Physics 30 Lesson 32 x-rays and the Compton Effect

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1 I. Disovery of x-rays Physis 30 Lesson 32 x-rays and the Compton ffet During all the researh on athode rays, several sientists missed their hane at some glory. Hertz narrowly missed disovering x-rays during his photoeletri effet researh. Fredrik Smith ame lose to the disovery, but he threw the hane away when he asd his assistant to move his photographi plates into another room as his athode ray tube seemed to ruin the plates. Wilhelm Roentgen ( ) finally made the disovery in Barium platinoyanide is a fluoresent material whih will emit visible light when illuminated with ultraviolet light. He notied that a piee of barium platinoyanide glowed when in the region of an operating athode ray tube. Roentgen immediately began to investigate and disovered that the tube was emitting some unknown radiation. Roentgen alled the radiation an x-ray (unknown ray). Roentgen disovered that x-rays passed through some materials but were stopped by other materials. This disovery was immediately put to use as a medial aid. The piture to the left is one of the earliest x-ray photographs made in the United States (1896). (The white dots are individual lead shot pellets in a man s hand who had been hit by a shotgun blast.) But what were x-rays? x-rays were tested to see if eletri or magneti fields would deflet them. Sine x-rays were not affeted by either eletri or magneti fields they were thought to be either neutral partiles or an eletromagneti wave. x-rays were known to penetrate objets, so it was thought that it might be an eletromagneti wave with a very small wavelength about 0.1 nm. Therefore if a 0.1 nm diffration grating were used, its wave nature ould be onfirmed. The atoms of rystals were thought to be separated by about 0.1 nm. In 1912, Bragg finally onfirmed that x- rays are a member of the eletromagneti spetrum when x-rays produed a diffration pattern through a rystal. In turn, Bragg diffration beame an important tool to understand the rystal struture of different minerals through x-ray rystallography. The easiest way to produe x-rays is by the rapid deeleration of eletrons as they stri the anode of a athode-ray tube. In the diagram to the right, a high potential differene is reated between the anode and athode of a athode-ray tube. letrons are aelerated toward a tungsten target. (Tungsten is used due to its eletrons emitted x-rays Dr. Ron Liht

2 exeptionally high melting point.) When eletrons stri the tungsten anode the kineti energy of the eletrons is onverted into x-ray radiation and heat energy. The minimum wavelength (maximum frequeny) of x-rays is when the kineti energy of the eletron is ompletely onverted into x-ray energy. Using the priniple of the onservation of energy we an say that the initial eletri potential energy is onverted into the kineti energy of eletron whih is onverted into x-ray energy when the eletrons deelerate to a stop. Therefore we an say eletri potential energy = maximum x-ray photon energy p qv hf or xray h qv xample 1 What is the maximum frequeny of the x-rays produed by a athode ray tube with an applied potential differene of 30 kv? p qv hf qv f h f xray e(30 10 V) C(30 10 V) or ev s Js 18 f Hz xample 2 A athode ray tube operates at 80 kv with a urrent of 875 A. What is the intensity (photons per seond) and minimum wavelength of the x-rays produed by the athode ray tube? For every eletron that deelerates when it hits the anode, one photon is produed. Therefore we alulate the number of eletrons using the urrent and a time of 1 s. q It 6 q ( A)(1s) 4 q C n e q q e C 15 n e e C x-ray intensity photons s p xray h qv h qv ev s( ms ) 3 1e(80 10 V) m Dr. Ron Liht

3 II. Properties of x-rays The properties of x-rays are rather unique beause of their ability to at as a wave or as a partile to a muh greater extent than visible light. Wave harateristis of x-rays x-rays an penetrate opaque objets suh as wood, paper, aluminum and human flesh. They will not penetrate bone. x-rays an be diffrated by rystals. Partile (photon) harateristis of x-rays x-rays will ionize a gas i.e. they ollide with eletrons and drive them off the moleules to produe ions. x-rays will ause eletron emission in water by the same proess as desribed above. x-rays will affet photographi plates. III. The Compton effet Arthur Holly Compton was born at Wooster, Ohio, and was eduated at the College of Wooster, graduating Bahelor of Siene in 1913, and he spent three years in postgraduate study at Prineton University reeiving his M.A. degree in 1914 and his Ph.D. in After spending a year as instrutor of physis at the University of Minnesota, he took a position as a researh engineer with the Westinghouse Lamp Company at Pittsburgh until 1919 when he studied under Rutherford at Cambridge University as a National Researh Counil Fellow. In 1920, he was appointed Wayman Crow Professor of Physis, and Head of the Department of Physis at the Washington University, St. Louis and in 1923 he moved to the University of Chiago as Professor of Physis. Compton returned to St. Louis as Chanellor in 1945 and from 1954 until his retirement in 1961 he was Distinguished Servie Professor of Natural Philosophy at the Washington University. In his early days at Prineton, Compton devised an elegant method for demonstrating the arth's rotation, but he would soon begin his studies in the field of x-rays. He developed a theory of the intensity of x-ray refletion from rystals as a means of studying the arrangement of eletrons and atoms, and in 1918 he started a study of x- ray sattering. Compton was intrigued by the idea that if photons have energy do they have momentum. He derived the equation(s) that desribed the momentum of a photon. Dr. Ron Liht

4 As we learned in Lesson 1, the momentum of a partile is given by p = m v but from instein s famous equation ( = m 2 ) we know that m 2 Substituting this for m we get v p 2 and sine the speed of a photon is the speed of light (v = ) thus If p 2 p p and we know from Plank s equation that h f p or h hf then p h (momentum of a photon) Compton initially experimented with x-ray photons to bombard atoms, but the effet was so small that it was not measurable. Compton then began to bombard eletrons rather than atoms. He measured the wavelength of the inoming x-ray ( i ) and the wavelength of the sattered x-ray ( f ) that sattered through an angle. The ollision between the x-ray photon and the eletron is a purely elasti ollision. (Reall from Lesson 2 that for an elasti ollision both momentum and kineti energy are onserved.) Therefore, in terms of the onservation of energy we have i i before ollision f (onservation of energy) And in terms of the onservation of momentum, the ollision between the inoming x- ray and the eletron yields p p p (onservation of momentum) i e f after ollision e e inoming = eletron s + sattered x-ray kineti x-ray energy energy energy h 2 h = 1 2 mv + f momentum of = eletron s + momentum of inoming momentum sattered x-ray x-ray Reall from Lesson 3 that the vetor equation translates into the vetor diagram shown to the right. Dr. Ron Liht

5 Utilising both the onservation of energy and the onservation of momentum, along with an appliation of instein s speial theory of relativity, Compton derived the following relationship for the hange in wavelength of the x-ray photon. h f i (1 os ) m where m is the mass of the eletron and is the angle through whih the x-ray satters. Compton s experiment beame known as the Compton effet and he won a Nobel prize for physis in 1927 for his efforts. Compton s experiments show that a photon of eletromagneti radiation an be regarded as a partile with a definite momentum and energy. Photons have momentum and energy (li a moving partile) but they also have a frequeny and a wavelength (li a wave). (Refer to Pearson pages 721 to 725 for a disussion about the Compton effet.) xample 3 Calulate the momentum of a photon that has a wavelength of 455 nm. h p p Js m -27 p kg m/s xample 4 A photon of light with wavelength nm hits a stationary eletron. The sattered x- ray has a wavelength of nm. What is the resulting speed of the eletron and the sattering angle of the x-ray? To find the speed of the eletron we use the onservation of energy. i h h i 1 1 h i f f f 34 8 m J s( s) m m J v v 2 m 16 2( J) kg v m s To find the sattering angle we use Compton s equation. h f i (1 os ) m 1 mf i os 1 h 1 os 1 o kg( m ) m m s Js Dr. Ron Liht

6 IV. Pratie problems 1. A olour TV operates using a potential differene of 60 kv. If eletrons are ompletely stopped by the anode, what is the wavelength and momentum of the radiation produed? (2.1 x m, 3.2 x kgm/s) 2. An inident x-ray auses an eletron, whih was initially at rest, to have a speed of 2.5 x 10 7 m/s. The sattered x-ray has a frequeny of 1.26 x Hz. What is the wavelength and momentum of the inident x-ray? (5.4 x m, 1.2 x kgm/s) V. Hand-in assignment x-rays 1. letrons are aelerated through a potential differene of 11.1 kv. What is the minimum wavelength of the radiation produed when the eletrons deelerate to a stop? What region of the spetrum is this? (1.12 x m) 2. X-rays with a wavelength of nm are produed in an X-ray tube. What is the potential differene used in operating the tube? (3.36 x 10 3 V) 3. An eletron traveling at 5.2 x 10 4 m/s stris a dense metal target and omes to rest. What is the emitted photon's frequeny? (1.9 x Hz) 4. In the CRT used for demonstrations in lass, the potential differene used is 50 kv. What is the frequeny of the radiation produed by the CRT? Is this radiation dangerous to your health? (1.2 x Hz) 5. An alpha partile is aelerated through a potential differene of 320 kv and stris a tungsten barrier. What is the wavelength of the emitted radiation? (1.94 x m) 6. A urrent of 5.0 A whih lasts for 4.2 s is produed when a stream of eletrons Dr. Ron Liht

7 from an eletron beam stri a metal target. If the eletron beam has an energy of 1.4 x J, find the average wavelength of the photons generated at the barrier. (Hint: Is the given energy the energy for one photon or for many photons?) (1.9 x 10-6 m) The Compton ffet 1. In any elasti ollision, what two things are onserved? 2. How did Compton explain that photons have a momentum if they have no mass? What is the mass equivalent of a photon? 3. Disuss how Compton sattering shows that it is impossible to see an eletron without disturbing the eletron. 4. What is the momentum of a photon with a frequeny of 9.65 x Hz? (2.13 x kgm/s) 5. If the energy of a photon is 225 V, what is its momentum? (1.20 x kgm/s) 6. A photon with a wavelength of 2.00 x m ollides with a stationary eletron. As a result the eletron moves off with a speed of 2.90 x 10 7 m/s. What is the wavelength and angle of the sattered photon? (2.08 x m, 48.0 o ) 7. A photon with a wavelength of x m ollides with a stationary eletron. If the sattered photon has a frequeny of x Hz, what is the resulting speed of the eletron? (4.68 x 10 7 m/s) 8. An x-ray ollides with a stationary eletron. What is the hange in the wavelength of an x-ray if it is sattered through an angle of 120 o? (3.64 x m) 9. A photon an undergo Compton sattering from a moleule suh as nitrogen (N 2 ) just as it does from an eletron. However, the hange in photon wavelength is muh less than when an eletron is sattered. Using Compton s equation for nitrogen instead of an eletron, explain why the maximum hange in wavelength for a sattered photon is less for nitrogen then for an eletron. ( m, m) max e maxn2 Dr. Ron Liht

8 Use the following information to answer this question. In an observation of the Compton effet a photon is inident on a free eletron. The sattered photon is deteted, as illustrated below. The sattered eletron is not shown. 10. Determine the veloity of the sattered eletron. As part of your response, sth the situational diagram showing the path of the sattered eletron, sth a vetor addition diagram onsistent with the vetor analysis method you are hoosing, and state all neessary physis priniples and formulas. Marks will be awarded based on your vetor diagrams, on the physis that you use, and on the mathematial treatment you provide. Dr. Ron Liht