that limits the operation of some experiments is heat generation. Even a modest experiment routinely creates plasmas with temperatures exceeding 1 million degrees. If you maintain a plasma for more than a few seconds, then you have to worry about the interior wall of you experiment heating up and melting. To avoid this you can actively cool your experiment. To save money most smaller experiments aren't not designed with cooling systems. Keep in mind that the characteristic time scale of a the plasma in most of these experiments is a few microseconds. So you can do a lot of meaningful experiments in a second.

The issue has to do with instabilities in the plasma. These instabilities make the reactor unable to contain part of the plasma, which is then able to touch the reactor walls. When the hot, ionized plasma touches the walls, it "scorches" the wall and heavy particles from the wall itself are thrown into the plasma. This introduction of heavy elements that cannot fuse into the plasma wrecks the reaction as a whole.

This really isn't true. Impurity control is an issue and so is plasma stability. An instability can cause the plasma to disrupt ending the discharge. But over the years we've gotten pretty good at avoiding these (they still happen but less frequently). And then they happen most frequently when we are pushing the limits of our experiments (and understanding). Impurities radiate away energy much more efficiently than hydrogen, cooling the plasma. Actually fusion of DT isn't part of most modern experiments. If fact no operating tokamak currently uses DT fuel. Again this saves a ton of money. A ton. As discussed above, DT fusion releases a ton of neutrons. Before you can put DT in an experiment it has to be designed and build with adequate neutron shielding. And then once you put DT in the experiment, you have to perform most maintenance remotely. Both of these come at an enormous cost. Not to mention that tritium isn't cheep.

Hope they are right because there is quite much hydrogen here on the planet. Even if the type of hydrogen (Deuterium?) is very rare but yet available in our seas.

Too bad we need tritium for it to work, which only occurs in very very very small amounts naturally, decaying with a half life of about 15 years I believe.

.
Everything in the reactor's surroundings gets struck by the high-energy particles produced, becoming radioactive.

The D-T reaction is the easiest reaction to achieve, and our first generation fusion power plants will likely use D-T for this reason. But there are numerous other reactions/fuel cycles including D+He3, D+D catalysed, p+B, etc. Once we get fusion working its a small step from D-T to one of these other reactions. All of these reactions produce fewer neutrons than D+T. In fact p+B produces none. (Technically D-He3 doesn't either, but a burning D-He3 reactor will also fuse some D+D which produces neutrons). The neutrons are what activate the reactor walls. Using a D+D catalysed reaction, there is enough deuterium in the worlds oceans to meet our current energy demands for ~26 billion years. For comparison the universe is only ~14 billion years old.

 

[Drakkith]

This really isn't true. Impurity control is an issue and so is plasma stability. An instability can cause the plasma to disrupt ending the discharge. But over the years we've gotten pretty good at avoiding these (they still happen but less frequently). And then they happen most frequently when we are pushing the limits of our experiments (and understanding). Impurities radiate away energy much more efficiently than hydrogen, cooling the plasma.

Ah, so that's how it works.

The D-T reaction is the easiest reaction to achieve, and our first generation fusion power plants will likely use D-T for this reason. But there are numerous other reactions/fuel cycles including D+He3, D+D catalysed, p+B, etc. Once we get fusion working its a small step from D-T to one of these other reactions. All of these reactions produce fewer neutrons than D+T. In fact p+B produces none. (Technically D-He3 doesn't either, but a burning D-He3 reactor will also fuse some D+D which produces neutrons). The neutrons are what activate the reactor walls. Using a D+D catalysed reaction, there is enough deuterium in the worlds oceans to meet our current energy demands for ~26 billion years. For comparison the universe is only ~14 billion years old.

Sure, but we have to get to the point where we can use something other than D-T first. Until then we're stuck with having to produce tritium and dealing with the activation of the reactor structure.

 

[rogerk8]

Hi Wolfman!

This was extremely interesting to read!

Thank you very much for putting down so much time and energy into explaining it for me!

I'm especially interested in tokamaks (MFE).

Another type of MFE device is the tokamak. They're operation is usually limited to a few seconds or less. This because use induction to drive large (MA) toroidal currents. As such there operational life time is limited by the number of V-S stored in the capacitor banks to drive these currents. In fact, non-solenoidal (non-induction) start up and current drive is a hot area of active tokamak research.

Mega Amps, incredible!

I'm just playing around now...

Using

$$m\frac{dv}{dt}=qvXB$$

Solving the differential equation yields

$$w_c=\frac{|q|B}{m}$$

which is the cyclotron frequency i.e the frequency the charged (observe) particles spin around a perpendicular magnetic field B and with the Langmor radius of

$$r_L=\frac{mv_p}{|q|B}$$

I had to look this one up but I have "always" known that this type of equation is the key to understanding why tokamaks may work at all.

I do however not fully understand what the velocity constant vp (as in perpendicular to B) really mean.

Current drive research sounds very interesting. Where can I apply :)

Another thing that limits the operation of some experiments is heat generation. Even a modest experiment routinely creates plasmas with temperatures exceeding 1 million degrees. If you maintain a plasma for more than a few seconds, then you have to worry about the interior wall of you experiment heating up and melting. To avoid this you can actively cool your experiment. To save money most smaller experiments aren't not designed with cooling systems. Keep in mind that the characteristic time scale of a the plasma in most of these experiments is a few microseconds. So you can do a lot of meaningful experiments in a second.

Very interesting indeed.

Impurity control is an issue and so is plasma stability. An instability can cause the plasma to disrupt ending the discharge. But over the years we've gotten pretty good at avoiding these (they still happen but less frequently). And then they happen most frequently when we are pushing the limits of our experiments (and understanding). Impurities radiate away energy much more efficiently than hydrogen, cooling the plasma. Actually fusion of DT isn't part of most modern experiments. If fact no operating tokamak currently uses DT fuel. Again this saves a ton of money. A ton. As discussed above, DT fusion releases a ton of neutrons. Before you can put DT in an experiment it has to be designed and build with adequate neutron shielding. And then once you put DT in the experiment, you have to perform most maintenance remotely. Both of these come at an enormous cost. Not to mention that tritium isn't cheep.

I seem to remember that there was much talk about instabilities when I took the course (96). But impurity considerations is new to me.

You may gladly tell me more about how you have solved those problems today.

The D-T reaction is the easiest reaction to achieve, and our first generation fusion power plants will likely use D-T for this reason. But there are numerous other reactions/fuel cycles including D+He3, D+D catalysed, p+B, etc. Once we get fusion working its a small step from D-T to one of these other reactions. All of these reactions produce fewer neutrons than D+T. In fact p+B produces none. (Technically D-He3 doesn't either, but a burning D-He3 reactor will also fuse some D+D which produces neutrons). The neutrons are what activate the reactor walls. Using a D+D catalysed reaction, there is enough deuterium in the worlds oceans to meet our current energy demands for ~26 billion years. For comparison the universe is only ~14 billion years old.

Would you mind explaining what a D+D catalysed reaction is?

Sounds very promising in the long run!

If you are up to it you may explain them all :)

In basic terms, of course.

It's a pity that D-T produces so many neutrons (which cannot be confined within the plasma due to no charge).

Take care!

Roger
PS
I am a bit confused though. At one moment you say that DT isn't really a part of modern research but later on you say that D-T is the easiest to achieve. Is there a difference?

 

[rogerk8]

Until the reactor is repaired and the activated part removed. Then it must be stored onsite, or shipped to a radioactive waste repository.

You might be interested in the following links:

http://www.frc.gatech.edu/Transmutation-Reactors.html

and

http://en.wikipedia.org/wiki/Nuclear_transmutation

But most of all I think you should read what the_wolfman above has to say.

Roger

 

[the_wolfman]

Lets start with the easy one:

Would you mind explaining what a D+D catalysed reaction is?

When you fuse D+D there are two possible outcomes with equal probability.
D+D = T+ p
D+D = He-3 + n

Next you can fuse the T and He-3 each with another D. Using the reactions:
D + T = He-4 + n
D + He-3= He-4 + p

When you sum up these for reactions you get the simplified reaction
6D = 2 He-4 + 2n + 2p +43 MeV
We call this chain of reactions catalysed D-D.

I am a bit confused though. At one moment you say that DT isn't really a part of modern research but later on you say that D-T is the easiest to achieve. Is there a difference?

A lot of MFE research is really understanding plasma physics related to confinement. In these experiments we use a surrogate plasma in lieu of DT (often a pure deuterium plasma). The basic underling physics is the same regardless of the "fuel" used. It is cheaper to not use tritium. It's like a engineer designing an air plane. If you want to study aerodynamics you could build a full scale plane and fly it or you could use a scaled replica and put it in a wind tunnel.

If you want to study a self heated burning plasma you have to use DT fuel. ITER which is designed to study just that is going to eventually use DT fuel. Like all other experiments, they are going to start with a surrogate fuel and work out all the kinks before they switch to DT.

I'm just playing around now...

Using

mdvdt=qvXB


Solving the differential equation yields

wc=|q|Bm


which is the cyclotron frequency i.e the frequency the charged (observe) particles spin around a perpendicular magnetic field B and with the Langmor radius of

rL=mvp|q|B


I had to look this one up but I have "always" known that this type of equation is the key to understanding why tokamaks may work at all.

I do however not fully understand what the velocity constant vp (as in perpendicular to B) really mean.

This one is a little tricky. If you're looking at an individual particle its the velocity of particle perpendicular to B. However in plasma physics with often use a fluid model where the individual particles have a distribution of energies (often close to a Maxwellian). In these cases we care about some averaged gory-radius. We often use the average thermal velocity of the particles, but in some cases its more appropriate to use the ion sound speed.

 

[rogerk8]

Hi Wolfman!

Please bear with me and treat me like your humble apprentice :)

When you fuse D+D there are two possible outcomes with equal probability.
D+D = T+ p
D+D = He-3 + n

Next you can fuse the T and He-3 each with another D. Using the reactions:
D + T = He-4 + n
D + He-3= He-4 + p

When you sum up these for reactions you get the simplified reaction
6D = 2 He-4 + 2n + 2p +43 MeV
We call this chain of reactions catalysed D-D.

Jesus, 43MeV!

Let's begin from the start by calculating the temperature and speed a proton would have at these energies.

$$E_{AV}=\frac{1}{4}mv^2=\frac{1}{2}kT$$

Solving for speed yields

$$v=\sqrt{\frac{2kT}{m}}$$

or

$$v=\sqrt{\frac{4E_{AV}}{m}}$$

Solving for temperature yields

$$T=\frac{2E_{AV}}{k}$$

I'm not sure of the details in the first equation because I think the kinetic energy should equal

$$Ek=\frac{mv^2}{2}$$

but according to Francis F. Cheng it doesn't (in the case of a Maxwellian distribution, anyway).

The first equation comes from the Maxwellian distribution of velocity or

$$f(v)=A\exp{(-\frac{mv^2/2}{kT})}$$

where, strangely enough, Ek is set different from kT.

Should they not always follow as has been described in the beginning?

Anyway, the number of particles per square meters is the velocity integration over f(v) (sorry, I do not know how to write integrals in tex)

$$n=\int_{-\infty,\infty}{f(v)dv}$$

Yielding

$$A=n\sqrt{\frac{m}{2\pi kT}}$$

But this means an integral of type

$$\int{\exp{(-ax^2)}dx}$$

will have to be calculated. And I do not know how!

Anyway, Beta tells me that the result should be as above.

The constant A then magically becomes

$$A=n\sqrt{\frac{m}{2\pi kT}}$$

Let's calculate the speed and temperature of a 43MeV "warm" proton (for simplicity).

$$k=1,38*10^{-23}$$

$$e=1,6*10^{-19}$$

$$m_p=1,67*10^{-27}$$

Using the above equations and Eav:

$$v=128*10^6 m/s$$

Which is relativistic...

And the temperature is

$$T=10^{12}K$$

This has to be wrong because we are not only talking millon degrees here, we are talking trillion degrees! :D

Finally, let's calculate the Magnetic Flux Density, B, in Teslas for confining such a fast particle while not touching the wall of the say 2 meters wide tokamak chamber:

The earlier formula is repeated here for convenience

$$R_{L}=\frac{mv}{|q|B}$$

which gives

$$B=\frac{mv}{|q|R_{L}}$$

Setting

$$q=+e$$

and

$$R_L=1$$

yields

$$B=1,34T$$

which however does not seem so much (ordinary transformer irons are often designed at some 1,6T).

But due to known istabilities we might not want the plasma to be so close to the tokamak wall.

Decreasing the Langmor radius some 10 times would yield a neccesary B of 16T.

How far from the truth am I?

A lot of MFE research is really understanding plasma physics related to confinement. In these experiments we use a surrogate plasma in lieu of DT (often a pure deuterium plasma). The basic underling physics is the same regardless of the "fuel" used. It is cheaper to not use tritium. It's like a engineer designing an air plane. If you want to study aerodynamics you could build a full scale plane and fly it or you could use a scaled replica and put it in a wind tunnel.

So this means that you are experimenting with a plasma that you dont' really expect to ignite, so to speak?

If you want to study a self heated burning plasma you have to use DT fuel. ITER which is designed to study just that is going to eventually use DT fuel. Like all other experiments, they are going to start with a surrogate fuel and work out all the kinks before they switch to DT.

This is very interesting.

This one is a little tricky. If you're looking at an individual particle its the velocity of particle perpendicular to B. However in plasma physics with often use a fluid model where the individual particles have a distribution of energies (often close to a Maxwellian). In these cases we care about some averaged gory-radius. We often use the average thermal velocity of the particles, but in some cases its more appropriate to use the ion sound speed.

I still don't understand the perpendicular velocity, vp as I have denoted it. How can there be a perpendicular velocity at all? Does it come from collisions or is it just a way to describe the x and y-components while gyrating?

The next bold part must say gyro-radius, right?

Finally, I tried Wikipedia regarding "ion sound speed" but found nothing.

So I would very much like an explanation of that.

Thank you for your reply!

Roger

 

[mfb]

Those 43 MeV are distributed on all reaction products (in 4 reactions!). You don't get a single proton with 43 MeV.
You are right that fusion products have an energy way above the thermal energy in the plasma - that's why they can heat the plasma.

So this means that you are experimenting with a plasma that you dont' really expect to ignite, so to speak?

Right. Most experiments are mainly interested in the plasma and its stability, you don't need fusion reactions to study that. With a pure deuterium plasma, you have nearly no neutron flux, so activation of the reactor materials is easier to handle.

How can there be a perpendicular velocity at all? Does it come from collisions or is it just a way to describe the x and y-components while gyrating?

The plasma is hot.

 

[Romulo Binuya]

ITER, at a cost of 15 Billion Euros wants to fuse half a gram of hydrogen isotopes for at least 1000 seconds. Plasma confinement could be relatively easy to achieve, but I guess they will find it difficult to stabilize the quantum tunneling process to achieve the desired rate of proton fusions.

 

[mfb]

"stabilize the quantum tunneling process" does not make sense. If you have a plasma with good macroscopic parameters (density, temperature, composition), you get fusion.

 

[rogerk8]

Let's try to calculate the current needed to generate 16T in vacuum (for simplicity):

Reusing the formula for an ordinary toroidal iron core

$$B=\mu_r\mu_0\frac{NI}{l_m}$$

where

$$\mu_r=1$$

and

$$\mu_0=4\pi*10^{-7}$$

Rearraging gives

$$I=\frac{l_mB}{N\mu_0}$$

Putting

$$B=16T$$

and the tokamak radius (chamder radius=1m)

$$R_t=1+2m$$

which gives the mean magnetic length

$$l_m=2R_t*\pi=20m$$

which yields

$$I=255MA/N$$

This also sounds wrong.

Even though the turns (N) might be quite few to avoid resistive losses.

Roger

 

[rogerk8]

Summarizing the formulas:

$$m\frac{dv}{dt}=qvXB$$

Solving the differential equation yields

$$w_c=\frac{|q|B}{m}$$

which is the cyclotron frequency i.e the frequency the charged (observe) particles spin around a perpendicular magnetic field B and with the Langmor radius of

$$r_L=\frac{mv}{|q|B}$$

Using

$$E_{AV}=\frac{1}{4}mv^2=\frac{1}{2}kT$$

Solving for speed yields

$$v=\sqrt{\frac{2kT}{m}}$$

or

$$v=\sqrt{\frac{4E_{AV}}{m}}$$

Solving for temperature yields

$$T=\frac{2E_{AV}}{k}$$

I'm not sure of the details in the first equation because I think the kinetic energy should equal

$$Ek=\frac{mv^2}{2}$$

but according to Francis F. Cheng it doesn't (in the case of a Maxwellian distribution, anyway).

The first equation comes from the Maxwellian distribution of velocity or

$$f(v)=A\exp{(-\frac{mv^2/2}{kT})}$$

the number of particles per square meters is the velocity integration over f(v)

$$n=\int_{-\infty,\infty}{f(v)dv}$$

Yielding

$$A=n\sqrt{\frac{m}{2\pi kT}}$$

Let's calculate the speed and temperature of a 43MeV "warm" proton (for simplicity).

$$k=1,38*10^{-23}$$

$$e=1,6*10^{-19}$$

$$m_p=1,67*10^{-27}$$

Using the above equations and Eav:

$$v=128*10^6 m/s$$

Which is relativistic...

And the temperature is

$$T=10^{12}K$$

Let's calculate the Magnetic Flux Density, B, in Teslas for confining such a fast particle while not touching the wall of the say 2 meters wide tokamak chamber:

The earlier formula is repeated here for convenience

$$r_{L}=\frac{mv}{|q|B}$$

which gives

$$B=\frac{mv}{|q|r_{L}}$$

Setting

$$q=+e$$

and

$$r_L=1$$

yields

$$B=1,34T$$

But due to known istabilities we might not want the plasma to be so close to the tokamak wall.

Decreasing the Langmor radius some 10 times would yield a neccesary B of 16T.

Remembering that

$$w_c=\frac{|q|B}{m}$$

and

$$r_L=\frac{mv}{|q|B}$$

yields that

$$v=w_c*r_L=v$$

Trying to calculate the current for generating 16T in vacuum (for simplicity):

$$B=\mu_r\mu_0\frac{NI}{l_m}$$

where

$$\mu_r=1$$

and

$$\mu_0=4\pi*10^{-7}$$

Rearraging gives

$$I=\frac{l_mB}{N\mu_0}$$

Putting

$$B=16T$$

and the tokamak radius (chamder radius=1m)

$$R_t=1+2m$$

which gives the mean magnetic length

$$l_m=2R_t*\pi=20m$$

which yields

$$I=255MA/N$$

 

[rogerk8]

Hi Wolfman!

What do you think of this procedure:

Starting up a proton-based fusion reactor:

We know that a proton will gyrate around a perpendicular magnetic field, B.

So we will need a magnetic field to confine the plasma.

We cannot however use the basic "cyclotron-equation" to determine speed.

Just note that the proton will gyrate with the Langmor radius depending on speed (and B).

But we can aim at a certain average kinetic energy (and thus temperature).

With the use of the energy-equation above we can then calculate speed.

Putting this speed into the "cyclotron-equation" then gives the Langmor radius.

Which we must make sure is less than the tokamak chamber radius.

Realising this gives the minimum B required.

It is interesting to note that the amount of B does not affect speed at all!

Only the cyclotron frequency and the Langmor radius.

"All" we have to do now is to generate this B (while injecting our fuel).

Use of the toroidal formula for B gives an obscene amount of Amps :)

How far from the truth am I?

Roger

 

[mfb]

A magnetic field of 1T leads to ~3m radius per GeV momentum for hydrogen. Hydrogen with an energy of 10 MeV has 137 MeV of momentum, so it has a radius of ~40cm. Should be fine. The wall is cooled anyway (this is how the reactor extracts useful energy!)
Helium is even easier to confine, as it has twice the charge and gets the a similar momentum in the reactions (exactly the same in D+D->He+p).
A magnetic field of 1 T is not hard to produce - the LHC experiments have stronger fields.

@rogerk8: Why do you put all those empty lines in your posts? This just makes the posts longer than they really are.

 

[rogerk8]

A magnetic field of 1T leads to ~3m radius per GeV momentum for hydrogen. Hydrogen with an energy of 10 MeV has 137 MeV of momentum, so it has a radius of ~40cm. Should be fine. The wall is cooled anyway (this is how the reactor extracts useful energy!)
Helium is even easier to confine, as it has twice the charge and gets the a similar momentum in the reactions (exactly the same in D+D->He+p).
A magnetic field of 1 T is not hard to produce - the LHC experiments have stronger fields.

@rogerk8: Why do you put all those empty lines in your posts? This just makes the posts longer than they really are.

Hi mfb!

Thanks for your reply!

This is very fun!

A magnetic field of 1T leads to ~3m radius per GeV momentum for hydrogen.

Using

$$v=\sqrt{\frac{4E_{AV}}{m}}$$

and

$$r_L=\frac{mv}{|q|B}$$

I get approximatelly

$$v=600,000km/s (>c)$$

and

$$r_L=6m$$

What am I doing wrong?

Hydrogen with an energy of 10 MeV has 137 MeV of momentum.

Please explain to me like I was a child the diffence between the "energy" and the "momentum".

Because I really don't get it.

Actually, the Maxwellian distribution equation

$$f(v)=A\exp{(-\frac{mv^2/2}{kT})}$$

really puzzles me because there is obviously a difference between

$$mv^2/2$$

and

$$kT$$

While at the same time

$$E_{AV}=\frac{1}{4}mv^2=\frac{1}{2}kT$$

seem to hold.

The wall is cooled anyway (this is how the reactor extracts useful energy!)

Thanks for telling me. I have actually not thought so much about that "trivial" part! :)

Finally, regarding complaints about my writing style I just want to say that I am a Von Neumann machine of flesh and blood ;)

Roger

 

[mfb]

What am I doing wrong?

You are using nonrelativistic formulas for relativistic particles.

Please explain to me like I was a child the diffence between the "energy" and the "momentum".

They are two different concepts, like mass and velocity. Their meaning cannot be shown by showing differences. "Where is the difference between mass and velocity?"
Non-relativisticly, $E=\frac{1}{2}mv^2$ and $p=mv$.

The Maxwell distribution is not relevant for fusion products. They are not in a thermal equilibrium.

 

[rogerk8]

You are using nonrelativistic formulas for relativistic particles.

They are two different concepts, like mass and velocity. Their meaning cannot be shown by showing differences. "Where is the difference between mass and velocity?"
Non-relativisticly, $E=\frac{1}{2}mv^2$ and $p=mv$.

The Maxwell distribution is not relevant for fusion products. They are not in a thermal equilibrium.

You are not that good of a teacher. :)

I fully understand the difference between E and p as you describe it.

But how is it relevant?

And how do I calculate the actual speed when it approaches or even "exceeds" the speed of light?

I find no clues at Wikipedia.

The Maxwell distribution is not relevant for fusion products.

This is obviously wrong according to Wolfman above.

Roger

 

[mfb]

You are not that good of a teacher. :)

That's your opinion :).

I fully understand the difference between E and p as you describe it.

But how is it relevant?

I have no idea why you think some "difference" is relevant here.
The radius of charged particles in the magnetic field depends on the momentum of those particles (and the charge). I calculated the momentum of a proton with an energy of 10 MeV (=typical energy after a fission reaction).

And how do I calculate the actual speed when it approaches or even "exceeds" the speed of light?

I find no clues at Wikipedia.

Did you try special relativity?
$E=\gamma m c^2$, $p=\gamma m v$, $E^2=p^2c^2 + m^2 p^4$ with $\displaystyle \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
E=energy, p=momentum, m=mass, c=speed of light
This allows to convert between energy, momentum and velocity.

This is obviously wrong according to Wolfman above.

It is not. The plasma as a whole follows the distribution (well, approximately) with a temperature of the order of 100 million K, the high-energetic fusion products immediately after fusion do not.

 

[rogerk8]

That's your opinion :).

I have no idea why you think some "difference" is relevant here.
The radius of charged particles in the magnetic field depends on the momentum of those particles (and the charge). I calculated the momentum of a proton with an energy of 10 MeV (=typical energy after a fission reaction).Did you try special relativity?
$E=\gamma m c^2$, $p=\gamma m v$, $E^2=p^2c^2 + m^2 p^4$ with $\displaystyle \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
E=energy, p=momentum, m=mass, c=speed of light
This allows to convert between energy, momentum and velocity.It is not. The plasma as a whole follows the distribution (well, approximately) with a temperature of the order of 100 million K, the high-energetic fusion products immediately after fusion do not.

Now you can be called a teacher! :)

Thank you!

The correct expression for E should however be

$$E=\sqrt{(pc)^2+(mc^2)^2}$$

Which for

$$v=0$$

and

$$p=\gamma mv$$

reduces to

$$E=mc^2$$

Right?

It would however be nice to fully understand the expression for E above.

But you have given me a new insight.

Due to the term

$$(pc)^2=(\gamma mvc)^2$$

in the expression for energy above, E can be related to speed, v, in a relativistic way.

Desperatelly wanting a complete analytic expression:

$$E=\sqrt{(pc)^2+(mc^2)^2}=\sqrt{(\gamma mvc)^2+(mc^2)^2}=\sqrt{(mc\frac{v}{\sqrt{1-\frac{v^2}{c^2}}})^2+(mc^2)^2}$$

Which means that the speed we observe, v', is

$$v'=\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}$$

or

$$v=\frac{v'}{\sqrt{1+\frac{v'^2}{c^2}}}$$

So if I calculated 600,000km/s it really should be 268,000km/s?

Roger

 

[mfb]

The correct expression for E should however be

$$E=\sqrt{(pc)^2+(mc^2)^2}$$

That is just the square root on both sides. It is equivalent to the formula I posted.

Which for $v=0$ and $p=\gamma mv$ reduces to $E=mc^2$
Right?

Right (I compressed the quote a bit).

E can be related to speed, v, in a relativistic way.

I would use $E=\gamma m c^2$ to do this.

Which means that the speed we observe, v', is

$$v'=\frac{v}{\sqrt{1-\frac{v^2}{c^2}}}$$

or

$$v=\frac{v'}{\sqrt{1+\frac{v'^2}{c^2}}}$$

I don't see why this should be true.

So if I calculated 600,000km/s it really should be 268,000km/s?

600,000km/s with non-relativistic formulas give E_kin=2mc² or E_total=3mc² => γ=3. This corresponds to v=sqrt(8/9)c or v=283,000km/s.

 

[rogerk8]

That is just the square root on both sides. It is equivalent to the formula I posted.

You don't give me much credit, do you? :)

If you scroll to your post, you can see that you happened to use p instead of c in the formula.

I just corrected that and added the square root for appearance.

600,000km/s with non-relativistic formulas give E_kin=2mc² or E_total=3mc² => γ=3. This corresponds to v=sqrt(8/9)c or v=283,000km/s.

This is interesting.

Above you say that you would use

$$E=\gamma mc^2$$

to calculate v from E in a relativistic way.

Then you thankfully use my example and simply put

$$\gamma=2$$

adding yet another gamma for the always present "rest energy", right?

But how can gamma equal 2 in the first place?

Looking at it from my point of view you seem to relate gamma to v/c in a linear way.

I can't see why this is possible.

By the way, I understand now that the masses above are "rest-masses" which gives the formulas.

But shouldn't the square root read

$$E=\sqrt{(pc)^2+(\gamma mc^2)^2}$$

Or is the second term always "rest energy" (gamma=1)

And where does the first term apply in our discussion?

I hope you don't feel that I'm asking too many questions.

Getting back to the Maxwellian distribution of speed I now think I understand what the distribution function actually means. Correct me if I'm wrong but doesn't it mean that the most probable speed is when

$$mv^2/2=kT$$

This while there are other speeds too, both higher and lower than "kT" and while their "temperature" could be the same, they are less probable to have that temperature.

Is something like this right?

Roger
PS

$$E=\gamma mc^2$$

Is actually news to me because this means that energy approaches infinity when the speed approaches the speed of light.

I have heard people telling me that the speed of light is impossible to reach energy-wise, and now I know why.

Thank you!

 

[mfb]

You don't give me much credit, do you? :)

If you scroll to your post, you can see that you happened to use p instead of c in the formula.

Oh sorry, you are right.

Then you thankfully use my example and simply put$\gamma=2$

No I don't. I calculate the total energy to be $E_{total}=\frac{1}{2}mv^2 + mc^2$ (with the non-relativistic formula to reconstruct your 600,000km/s). Afterwards, I calculate γ as $\gamma=\frac{E_{total}}{mc^2}$.

Looking at it from my point of view you seem to relate gamma to v/c in a linear way.

No.

By the way, I understand now that the masses above are "rest-masses" which gives the formulas.

"Mass" is always rest mass, apart from ancient textbooks and bad TV documentations.

But shouldn't the square root read
$$E=\sqrt{(pc)^2+(\gamma mc^2)^2}$$
Or is the second term always "rest energy" (gamma=1)

The second term is always the rest energy, exactly.

And where does the first term apply in our discussion?

That leads to the correct relativistic kinetic energy.

Getting back to the Maxwellian distribution of speed I now think I understand what the distribution function actually means. Correct me if I'm wrong but doesn't it mean that the most probable speed is when

$$mv^2/2=kT$$

This while there are other speeds too, both higher and lower than "kT" and while their "temperature" could be the same, they are less probable to have that temperature.

Is something like this right?

Right - for particles in thermal equilibrium.
Fusion products are not in thermal equilibrium.

$$E=\gamma mc^2$$

Is actually news to me because this means that energy approaches infinity when the speed approaches the speed of light.

I have heard people telling me that the speed of light is impossible to reach energy-wise, and now I know why.

Thank you!

:)

 

[rogerk8]

I calculate the total energy to be $E_{total}=\frac{1}{2}mv^2 + mc^2$ (with the non-relativistic formula to reconstruct your 600,000km/s). Afterwards, I calculate γ as $\gamma=\frac{E_{total}}{mc^2}$.

This is beautiful!

You actually calculate gamma as a "deviation" over maximum "rest-energy" content, i.e

$$\gamma=\frac{E_{kin}+E_{stat}}{mc^2}$$

Interesting.

"Mass" is always rest mass, apart from ancient textbooks and bad TV documentations.

I like your convention!

That leads to the correct relativistic kinetic energy.

Let's analyse this.

$$E_{tot}=\sqrt{(\gamma mvc)^2+(mc^2)^2}$$

If the first term relates to kinetic energy, why have you just added E_kin with E_stat above?

Fusion products are not in thermal equilibrium.

Please explain why this is not the case and what that mean.

Because if I understand you correctly, my thoughts about the Maxwellian distribution is right.

Roger

 

[mfb]

If the first term relates to kinetic energy, why have you just added E_kin with E_stat above?

That was the nonrelativistic formula, which is wrong for energies we discuss here. I just wanted to use your example, so I used the wrong formula to work backwards.

Please explain why this is not the case and what that mean.

The plasma has a temperature of ~100 million K, corresponding to an average kinetic energy of the order of 10keV and the most likely energy is similar to that. Those particles are in equilibrium, they follow the Maxwell-Boltzmann distribution.
In addition to the plasma, there are a few (!) fusion products with energies of MeV. They are not in equilibrium - they quickly lose their energy in the plasma until they reach ~10 keV.

 

[the_wolfman]

Roger if you want to do some back of the envelope calculations for ITER I recommend using a magnetic field of ~5T and a bulk plasma temperature of 10keV.

For most fusion applications the energies are low enough that non-relativistic formulas for the speed and gyroradius are ok (You can check this of course). There are exceptions but they're not important for the current discussion.

Also a word of caution. The idea that a charged particle gyrates around a "magnetic field line" is the basis for magnetic confinement but it is far from the end all be all. It is often important that the gyroradius be much smaller than the minor radius of a device (if you want you can check this for ITER which has a minor radius of 2m), but the size of a fusion reactor is ultimately determined by radial transport and stability. Remember we want to reach 100 million degrees in the core of the plasma. However stability and transport limit how steep of a pressure gradient we can have. A very crude way to determine the minimum size of a reactor would be to divide 100 million degrees by the maximum gradient.

 

[rogerk8]

That was the nonrelativistic formula, which is wrong for energies we discuss here. I just wanted to use your example, so I used the wrong formula to work backwards.

And I thank you for this!

The plasma has a temperature of ~100 million K, corresponding to an average kinetic energy of the order of 10keV and the most likely energy is similar to that. Those particles are in equilibrium, they follow the Maxwell-Boltzmann distribution.
In addition to the plasma, there are a few (!) fusion products with energies of MeV. They are not in equilibrium - they quickly lose their energy in the plasma until they reach ~10 keV.

This sounds reasonable. I think I now understand what you mean.

Skipping more relativistic questions I will move on to trying to understand how to confine a fusionable plasma and what is required.

You mentioned that a magnetic field of 1T was not so hard to make.

I think I have used the wrong formula.

But in principle

$$B=\mu_0\frac{NI}{l_m}$$

should hold.

Visualising the tokamak or LHC

$$l_m$$

might not need to be more than a couple of decimeters while you could use several coils in series.

So let's say 0.5m (and B=1T) then

$$I=\frac{l_mB}{N\mu_0}=400kA/N$$

Keeping the resistive losses relatively low we could easily use 100 turns which equals to

$$I=4kA$$

How close am I to the truth?

Roger

 

[mfb]

Some kA is similar to the current in the LHC magnets, they are designed to provide a field of ~8T.

 

[rogerk8]

Roger if you want to do some back of the envelope calculations for ITER I recommend using a magnetic field of ~5T and a bulk plasma temperature of 10keV.

For most fusion applications the energies are low enough that non-relativistic formulas for the speed and gyroradius are ok (You can check this of course). There are exceptions but they're not important for the current discussion.

Also a word of caution. The idea that a charged particle gyrates around a "magnetic field line" is the basis for magnetic confinement but it is far from the end all be all. It is often important that the gyroradius be much smaller than the minor radius of a device (if you want you can check this for ITER which has a minor radius of 2m), but the size of a fusion reactor is ultimately determined by radial transport and stability. Remember we want to reach 100 million degrees in the core of the plasma. However stability and transport limit how steep of a pressure gradient we can have. A very crude way to determine the minimum size of a reactor would be to divide 100 million degrees by the maximum gradient.

Hi Wolfman!

Thank you for these data.

I am much obliged!

First, my calculations regarding B(I) is somewhat wrong. A formula between

$$B=\mu_0\frac{NI}{lm}$$

inside a long solenoid or toroid and

$$B=\mu_0\frac{NIA}{2\pi r^3}=\mu_0\frac{NIR^2}{2r^3}=\mu_0\frac{NI}{2R}$$

in the center of a short solenoid where the distance from top of coil and onto z-axis, r, has been set to R (which might not be possible).

Anyway, usinq R=2m and my estimated l_m=0,5m gives

$$\frac{B(long solenoid)}{B(short solenoid)}=8$$

Let's say that the constant is 4. Then I(5T) would equal

$$I=\frac{B4l_m}{\mu_0N}=8MA/N$$

Now I will leave this part once and for all (maybe except for calculating an exact formula).

Plasma pressure, p, is defined by

$$p=nkT$$

where n is the particle density.

For an isoterm (whatever that means) plasma we then have

$$\nabla p=kT\nabla n$$

More of this later...

kT seems related to eV according to Cheng:

$$eV=kT$$

So calculating the temperature of a 10keV hot plasma yields:

$$T=\frac{10^3*1,6^{-19}}{1,38^{-23}}$$

or 116MK~100MK

Returning to the Maxwellian distribution the average energy however relates to kT and v:

$$E_{av}=mv^2/4=kT/2$$

where there is a kT/2 for each degree of freedom (three in my amateur book).

And the most probable speed for two protons (~deuterium) is

$$v=\sqrt{\frac{2kT}{2m_p}}=\sqrt{\frac{kT}{m_p}}$$

or 980000m/s~1000km/s

Which indeed is non-relativistic...

The Larmor radius is

$$r_L=\frac{2m_pv}{|2q|B}=\frac{m_pv}{|q|B}$$

or 2mm(?)

And the cyclotron frequency is

$$w_c=\frac{|2q|B}{2m_p}=\frac{|q|B}{m_p}$$

or 480Mrad/s.

And now I will have to go to bed :)

Tomorrow I will wright about drifts in a plasma (=transport?)

Roger

 

[rogerk8]

My calculations regarding B(I) is still somewhat wrong while using a formula for a long solenoid or toroid (where l_m is included).

The below formula is for a short solenoid where r indicates the distance from top of coil to z-axis

$$B=\mu_0\frac{NIA}{2\pi r^3}=\mu_0\frac{NIR^2}{2r^3}=\mu_0\frac{NI}{2R}$$

the latest term comes from the assumption that r can be set to R in the middle of the short solenoid (which might not be possible).

Anyway, usinq R=2m I(5T) would equal

$$I=\frac{2RB}{\mu_0N}=16MA/N$$

As far as I have seen in pictures the coils are stacked some 1m apart around the tokamak and supposing each coil do not have more turns than some 10 (due to resistive losses) the current through them all connected in series would be some 2MA.

However, this is not the "important" part. The important part is that the B-field will be curved and non-uniform along the

$$\phi-axis$$

Which introduces a gradient in the B-field. I will get back to this later on.

Now I will continue to work :)

Roger
PS
It was too late to edit the above...

 

[rogerk8]

Now I wish to somewhat prove the above formula.

Consider a current carrying loop in vacuum then these four equations apply:

$$\nabla \cdot B=0$$
$$B=\nabla XA$$
$$A=\frac{\mu_0}{4\pi}\int_v{\frac{J}{R}dv}$$

and while

$$Jdv=JSdl=Idl$$

$$B=\frac{\mu_0I}{4\pi}\oint_c\frac{dlXa_R}{R^2}$$

where the first is from Maxwell's Equations, the second is a consequence while using an arbitrary vector A, the third is a definition of the vector magnetic potential A and the fourth is the Biot-Savart Law.

Using b as the loop radius and R as the distance from dl to point we get

$$dl=bd\phi a_{\phi}$$
$$R=a_zz-a_rb$$
$$R=\sqrt{z^2+b^2}$$

then

$$dlXR=a_{\phi}bd\phi X (a_zz-a_rb)=a_rbzd\phi + a_zb^2d\phi$$

and while the r-components cancel out we get using the Biot-Savart Law above

$$B=\frac{\mu_0I}{4\pi}\int_{0 2\pi}a_z\frac{b^2d\phi}{(z^2+b^2)^{3/2}}$$

or

$$B=\frac{\mu_0I}{2}\frac{b^2}{(z^2+b^2)^{3/2}}=\frac{\mu_0I}{2}\frac{b^2}{R^3}$$

It is interesting to note that z=0 is valid.

Roger
PS
I have used my book's (Field and Wave Electromagnetics by David K. Cheng) convention due to simplicity which means I thereby hopefully state the correct equations.

 

[rogerk8]

This chapter of my interest is about different drift mechanisms in a plasma.

Using

$$m\frac{dv}{dt}=q(E+vXB)$$

and the fact that

$$F=qE$$

we get the common expression

$$v_{force}=\frac{1}{q}\frac{FXB}{B^2}$$

Where we can put

$$F_E=qE$$

or

$$F_g=mg$$

or

$$F_{cf}=a_r\frac{mv_{//}^2}{R_c}$$

Where the first is the force from a electric field upon a charge and the second is the force on the particle due to gravity and the third is the centrifugal force as the particles move along a curved line of force, B.

Therefore

$$v_E=\frac{EXB}{B^2}$$
and
$$v_g=\frac{m}{q}\frac{gXB}{B^2}$$
and
$$v_R=\frac{1}{q}\frac{F_{cf}XB}{B^2}=\frac{mv_{//}^2}{qB^2}\frac{R_cXB}{R_c^2}$$

where Rc is the curvature radius of the lines of force.

It is interesting to note that

$$|v_E|=|\frac{E}{B}|$$

It is harder to explain why

$$F_{\nabla B}=-/+\frac{qv_pr_L}{2}\frac{dB}{dy}a_x$$

where B is along the z-axis and grad B is along the y-axis and the sign denotes ion drift direction.

Which put in the v_force equation above gives

$$v_{\nabla B}=-/+\frac{v_pr_L}{2}\frac{\nabla BXB}{B^2}$$

p is here denoting perpendicular (to B).

In a curved vacuum field we may add

$$v_R$$

to

$$v_{\nabla B}$$

thus

$$v_{cv}=v_R+v_{\nabla B}=\frac{m}{qB^2}\frac{RcXB}{Rc^2}(v_{//}^2+\frac{1}{2}v_p^2)$$

Now I quote Francis F. Chen:

"It is unfortunate that these drifts add. This means that if one bends a magnetic field line into a torus for the purpose of confining a thermonuclear plasma, the particles will drift out of the plasma no matter how one juggles the temperature and magnetic fields".

Finally we have the diamagnetic drift (considering the plasma as a fluid)

$$v_D=-\frac{\nabla pXB}{qnB^2}$$

where

$$F_D=-\frac{\nabla p}{n}$$

where p denotes pressure and n volume particle density.

For an isoterm plasma

$$\nabla p=kT\nabla n$$

this gives

$$v_D=-\frac{kT}{qn}\frac{\nabla nXB}{B^2}$$

But the total drift perpendicular to B is actually

$$v_{ED}=v_E+v_D$$

Roger
PS
Most of the time I am using the book Plasma Physics and Controlled Fusion by Francis F. Chen. Please note that there is a certain Cheng here inside this thread also...