HW Electron Optics PHY 410/510

HW1 due 4/17/03

1)
A block of glass, x, y, z dimensions, where d is the length in the y  direction. A photon ray is incident on the xz face at an angle. Suppose the index of refraction n of the glass increases from n1 to n2 by a factor 2 in the y-direction, and  n is constant in x and z directions.

A: How will the ray behave qualitatively in the glass?
B: Sketch the analoguous case for electron optics..


2)
A pair of deflecting plates are separated by a distance d, and have a length l. The deflection voltage is Vd. An electron having a beam voltage Va enters the deflector midway between the top and bottom plates.

A: Derive an expression for the angle delta through which the electron is deflected.
B: Evaluate the deflection angle for: d=2cm, l=5cm, Vd=100 volts, and Va=1000 volts.
C: What is the deflection angle for a proton with the same kinetic energy?

3)
The rest mass, moc2, for electrons is 511 keV. In a TEM an electron is accelerated to 400 keV. Calculate its de Broglie wavelength. A typical aperture for a TEM is 0.01 rad. Give an estimate for the resolution based on the diffraction limit. What is the real resolution of a very good TEM? What prevents us from reaching the diffraction limit?

4)
The mean velocity of a gas atom is v=(2kT/m)1/2. Determine its free mean path at a gas pressure of 10-4 bar.

5)
In the earth's magnetosphere electrons with energy of  ~100keV enter a magnetic field of 10-5 T. Their orbits have a typical radius of ~100m.  What would be the orbit radius of a proton from the solar wind?

Solutions  to HW#1



 
HW#2  due 4/24/03

1)
Bring the program "cyclotron.iob" on the screen. It is stored in "homogeneous fields". Fly the electrons, produce an image in the x-y-plane and make a print of the array and the trajectory. Determine the value of the potentials in the array and enter them on your print out by hand.

Switch off the electric field and define the particles as protons. Fly them. The radius of their orbit is very large. How many times larger than the electron radius is it? –write the results on the print out.

Multiply the magnetic field by a factor 42.8. Fly the protons again. What do you find? Explain on the print out!

2)
Go back to the original "cyclotron"-program, now switch off the magnetic field. Fly the electrons, then use protons under the same conditions, but change their charge to -1. Fly them. What do you find? Explain the result.



HW#3 due 4/29/03

1)
Discussion of the linearized paraxial orbit equation:
We have assumed rotational symmetry. What does that mean for the magnetostatic and electrostatic electrode configurations?
Electrostatic case only:
For the orbit equation we have retained only first order terms of the potential expansion. What is the physical significance of that?
We have assumed (r')2 << 1. What does this imply?
Find the expression for V5 and V6 for the electrostatic potential using the recursion formula.

2)
Use the electrostatic electrode configuration " elens.iob" from the e-mailed file "Real-EM-Lens" for a simulation in SIMION7.
The configuration consists of a so-called Einzel-lens
Fly electrons, then anti-protons. What are the focal distances in these two cases? The focal distance is the distance between the lens center (here at x = -80mm) and the focus.
Double the potentials on the lens and double the kinetic energy of the electrons. What is the focal distance now? Explain!

Reverse the potentials of the lens, keep the kinetic energy of the electrons at the original value (1keV). Fly them! What is the focal distance now? Switch to protons, and note the focal distance.
The lens is always positive. Is that correct?


Solutions discussed in class and not shown here

HW#3 due 4/30/03

 

1)

Discussion of the linearized paraxial orbit equation:

We have assumed rotational symmetry. What does that mean for the magnetostatic and electrostatic electrode configurations?

Electrostatic case only:

For the orbit equation we have retained only first order terms of the potential expansion. What is the physical significance of that?

We have assumed (r')2 << 1. What does this imply?

Find the expression for V5 and V6 for the electrostatic potential using the recursion formula.

 

2)

Use the electrostatic electrode configuration " elens.iob" from the e-mailed file "Real-EM-Lens" for a simulation in SIMION7.

The configuration consists of a so-called Einzel-lens

Fly electrons, then anti-protons. What are the focal distances in these two cases? The focal distance is the distance between the lens center (here at x = -80mm) and the focus.

Double the potentials on the lens and double the kinetic energy of the electrons. What is the focal distance now? Explain!

 

Reverse the potentials of the lens, keep the kinetic energy of the electrons at the original value (1keV). Fly them! What is the focal distance now? Switch to protons, and note the focal distance.

The lens is always positive. Is that correct?


Solutions to HW #3

HW #4 due 5/16 2003

 

1) Use "maglens" in SIMION. Determine the following lens properties:

L          - length of the lens

B(z)      - profile of the magnetic induction along the z-axis, use 5-8 points

Fly electrons with 30 eV, determine from the plot: focal distance g and the angular position of the image, Q(L), at the exit of the lens.

What is a thin lens? Is this a thin lens?

 

For thin lenses one can derive from Picht's approximate solution to the paraxial equation the following expressions for the focal length and the turning angle of the image:

 

 

 

a) Using the B(z) profile, approximate B(z) by the function B = Bmax/(1+(z/a)2)) in such a way that the B(z) values  at the maximum and at the lens exit and entrance points is approximately  the same for the profile and the function B = Bmax/(1+(z/a)2).

What are the values for Bmax, a, A, B?

Carry out the integrals and find f2 and Q(L).

 

b) Using the B(z) profile, approximate B(z) by a homogeneous field, ensuring that the integral over B(z) along the z-axis  is approximately the same for the SIMION case and the assumed homogenous field case, i.e.

 

                                    L x Bhom  =  Integral B(z)dz

 

Use the homogeneous field approximation to determine Q(L), f2, zH2. Compare the focal distance and the focal length to the SIMION solution and compare the focal length value to that of part a)

 

What is the value of B=Bhom in this case? How many focal points would the lens produce if it its length was L = 100mm?

 

c) From an integration of Biot-Savart's law one finds that the electric current from a coil  produces a magnetic field,

            B = moIR2/2(R2+z2)3/2

along the z axis.

Assuming a radius of R=10mm for a coil and 100 windings, what current would be needed to get a magnetic induction of 17 Gauss at z=0, i.e. in the symmetry plane of the coil?

d) The magnetic induction is now of the form,

            B= Bmax/(A2 + z2)3/2

What is the indefinite integral of that function?