22.101 Applied Nuclear Physics (F all 2006)

Lecture 7 (10/2/06) Overview of Cross Sect ion Calcula t ion

References –

P. Rom a n, Advanced Quantum Theory (Addison-W e sley, Reading, 1965), Chap 3.

A. Foderaro, The Elements of Neutron Interaction Theory (MIT Press, 1971), Chap 4.

The interaction of radiation with m a tter is a cornerstone topic in 22.101. Since all radiation interactions can be described in term s of cross sections, cl early the concep t of a cross section and how one can go about cal culating such a qu an tity are importan t considerations in our studies. To keep th e details, which are n ecessary for a deeper understand ing, from getting in the w a y of an ove rall picture o f cross section calcu lations, we attach two appendices at th e end of this Lecture Note, Appendix A Concepts of Cross Section and Appendix B, Cross Section Calc ulation: Method of Phase Shifts . The students regard the discussions there as part of the lecture m a terial.

We begin by referring to the definition of the cross section  for a general

interaction as discussed in Appendix A. One sees that  appear s as the p r op ortion a lity constant between the probability for the react ion and the number of target nuclei exposed per unit area, (which is why  has the dim e nsion of an area). We expect that  is a dynam i cal quantity which depends on the nature of the interaction forces between the radiation (which can be a partic le) and the target nucleus. Since these forces can be very com p licated , it m a y then appear that any calcu lation of  will b e quite complica t ed as well. It is therefore fortun ate that m e thods for calculating  have been developed which are within the grasp of student s who have had a few lectures on quantum m echanics as is the case of students in 22.101.

The m e thod of phase shifts, which is the subject of Appendix B, is a well-known elem entary m e thod worthy of our attention.. He re we will dis c uss the key steps in this

m e thod, going from the introdu ction of the scattering am plitu de f (  ) to the ex pression for the angular differential cross section  (  ) .

Expressing  (  ) in terms of the Sc attering Amplitu d e f (  )

W e consider a scattering scenario sketched in F i g. 7.1.

Fig.7.1 . Scattering of an incom i ng plane wave by a potential f i eld V(r), resulting in spherical ou tgoing wave. The scattered cu rrent crossing an elem ent of sur f ace area d  about the direction  is used to define the angul ar differential cross section

d  / d    (  ) , where the s cattering an gle  is the an gle between the direction of inciden ce an d direc tion of scattering.

W e write th e incid e nt plane wave as

  be i ( k  r   t ) ( 7 . 1 )

where the w a venum ber k is set by the energy of the incom i ng effective particle E, and the scattered spherical outgoing wave as

  f (  ) b

e i ( k r   t )

r

( 7 . 2 )

where f (  ) is the scattering amplitude. T h e a ngular differential cross section for scattering through d  about  is

 (  )  J sc   

J in

where we have used the expression (see (2.24)),

f (  ) 2

(7.3)

J  2  i

   * (   )   (   * )  

(7.4)

Calculating f (  ) from the Schrödinger w ave equati on

The Schrödinger equatio n to be solv e d is of the f o rm

 ħ 2 

  2  V ( r )  ( r )  E  ( r )

 2  

( 7 . 5 )

where   mm /( m  m ) is th e reduced m a ss, and E   v 2 /2 , with v being the relative

speed, is pos itiv e. To obtain a solution to our pa rticula r sca t tering se t-up, we im pose the boundary condition

 ( r ) 

e ik z  f (  )

e ik r

( 7 . 6 )

k r  r o r

where r is th e range of force, V(r) = 0 for r > r o . In the region beyond the force range the

wave equation describes a free particle. This fr ee-particle solution to is what we want to m a tch up wi th the RHS of (7.6). The m o st convenient form of the free-particle wave function is an expansion in te rm s of partial waves,

 ( r ,  )   R ℓ ( r ) P ℓ ( c o s  ) ( 7 . 7 )

ℓ  0

where P ( c os  ) is the Legendre polynom ial of order ℓ . Ins e rting (7.7) into (7.5), an d setting u ( r )  r R ( r ) , we obtain

 d 2

 k 2 

2  V ( r ) 

ℓ ( ℓ  1 ) 

u ( r )  0 , (7.8)

 dr 2 ħ 2

r 2  ℓ

 

Eq.(7.8) describes the w a ve function everyw here. Its solution clearly depends on the form of V(r). Outside of the in tera ction region, r > r o , Eq.(7.8 ) reduces to the rad i al w a ve equation for a free particle,

 d 2

 k 2 

ℓ ( ℓ  1 ) 

u ( r )  0 (7.9)

 dr 2

r 2  ℓ

with gener a l solution

u ( r )  B r j ( k r )  C r n ( k r ) ( 7 . 1 0 )

ℓ ℓ ℓ ℓ ℓ

where B

and C

are integration constants, and j

and n

are spherical Bessel and

Neum ann functions respectively (see A ppendix B for their properties).

Introductio n of the Phase Shift  ℓ

W e rewrite the genera l s o lution (7.1 0) as

u ( r ) 

kr  1

( B / k ) s i n ( kr  ℓ  / 2 )  ( C

/ k ) c o s ( kr  ℓ  / 2 )

 ( a ℓ / k ) s i n [ k r  ( ℓ  / 2 )   ℓ ] ( 7 . 1 1 )

where we have replaced B and C by two other co nstants, a an d  , the la tte r is seen to be a phase shift . Com b ining (7.11) with (7.7) the p a rt ial-wave expansion of the free-particle wave function in the asymptotic region becom e s

 ( r ,  ) 

 a sin[ kr  ( ℓ  / 2)   ℓ ] P (co s  )

(7.12)

kr  1 ℓ ℓ

ℓ

This is th e L H S of (7.6). Now we pr epare the RHS of (7.6) to have the sam e for m of partial wave expansion by writing

f (  )   f ℓ P ℓ ( c o s  ) ( 7 . 1 3 )

ℓ

and

e ikr cos    i ℓ (2 ℓ  1 ) j ( k r ) P ( c o s  )

ℓ ℓ

ℓ

  i ℓ (2 ℓ  1 ) si n( k r  ℓ  / 2 ) P ( c o s  ) (7.14)

kr  1

ℓ

kr ℓ

Inserting both (7.13) and (7.14) into the RH S of ( 7 .6), we m a t c h the coefficients of exp(ikr) and exp(-ikr) to obtain

f  1 2 ik

(  i ) ℓ [ a e i  ℓ  i ℓ ( 2 ℓ  1 ) ] ( 7 . 1 5 )

a  i ℓ (2 ℓ  1 ) e i  ℓ ( 7 . 1 6 )

Com b ing (7.15) and (B.13) we obtain

f (  )  ( 1 / k )  ( 2 ℓ  1 ) e i  ℓ s i n  P ( c o s  ) (7.17)

ℓ  0

Final Expressions for  (  )

and 

In view of (7.17), the angular diffe rentia l cros s s ection (7.3) becom e s

 2

2 i  ℓ

ℓ ℓ

(7.18)

ℓ  0

where D  1/ k is the reduced wavelength. C o rres pondingly, the total cross section is

  

d   (  )  4  D 2  ( 2 ℓ  1 ) s i n 2 

ℓ  0

( 7 . 1 9 )

S- w ave scattering

We have seen that if kr o is apprec iably less th an u n ity, th en on ly the ℓ = 0 term contributes in (7.18) and (7. 19). The differential and tota l cro ss sections for s-wave scattering are therefore

 (  )  D 2 s i n 2  ( k ) ( 7 . 2 0 )

  4  D 2 s i n 2  ( k ) ( 7 . 2 1 )

Notice th at s - wave sca tte ring is spher i cally sym m e tr ic, or  (  ) is independent of the

scatter i ng an gle. This is true in CMCS , but not in LCS. From (7.15) we see

 ( e i  o s i n  ) / k . Since the cross sectio n m u st be finite at low energies, as k  0 f o has

to rem a in f i nite, o r  ( k )  0 . We can set

li m k  0 [ e i  o ( k ) s i n  o ( k ) ]   o ( k )   a k ( 7 . 2 2 )

where the co nstant a is c a lled the sca tter i ng leng th . Thus for low-energy scattering, the differential and total cross s ections depend only on knowing th e scattering length of the target nucleus,

 (  )  a 2 ( 7 . 2 3 )

  4  a 2 ( 7 . 2 4 )

Physical significance of the si gn of the scattering length

Fig. 7.2 shows two sine wave s, one is the referen ce wave sin kr which has not had

Fig. 7.2 . Comparison of unscattered and scat tered waves showing a phase shift  in the asym ptotic region as a re sult of the s c atte ring.

any interaction (unscattered) a nd the other one is the wave sin( kr   o ) which has suffered a phase shift by virtue of the scattering. The en tire effect of the scattering is seen to be represented by the phase shift  o , or equ i valen tly the scatter i ng length through (7.22).

In the v i cin i ty of the pote n tia l, we tak e kr o to b e s m all (this is again th e co ndition of lo w- energy scattering), so th at u o ~ k ( r  a ) , in which cas e a becomes the distance at which

the wave fun c tion e x trap olates to zer o from its value and slope at r = r o . T h ere are two ways in which this extrapolation can take place, depending o n the valu e of kr o . As shown in Fig. 7.3, when kr o >  /2 , the wave function has reached m o re than a quarter of its wavelength at r = r o . So its slope is d o wnward and the extrap olation gives a distance a

which is pos itiv e. If on the other h a n d , kr o <  /2 , then the extrap olation gives a distance a which is n e gativ e. The signif i c a nc e is th at a > 0 m eans the potential is s u ch that it c a n have a bound state, whereas a < 0 m eans that th e potential can only give rise to a virtual state.

Fig. 7.3. Geom etric inte rpretation of positiv e and negativ e sc atte ring leng ths as th e distan ce of extrapo l ation of the wave func tion at the interface be tween in terior and exterior solutions, for potent ials which can have a bound state and which can only virtual state resp ectively.

22.101 Applied Nuclear Physics (Fall 2006)

Lecture 7 (10/2/06)

Appendix A: Concepts of Cross S ections

It is instructive to review the physical m eaning of a cross section , which is a m easure of the probability of a reaction. Im agine a beam of neutrons incident on a thin sam p le of thickness  x covering an area A on the sam p le. See Fig. A.1. The intensity of the beam hitting the area A is I neutrons pe r second. The incident flux is therefore I/A.

Fig. A.1 . Schem a tic of an inciden t b eam striking a thin targe t with a par t icle em itted into a cone subtending an angle  relative to the d i rec t ion of in cidence, the ' s cattering' angle. The elem ent of solid ang l e d  is a sm all piece of the cone (see also Fig. A.2).

If the nuclear density of the sam p le is N nuclei/cm 3 , then the no. nuclei exposed is

NA  x (assum i ng no effects of shadowing, i.e ., the nuclei do no t cover each other with respec t to th e incom i ng neutrons ). W e now write down the probability f o r a collis io n- induced reaction as

{reaction probability} =  / I   NA  x    ( A . 1 )

 

 

where  is th e no. reaction s occurring per sec. Notice th at simply appears in the definition of reaction probability as a proportionality constant , with no further justification. Som e tim e s this sim p le fact is overlooked by the stude nts. There are other ways to introduce or m o tivate the m eaning of th e cross s ection; they are essentially all

equivalent when you think about the physical situation of a beam of particles colliding with a ta rget of atom s. Rewriting ( A .1) we get

 = {reaction probability } / {no. exp o sed per unit area}

=   1   

( A . 2 )

IN  x I   N  x    x  0

Moreover, we define   N  , which is ca lled the macr oscopic cro s s section . T h en (A.2) becom e s

 x   , ( A . 3 )

I

or   {probability per unit pa th for sm all pa th tha t a reaction will occur} (A.4)

Both the microscopic cro ss section  , which has the dim e nsion of an area (unit of  is the barn which is 10 -24 cm 2 as already noted above) , and its counterpart, the m acroscopic cross sectio n  , which has the dim e nsion of recip r ocal length, are funda m e ntal to our study of radiation inter actions. Notice that this discussion can be applied to any radiation or pa rtic le, there is no thing that is sp ecif i c to n e u t rons.

W e can read ily ex tend th e presen t dis c ussion to an angular differential cross section d  / d  . Now we i m agine counting the reacti ons per second in an angular cone subtended at angle  with respec t to th e dire ction o f incidence (incom ing particles), as shown in Fig. A.1. Let d  be the elem e n t of solid angle, which is the sm all area through

which the u n it vec t or  passes through (see Fig. A.2). Thus, d   sin  d  d  .

Fig. A.2 . The unit vector  in spherical coordinates, with  an d  being the polar and azim u thal angles respectively (R would be unity if the vector ends on the sphere).

W e can write

1  d    N  x  d  

( A . 5 )

   

I  d    

Notice th at again d  / d  appear s as a propo r tiona lity con s tant b e twee n the re actio n rate per unit solid angle and a product of two sim p le factors specifying th e interacting system

- the incient flux and the num b er of nuclei exposed (or the nu mber of nuclei available for reaction). T h e norm a lization condition of the angular differential cross section is

 d  ( d  / d  )   , which m a kes it clear why d  / d 

section .

is calle d the angular differential cross

There is another differential cross secti on which we can introduce. Suppose we consider the incom i ng particles to have ener gy E and the particles after reaction to have energy in dE ' about E'. One can define in a sim ilar way as above an ener gy differen tial cross sectio n , d  / d E ' , which is a m easure of the proba bility of an incom i ng particle with energy E will have as a resu lt of the re action outgoing en ergy E' . Both d  / d  and d  / d E ' are distribution functions , the f o rm er is a dis t r i bu tion in the v a riab le  , the

solid ang l e, whereas th e latter is a dis t ribu tion in E' , the energy af ter sca tte ring. Their dim e nsions are barns per steradian and barns per unit energy, respectively.

Com b ining the two extensions above from cross section to dif f erential cross sections, we can further extend to a double differential cross section d 2  / d  d E ' , which is a quan tity that ha s bee n studied extensive l y in therm a l neutr on scatter i n g . This cros s section contains the m o s t fundam e nt al info rm ation about the structure and dynam i cs of the sca tte rin g sam p le. W h ile d 2  / d  d E ' is a distribution in two variables, the solid angle and th e energy after scattering , it is not a distribution in E, the energy before scattering.

In 22.106 we will be co ncerned with all three ty pes of cross sections,  , the two differential cross sections., a nd the double differential cross s ection for neutrons, whereas the double differential cross section is beyond the scope of 22.101.

There are many im portant applications wh ich are based on neutron interactions with nucle i in various m e dia. W e are inte res t ed in both the cross sections and the use of these cross sections in various ways. In diffraction and spectroscopy we use neutrons to probe the structure and dynam i cs of t h e sam p les being m easured. In cancer therapy w e use neutron s to preferen tially kill th e cancerou s cells. Both in volve a single collision event between the neutron and a nucleus, for which a knowledge of the cross section is all that required so long as th e neutron is concerned. In contrast, for reactor and other nuclear applications one is in terested in the effects of a sequence of collisions or multiple collisions , in which case knowing only th e cross section is not su fficient. One needs to follow the n e utrons as th ey undergo m a ny collisi ons in the m e dia of interest. This then requires the study of neutron transport - the distribution of neut rons in configuration space, direction of travel, and energy . In 22.106 we will treat transpo r t in two ways, theoretical discussion and direct sim u lati on using the Monte Carlo m e thod, and the general purpose code MCNP (Monte Carlo Neutron and Photon).

22.101 Applied Nuclear Physics (Fall 2006)

Lecture 7 (10/2/06)

Appendix B: Cross Section Calc u l ation: Method of Phase Shifts

References --

P. Ro m a n, Advanced Quantum Theory (Addison-W e sley, Reading, 1965), Chap 3.

A. Foderaro, The Elements of Neutron Interaction Theory (MIT Press, 1971), Chap 4.

W e will stud y a m e thod of analyzing poten tial s c attering ; it is called the m e thod of partial waves or the m e thod of phase sh ifts. This is the quantum m echa n ical description of the two-body coll ision process. In the cente r-of- m a ss coordinate system the prob lem is to des c r i b e the m o tion of an ef f ective particle with m a ss  , the reduced

m a ss, m oving in a cen tra l poten tia l V(r), wher e r is the sepa ra tion distance between the two collidin g particle s. W e will solv e the Schrödinger wave equation f o r the spa tia l distr i bution of this ef f ective par tic le, and extr act f r om this solution th e inf o rm ation needed to determ ine the angul ar differential cross section  (  ) . For a discussion of the

concepts of cross sec tions, see Appendix A.

The Scattering Amplitude f (  )

In treating the potential scattering probl em quantum m echanically the s t andard

approach is to do it in two steps, f i rs t to def i ne the cross se ctio n  (  ) in te rm s of the

scattering amplitude f (  ) , and then to calculate f (  ) by solving the Schrödinger equation. For the f i rs t step we visu aliz e the s c a tte ri ng process as an incom i ng bea m i m pinging on a potential fie l d V(r) cen te red at the origin (C MCS), as shown in Fig. B.1. The incid e nt

beam is represented by a traveling p l ane wave,

  be i ( k  r   t ) ( B . 1 )

where b is a coefficient d e term ined by the norm a lization condition, and the wave vector

k  k z ˆ is directed along the z-axis (direction of incidence). The m a gnitude of k is set by

Fig.B.1 . Scattering of an incom i ng plane wave by a potential field V(r), resulting in spherical ou tgoing wave. The scattered cu rrent crossing an elem ent of sur f ace area d  about the direction  is used to define the angul ar differential cross section

d  / d    (  ) , where the s cattering an gle  is the an gle between the direction of inciden ce an d direc tion of scattering.

the energy o f the effective particle E  ħ 2 k 2 / 2   ħ  (the relativ e en ergy of the c o llid ing particle s). F o r the sc atte red wave which resu lts f r om the inte r action in the region of th e potential V( r), we will w r ite it in the f o rm of an outgoing sph e ric a l wave,

  f (  ) b

e i ( k r   t )

( B . 2 )

sc r

where f (  ) , which has the dim e nsion of l e ngth, de notes the am plitude of scattering in a direction indicated by the polar angle  rela tive to the dire cti on of incidence (see Fig. B.1). It is clear that by representin g the scattered wave in the for m of ( B .2) our

inten tion is to work in sp heric a l coo r dinate s.

Once we have expressions for the incide nt and scattered waves, the corresponding curren t (o r flux) can be o b tained fro m the rela tio n (see (2.24 ) )

J  ħ

2  i

   * (   )   (   * )  

(B.3)

The inciden t curren t is J  v b 2 , where v  ħ k /  is th e speed of the ef f ective pa rticle.

For the number of particles per sec scattered through an elem ent of surface area d  about the direction  on a unit sphere, we have

J   d   v f (  ) 2 d  (B .4)

The angular differential cross section for scattering through d  about  is the r ef ore (se e Appendix A),

 (  )  J   

J in

f (  ) 2

(B.5)

This is th e fundam e ntal express i on relating th e s cattering amplitud e to th e cross s ection; it has an analogue in the anal ysis of potential scattering in classical m echanics.

Method of Partial Waves

To calculate f (  ) fro m t h e Schrödinger w a ve equation we note that since this is

not a tim e -dependent p r oblem , we c a n l ook for a separab l e solution in sp ace and

tim e,  ( r , t )   ( r )  ( t ) , w ith  ( t )  e x p (  itE / ħ ) . The Schrödinger equation to be solved then is of the f o rm

 ħ 2 

  2  V ( r )  ( r )  E  ( r )

 2  

( B . 6 )

For two-body scattering through a central poten tial, this is the wave equation for an ef f ective par tic le w ith m a ss equal to the redu ced m a ss,   mm /( m  m ) , and energy E

1 2 1 2

equal to the sum of the kineti c energies of the two particle s in CMCS, or equivalently

E   v 2 /2 , w ith v being the relative speed. The re duction of the two-body problem to the effective one-body problem (B.6) is a useful exercise, which is quite standard. For those in need of a review, a discussion of th e red u ction in classical as well as quan t um

m echanics is given at the end of this Appendix.

As is well known, there ar e two kinds of solutions to (B.6), bound-state solutions for E < 0 and scattering solutions for E > 0. W e are conc erne d w ith the la tte r situation . In view of (B.2) and Fig. B.1, it is conventi onal to look for a particul ar solution to (B.6), subject to the boundary condition

 ( r )  e

ik z

 f (  )

e ik r

( B . 7 )

k r  r o r

where r is th e range of force, V (r) = 0 for r > r o . T h e subscr ipt k is a rem i nder tha t the

entire ana l ys is is c a rr ied out at co nstant k, or at fixed incom i ng energy E  ħ 2 k 2 / 2  . It

also m eans that f (  ) depends on E, althou gh this is co m m only not indica ted ex plic itly.

For sim p licity of notatio n, w e w ill su ppress th is s ubscrip t hen cef orth.

According to (B.7) at d i stances far aw ay from the region of the scattering potential, the wave function is a superpos ition of an incident plane wave and a spherical outgoing scattered wave. In the far-away region, the w a ve eq uation is the r ef ore tha t o f a free particle since V(r) = 0. This f r ee -partic l e solution is w h a t w e w a nt to m a tch up w ith (B.7). The f o rm of the solution that is m o st convenient f o r th is purpose is the expansion

of  ( r ) into a set of partia l w a ves. Sin ce we are cons idering central poten tials,

inte ractions w h ich are sp heric a lly sy m m e tric, or V depends only on the separation distance (m agnitude of r ) of the tw o collid ing par t icle s, the na tural coo r din a te sys t em in

w h ich to f i n d the solu tio n is spher i ca l coord i nate s, r  ( r ,  ,  ) . The azimuthal angle  is an ignorable coordinate in this case, as the wave function depends only on r and  . The partial wave expansion is

 ( r ,  )   R ℓ ( r ) P ℓ ( c o s  ) ( B . 8 )

ℓ  0

where P ( c os  ) is the Legendre polynom ial of order ℓ . Ea ch term in th e sum is a partial wave of a definite orbita l angular m o m e ntum , with ℓ being the quantum number. The set of functions { P ( x ) } is known to be orthogonal and com p le te on the interval (-1, 1). Som e

of the properties of P ( x ) are:

1

 ℓ ℓ '

2

2 ℓ  1

ℓ ℓ '

dx P ( x ) P ( x )  

 1

P (1)  1 , P (  1 )  (  1 ) ℓ ( B . 9 )

ℓ ℓ

P ( x )  1 , P ( x )  x P ( x )  ( 3 x 2  1 ) / 2 P ( x )  ( 5 x 3  3 x ) / 2

Inserting (B. 8 ) into (B.6), and m a king a cha nge of the dependent va riable (to put the 3D problem into 1D for m ), u ( r )  r R ( r ) , we obtain

ℓ ℓ

 d 2

 k 2 

2  V ( r ) 

ℓ ( ℓ  1 ) 

u ( r )  0 , r < r o (B.10)

 dr 2 ħ 2

r 2  ℓ

This resu lt is called the r a dial w a ve e quati on for rather obvious reasons; it is a one- dim e nsional equation w hose solution dete rm ines the scattering process in three dim e nsions, m a de possible by the p r o p erties of th e central po tentia l V(r). U n less V (r) has a special form that adm its analy tic so lutions, it is of ten m o re ef f ective to in tegra t e (B.1 0) num e rically. H o w e ver, w e w ill not be concern e d w ith such calcu lations since our interest is not to solve the m o st general scattering problem.

Eq.(B.10) describes the wave functi on in the interaction region, r < r o , w h ere V(r)

= 0, r > r o . The solution to this equa tion c l early depends on the form of V (r). O n th e other hand, outside of the inte rac tion region, r > r o , Eq.(B.10) reduces to the radial wave equation for a free particle. Si nce this equation is universal in that it app lie s to all scattering problem s whe r e the inte ra ction po tential has a f i n i te range r o , it is worthwhile