The Bremsstrahlung Effect

Ever had a bone fracture or a really bad injury that made you end up in a hospital? We all know what comes next: the X-Ray procedure, right? The physician places our injured area right below the X-Ray machine and captures the hidden images of our bones beneath our skin. Remember the first time you had an X-Ray? Several questions that might have burst into your mind are, "What is this X-Ray thingy?" "Why is it called specifically an 'X' ray?" "How does this weird invisible thing come out of the machine?" "How does it capture images of something we can't see?" etc., etc., etc.

Basics

Well, for starters, X-Ray is an electromagnetic wave, that is, it has both an electric as well as a magnetic component. Learn more about electromagnetic waves here: Electromagnetic Waves. So X-Ray happens to be an EM wave which has a very high frequency, just behind gamma rays and a very small wavelength, just behind the same. Now if you happen to be a physics enthusiast, you might even know a small relation between the terms wavelength, frequency and energy.

Energy from the above relation is inversely proportional to the wavelength of an atomic particle, say electron. Frequency (f) is nothing but the inverse or reciprocal of "Time" (s), which implies f = 1/T. From our basic physical formulas, we know Time = Wavelength/Speed. Hence Frequency f = Speed/Wavelength. 

From here we can say that frequency is inversely proportional to lambda or the wavelength. By definition, frequency is the number of oscillations in 1 time period. Learn more about wave mechanics here: Sound Waves.

X-Ray

Having basics understood, lets delve deeper into X-Rays now. So in the introduction I mentioned how X-Rays are used to capture images of our bones, right? They are explicitly used over other EM waves because it has a high penetration capacity and is less harmful than gamma rays to our skin. (Nope, you are not gonna become the hulk if you think gamma rays enter your body. Its highly harmful and can even mutate or destroy the DNA)

The wavelength of X-Rays lie in between 0.01 Angstrom to 100 Angstrom (1 Angstrom = 10^-10 meters) in air/vacuum.

Did you know?

Did you know that in 1895, a German scientist W.C Roentgen while testing electric currents in a glass tube accidentally discovered some mysterious rays which were causing a nearby screen to glow. Since he did not know what these rays were, he simply named them "X-Radiation". Later this name popped up and these rays were then called by the name "X-Rays". This is the story behind its mysterious and odd name.

Types of X-Rays:
There are essentially two types of X-Rays
1) Soft X-Rays: Their wavelength is greater than 10 Angstrom, and have low energy as well as low penetration capacity.

2) Hard X-Rays: Their wavelength is less than 1 Angstrom, and are highly energetic and have high penetration capacity.

Lets now understand the phenomena behind the "production" of these X-Rays. Lets think intuitively instead of jumping into facts and conclusions. What do we really mean when we say we want to produce X-rays? In simple language, we mean to produce some wave using some physical interaction or process which happens to physically fit into all the properties X-ray has. How do we do this then? Lets create a picture in our mind keeping the fact that we want to produce a wave whose wavelength is in the order of some Angstroms and whose energy will therefore be in the order of kilo electron volts (Here by energy we imply kinetic energy). What physical interactions can we organize such that we start getting some waves? A simple answer would be, collision. You see, collision is a beautiful physical phenomena which produces energy in some form, say sound, light or even heat. Lets take photoelectric effect for example. An outstanding phenomenon by Einstein which got him the Nobel prize in physics, explains that when photons of significant frequency or energy (threshold frequency) strike metal surfaces, they eject electrons which can then be used for various applications such as generating electricity using solar panels. Now coming back to our case, we need to produce a wave, or more specifically, an electromagnetic wave, right? What if, analogous to the photoelectric effect by Einstein, we bombarded a metal surface with electrons, instead of photons?

The Bremsstrahlung Effect

In 1909, Arnold Sommerfeld coined the German word "Bremsstrahlung" meaning "braking radiation" (Note: Its braking, not breaking) to describe a phenomena caused by the sudden slowing down of electrons. Who's slowing electrons down? Lets understand the Bremsstrahlung effect and how it is used to produce X-Rays today.

Bremsstrahlung effect states that if highly energetic electrons are bombarded on a target metal (usually tungsten or molybdenum), then the nucleus of the target metal retards the motion of the electrons which results in the loss of kinetic energy of electrons, which comes out in the form of photons.

Consider the above picture, say a highly energetic electron with a kinetic energy say E1 and velocity V1 is bombarded on a target metal. As the electron approaches the nucleus of the target metal atom, it experiences retarding motion due to the attractive pull of the nucleus obeying the basic laws of electromagnetism where unlike attract. This causes a change in velocity (ΔV = V1 - V2) or an acceleration (here deceleration) which leads to the production of radiation. From Maxwell's laws, an accelerated electron emits electromagnetic radiation. We need to ensure that the energy of this radiation which comes from the energy loss (ΔE = E2 - E1) of the original electron caused due to the retarding motion due to the attractive pull of the nucleus, is in the order of some kilo electron volts which basically makes wavelength in the order of some Angstroms (you can check the math by inputting values in the formula E = hc/lambda where hc = 12420 eV). This energy loss in kilo electron volts is actually the energy of continuous X-Ray spectrum.

The photons thus emitted have the wavelength which are numerically equal to the wavelength of X-Rays, and this is how X-rays are produced, a basic idea. So, now, how is the Bremsstrahlung effect used practically? Lets have a look.

Coolidge Tube

This is a tube, also known as X-ray tube, which is used to produce X-Rays. It is the experimental setup for the Bremsstrahlung effect.

Lets understand this in a stepwise and simple way.

1) A spherical shape of transparent glass is taken to be the tube in which a filament is connected in one end with a battery of high voltage. Near the filament, a metal which acts as the source of electrons is placed, which when gets heated up because of the filament, produces electrons via thermionic emission.

2) The metal which produces electrons is connected to an accelerating voltage (Va) which is in kilovolts. This is in turn connected to the target metal which is located at the other end of the Coolidge tube.

3) The target metal is oriented in such a way that maximum electrons can hit the surface of the metal. It is essentially made of 2 parts, the one which will be colliding with the electrons (the real target metal) and the other which is connected to the target metal to absorb the heat which gets produced upon collision of electrons.

4) Cooling water is supplied to the supporting target metal (usually copper) because copper will absorb heat from the main target metal (usually tungsten / molybdenum as mentioned earlier as they have really high melting points) and water absorbs this heat and comes out warm water. This, however is not the primary purpose of the Coolidge tube.

5) The accelerating voltage which is connected to the target metal sets it up at a very high potential thus it acts as the anode and the source of electron acts as the cathode. Hence an electric field is generated which causes electrons from the cathode to flow to the anode or the target metal.

6) Once the electrons hit the target metal, they produce X-Rays, this is because the accelerating potential is set up very high which causes electrons to accelerate really quickly only to lose a lot of energy when they collide to the nucleus of the target metal which ultimately results in the energy of the X-Rays, as stated by the Bremsstrahlung effect. This is the direct application or result of this effect.

7) A collimator is introduced in front of the metal so that all the electrons pass through a single path and form a beam of electrons and hit the target efficiently. Same is done at the other end of the tube to produce a beam of X-rays. 

This setup is credited to William Coolidge who invented the Coolidge tube in 1913. A simplified setup of the Coolidge tube along with a real life Coolidge tube are as shown below.

So we see that the Bremsstrahlung effect is very much the opposite of the photoelectric effect where we are bombarding electrons on metal surface to eject photons of suitable frequency, and henceforth produce X-Rays. Sometimes we are interested in finding the kinetic energy (K) of electrons striking the metal surface. To calculate this, we use total mechanical energy conservation at the anode and cathode since all the energy is being carried by the electrons.

Applying mechanical energy conservation at the cathode and anode, assuming initial kinetic energy to be zero, we get:

PE1 + KE1 = PE2 + KE2

(-e)0 + 0 = (-e)Va + K

Hence, K = (e)Va

(Potential energy is given as U = qV where q is the charge of the particle and V is the potential of the particle, here Va is the accelerating potential)

The most energetic photon emitted will have the energy equal to its kinetic energy whose wavelength (cutoff wavelength/minimum wavelength) can then be calculated by simply equating both the energies. Thus we get the formula:

Where Lambda(min) is the cutoff wavelength. Lambda will be minimum because it is the wavelength of the most energetic electron and by the inverse relation we saw in the beginning, the most energetic electron will automatically have the least wavelength. The below values can be used to solve problems involving calculation of the cutoff wavelength.

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