Free electron border between relativity and non-relativity

In summary, the conversation discusses the concept of a "free" electron and the calculation of kinetic energy in both non-relativistic and relativistic cases. It also addresses the meaning and significance of the quantity T/mc^2, and clarifies the role of the Lorentz factor gamma in determining the energy of a particle.
  • #1
Odyssey
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"Free" electron...border between relativity and non-relativity

Hello guys...I have a couple of issues that I'm not exactly sure about...

Homework Statement



a) What does it mean to say that an electron is "free"?
b) To figure out the kinetic energy of a particle...one must first have to check if it's a non-relativistic case, or an ultra-relativistic case...my notes have it one should compute the value of [tex] T/mc^2[/tex] (T=kinetic energy)...but why do we have to do that?

Homework Equations


[tex] T/mc^2[/tex]
[tex]T=p^2/2m[/tex]
[tex] E=pc[/tex]
[tex]E=T=pc[/tex]

The Attempt at a Solution



a) I think "free" means the electron is moving under no potential? So...in that case...the total energy equals the kinetic energy?

b) I don't know why we have to compute the value of [tex] T/mc^2[/tex]...but I know if it's <<1 then it's a non-relativistic case, and we can use the formula [tex]T=p^2/2m[/tex] to find the Kinetic energy. If [tex] T/mc^2[/tex] is >>1 then it's a relativistic case, and we use [tex]E=T=pc[/tex] to solve for the kinetic energy? What is the definition of the border between non-relativistic and relatiticistic cases? :confused:
 
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  • #2
a) correct, although Total energy may include its mass?
b) Well, Energy of a particle is given by [tex]m_0c^2\gamma[/tex] whilst its energy at rest is given by the same expression without the Lorentz factor. So the kinetic energy is the difference of these two. So Replace T with [tex]m_0c^2(\gamma -1) [/tex] and see what happens :) Hope it helps, you'll find it makes sense because small v makes gamma equal 1, which is correct.
 
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  • #3
Thanks for the help. :)
Yes...I plugged in some numbers for v...makes sense! I got one more question...what does the quantity [tex] T/mc^2[/tex] represent? It has no dimensions...I replace T with [tex]m_0c^2(\gamma -1) [/tex]...so it becomes [tex]m_0c^2(\gamma -1)/m_0c^2 [/tex]...what does it means do have [tex]\gamma -1 >> 1[/tex]? Why does it represent a relativistic case? [tex]\gamma[/tex] can only take on values between 0 and 1...but with [tex]\gamma -1 >> 1[/tex]...we can actually have [tex]\gamma[/tex]be >1? I think I'm getting a bit mixed up with gamma. =\
 
  • #4
It represents the ratio of the kinetic energy to rest energy.

[tex]\gamma=\frac{1}{sqrt{1-\frac{v^2}{c^2}}}[/tex]. Say we subbed v=0 into gamma, gamma = 1. That means our T/mc^2 ratio is equal to zero, which makes sense, because the ratio of zero kinetic energy to rest energy should be zero.

What it means for gamma - 1 to approach one is quite simple. Its the same as for gamma to approach 2. Let's set gamma=2 and solve for v. Some simple algebra will show you that v/c = (the square root of 3)/2, which is a good portion of the speed of light. That is why it represents a relatavistic case :)

Your a little confused with the gamma. Yours thinking about [tex]sqrt{1-\frac{v^2}{c^2}}[/tex], which can not exceed 1. But gamma is the recipricol of that, and that can exceed 1, because gamma =[tex]\frac{1}{sqrt{1-\frac{v^2}{c^2}}}[/tex]. Sub in any case where v>0 you will see that whilst the bottom can not exceed 1, the entire gamma can.

This is exactly what you want to see, since gamma is there to sort of make up for the extra energy.

EG- Rest energy is mc^2, whilst Rest+Kinetic is the same, multiplied by gamma. Since we want REST+KINETC to be more than REST alone, gamma should be more than one.

Sorry if i haven't explained it very well.
 
  • #5
Thanks! =) gamma is the recipricol...now I understand!
 
  • #6
Good Work :)
 

1. What is the concept of a free electron border?

The free electron border, also known as the Fermi surface, is the boundary between the filled and unfilled energy levels of an atom's electrons. It marks the maximum energy that an electron in a solid can have while still being bound to the atom's nucleus.

2. How is the concept of a free electron border related to relativity?

The concept of the free electron border is related to relativity through the theory of special relativity. This theory states that the energy of a particle increases as its velocity approaches the speed of light. The electrons near the Fermi surface have a high velocity, thus their energy is affected by the principles of relativity.

3. Can the free electron border be observed in non-relativistic systems?

Yes, the free electron border can be observed in non-relativistic systems. While the concept is related to relativity, it also applies to non-relativistic systems such as metals. In these systems, the electrons near the Fermi surface have a high energy due to their close proximity to the nucleus, rather than their velocity.

4. How does the free electron border affect the properties of a material?

The free electron border plays a crucial role in determining the properties of a material. It affects electrical conductivity, thermal conductivity, and other properties related to the movement of electrons. The size and shape of the Fermi surface can also influence the behavior of a material under certain conditions.

5. Can the free electron border be manipulated in a material?

Yes, the free electron border can be manipulated in a material through various methods such as applying an electric or magnetic field. This can change the shape and size of the Fermi surface, altering the properties of the material. This manipulation is the basis for many electronic devices and technologies.

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