22.101 Applied Nuclear Physics (F all 2006)

Lecture 20 (11/27/06)

Gamma In t eract ions: Photoelec t r i c Effect an d Pair Production

References :

R. D. Evans, Atom ic Nucleus (McGraw-Hill New York, 1955), Chaps 24, 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.

Photoelectric Effect

This is the predom inant mode of - in tera ction in all k i nds of m a tter, espe cially the high-Z absorbers, at ener gies less than ~ 0.1 Mev. The process, sketched in Fig. 18. 1, is the interaction betw een the gam m a and the entire atom (the electron cloud) that

results in the absorption of the gamma and th e ejection of an electr on with kinetic energy

T ~ ħ   B e (18.1)

where B e is the electron binding energy. The recoil ing atom is left in an excited state at an excitatio n energy of B e ; it can then de-exc ite by em itting x-r a ys or Auge r ele c tron s.

Fig. 18.1. Schem a tic of photoelectric effect, abso rption of an incident gamma ray and the ejection of an atom ic electron a nd excitation of th e recoiling atom .

1

Notice this is atom ic and not nuclear ex citation. One can show that the incident cannot be totally absorbed by a free electron because m o m e ntum and energy conservations cannot be satisfied sim u ltaneou sly, but total absorption can occur if the

electron is initially bound in an atom . The m o st tightly bound electrons have the greatest

probability of absorbing the . It is known theoretically and experim e ntally that ~ 80% of the absorption occurs in the K (innerm o st) shell, so long as ħ   B e ( K  shell ) .

Typical ionization po ten tia ls of K elect rons are 2.3 kev ( A ℓ ), 10 kev (Cu), and ~ 100 kev

(Pb).

The conservations equations for photoelectric effect are

ħ k  p  p

a

(18.2)

ħ   T  T a  B e (18.3)

where p a is the m o m e ntum of the recoil ing atom whose kinetic energy is T a . Since T a ~

T ( m e / M ) , where M is the m a ss of the atom , one can usually ignore T a .

The theory describing the photoelectric effect is essentia lly a first-order perturb a tion theory c a lc ulation (cf . Heitle r, S ec. 21). In th is case th e tr an sition tak i ng place is between an initial state con s isti ng of two particles, a bound electron, wave function ~ e  r / a , with a  a / Z , a being the Bohr radius (= ħ 2 / m e 2 = 0.529 x 10 -8

cm ), and an incident photon, wave function e i k  r , and a final state consisting o f a free electron whose wave function is e i p  r / ħ . The inte rac tio n potential is of the f o rm A  p , where A is the vecto r po tential of the electrom agnetic radiation and p is th e electron m o m e ntum . The result o f this calcula tion is

d  r 2 Z 5  m c 2  7 / 2 sin 2  cos 2 

  4 2 e  e 

(18.4)

d  ( 137 ) 4  ħ    v  4

   1 

cos  

 c 

2

where v is the photoelectron speed. The pol ar and azim u thal angles specifying the direction of the photoelectron ar e def i ned in Fig. 18.2. On e should pay particular notice to the Z 5 an d ( ħ  )  7 / 2 variation. As for the angular dependence, the num e rator in

Fig. 18.2 . Spherical coordinates defining the pho toelectron direction of em ission relative to the incoming photon wave vector and its polarization vector.

(18.4) suggests an origin in ( p   ) 2 , while the denom inator suggests p  k . Integr ation of (18.4) gives the to tal c r o ss section

   d  d  

 4 2  o

Z 5  m c 2  7 / 2

4

d  ( 137 )

 ħ  

where one h a s ignored the angular d e penden ce in the denominato r and h a s m u ltiplied the result by a factor of 2 to account for 2 el ectrons in the K-shel l [Heitler, p. 207].

In practice it has been found that the charge and energy dependence behave m o re like

   Z n /( ħ  ) 3 (18.6)

with n varying from 4 to 4.6 as ħ  varies from ~ 0.1 to 3 Mev, and with the energy exponent decreasing from 3 to 1 when ħ   m c 2 . The qualitative behavior of the photoelectric cross section is illu strated in the figures below.

3

4.6

4.4

n

4.2

4.0

0.1 0.2 0.3 0.5

1 2 3

Photon Energy in Mev

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

Fig. 18.3. Variation with energy of in cident photon of the exponent n in the total cross section for photoelectric effect. (from Evans)

1000

100

10

1.0

0.1

0.01

0.1

1.0

m 0 c 2 / h 

10 10

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

Fig. 18.4. Photoelectric cross sectio ns showing approxim a te inverse pow er-law behavior which varies with the energy of th e incident photon. (from Evans)

4

60

40

20

0

0 o 20 o

40 o

60 o

80 o 100 o

120 o

140 o

160 o 180 o

 = angle photoelectrons make with direction of  rays

Figure b y MIT OCW .

Fig. 18.5. Angular distribution of photoelectrons fo r various incident photon energies. The peak moves toward forward direction as the energy increases, a behavior which can be qualitatively obtained from the  -dependence in Eq. (18.4). (from Meyerhof)

Edge Absorption

As the photon energy increases the cross section   can show discontinuous jum p s whic h are known as edges . T hese correspond to the onset of additional contribution s when the energy is sufficien t to e j ec t an inne r sh ell e l ectron. The effect is more pronounced in the high-Z m a terial. See Fig. 18.13 below.

Pair Production

In this process, which can occur only when the energy of the incident exceeds 1.02 Mev, the photon is absorbed in the vicinity of the nucleus, a positr on-electron pair is produced, and the atom is left in an excited state, as indicated in Fig. 18.6. The

Fig. 18.6. Kinem a tics of pair production.

5

conservation equation s are therefore

ħ k  p  p

 

( 1 8 . 7 )

ħ    T  _ m c 2    T  m c 2  (18.8)

One can show that these conditions cannot be satisfied sim u ltaneously , the presence of an atom ic nucleus is required. Since the nucle us takes up som e m o m e ntum , it also takes up som e energy in the form of recoil.

The existence of positron is a consequence of the Dirac’s re la tivis tic theo r y of the electron which allows for negative energy states,

E    c 2 p 2  m 2 c 4  1 / 2

(18.9)

One assum e s that all the negativ e states are filled ; these repres ent a “s ea” o f electrons which are generally not observable because no transi tions in to th ese sta t es ca n occur due to the Exclus ion Princip l e. W h en one of the electrons m a kes a transition to a positiv e ene r gy leve l, it leaves a “ho l e” (positron ) which beha ves lik es an elec tron but with a positive charg e. This is illu strated in Fig. 18.7. In other words, holes in the

W

m 0 c 2

- m 0 c 2

Figure b y MIT OCW .

Fig. 18.7. Creation of a positron as a “hole” in th e f illed (neg ative ) ene r g y states and an elec tron as a particle in the unf ille d e n ergy sta t e. (f rom Meyerhof )

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negativ e ene r gy sta t es ha ve positive c h arge, wher eas an e l ectr ons in the p o sitiv e ene r gy states have n e gativ e char ge. The “ho l e” is unstab l e in that it w ill r ecom b ine with an elec tron wh en it loses m o st of its k i n e tic energy (therm alized). This reco mbination process is called pair annihilati on. It usually produces two (annihilation radiation),

each of energy 0.511 Mev, em itted back-to - back . A positron and an electron can form an atom called the positron i um . It’s lif e tim e is ~ 10 -7 to 10 -9 sec depending on the relative spin orientation, the shorter lifetim e co rresponding to antipara llel orientation.

Pair production is intim ately related to the process of Bremsstrahlung in which an electron undergoes a transition from one posit ive energy state to another while a pho to n is em itted. One can tak e over direc t ly the theory of Bremsstrahlung for the trans i tion probability with the incident par tic le being a pho ton instead o f an electron and using the appropriate density of states f o r th e em ission of the positron - elec tron pa ir [ H eitler, Se c. 26]. For the positron the energy differential cross section is

T 2  T 2  2 T T

d  

2   3

    2 T  T   1 

dT  4  o Z

 ħ   3

 ℓ n  ħ  m c 2   2  (18.10)

    e   

where  o

 r 2 / 137 = 5.8 x 10 -4 barns. This result holds under the conditions of Born

approxim a tion, which is a high-energy condition ( Ze 2 / ħ v   1 ), and no screening, which requires 2 T  T  / ħ  m c 2  137 / Z 1 / 3 . Here screening m eans the partial reduction of the nuclear charge by the poten tia l of the inne r-she ll e l ec tr ons. As a re sult of the B o rn approxim a tion, the cross s ection is s y mm etric in T + and T - . Screen ing effects will lead to a lower cros s section.

To see the energy distribution we rewrite (18.10) as

d  

dT 

 Z 2

 o P (18.11)

ħ   2 m c 2

7

where the d i m ensionless factor P is a rather com p licated function of ħ  and Z; its behavior is depicted in Fig. 18.8. W e see the c r oss sec tion incre ases with inc r eas in g

8

6

P

4

2

0

0 0.2 0.4 0.6

T + / (h  -2m o c 2 )

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

0.8 1.0

Fig. 18.8. Calculated en ergy dis t r i bu tion of positron em itted in pair produ ction, a s expressed in Eq.(18.11), with correction for screening (dashed curves) for photon energies abo v e 10 m c 2 . (from Evans)

energy of the incident photon. Sinc e the va lue is sligh tly h i gher f o r Al than f o r Pb, it m eans the cross section v aries wi th Z som e what weaker than Z 2 . At lower energies the cross section favors equal distribution of energy between the positr on and the electron , while at h i g h energies a s light tendency toward unequal di stribution could be noted.

Intuitively we expect the energy distribution to b e biased toward m o re energy for the positron than for the electron s i m p ly because of Coulom b repulsion of th e positron and attraction of the elect ron by the nucleus.

The tota l cro ss section f o r pair p r odu c tion is obtained by in tegrating (18.11). Analytical integration is possi ble only for extrem ely relati vistic cases [Evans, p. 705],

  dT  d      Z

dT

2  28  2 ħ  

 ℓ

9 m c 2

218 

27 

, no screening (18.12)

  

   e   

=  Z 2  28 ℓ n  183  

2  , complete screening (18.13)

o  9  Z 1 / 3 

27 

   

8

Moreover, (18.12) and (18.13) are valid only for m c 2  ħ   137 m c 2 Z  1 / 2 and

ħ   137 m c 2 Z  1 / 2 respectively. The behavior of th is cross section is shown in Fig.

18.9 , along with the cross section for Com pton scattering. N otice that screening leads to a saturation effect (energy independence) ; however, it is no t s i gnif i can t be low ħ  ~ 10 Mev

1

1

1

h  /mc 2

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

Fig. 18.9. Energy variation of pair produc tion cross section in units of  Z 2 .

Mass Atten u ation Coef fic i ents

So long as the different processes of photon interaction are not correlated, the tota l lin ear a ttenua tion c o ef f i cient for -in t eraction can be taken as the s u m of contributions from Compton scattering, photoe lectric effect, and pa ir production, with

  N  and N being the num be r of atom s per cm 3. Since N  N o  / A , where N o is

Avogardro’s num b er and the m ass density of the absorber, it is aga i n usef ul to exp r ess the in tera ction in te rm s of the mass attenua tion coefficient  /  . As we had discussed in Lecture 14, this quantity is e ssentially independent of the de nsity and physical state of the absorber. R ecall our obs ervation in the case of charged particle in teractions that Z / A is

approxim ately constant for all elem ents . Here w e see that in the product N  we get a

factor of 1/A from N, and we can get a facto r of Z from any of th e three cross sectio ns  , so we can also take advantage of Z/A be ing roughly constant in describing photon

9

interaction (the sam e argum e nt does not hol d for neutrons even though it is useful to think of neutron interactions in te rm s of the m acr oscopic cross section  ). W e write

   C      			( 1 8 . 1 4 )
 C /    N o / A  Z  C ,	 C ~ 1 / ħ 	per electron	(18.15)
  /    N o / A    ,	  ~ Z 5 /  ħ   7 / 2	per atom	(18.16)

  /    N / A   ,  ~ Z 2 ℓ n  2 ħ  / m c 2  per atom (18.17)

It should be quite clear by now that th e three processes we have studied are not equally im portan t f o r a given region of Z and ħ  . Generally speaking, photoelectric effect is im portan t at low energies an d high Z, Compton scatte ring is im portan t at interm ediate energies ( ~ 1 – 5 Mev) and a ll Z, and pair production becomes at higher energies and high Z. This is illustrated in Fig. 18.10.

We also show several m a ss attenuation coefficients in Figs. 18.11 – 18.13. One should m a ke note of the m a gnitude of th e attenuation coeffi cients, their energy dependence, and the con t ribu tion ass o ciated with each proces s. In com p aring th eory with experim e nt the agreem ent is good to about 3 % for all elem ents at ħ  < 10 m c 2 . At

higher en erg i es dis a greem ent sets in at high Z (can reach ~ 10 % for Pb), which is du e to the use Born approxim a tion in calculating   . If one corrects for this, th en ag reem ent to

within ~ 1% is obtained out to energies ~ 600 m c 2 .

10

120

80

40

0

0.01

0.1 1

h  in Mev

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

10 100

Fig. 18.10. Regions where one of the three -interactions dominates over the other two. (from Evans)

10

1

0.1

0.01

0.001

0.01

0.1

1

Mev

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

10 100

Fig. 18.11. Mass attenuation coefficient for photons in air computed from tables of atom ic cross sections. (from Evans) 11

10

1

0.1

0.01

0.001

0.01

0.1

1

Mev

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

10 100

Fig. 18.12. Mass attenu ation co efficients for pho tons in water. (from Evans)

12

Figure b y MIT OCW .

Fig. 18.13. Mass attenu ation co efficients for pho tons in Pb. (from Evans)

13