22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 9 (11/22/06)

Gamma In teractions: Compton Scatte ring

References :

R. D. Evans, Atom ic Nucleus (M cGraw-Hill New York, 1955), Chaps 23 – 25..

W. E. Meyerhof, Elements of Nuclear Physics (McGraw-Hill, New York, 1967), pp. 91 - 108.

W . Heitler, Quantum Theory of Radiation (Oxford, 1955), Sec. 26.

W e are inte r e sted in the intera ction s of ga mm a ra ys, electrom agnetic radiations produced by nuclear tran sition s . These are t ypically photon s with energ i es in the rang e of ~ 0.1 – 10 Mev. The attenuation of the in tensity of a beam of ga mma rays in an absorber, sketched in Fig. 17.1, follows a true exponential variation wi th the distance of penetration, which is unlike th at of charged particles,

Fig. 17.1 . Attenuation of a beam of gamma radi ation through an absorber of thickness x.

I ( x )  I e   x (17.1)

The interaction is expressed through the linear attenuation coefficient which does not depend on x but does depend on the energy of the incident gamma. By at tenuation we m ean either scattering or absorption. Since either process will rem ove the gamma from

the beam , the probability of penetrating a dist ance x is the sam e as the probability of

traveling a distance x without any interaction, exp(- x). The a ttenua tion c o ef f i cient is

therefore the probability per unit path of interac t ion; it is wha t we would call the m acroscopic cross sectio n   N  in the case of neutron interaction.

There are several d i fferent proces ses of ga mm a interaction. E ach process can be treated as occurring independently of the othe rs; f o r this reaso n is the sum of the

individual contributions. Thes e contributions, of course, are not equally important at any given energy. Each process has its own en ergy variation as well as dependence on the atom ic num b er of the ab sorber . W e will f o cus o u r discus sio n s on the th r ee m o st im portant processes of gamma interaction, Com p ton scattering, photoelectric effect, and pair production. These can be classif i ed by the o b ject with which the pho ton interac t s and the type of process (absorption or s catte ring), as shown in the m a trix b e low.

Photoelec t r i c is th e abso rption of a photon followed by the ejecti on of an atom ic electron. Com p ton scatte ring is th e inel astic (photon loses energy) rela tivistic scattering by a free electron. Fo r Com p ton scat tering it is im plied th at the pho to n energy is a t lea s t com p arable to the rest m a ss energy of the el ectron. If the photon energy is m u ch lower than the res t m a ss energ y , the scattering by a free electron th en becom e s elastic (no energy los s ), a process th at we will call T hom son scattering. W h en the photon energy is greater than twice the rest m a ss energy of electron, the photon can be absorbed and an electron-po s itron p a ir is em itted. This is the pro c ess of pair p r oduction. Other com b inations of interaction and process in th e matrix (m arked x) c ould be discussed, but they ar e of no inte rest to us in this cla ss.

Inter actio n with	\	absorption	elas ti c s cattering		inela s ti c sca tte ring
atom ic electron		photoelectric	Thomson		Compton
nucleus		x		x	x
electric field aro the nucleus	und	pair productio	n	x	x

Given what we have just said, th e attenuation coefficient beco m e s

   C       (17.1)

where the su bscrip ts C, , and will be used hencef orth to d e note Com p ton scattering, photoelectric effect, and pair production respectively.

Compton Scatter i ng

The trea tm ent of Com p ton scatter i n g is sim ilar to our analysis of neutron scattering in several ways. This analogy shoul d be noted by the student as the discussion unfolds here. The phenom enon is the scatte ring of a photon with incom i ng m o m e ntum ħ k by a f r ee, stationa ry ele c tron, which is tr eated re lativis tica lly . Af ter scattering a t

angle  , the photon has mom e ntum ħ k ' , while the electron m oves off at an angle with

mo ment u m p and kinetic energy T, as shown in Fig. 17.1.

Fig. 17.1 . Schem a tic of Com p ton scattering at angle  with mom e ntum and energy transferred to the free electron.

To analyze the kinem a tics we write the m o mentum and energy conservation equations,

ħ k  ħ k '  p (17.2)

ħ ck  ħ ck '  T (17.3)

where the relativis tic en ergy-m o m e nt um relation for the electron is

cp 

(17.4)

with c being the speed of light. One should also recall the relations,   ck and    c , w ith and being the circular and linea r frequency, respectively (   2   ), and  the wavelength of the photon. By algebr aic m a nipulations one can obtain the following results.

 '    c  c  h  1  cos   (17.5)

 '  m e c

 '  1

(17.6)

 1   ( 1  cos  )

T  ħ   ħ  '  ħ   ( 1  cos  ) (17.7)

1   ( 1  cos  )

cot   ( 1   ) tan 

2

(17.8)

In (17.5) the factor h/m e c = 2.426 x 10 -1 0 cm is called the Compton wavelength. The gain in w a veleng th af ter s ca ttering at an angle of  is k now n as the C o m p ton shif t. This shif t in w a veleng th is ind e pen d ent of the incom i ng photon energy, whereas the shift in energy (17.7) is dependent on energy. In (17.6) the param e ter   ħ  / m e c 2 is a m easure of the photon energy in units of the electron rest m a ss energy (0.511 Mev). As   1 ,  '  

and the pro c ess goes fro m inelastic to elas tic. L o w-energy p hotons are s cattered with only a m ode rate energy change, while high-en ergy photons suffer large energy change.

For exam ple, at    / 2 , if ħ  = 10 kev, then ħ  ' = 9.8 kev (2% change), but if ħ  =

10 Mev, then ħ  ' = 0.49 Mev (20-fold change).

Eq.(17.7) gives the en ergy of the recoiling electron which is of interes t b ecause this is of ten the quantity that is m easured in C o m p ton sca tte ring. In the lim it of energetic gammas, >> 1, the scattered g a mma energy beco m e s only a function of the scattering

angle; it is a m i nim u m f o r backward s cattering (    ), ħ  '  m c 2 / 2 , w h ile f o r 90 o scattering ħ  '  m c 2 . The m a xi m u m energy transfer is given by (17.7) with    ,

T max

 ħ 

1  1

(17.9)

2 

Klein-Nishina Cross Section

The proper derivation of the angular differential cross section for Com p t on scattering requires a qu antum m echanical calcu la tion using th e D i rac ’ s re lativ istic the o ry of the electron. This was first published in 1928 by Klein and Nishin a [for details, see W . Heitle r]. W e will s i m p ly quote the f o rm ula a nd discuss som e of its im plication s . The cross sectio n is

d  r 2   '  2    ' 

C  e    

 2  4 cos 2  

(17.10)

d  4      '  

where  is th e angle between the electric vector  (polarization) of the incident photon and that of the scattered photon,  ' . The diagram s shown in Fig. 17.2 are helpful in visualizing the various v ectors involved. Reca ll that a pho to n is an electrom a gnetic wave

Fig. 17.2 . Angular relations am ong i n co m i ng and outgoing wave vectors, k and k ' , of the scattered photon, and the electric vectors,  and  ' , which are transve r se to the corresponding wave vectors.

characterized by a wave vector k and an electric v ector  which is perpend i c u lar to k . For a given incident photon with ( k ,  ) , shown above, we can decompose the scattered photon electric vector  ' in to a com ponent  '  perpend i cular to the plane con t aining k and  , and a parallel com p onent  ' ∐ which lies in th is plane. For the perpend i cular com ponent cos   = 0, and for the parallel com ponent we notice that

cos   cos    

2

  sin 

 sin  cos  (17.11)

 

Therefore,

cos 2   1  sin 2  cos 2  (17.12)

The decom p osition of the scattered photon el ectric vector m e a n s that the angular dif f e rentia l cross se ctio n can be written a s

d  C

 d  

 

 d  

 

d   d     d   ∐

r 2   '  2    ' 

= e 

  

 2 sin 2  cos 2  

(17.14)

2      '  

This is because the cross section is proportiona l to the total scattere d intensity which in turn is p r opo rtion a l to (  ' ) 2 . Since  '  and  ' are orthogonal, (  ' ) 2 = (  '  ) 2 + (  ' ) 2 and the cross section is the sum of the c ontributio ns from each of the components.

In the low - e n ergy (non- r e lativis tic ) lim it, ħ   m c 2 , we have  '   , then

 d  C 

~ 0 ,

 d  C 

~ r 2  1  sin 2  cos 2   (17.15)

 d    d   e

     ∐

This m eans that if the incident radiation is polarized (photons have a specific polarization vector), then the scattered radi ation is also polarized. But if the incident radiation is unplolarized, then we have to average d  C / d  over all the allowed directions of  ( r e me mb e r  is perpend i c u lar to k ). Since d  C / d  depends only on angles 

and , we can obtain the result for unpolarized radiation by averaging over . Thus,

 d   1 2   d  

 C   d   C 

 d   unpol 2  0  d   ∐

r 2

= e 1 

2

cos 2  

(17.16)

2 2 -1 3

w ith r e  e / m c = 2.818 x 10 cm being the classical ra dius of the electron (cf.

(14.1)). T h is is a well-known expressi on for the angular differential cross section for Thom son scattering. Integrating th is over all solid angles g i v e s

   d   d  C 

 8  r 2   o (17.17)

C  d   3 e

  unpol

which is known as the Thom son cross section.

Returning to the general result (17. 14) we have in th e case of unpolarized radiation,

d  r 2   '  2    ' 

C  e    

 sin 2  

(17.18)

d  2      '  

W e can rew r ite this result in term s of and cos  by using (17.6),

d  r 2

 1  2   2 ( 1  cos  ) 2 

C  e  1  cos 2   

  1 

 (17.19)

d  2

 1   ( 1  cos  )  ( 1  cos 2  )  1   ( 1  cos  ) 

   

The behavior of d  C / d  is shown in Fig. 17.3. Notice that at any given the angu lar distribution is peaked in th e forward direction. A s increases, the forward peaking becom e s m o re pronounced. The deviation from Thom son scattering is largest at large

scattering an gles; even at ħ  ~ 0.1 Mev the assum p tion of Thom son scattering is not

1.00

0.75

0.50

0.25

0

0 o 30 o

60 o

90 o 120 o 150 o 180 o

Photon Angle 

F i g u r e b y M I T O C W .

Fig. 17.3 . Angular distribution of Compton scat tering at various incident energies E r . All curves are norm alized at 0 o . N o te the low - en ergy lim it of Thom son scattering. (from H e itle r)

valid. In practice the Klein-Nishina cross section has been found to be in excellent agreem ent w ith exper i m ents at lea st out to ħ   10 m c 2 .

To find the total cross section per el ectron for Compton scattering, one can

integrate (17.19) over solid angles. The analytic al result is given in Evans, p. 684. W e w ill note only the tw o lim iting cases ,

 C  1  2   26  2  ...    1

 o 5

3 m c 2  2 ħ  1 

( 1 7 . 2 0 )

= e  ℓ n     1

8 ħ   m c 2 2 

W e see that at high en ergies (  1 Mev ) the Com p ton cross section decreases with energ y like 1 / ħ  .

Collis ion, Scatte ring an d Absorptio n Cross Sections

In discus sin g the Com p ton ef f ect a distin ction sh ould be m a de between c o llis ion and scattering. Here collision refers to ordina ry scattering in the sense of rem oval of the photon from the beam . This is what we have been discussing above. Since the electron

recoils, not all the origin al energy ħ  is scattered, on ly a fraction  ' /  is. Thus on e

can define a scattering cross section,

d  sc   ' d  C (17.21)

d   d 

This lead s to a sligh tly d i f f e rent tota l cross se ctio n,

 sc ~ 1  3   9 . 4  2  ...    1 (17.22)

 o

Notice that in the case of Th om son scattering all the ener gy is scattered and none are absorbed. T h e difference between  C and  sc is ca lled the Com p ton absorption cross section.

Energy Distribution of Comp ton Electrons and Photons

We have been discussing the angular distribution of the Compton scattered

photons in term s of

d  C

d 

. To transform the angular distri bution to an energy distribution

we need first to reduce the angular distribution of two angle variables,  and , to a distribution in  (in the sam e way as we had done in Chapter 15). W e therefore define

d  2  d  d 

C  d  C sin   C 2  sin  (17.23)

d  0 d  d 

and write

d  C 

d  '

d  d  '

(17.24)

with  ' and  being related through (17.6). Since w e can also relate the scattering angle

 to the ang l e of electron recoil  through (17.8), we can obtain the distribution of electron energy by perform i ng two transform a tions, from  to  f i rst, th en f r om  to T by using the relation

T  ħ 

2  cos 2 

( 1   ) 2   2 cos 2 

(12.24)

as found by com b ining (17.7) and (17.8). Thus,

d  C 2  sin  d   d  C 2  sin  d  (12.25)

d  e d 

d  C  d  C 2  sin  (12.26)

d  d  e

d  C 

dT

d  dT

(12.27)

These resu lts show that all the dis t r i butions a r e related to one another. In Fig. 17.4 we show severa l calcula te d electron recoil energy distributions which can b e com p ared with the experimental data shown in Fig. 17.5.

16

12

8

4

0

0 0.5

1.0

1.5

2.0

2.5

Electr on Energy T c , Mev

Figure by MIT OCW .

Fig. 17.4. Energy distribution of Compton elect rons for several incident gamma-ray energies. (from Meyerhof)

1,000

500

0

0 20 40

60 80 100

Pulse Height, V olts

Figure b y MIT OCW .

Fig. 17.5 . Pulse-height spectra of Compton el ectrons produced by 0.51- and 1.28-Mev gam m a rays. (from Meyerhof)

For a given incident gamm a energy the reco il energy is m aximum at    , where ħ  ' is

sm allest. We had seen previously that if the photon energy is hi gh enough, the outgoing photon energy is a constant at ~0.255 Mev. In F i g. 17.4 we see that for incident photon energy of 2.76 Mev the m a xi m u m electron r ecoil energy is approxim a tely 2.53 Mev, which is close to the value of (2.76 – 0.255). This correspondence should hold even

better at h i g h er energies, and not as well at lower energies, such as 1.20 and 0.51 Mev.

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We can also see by comparing Figs. 17.4 a nd 17.5 that the relative m a gnitudes of the distributions at the two lower incident ener g i es m a tch quite well betwee n calcu latio n and experim e nt. The distributi on peak s n ear th e cutof f T ma x because there is an appreciable range of  near    , where cos  ~ 1 (cosine changes slowly in this region) and so

ħ  ' rem a ins clo s e to m e c 2 /2. This f eatur e is rem i nisc ent of the Bragg curve depicting the

specific ionization of a charged particle (F ig. (14.3)).

From the electron energy distribution d  C we can deduce directly the photon

dT

energy distribution d  C fro m

d  '

d  C 

d  '

 ħ d  C

dT

(12.28)

since ħ  '  ħ   T .

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