22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 5 (11/6/06)

Charged-Particle Interactions: Radiation Loss, Range

References :

R. D. Evans, The Atomic Nucleus (M cGraw-Hill, New York, 1955), Chaps 18-22.

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

The sudden deflection of an electron by th e Coulom b field of nuclei can cause the electron to radiate, producing a co ntinuous spectrum of x-rays called bremsstrahlung . The fraction of electron energy conv erted into brem sstrahlung increases with increasing electron energy and is greater for m e dia of high atom ic num b er. (This process is im portant in the production of x-ra ys in conventional x-ray tubes.)

According to the classical theory of electrodynam i cs [J. D. Jackson, Classical Electrod y na mics (W iley, New York, 1962), p. 509], the acceleration prod uced by a nucleus of charge Ze on an incident partic le of charge ze and m a ss M is proportional to Zze 2 /M. Th e inten s ity o f radiation e m itted is pro portion a l to (ze  acceleration) 2 ~ (Zz 2 e 3 /M) 2 . Notice th e ( Z /M) 2 dependence; this shows that bremsstrahlung is m o re im portant in a high-Z medium . Als o it is m o re important for electrons and positrons than for protons and -particles. Another way to understand the (Z/M) 2 dependence is to recall the derivation of stopping power in Lec13 where th e m o m e ntum change due to a collision between the incident part icle and a target nucleus is (2ze 2 /vb) x Z. The factor Z represents the Coulom b field of the nucleus (in L ec13 this was unity since we had an atom ic electron as the target). Th e recoil v e locity of the target nucleus is therefore proportional to Z/M, and the recoil energy, whic h is the intensity of the radiation em itted, is therefore proportional to (Z/M) 2 .

In an individual deflection by a nucleus , the electron can radiate any am ount of energy up to its kinetic energy T. The spectrum of bremsstrahlung wavelength for a thick targe t is of the f o rm sketched below, with  min  hc / T . This converts to a frequency spectrum which is a cons tant up the m a xim u m frequency of  max  T / h . The

shape of the spectrum is independent of Z, a nd the intens ity va ries with electron energy like 1 / T.

In the quan t um m echanical th eory of bremsstrahlung a plane wave representing the electron enters the nu clear field and is scattered. There is a sm all but finite chance that a pho to n will be em itted in th e p r ocess. Th e theory is in tim a tely rela ted to the the o ry of pair production where an el ectron-positron pair is produce d by a photon in the field of a nucleus. Because a radiative process invol v e s the coupling of th e electron with the electrom agnetic field of the em itted photon, the cross sectio ns for radiation are of the

order of the f i ne-stru c tu r e constan t [ D icke and W ittke, p. 11] , e 2 / ħ c ( = 1/137), tim es the cross section for elastic scatte ring. This m eans that m o st of the deflections of electrons by atom ic nuclei result in elas tic scattering, only in a sm all num b er of instances is a photon em itted. Since th e class i cal th eory of bremsstrahlung predicts the em ission of radiation in every colli sion in which the electron is defl ec ted, it is inco rrec t . However, when averag ed over a ll c o llis ions the class i ca l and quantum mechanic al c r oss sections are of the sam e order of m a gnitude,

 rad ~

2  2  2

 

2

cm 2 /nucleus (14.1)

137  m e c 

where e 2 / m e c 2 = r e = 2.818 x 10 -1 3 cm is the class i cal rad i us of electron. In the few collisions w h ere photon s are em itted a relativ el y large am ount of energy is radia t ed. In this w a y th e quantum theory replaces the m u ltitud e of sm all-e n ergy loss es predic ted b y the classical theory by a m u ch sm alle r num ber of large r-en e rg y losses. Th e spectral

distributions are therefore di f f e rent in the tw o th eories, with the quantum description being in better agreem ent w ith expe rim e nts.

Given a nucleus of charge Ze and an in cident electron of kinetic energy T, the quantum m e chanical differential cross section for the em ission of a photon with energy in d ( h  ) about h  is

 d   T  m c 2 1

  BZ 2 e

(14.2)

  o

  rad

T h 

where    e 2 / m c 2  2 / 137 = 0.580 x 10 -3 barns and B ~ 10 is a very slowly varying dim e nsionless function of Z a nd T. A general relation betw e e n the ene r g y dif f e rentia l cross section, such as (14.2), and th e energy loss per unit path length is

dT T d 

 dx  n  dE E dE

(14.3)

where d  / dE is th e dif f e rentia l cross se ctio n f o r en ergy loss E. Applying this to (14.2) we have

 dT  T  d  

  

dx

 n  d ( h  ) h   d ( h  ) 

  rad 0   rad

= n ( T  m c 2 )  ergs/cm (14.4)

1  h  

where  rad   o Z 2  d   B   Z B (14/5)

2

0  T 

is the tota l b r emsstrahlu n g cross section. The variation of B , the bremmstrahlung cross section in units of  Z 2 , with the kinetic en ergy of an in cident electron is shown in the sketch for media of various Z [Evans, p. 605].

20

10

0

0.1

1 10 100 1000

Kinetic Ener gy , T , of Electrons in Mev

Figure b y MIT OCW . Adapted f rom Evans.

Comparison of Various Cross Sections

It is instructive to com p are the cross s ections describing the interactions that we have consid ered between an incid e n t el ectron and the atom s in the m e dium. For nonrelativistic electrons, T  0.1 Mev and  v / c  0 . 5 , we have the following cross sections (all in barns/atom ) [Evans, p. 607],

 ion 

2  Z

 4



ℓ n

 I



 ionization (14.6)



 nuc

  Z 2

4  4

backscattering by nuclei (14.7)

 el

 2  Z

 4

elas tic s cattering by at omic electrons (14.8)

'

rad

 8  1 Z 2

3  137  2

bremsstrahlung (14.9)

4

where   4  ( e 2 / m c 2 ) 2 = 1.00 barn. The values of thes e cross sections in th e case of

0.1 Mev electrons in air (Z = 7.22, I ~ 100 ev) and in Pb (Z = 82, I ~ 800 ev) are given

in the follow i ng table [from Evans, p.608]. The difference between 

rad

and  '

is tha t

the form er corresponds to fractional loss of total energy, dT /( T  m c 2 ) , while the la tter corresponds to fractional loss of kinetic energy, dT /T.

Appr oximate Cr oss Sections in Barns per Atom of Pb, and of Ai r , for Incident 0.1-Mev Electr ons

Approximate variation with Z and  ...............	Z /  4	Z 2 /  4	Z /  4	Z 2	Z 2 /  2
Air ........................ Pb.........................	1,200 9,400	150 19,000	160 1,800	0.16 21	1.0 130

Mass Absorption

Figure b y MIT OCW . Adapted f rom Evans.

Ionization losses per unit distance are proportional to nZ, the num b er of atom ic electrons per cm 3 in the absorber (medium ) . W e can express nZ as

nZ  (  N o / A ) Z   N o ( Z / A ) (14.10)

where is the m a ss density, g/cm 3 , and N o the Avogadro’s number. Since the ratio (Z/A) is ne a r ly a cons tan t f o r all ele m ents, it m e ans tha t nZ /  is also approx im ately

constant (except for hydrogen). Therefore, if the distan ce alo ng the path of the charg e d particle is m easured in u n its of  dx  dw (in g / cm 2 ), then the ioniz atio n losses, -dT/dw (in erg s cm 2 /g) becom e more or less independent of the m a terial. W e see in Fig. 14.1, the expected behavior of energy loss being m a te rial independent holds only approxim a tely,

as -dT/dw actually decreases as Z in creas es. This is due to two reasons, Z/A decreasing slightly as Z incre a ses an d I incre a sing ly lin ear ly with Z.

6

4

2

Ionization Loss Radiation Loss

0

0 2

Air

Al

Pb Pb

Al Air

4

Kinetic Energy of Electr on, Mev

Figure b y MIT OCW . Adapted f rom Evans.

Fig. 14.1. Mass abso rp tion energy losses, -dT/d w , for electrons in air, Al, and Pb, ionization losses (upper curves) versus brem sstrahlung (lower cu rves). All curves refer to energy losses along the actual path of the elec tron. [Evans, p.609]

We have seen that ionization losses per path length vary m a inly as 1/v 2 wh ile radiative losses increase with increasing energy. The tw o becom e roughly com parable when T >> Mc 2 , or T >> m e c 2 in th e case of electrons. The ratio can be approxim ately expressed as

 dT / dx   m  2  T 

ra d  Z  e   

(14.11)

 dT / dx   M   1400 m c 2 

ion

 e 

where for electrons, M  m e . The two losses are therefore eq ual in the case of electro ns for T = 18 m e c 2 = 9 Mev in Pb and T ~ 100 Mev in water or air.

Range, Range-Energy Relations, and Track Patterns

W h en a charged particle enters an ab sorbing m e dium it imme diately interacts

any indiv i du al encoun ter is sm all, so the track of the heavy charged particle tends to be quite s t ra igh t excep t at the very end of its tr av el when it has lost prac tically all its kin e tic energy. In this case we can estim a te the ra nge of the particle, the distance beyond which it cannot penetrate, by inte grating the stopping power,

R 0  dx 

T o 

 1

dT 

R   dx    dT  dT     dx 

dT (14.12)

0 T 

 0  

where T o is the initia l kin e tic ener gy of the particle. An estim ate of R is given by taking the Bethe form ula, (13.7), for the stopping power and ignoring the v-dependence in the logarithm . Then one finds

T o

R  

0

TdT = T 2 (14.13)

This is an exam ple of a r a nge -energy relation. Given what we have said about the region of applicability of (13.7) one m i ght expect th is behavior to hold at low energies. At high energies it is m o re reaso n able to take the stopping power to be a constant, in which case

T o

R   dT = T o

0

(14.14)

W e will retu rn to see wh ether su ch b e havior are s een in expe r i m e nts.

Experim e ntally one can determ ine the energy loss by the number of ion pairs produced from an ionization event. The am ount of energy W required for a particle of certain energy to produce an ion pair is known. The num ber of ion pairs, i, produced per unit pa th len g th ( spec ific ionization ) of the charg e d particle is then

i  1   dT 

(14.15)

 dx 

The quantity W depends on com plicated proc esses such as ato m ic excitation and secondary io nization in additi on to prim ary ionization. On the other hand, for a given m aterial it is approxim ately indep en d ent of the nature of the particle or its kinetic energy. For exam ple, in air the values of W a r e 35.0, 35.2, and 33.3 ev for 5 kev electrons, 5.3 Mev alphas, and 340 Mev protons respectively.

The specific ionization is an appropria te m easure of the ionization processes taking p l ace along the p a th (track len g th) of the charged particle. It is u s eful to regard (14.15) as a function of the distance traveled by the particle. Such results can be seen in Fig. 14.2, where one s e es a characteristi c sh ape of the ionization cu rve for a heavy charged particle. Ionization is constant or in crea s i ng slowly d u ring th e ear ly to m i d stages of the total travel, then it rises more quickly and reach es a peak value at th e end of the range before dropping sharply to zero.

0.8

0.4

0

3.0

2.0

1.0 0

Residual Range (  -particle), cm air

Figure b y MIT OCW . Adapted f rom Meyerhof.

Fig. 14.2. Specific ionization of heavy particles in air. Resid u al rang e ref ers to the distan ce still to travel bef o re com i ng to rest. Proton range is 0.2 cm shorter than that of the -particle [Meyerhof, p.80].

We have already m e ntioned that as th e charged particle loses energy and slows down, the probability of capturi ng electron increases. So the m ean charg e of a beam of particles will decrease w ith th e decrease in their s p eed (cf. Fig . 13.3). This is the reaso n why the specific ionization shows a sharp dr op. The value of –dT/dx along a particle track is also called specific energy los s . A plot of –dT/dx along the track of a charged

from a plot of –dT/dx for an individual partic le in that the form er is an average over a large num ber of particles. Hence the Bra gg curve includes the effects of straggling (sta tis tic al d i str i bution o f range valu es f o r particles having th e sam e initia l veloc ity ) a nd has a pronounced tail beyond the extrapolated range as can be seen in Fig. 14.3.

r x r R i

Figure b y MIT OCW . Adapted f rom Evans.

Fig. 14.3 . Specific ion i zation for an individual particle vers us Bragg curve [Evans, p. 666].

A typical experim e ntal arrangem e nt for de term ining the rang e of charged particles is shown in Fig. 14.4. The m ean range R is defined as the abso rber th icknes s at which the in tensity is reduc ed to one-half o f the initial value. The extrapo l ated range R o is obtained by linear extrapolation at the inflection point of the tr ansm ission curve. This is an exam ple that I/Io is not always an exponent ial. In charged partic le interactions it is not sufficient to think of I / I  e   x , one should be thinking about the range R.

I/I o

R = R -    ,  = straggling parameter 2

0.5

Straggling ef fects

roughly ,

 ~ .015 R

R o t

Figure b y MIT OCW . Adapted f rom Knoll.

An alpha particle transmission experiment. I is the detected number of alphas through an absorber thickness t , whereas I o is the number detected without the absorber . The mean range R and extrapolated range R o are indicated.

Fig. 14.4 . Determ ination of range by transm ission experim e nt [from Knoll]. 9

In practice o n e uses rang e-energy re lations that a r e m o stly em pirica lly determ ined. For a rough estim ate of the range one can use the Bragg-Kleem a n rule,

R   1

R 1

(14.16)

where the subscript 1 denotes the ref e rence m e dium which is conventionally taken to be

air at 15 o C, 760 mm Hg (

= 3.81,  1

= 1.226 x 10 -3 g/cm 3 ). Then

R  3 . 2 x 10  4



x R air (14.17)

with in g/c m 3 . In general such an estim ate is good to within about  15 percent. Figs. 14.5 and 14.6 show the range-energy relations for protons and -par tic les in air respectively. Notice th at at low energy the va riation is quadratic, as predicted by (14.13),

and at high energy the relation is m o re or le ss linear, as given by (14.14). The sam e trend

is also seen in the resu lts for el ectrons, as shown in Fig. 14.7.

Image removed due to copyright restrictions. Please see Figure 3.2 in Evans . p. 650.

Fig. 14.5 . Range-energy relations of -particles in air [Evans, p. 650].

Image removed due to copyright restrictions.

Please see Figure 3.3 in Evans. p. 651.

Fig. 14.6 . Range-energy relation for pr otons in air [Evans, p. 651].

Image removed due to copyright restrictions.

Please see Figure 3.3 in Evans. p. 624 .

Fig. 14.7. Range-energy relation for electr ons in alum inum [Evans, p. 624].

We have m e ntioned that heavy charged part icles travers e esse ntia lly in a s t ra ight line un til rea c hing the en d of its rang e where st ra ggling ef f ects m a nif e st. In the c a se o f

electrons large deflectio ns are quite like l y durin g its trave r sa l, so the trajectory of electron in a th ick ab sorber is a s e ries of zigzag path s. W h ile one can still spe a k of the rang e R, the concep t of path length is now of little va lue. This is illu strated in Figs . 14.8 and 14.9. The total path length S is apprecia bly greater th an the rang e R

Image removed due to copyright restrictions.

Please see Figure 1.1 i n Evans. p. 612

Fig. 14.8. Distinction between to tal path length S and range R [Evans, p. 612]

Image removed due to copyright restrictions.

Please see Figure 1.2 i n Evans. p. 612

Fig. 14.9. Com p aring distributions of total path length and range for electrons in oxygen [Evans, p. 612].

The transm ission curv e I/I o for a heavy charged particle was shown in Fig. 14.4. The curve has a different characteristic shape fo r m onoenergetic electron s, as indicated in Fig. 14.10, and a still different shape for  -rays (electrons with a distribution of energies), seen in Fig. 14.11. Although the curve for m onoenergetic electrons depends to som e extent on experim e ntal arrangem ent, one may regard it as roughly a linear variation which is characteristic of singl e inte r action event in rem oving the electron. That is, the fraction of electrons getting through is proportional to 1 – P, where P is the interaction probability which is in turn propor tional to the thickness. F o r the  -ray transm ission curve which essentially has the form of an exponential, the shap e is an accidental consequence of the  -ray spectrum and of the differences between the scattering and absorption of electrons which have various initial energies [ c f. Evans, p. 625]. It is found em pirically that R m is the sam e as R o if the m onoe nergetic electrons are given the energy E = E ma x , the m a xi m u m energy of th e  -ray spectrum (the end-point energy).

Image removed due to copyright restrictions.

Please see Figure 3.2 i n Evans . p. 623

Fig. 14. 10 . Transm ission curve of monoenergetic electrons (sensitive to experim e ntal arrangem ent) [Evans, p. 623].

Image removed due to copyright restrictions. Please see Figure 3.4 i n Evans . p. 625

Fig. 14.11. Transm ission curve for  -rays [Evans, p.625].

Cerenkov Radiation

Electrom a gnetic radiatio n is em itted wh en a char ged particle passes th rou gh a m e dium under the condition

v group   c  v phase  c / n (14.15)

where n is the index of refraction of the m e dium . W h en  n >1, there is an angle (a direc tion ) where cons tru c tive in terference occurs . This rad i ation is a p a rticular form of energy los s , due to soft collis ions, an d is not an addition a l am ount of energy loss. Soft collisions involve sm all energy transfers from charged particles to distant atom s which becom e excited and subsequently emit c oherent radiation (see Evans, p. 589).