Video: Purdue PHYS 342 L1.6: Classical Models: Compton Effect

Transcript

0:01

check it out now

0:04

y'all nanohub U online

0:09

instruction hi we're back again and

0:11

we're we're here to discuss uh this uh

0:14

experiment that was performed by Compton

0:17

and it's referred to as Compton effect

0:19

or Compton

0:20

scattering and U it uh it had

0:23

historically it was a very important

0:25

result and I like to explain the

0:27

experiment and try to explain why it was

0:30

so

0:32

important so first of all uh the

0:36

experiment was performed in

0:38

1923 by an American physicist by the

0:41

name of Compton he he eventually

0:43

received a Nobel Prize for this

0:45

experiment and what he basically did was

0:48

he took uh uh an x-ray tube right this

0:52

this produced energetic uh photons right

0:56

so it's a source of high energy photons

0:59

and he directed those x-rays onto a

1:02

carbon block uh which is in this diagram

1:05

represented by by this little uh uh

1:09

Square symbol and he looked at the uh El

1:13

uh he looked at the photons that were

1:15

scattered by this carbon block he used

1:19

a def frct tometer the details of which

1:22

we don't need to uh um go into but what

1:26

he was able to do was he was able to

1:28

look at the energy of the scattered uh

1:31

uh x-rays as a function of this

1:34

scattering angle here and the scattering

1:37

angle is always defined with respect to

1:39

the incident beam of x-rays right and he

1:43

he measured the energy of the scattered

1:45

x-rays is a function of the scattering

1:48

angle Theta so what was expected at the

1:51

time well what was expected at the time

1:54

was that um if if these uh uh this

1:59

incident x-ray radiation were

2:01

represented by photons there would be a

2:05

photon incident onto the carbon block

2:07

and the energy of each Photon would have

2:09

a have a value given by Plank's constant

2:13

H times the frequency F of the of the

2:17

X-ray this energy of the X-ray was

2:19

basically controlled by the X-ray tube

2:22

that he

2:23

used if that Photon interacts with an

2:26

electron then uh inside the carbon block

2:30

then the electric field of the photon uh

2:33

will cause the electron to oscillate up

2:35

and down if the electron oscillates up

2:38

and down it's going to oscillate at the

2:40

same frequency as the electric field uh

2:44

of the photon so when it uh when the

2:49

um when the electron uh emits radiation

2:53

it's going to emit radiation at the same

2:56

frequency uh so that means a photon

2:58

should come out at a different angle but

3:01

it should have exactly the same energy

3:03

as the incident Photon because the

3:06

incident Photon causes an oscillation of

3:08

the electron at the same frequency as

3:10

the electric field in that the photon

3:14

has so there should be no energy loss

3:16

that's the bottom bottom line it's

3:19

referred to as an elastic scattering

3:21

process and uh that's what

3:24

experimentally you would expect to

3:25

observe you would expect to observe that

3:27

the x-rays scattered from the this

3:30

carbon block as a function of the

3:32

scattering angle uh should have exactly

3:35

the same energy as the incident uh x-ray

3:38

beam well experimentally that was not

3:41

observed to be true in fact uh what

3:44

Compton observed was that as the

3:46

scattering angle was increased right a

3:50

significant number of the scattered

3:52

electrons uh lost energy they came out

3:55

at longer wavelengths right if they come

3:57

out at longer wavelengths they fre

3:59

quency is smaller and uh if you multiply

4:03

the frequency times H you get the energy

4:05

of the the photons so uh longer

4:09

wavelengths corresponds to a loss of

4:12

energy and uh that was completely

4:16

contradicted by this simple classical

4:18

argument that uh was was expected to be

4:23

observed so uh what's going on um what

4:28

Compton did was he analyzed IED his

4:30

experimental data by applying energy and

4:33

momentum principles uh to explain the

4:36

data so he took he took the results that

4:39

were well known from Collision Theory

4:41

right where you've got uh an object of

4:44

mass M1 colliding with an object of mass

4:47

M2 if you know the initial velocities

4:50

then you can predict the the final

4:52

velocities and the final directions uh

4:55

just based on conservation of momentum

4:57

and conservation of energy right so

5:00

these were the two uh uh uh conservation

5:03

principles that that he used uh to

5:06

explain his uh data the only puzzling

5:10

thing the only significant difference

5:13

between these classic billiard ball type

5:15

collisions and Compton's experiment was

5:19

that uh one of these billiard balls was

5:22

now replaced by a photon right and that

5:25

Photon had a particular energy given by

5:28

H * the frequency F uh it was a fair

5:32

question what the momentum of a photon

5:34

was right you you had to come up with an

5:36

expression for the momentum now why is

5:39

that why is that puzzling well um it's

5:42

puzzling because classically the

5:43

momentum is defined as mass times

5:46

velocity and what we're trying to do is

5:48

we're trying to attribute a momentum to

5:51

a photon uh but the photon has no Mass

5:54

right it's electromagnetic radiation so

5:56

the mass is equal to zero so you know

5:59

you can't can't just use this simple

6:01

expression for momentum of a photon and

6:03

expect to to get anywhere so what you

6:07

had to do is you had to come up with a a

6:09

general expression for momentum of a

6:11

photon the argument for that is um it's

6:15

not rigorous but uh it it it gives the

6:19

uh the basic idea you know from plon we

6:22

know that the energy of a photon is

6:24

equal to H * the frequency F right but

6:27

from Maxwell we know that uh the mo that

6:30

when an electromagnetic wave interacts

6:32

with the surface there's a momentum

6:34

transfer and that momentum transfer is

6:37

equal to the uh the energy U uh

6:40

delivered to the surface divided by the

6:42

speed of light C right so if you

6:46

substitute the energy of a photon in for

6:50

the uh expression U that that Maxwell uh

6:54

would attribute to an electromagnetic

6:56

wave then you can find an expression for

6:59

the moment of a photon in terms of the

7:01

frequency or or the wavelength Lambda of

7:04

the photon right so this is this is the

7:07

expression that we uh the Compton would

7:10

have to use in order to characterize the

7:12

momentum of the incident Photon as it as

7:15

it is it interacts with this block of

7:19

carbon so uh with with Compton imagine

7:23

was an incident Photon uh coming into a

7:27

carbon block the incident photon has a

7:30

wavelength Lambda subbi uh in the

7:33

process of penetrating into the carbon

7:35

block that electron or that Photon

7:38

interacts with an electron the electron

7:40

has a mass m subb and the electron is

7:43

assumed to be at rest right uh you then

7:47

apply conservation of energy and

7:49

momentum principles to this collision

7:52

and what happens is the electron then

7:54

recoils right so inside the block the

7:57

electron uh acquires uh a certain energy

8:01

and the uh exiting Photon has to lose

8:04

that amount of energy in order to

8:06

satisfy conserv conservation of energy

8:10

considerations so the energy loss of

8:13

this scattered photon is attributed to

8:15

the recoil of the electron inside the

8:18

carbon block now of course

8:20

experimentally you can't measure the uh

8:22

The Recoil of the electron that's buried

8:24

inside the block but what you can do is

8:27

you can measure the uh the weight length

8:30

of the scattered Photon and then you can

8:32

infer how much energy is lost to the

8:35

electron so Compton pursued this

8:38

analysis uh we we're not equipped at

8:41

this point in time to go through his

8:43

derivation because he he uh he was

8:47

required to use relativistic mechanics

8:49

and we have not yet covered that in this

8:51

course so when we discuss relativity

8:54

towards the end of the semester uh we'll

8:57

be able to come back and uh and and

8:59

derive Compton's formula at this point

9:02

in time we're just going to State it uh

9:05

and and and the statement of Compton

9:07

formula is that the the relationship

9:10

between the final wavelength that's been

9:12

scattered and the initial wavelength of

9:15

the x-rays incident onto the carbon

9:17

block is equal to a constant H over m m

9:21

is the mass of the electron C is the

9:24

speed of light time 1us cosine Theta

9:27

where Theta is the scattering angle

9:29

Define with respect to the direction of

9:32

the incident Photon beam right so this

9:36

proportionality constant H over MC has

9:40

to have units of wav has to have units

9:43

of uh meters right because Lambda Lambda

9:46

F and Lambda I are or wavelengths those

9:50

are those have units of meters so H over

9:53

m c has to have units of meters and I

9:56

just work out uh evaluate what what that

10:00

uh that proportionality constant is it's

10:03

referred to as the Compton wavelength of

10:05

an electron and it turns out to be about

10:08

2.43 picometers a very important number

10:12

in the in the development of quantum

10:14

theory and this will this this this this

10:18

will become clear as the uh uh semester

10:21

unfolds so this is um Compton's

10:25

prediction I wouldn't be up here talking

10:27

about it if Compton in fact didn't

10:30

Compton's analysis did not in fact

10:32

explain his experimental data very well

10:35

so there was some validity to this

10:38

conservation of energy momentum argument

10:41

uh of a of a photon interacting with an

10:44

electron that allowed him to explain his

10:46

experimental data so we can sort of

10:49

summarize uh the the experimental

10:52

results in this diagram um what I show

10:56

here is I show an electron at rest I

10:58

show incident photons striking that

11:01

electron and then I show different

11:03

scattering angles right if if the x-rays

11:07

uh uh interact with the electron and and

11:10

continue on in the in a direction uh

11:14

that's unperturbed from their initial

11:16

Direction indicated by this black lrow

11:19

Black Arrow then the angle Theta is

11:22

equal to zero and you can see from this

11:25

equation that the the scattered

11:28

wavelength Lambda sub is going to equal

11:30

to the uh incident wavelength Lambda

11:32

subi and so there's no energy loss and

11:35

this is a referred to as an elastic

11:37

scattering event if you start to measure

11:40

the photons scattered at different

11:43

angles then the inelastic scattering

11:46

increases right and the inelastic

11:48

scattering is then measured as the

11:50

difference between the final wavelength

11:52

Lambda subf and the initial wavelength

11:55

Lambda subi so I just schematically

11:58

indicate how that separation increases

12:01

it reaches a maximum value which is

12:04

equal to two times Compton's wavelength

12:07

uh when when the photons are actually

12:10

back scattered from this electron so uh

12:13

qualitatively this diagram summarizes

12:16

Compton's experiment and it and it shows

12:19

very nicely how you go from elastic

12:21

scattering to inelastic scattering the

12:25

uh where the uh energy lost by the

12:28

photon increases as the scattering angle

12:30

increases so it's a it's a it's a very

12:34

nice diagram to uh understand Compton's

12:38

experiment uh as an as an example um uh

12:42

I just worked through uh some numbers

12:44

here to give you some uh feeling for

12:47

what what what what's actually happening

12:50

uh I like to like to consider the case

12:53

where the incident wavelength spans a a

12:56

range of different values right so uh

12:59

here I've got incident wavelengths of

13:01

420 NM which is in the blue region of

13:04

the electromagnetic spectrum I go from

13:07

420 nanometers all the way down into the

13:10

X-ray region of the electromagnetic

13:12

spectrum where the wavelength is uh

13:15

04 uh nanometers and uh I in this

13:20

example I just consider what happens if

13:22

the photon is uh scattered through a

13:25

scattering angle of

13:26

37° so once I set 30 7° then I can

13:31

evaluate the the right hand side of

13:33

Compton's formula right it's it's the

13:36

Compton wav length which is this 2.4

13:38

Peters * 1 minus the cosine of of this

13:42

scattering angle 37° turns out that's

13:45

equal to 0489 picometers and what that

13:48

just simply means is that just simply

13:50

means that the final wavelength is

13:52

related to the initial wavelength by

13:54

this quantity okay so the moral of the

13:58

story is all electromagnetic radiation

14:01

in principle uh will comp and Scatter

14:04

from an electron right but the shift in

14:08

wavelengths is going to be so small that

14:11

when the initial wavelength is up here

14:12

in the 400 to to 40 to 4.2 nanometer

14:16

range right the shift in wavelength is

14:19

going to be parts per million or or or

14:22

or tens of parts per million and it'll

14:24

be extremely difficult to measure that

14:26

fractional shift in wavelength so

14:29

experimentally even though the

14:31

wavelength of the incident light is

14:33

Shifting experimentally you don't have a

14:36

chance of actually uh making a

14:38

measurement of that energy shift it's

14:40

only when you get down into the

14:42

ultraviolet region or into the X-ray

14:44

region of the electromagnetic spectrum

14:47

that the shift in the wavelength from

14:49

the initial to the final wavelength is

14:51

significant so here it starts to be

14:53

about 11

14:54

12% right uh uh that that you can start

14:58

to make an accurate measurement so the

15:00

moral of this discussion is that uh all

15:03

electromagnetic radiation should suffer

15:06

Compton scattering uh but experimentally

15:10

it's very difficult to measure the

15:11

energy loss when the wavelength of of

15:14

the electromagnetic radiation is large

15:16

and so for that reason you have to

15:18

confine the uh the measurements

15:20

experimentally to the X-ray region of

15:23

the electromagnetic

15:25

spectrum another assumption that you uh

15:28

might um wonder about is uh why do we

15:32

assume that the uh electromagnetic

15:34

radiation is scattered just from an

15:37

electron right uh this incident

15:39

electromagnetic radiation penetrates a

15:42

carbon block the carbon block contains

15:45

carbon atoms that could scatter the

15:47

photons uh the carbon block contains

15:50

protons in the nucleus of the carbon

15:53

atom that could scatter the uh the

15:55

photons there's neutrons inside the

15:57

carbon block also confined to the

15:59

nucleus they could scatter uh uh the

16:03

incident electromagnetic radiation why

16:05

is it that it's only the electrons that

16:07

are considered in Compton's

16:10

analysis and so what I what I do to

16:13

address that issue is I I just calculate

16:15

what the Compton wavelength is for a

16:17

variety of different objects so if you

16:20

assume that the scattering of the uh

16:23

incident x-rays is due to the carbon

16:26

atom then in this formula you've got to

16:28

put in the mass of the carbon atom so

16:31

the mass of the carbon atom is about 2 *

16:33

10us 26 kg and when you calculate the

16:37

Compton wavelength for scattering uh

16:40

from a a carbon atom you'll find that

16:42

the Compton wavelength this quantity H

16:45

over MC is about 10 Theus 16 M well

16:49

that's a that's a that's a number that's

16:51

extremely small and again experimentally

16:54

would not be uh measurable so if um if

16:58

in in fact uh uh the incident photons

17:01

are scattered from a carbon atom um the

17:05

exiting photons would have a a a

17:07

wavelength shift that would be

17:09

undetectable simply because it would be

17:11

shifted by a quantity that's too small

17:14

to be measured you can come to the same

17:17

conclusion for a proton right uh the

17:20

Situation's a little bit better but you

17:22

still get numbers on the order of 10

17:24

Theus 15 M and wavelength shifts on the

17:27

order of 10 Theus 15 m i don't believe

17:30

can be measured experimentally so it's

17:32

only when you get the mass of an

17:33

electron which is which is really small

17:37

right that you start to measure uh uh a

17:40

significant uh effect in the uh uh

17:44

energy loss from the incident Photon to

17:47

the uh scattered Photon so that's

17:50

something that uh hopefully is

17:52

reasonably clear and shouldn't cause

17:54

confusion in your in your uh attempt to

17:57

try and understand the the uh importance

17:59

of this Compton

18:01

scattering so uh is there a physical

18:05

significance for this Compton wavelength

18:08

uh it's a it's a it's a number that

18:11

keeps cropping up and we'll we'll see it

18:13

U later on in in the semester when we

18:16

discuss uh uh the energy levels for

18:20

let's say a hydrogen atom um and the the

18:23

physical significance of that Compton

18:25

wavelength is that uh if just by just by

18:31

manipulating uh Compton's formula right

18:35

you can show that the Compton wavelength

18:38

one over the Compton wavelength is equal

18:40

to mc^ 2 with a proportionality constant

18:44

given by Plank's constant times the

18:46

speed of light and so what this

18:48

basically means is this means that when

18:50

the rest energy of of the of a particle

18:54

uh is equal to uh let's say the the the

18:58

frequency the the the uh the photon

19:01

energy right when those two things are

19:04

equal then then you've got a

19:06

relationship that ties the Compton

19:08

wavelength to this rest energy uh MC

19:11

squar for the

19:13

particle we are not in a position to

19:16

discuss rest energies at this point that

19:18

will come much later in the course but I

19:22

wanted to uh at least uh make this um uh

19:26

connection now because uh this this is

19:29

Einstein's famous formula equals

19:32

mc^2 U and we'll derive that uh uh later

19:36

on and and hopefully you remember that

19:39

and tie it back to this

19:41

discussion uh the last the last point I

19:43

want to discuss is why historically is

19:46

Compton's experiment so important and

19:49

the answer is it it was really the

19:52

experiment that convinced the last

19:54

Skeptics uh for this Photon theory of

19:57

light right so you recall uh photons are

20:01

packets of energy uh they had been

20:04

initially introduced in 1900 by PL in

20:07

order to to uh uh discuss the black body

20:11

emission spectrum from heated objects uh

20:15

the idea that you could treat light as

20:17

quantized packets of energy was rather

20:20

uh remarkable and uh there were still

20:23

Skeptics up until

20:25

1926 that thought this uh this uh

20:28

hypothesis of photons was uh probably

20:32

not correct but when Compton analyzed

20:35

his data by treating light uh with a a

20:40

frequency F to have a specific energy

20:43

and a specific momentum and when he was

20:45

able to uh do the kin kinematics of that

20:50

uh collision between the photon and the

20:52

electron it was you know the the

20:55

agreement between his theory and

20:57

experiment that finally convinced the

20:58

last Skeptics that yeah light is in fact

21:02

a particle it does uh it does have a

21:05

momentum and it its energy is quantized

21:08

according to the simple formula eal H *

21:11

F so historically that's why Compton's

21:14

experiment is is so critical and um it

21:19

sort of uh sets the stage for for the

21:23

developments of of modern quantum

21:25

physics uh for the next uh uh

21:29

75 years or so right it uh it uh it it

21:33

finally convinced everybody that light

21:35

does behave as a particle so that in

21:39

itself is a puzzle and uh it will form

21:42

the uh topic of discussion for uh the

21:45

lectures that are going to follow next

21:47

week so uh come back for those and uh

21:51

we'll uh try to set up the uh dilemas

21:54

that uh uh light poses uh when we try to

21:58

understand uh its properties so uh I

22:02

think that's all I got to say for this

22:03

week uh thank you for uh listening and

22:06

we'll see you uh later on