22.101 Applied Nuclear Physics (F all 2006)

Lecture 5 (9/20/06) Barrier Pen e tration

References --

R. D. Evans, The Atomic Nucleus , McGraw-Hill, New York, 1955), pp. 60, pp.852.

We have previously observed that one can look f o r different types of solutions to the wave eq uation. An application which will turn out to b e useful for later d i scuss i o n of nuclear decay by -particle em ission is the problem of barrier penetration. In this case one looks for positive-energy solutions as in a scattering problem . W e c onsider a one- dim e nsional system whe r e a particle w ith m a ss m and energy E is incident upon a potential barrier with width L and height V o (V o is greater than E). Fig. 1 shows that with the par tic le approach ing f r om the lef t , the p r obl e m separate s into th r ee regions, left of the barrier (region I), inside the barrier (region II), and right of th e barrier (re gion III).

Fig. 1. Particle with energy E penetra ting a square barrier of height V o (V o > E) and width L.

In region s I and III the p o tent ial is zero, so the wave e quation (3.1) is of th e for m

d 2  ( x ) 

dx 2

k 2  ( x )  0 , k 2

 2 mE / ħ 2

(5.1)

where k 2 is positiv e. Th e wave f unctions in th ese two regions are th eref or e

  a e ikx  b e  ikx    

(5.2)

1 1 1

1  1 

  a e ikx  b e  ikx  

(5.3)

3 3 3 3 

where we have set b 3 = 0 by im posing the boundary condition that there is no particle in region I II tra v eling to the lef t (s ince there is nothing in this regi on that can reflect the particle). In contrast, in re gion I we allow for ref l ection of the incident particle by the barrier; this m eans b 1 wi ll be nonzero. The subscripts  and  denote the wave functions traveling to the right and to the lef t resp ectiv ely.

In region II, the wave equation is

d 2  ( x )   2 

dx 2

( x )  0 ,  2

 2 m ( V o  E ) / ħ

(5.4)

So we write the solu tion in the f o rm

  a e  x  b e   x (5.5)

Notice that in region II the kinetic energy, E – V o , is negative, so the wavenum b er is im aginary in a propagating wave (another way of saying the wave function is monotonically decay ing rather than o s cillato r y). What this means is there is no wave-like solution in this region. By introducing we can think of it as the wavenumber of a hypothetical particle whose ki netic energy is positive, V o – E.

Having obtained the wave function in all three regions we proceed to discuss how to organize this inform ation into a useful for m , nam e ly, the transm ission and reflection coefficients. W e recall th at given the wave function  , we know i m m e diately the particle density (num ber of pa rticles per unit volum e, or the probability of the finding the

particle in an elem ent of volum e d 3 r about r ),  ( r ) 2 , an d the net current, g i ven b y (2.24),

j  ħ   *       *  ( 5 . 6 )

2 mi

Using the wave functions in regions I and III we obtain

j ( x )  v  a

2  b

2  ( 5 . 7 )

j ( x )  v a 2 ( 5 . 8 )

where v  ħ k / m is th e particle spe e d. W e see f r om (5.7) that j 1 is th e net cur r ent in region I, the difference between the c u rrent go ing to the right and that going to the left. A l so, in reg i on III there is only the cu rrent go i ng to the right. N o tice th at c u rrent is like a flux in that it has the dim e nsi on of num b er of particles per unit area per second. This is

consistent w ith (5.7 ) and (5.8) sin ce a 2 and b 2 are pa rticle d e nsitie s w ith the d i m e nsion

of num b er of particles per unit volum e. From here on we can regard a 1 , b 1 , and a 3 as t h e am plitudes of the incide nt, ref l ec ted, and tr ansm itted w a ves, r e spectively. W ith this inte rpre tatio n w e def i ne

2 2

T  , R  (5.9)

Since particles cannot be abso rbed or created in region II and there is no reflection in region III, th e net cu rren t in reg i on I m u st be equ a l to the net curren t in re gion II I, or j 1 = j 3 . It then follows that the condition

T  R  1 ( 5 . 1 0 )

is always satisfied (as one would expect). The transm ission coefficient is som e ti m e s also called the Penetration Factor and denoted as P.

To calculate a 1 and a 3 , w e apply th e boundary co nditi ons at the interfaces , x = 0 and x = L,

 1   2

, d  1  d  2 x = 0 (5.11)

dx dx

 2   3

, d  2  d  3 x = L (5.12)

dx dx

These 4 con d itions allo w us to elim inate 3 of the 5 integration constants. For the purpose of calculatin g the transmission co efficient we need to keep a 1 and a 3 . Thu s we will elim inate b 1 , a 2 , and b 2 and in the p r o cess arriv e at the ratio of a 1 to a 3 (after about a page of algebra),

a 1  e ( ik   ) L  1  i    k    e ( ik   ) L  1  i    k   (5.13)

a    

   

3  2 4  k     2 4  k   

This resu lt then leads to (af t er ano t her half-page o f algebra)

2

  1

a 2 V 2

 P ( 5 . 1 4 )

1

1  o sinh 2  L

4 E ( V o  E )

with sinh x  ( e x  e  x ) / 2 . A sketch of the variation of P with  L is shown in Fig. 2.

Fig. 2. Variation of tran sm ission coefficient (P enetration Factor) with the ratio of barrier width L to  , the ef f ective wavelength of the incide nt par tic le.

Using the leading expression of sinh(x) for sm all and larg e argum e nts, one can readily obtain s i m p ler expr essio n s f o r P in the lim it of thin and th ic k barriers,

V 2 ( V L ) 2 2 m

P ~ 1  o (  L ) 2  1  o  L << 1 (5.15)

4 E ( V o

 E )

4 E ħ 2

16 E 

E   2  L

P ~  1   e V o  V o 

 L >> 1 (5.16)

Thus the transm ission coefficient decreas es m o notonica lly w ith in crea sin g V o or L, rela tive l y slowly f o r thin barriers an d m o re rapidly for thick barriers.

W h ich lim it is m o re appropria te f o r o u r interest? Consider a 5 Mev proton incident upon a barrier of height 10 Mev and width 10 F. This gives ~ 5 x 10 12 cm -1 , o r

 L ~ 5. Using (5.16) we find

P ~ 16 x 1 x 1 xe  10 ~ 2 x 10  4

2 2

As a further sim p lification, one som e tim es even ignores the prefactor in (5.16) and takes

P ~ e   ( 5 . 1 7 )

with

  2  L 

( 5 . 1 8 )

W e show in Fig. 3 a schem a tic of the wave f unction in each region. In regions I and III,

 is com p lex, so we plot its real or im aginary part. In region II  is not oscillatory. Although the wave function in region II is nonzero, it does not appear in either the transm ission or the reflection coefficient.

Fig. 3. Particle penetration through a square barrier of height V o and width L at energy E (E < V o ), schem a tic behavior of wave functions in the three regions.

When the potential varies continuously in space, one can show that the attenu ation coefficient is given approxim a tely by the expression

2 x 2 

 1 / 2

   dx 2 m { V ( x )  E }

x 1

( 5 . 1 9 )

where the lim its of integrati on are indicated in Fig. 4; th ey are known as the ‘classical turning po in ts’. This res u lt is f o r 1D. For a spher i cal b a rr ier ( ℓ  0 or s-wave s o lution ) ,

Fig. 4. Region of integration in (5.19) for a variable potential barrier.

one has

2 r 2 

 1 / 2

   dr 2 m { V ( r )  E }

r 1

W e will use this exp r ess i on in the d i s c ussion of -decay.

( 5 . 2 0 )