22.101 Applied Nuclear Physics (Fall 2006)

Lecture 1 4 (11/1/06)

Charged-Partic le Int e ractions : Stopping Pow er, Collisions and Ioniz ation

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).

When a swift charg e d particle en ters a m a terials m e dium it intera cts with the electrons and nuclei in the m e dium and begins to lose ene r g y as it p e netrates into th e m a terial. T h e interactions can be thought of as individual co llis ions between the charged particle and the atom ic electrons surrounding a nucleus or the nucle us itself (considered separa tely ). The energy given of f du ring th ese co llis ions will result in ionization , the production of ion-electron pair s, in the m e dium ; a l so it can appear in the form of electromagnetic radiation , a process known as bremsstrahlung (braking radiation). We are interested in describing the energy lo ss per unit distance tr aveled by th e charged particle, and the rang e of the par tic le in var i ous materials. The range is defined as the distance traveled from the point of entry to the point of essentially com i ng to rest.

A charged p a rticle is ca lled ‘heavy ’ if its r e st m a ss is la rge co mpared to th e rest m a ss of the electron. Thus, m e sons, protons, -particles, and of course fission fragm e nts, are all heavy charge d particles. By the sam e token, electrons and positrons are ‘ ligh t ’ p a rticle s.

If we ignore nuclear forces and consid er only the interact ions arising from Coulom b forces, then we can speak of f our principal types of charged-particle inte ractions:

(i) Inelas tic Collision with Atom ic Elec trons . This is the principal process of energy trans f er, particula r ly if the p a r tic le velo city is below the leve l where brem sstrahlung is significant. It leads to the excitation of the atom ic electrons (still bound to the nu cleus) and to ionization (electron stripped off the nucleus). Inelas tic here refers to the excitatio n of electronic levels.

(ii) Inelas tic Collision with a Nucleus. This process can leave the nucleus in an excited state or the partic le can radiate (brem sstrahlung).

(iii) Elastic Collision with a Nucleus. This process is known as Rutherford scattering. There is no excitation of th e nucleus, nor em ission of radiation. The particle loses energy only th rough the recoil of the nucleus.

(iv) Elastic Collision with Atom ic Electrons . The process is elastic deflection which results in a sm all am ount of e n er gy transfer. It is significant only for charged particles that are low-en ergy electro ns.

In general, interaction of type (i), whic h is som e tim e s si m p ly called collision, is the dom inant process of energy loss, unless the charged particle has a kinetic energy exceeding its rest m a ss energy, in wh ich case th e radiation p r ocess, type (ii), b ecom e s im portant. For heavy particles, radiation occurs only at such kinetic energies, ~ 10 3 Mev, that it is of no practical interest to nuclea r engineers. The characteristic behavior of electron and proton energy loss in a high-Z m e dium like lead is shown in Fig. 13.1 [Meyerhof Fig. 3.7]. This is an im portant re sult to keep in m i nd for the discussions to f o llow.

15

10

5

0

0.005

0.05

0.5

5 50 500

Particle Energy T , Mev

Figure b y MIT OC W . Adapted from Burcham, 1963.

Fig. 13.1. Stopping power (vertical ax is in units of Mev/g-cm 2 ) showing the contribution s from collis ions and rad i ation.

Stopping Power: Energy Loss of Charged Particles in Matter

The kinetic energy loss per unit distan ce suffered by a charged particle, to be denoted as –dT/dx, is conventionally known as the stopping power . This is a positive quantity since dT/dx is <0. Th ere are quantum m echanical as well as classical th eories for calculating this basic quantity. One wa nts to express –dT /dx in term s of the properties specifying the incident charged partic le, such as its velocity v and charge ze, and the properties pertaining to the atom ic m e dium, the charge of the atom ic nucleus Ze,

the density of atom s n, a nd the average ionization potential I .

We consider only a crude, approxim a te derivation of the formula for –dT/dx. W e begin with an estim ate o f the energy loss suffe red by an incident charged particle when it interacts with a free and initia lly stationary electron. Refe rring to a collision cylinder whose radius is the im pact param e ter b and whose length is the sm all distance traveled dx shown in Fig. 13.2 [Meyerhof Fig. 3-1], we see th at the net mom e ntum t r ansferred to the

Figure b y MIT OCW . Adapted f rom Meyerhof.

Fig. 13.2. Collis ion cy linder f o r der i ving the en e r gy loss to a n atom ic ele ctron by an incident charged particle.

electron as the particle moves from one end of the cylind er to the othe r e nd is essen t ially entirely d i rected in the p erp endicular direction (b ecause F x changes sign, so the net mom entum along the horizontal direction va nishes) along the negative y-axis. So we write

 dtF x ( t )  0 ( 1 3 . 1 )

3

p e   dt F y ( t )

ze 2 b dx x 2  b 2  x 2  b 2  1 / 2 v

ze 2 b  dx 2 ze 2

 v   x 2  b 2  3 / 2 = vb

(13.2)

The kinetic energy trans f erred to the electron is therefore

2 2 ( ze 2 ) 2

(13.3)

2 m m b 2 v 2

If we assum e this is equ a l to the ene r gy loss of the charged p a rticle, then m u ltiply ing (13.3) by nZ ( 2  bdbdx ) , the num ber of electron s in th e collision cylinder, we obtain

dT b max 2  ze 2  2

 dx   nZ 2  bdb m  vb 

b min

e  

4  ( ze 2 ) 2 nZ  b 

= ℓ n  ma x  (13.4)

m v 2

 

 min 

where b ma x a nd b mi n are the m a xi m u m and m i nimum i m p act param e ters which one should specify acco rding to the physical de scrip tion he wishes to treat.

In reality the atom ic electrons are of course not free elec trons, so the charged particle m u st transfer at leas t an am ount of energ y equal to th e first ex cited state of the atom . If we take th e tim e inte rval of energy trans f er to be  t  b / v , then (  t ) max ~ 1 /  , where h   I is th e m ean ioniza tion potentia l . This giv e s an estim a te of the m a xim u m im pact para m e ter,

b max  hv / I (13.5)

An e m pirical expression for I is I  kZ , with k ~ 19 ev for H and ~ 10 ev for Pb. Next we estim ate b mi n by using the uncertainty principle to say that the electron position cannot be specified more precis e ly than its de Br oglie wavelength (in th e relative coordin a te system of the electron an d the charged particle ). Since electron m o m e ntum in the re la tive coordinate system is m e v , we have

b min  h / m e v (13.6)

Com b ining these two estim a tes we obtain

 dT

 4  z 2 e 4 nZ

 2 m v 2 

ℓ n  e 

(13.7)

dx m v 2  

In (13.7) we have inserted a factor of 2 in th e a r gu m e nt of the logarithm , this is to m a ke our form ula agree with the result of quant um m e chanical calculation which was first carried out by H. Bethe using the Born approxim a tion.

Eq.(13.7) describes the energy loss due to particle collisions in the nonrelativistic regim e . One can includ e relativ istic effects by replacing the logarithm by

 2 m v 2   v 2  v 2

ℓ n  e   ℓ n  1   

  

 

c 2  c 2

This correction can be important in the cas e of electrons and positron s.

Eq.(13.7) is a relatively sim p le expression, yet on e can gain m u ch insigh t into the factors th at govern the energy loss o f a char ged particle by collis ions with the atom ic electrons. We can see why the usual neglect of the contributions due to collisions with nuclei is ju stified. In a collis ion with a nucleus th e stopping p o wer would increase by a

factor Z, because of the charge of the target with which the in cident charg e d particle is colliding, an d decrease b y a factor of m e /M(Z), where M(Z ) is the m a ss of the atom ic nucleus. Th e decrease is a result of the larg er m a ss of the recoiling targ et. Since Z is always les s than 10 2 whereas M ( Z) is at least a factor 2 x 10 3 greater th an m e , the m a ss factor alway s dom inates over the charge factor . Another useful obser vation is that (13.7) is independent of the m a ss of the incide nt charged particle. T h is m eans that nonrelativistic electrons and prot ons of the sam e velocity w ould lose energy at the sam e rate, or equivalently the stopper pow er of a prot on at energy T is about the sam e as that of an electron at energy ~ T / 2000. That this scal ing is m o re or less correct can be seen in Fig. 13.1.

Fig. 13.3 shows an experim e ntally de term ined energy loss curve (stopping power) for a heavy charged particle (proton), on two energy scales, an expanded low-energy region where the stopping power decreases sm oothly with increasi ng kinetic energy of the charged particle T below a certain p eak centered about 0.1 Mev, and a m o re com p ressed high-energy region where the st opping power reache s a broad m i nim u m around 10 3 Mev. Notice also a s lig ht upturn as one goes to highe r en erg i es pas t the broad m i ni m u m which we expect is associated with relativistic corrections. One should regard Fig. 13.3 as the extension, at both ends of the energy, of the curve for proton in Fig. 13.1.

700

600

500

400

300

200

100

0

0

0.1 0.2 0.3 0.4

0.5 0.6 0.7 0.8 0.9 1.0

Proton ener gy (T), Mev

7

6

5

4

3

2

1

10 2 10 3

Proton ener gy (T), Mev

Figure b y MIT OCW . Adapted f rom Meyerhof.

10 4

Fig. 13.3. The experim e ntally determ ined stoppi ng power, (-dT/dx), for protons in air,

(a) low-energy region w h ere th e Bethe for m ula applies down to T ~ 0.3 Mev with I ~ 8 0 ev. Below this range ch arge los s du e to electron capture cau ses the s t opp er power to reach a p eak and start to decrease (s ee al so Fig. 1 3 .4 below), (b) high-en ergy region where a broad m i ni m u m occurs at T ~ 1500 Mev. [from Meyerhof]

Experim e ntally, collisional energy loss is m easured through the num ber of ion pairs for m ed along the trajectory path of the charged particle. Suppose a heavy charged particle loses on the average an am ount of energy W in producing an ion pair, an electron and an ion. Then the number of ion pairs produced per unit path is

i  1   dT 

(13.8)

 dx 

W e will say more about the energy W in the nex t lecture. 7

Eq.(13.7) is valid on ly in a certain en ergy rang e b ecause of th e assum p tions we have m a de in its derivation. We have seen fr om Figs. 13.1 and 13.3 that the atom ic stopping power varies with energ y in the m a nner sk etched below. In the interm ediate

Fig. 13.4. Schem a tic of overall behavior of st opping power of a charged particle with rest m a ss M.

energy region, 500 I  T  Mc 2 , where M is the m a ss of the charg e d particle, the stopping power behaves like 1/T, which is roughly what is predicted by (13.7). In this region the re lativis tic co r r ection is sm all a nd the logarithm factor varies slowly. At higher energies the logarithm f actor along with the relativisti c correction term s give rise to a gradual increase so th at a broad m i ni m u m is set up in the neighborhood of ~ 3 Mc 2 . At energies below the ma xim u m in t h e stopping power, T  500 I , Eq.(13.7) is not valid because the charged particle is m oving slow enough to captu re electrons and begin to lose its charge. Fig. 13.5 [Meyerhof Fig. 3-3] s hows the correlation between the m ean charge

2.0

1.6

1.2

0.8

0.4

0

0 0.4

0.8

1.2 1.6 2.0

Speed  , 10 9 cm/sec

Figure b y MIT OCW . Adapted from Evans, 1955.

Fig. 13.5. Loss of charg e with speed as a charged particle slows down.

of a charged particle and its ve locity. This is a d i f f i cult regio n to analy ze theoretically . For -particles and protons the ra nge begins at ~ 1 Mev and 0.1 Mev respectively [H. A. Bethe and J. Ashkin, “Passage of Radiation Through Matter”, in Experimental Nuclear Physics , E. Segrè, ed (W iley, New York, 1953), Vol. I, p.166].

Eq.(13.7) is generally know n as the Bethe form ula. It is a qu antum m echanical result derived on the basis of the Born appr oxim a tion which is essentially an assum p tion of weak scattering [E. J. W illiam s , Rev. Mod. Phys. 17 , 217 (1945)]. The result is valid provided

ze 2  e 2  z z

   

ħ v ħ c v / c 137 ( v / c )

<< 1 (13.9)

 

On the other hand, Bohr has used classica l theo ry to der i ve a s o m ew h at sim ilar expression for the stopping power,

 dT  4  z 2 e 4 nZ  M ħ v 2 m v 2 

    ℓ n  e  (13.10)

 dx 

class

m v 2

 2 ze 2 ( m

 M ) I 

ze 2

ħ v

>> 1 (13.11)

Thus (13.7) and the c l ass i cal form ula a pply under opposite con d itions. No tice th at th e two express i ons agree w h en the argu m e nts of the logar ithm s are equa l, th at is,

2 ze 2 / ħ v  1 , which is another way of saying that th eir regions of validity do not overlap. According to Evans (p. 584), the error of either approxim a tion tends to be an overestim ate, so the expression th at g i ves th e sm alle r energy loss is like ly to be the m o re correct. This turns out to be the classical expression when z > 137(v/c), and (13.7) w h en 2z < 137(v/c). Knowing the charge of the inci dent particle and its velocity, one can use this criterion to choose the appropriate stoppe r power for m ula. In the case of fission fragm e nts (h igh Z nuclid es) the classical result should be used. Also, it should be noted that a qu antum m echanical th eory h a s been developed by Bloch that gives the Bohr and Bethe resu lts as lim iting cases.

The Bethe form ula,(13.7), is app r op riate for heavy charged particles. For fast electrons (relativis tic) one should use

dT 2  e 4 nZ   m v 2 T  

   ℓ n  e    2 

(13.12)

dx m v 2   I 2 ( 1   2 )  

e    

where   v / c . For further discussions see Evans.