Barrier Draft2

7/31/2019 Barrier Draft2

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PROJECT REPORT

ON

Tr a nsmission Thr ough one-dimensiona l Compl ex pot ent ia l s

BY

Manoj K ishore Pradhan

Sapan K umar Behera

Subrasmita Pradhan

Shakatimayee Jena

Abhilash Patra

3rdsemester

P.G.DEPARTMENT OF APPLIED PHYSICS AND BALLISTICS

GUIDE:

DR. SANT OSH K UM AR AGARW ALLA P.G.DEPARTM ENT OF APPLIED PHYSICS AND BA LLI STICS,

FAKIR MOHAN UNIVERSITY, VYASA VIHAR, BALASORE-19.


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Abstract:

We have calculated the Reflection (RC), Transmission (Tc) and Absorption (A) Co-efficient for one-

dimensional complex potential barrier. The results are computed by using MATLAB software.

Finally we study the variation of Tc, RC and A for various combinations of barrier height (V0),complex strength (W0) and width of the barrier (a).

Introduction:

Nucleus-Nucleus elastic scattering results in model with an added finite absorptive part to the

short-ranged attractive nuclear interaction potential are better understood than results from

models without such an absorptive term. The absorptive part of the potential is represented by a

purely imaginary potential of the short range type or the one that rapidly converges to zero. This

absorptive potential is supposed to represent, in a crude way, the unknown (non-elastic)

channels, which preferably remove some flux and reduce the elastically scattered flux. The

relevant model, called the optical model, is found to be phenomenological suitable for

measuring the elastic scattering.

Nucleus-Nucleus potential mainly consists of an absorptive well followed by a barrier.

This structure is generated by nuclear potential, centrifugal potential and coulomb potential. The

role of barrier is important in nuclear fusion. The absorptive region with barrier gives information

about elastic scattering.

In this report, we have taken a one-dimensional rectangular barrier with constant

imaginary part. It gives some information about the missing flux during transmission of particle

through the rectangular barrier. In the section-I, we have calculated transmission, reflection and

absorption co-efficient. In section-II, we have analyzed the result with the help of MATLAB

Programming. Section-III contains conclusion.


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SECTION-I:

Formulation:

Consider the potential on shown in the figure V(x).

V0

I II III

x=0 x=a

-w 0

V(x) = -i ; 0< x< a=0 ; xa

The Particle is incident from left of the barrier, the corresponding Schrodinger equations are given

by

+

=0 (for xa) (3)

Where, and are the corresponding wavefunction in the region I, II, and III respectively;m = mass of the particle,

E = energy of the particle and E>v

Equation-(1) can be written as,

+ = 0 .. (4)

Where; =


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The solution of the above equation

= +B (5)Equation-2 can be written as,

+ =0 (6)

Where; = (E- + ) (7)The solution of the above equation is

=C + .. (8)The equation (3) can be written as,

+ = 0 (9)

Where= EThe solution of equation is

= Fe+GeLet us take G= 0 because the par t i c le does not su f fer any r e f lec t ion f r om in f in i t y .

= Fe ... (10)Now by using boundary condition at x=0 and x=a.

i.e.

x= 0 = x= o ..

(11)

x=0 =

x=0 . (12)

x=a = x=a (13) x=a =

x=a (14)

We get,


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C= (1+

) Fe

e ............................... (15)

D=(1-

) Fe

e ...................................... (16)

B=

(

)

[() ]

....................... (17)

And

F=() (

)

[() () () ]

.. (18 )

Reflection Co-efficient = R = =

J=

J= | B |

So, R=| B | .. (19)

Transmission co-efficient, T=

T=

J= | F|

T=| F| .. (20)

Case-I: (w = 0 , Real potential case):

The Schrodinger equation in region-II becomes,

+

( E v)=0 .. (21)

The equation of continuity or equation for conservation of particle flux is

. J + = 0 (22)

I n 1 D


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+

= 0 . (23)

Where

J=current density

=position probability density = || But we are analyzing the stationary state solution i.e. is independent of times, we get

=

|| = 0 .... (24)

Equation for continuity for 1D stationary state solution is

= 0 .... (25)

Or J=constant.

This indicates the total particle flux remain conserved.

Flux incident = flux reflected + flux transmitted.

=

| B |

+ | F|

1 = R

+ T

(26)

CASE-II: w is finite:

The current density is defined as

J =

[]In 1D

J =

[

] . (27)

Differentiating the above equation w.r.t x we get

=

[

]


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By using equation (2) and its conjugate equation, we have

=

w ||

We know w >0, so the right hand side quantity is less than zero. It indicates that some loss ofparticle flux during transmission through the complex barrier.

J = w ||

d x

Absorption co-efficient (A ) =| || | =

||

d x . (28)

By using equation (8)

|| d x = [ | C|

+ | D|

+ DC

+ CD(

)

] (29)

Where = + i > 0 > 0

C and D are calculated by using equation (15) and (16).

Hence the absorption co-efficient for a complex barrier transmission is calculated by using

equation(29).

Now one can check analytically that

R + T < 1

1-R-T = A (30)

In the section II, we have shown that the A calculated by using equation (29) and (30) are equal.


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SECTION-II: (Computation with MATLAB)

(I)M ATLAB SCRIPT FOR Rc& Tc FOR REAL BARRIER

function energyvariation_real

m=1;h=1;e=[0:10:200];a=1.5;v0=4;w0=0.0;k=sqrt(e);alpha=sqrt(e-v0+i*w0);q=(1-k./alpha).^2;p=(1+k./alpha).^2;b=((1-e./alpha.^2).*(1-exp(-2*i.*alpha.*a)))./(p.*exp(-2*i.*alpha.*a)-q);disp('displaying the value of b:');disp(b);f=(q-p)./((exp(i.*(k+alpha).*a).*(q))-(exp(i.*(k-alpha).*a).*(p)));disp('displaying the value of f:');disp(f);r=(abs(b)).^2;disp('displaying the value of Reflection coefficient R:');disp(r');t=(abs(f)).^2;disp('displaying the value of Transmission coefficient T:');disp(t');disp('displaying the value of R+T:');disp((r+t)');plot(e,t,'r');hold on;

plot(e,r,'g');hold on;legend ('T','R')xlabel('E[fm^-2]');ylabel('T,R');title('R & T with varing energy for real barrier');end

TABLE-I:

e= [0:10:200]; a=1.5; v0=4; w0=0.0;

Energy in fm ^{-2} Reflect ion coeff icient Rc: Transm ission coeff icient Tc: Rc+Tc:

0.0 1.0000 0 1.0000

10.0 0.0169 0.9831 1.0000

20.0 0.0010 0.9990 1.0000

30.0 0.0049 0.9951 1.0000

40.0 0.0005 0.9995 1.0000

50.0 0.0008 0.9992 1.0000

60.0 0.0011 0.9989 1.0000


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70.0 0.0001 0.9999 1.0000

80.0 0.0002 0.9998 1.0000

90.0 0.0005 0.9995 1.0000

100.0 0.0003 0.9997 1.0000

110.0 0.0000 1.0000 1.0000

120.0 0.0001 0.9999 1.0000

130.0 0.0002 0.9998 1.0000

140.0 0.0002 0.9998 1.0000

150.0 0.0001 0.9999 1.0000

160.0 0.0000 1.0000 1.0000

170.0 0.0000 1.0000 1.0000

180.0 0.0001 0.9999 1.0000

190.0 0.0001 0.9999 1.0000

200.0 0.0001 0.9999 1.0000

[Figure 1: Plot of Transmission coefficient and Reflection coefficient as a function of energy for a real

barrier]


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(II) MATLAB SCRIPT FOR RC, Tc, A FOR COMPLEX BARRIER:

function barrierenergy_varitionm=1;h=1;%we have choosen constant parameters as 1.

a=2;v0=2; %we can choose our required potential height.w0=1;%we can choose our required absorption strength.e=[0:0.005:10];alpha=sqrt(e-v0+i*w0);k=sqrt(e);p=(1+k./alpha).^2;q=(1-k./alpha).^2;b=((1-e./alpha.^2).*(1-exp(-2*i.*alpha.*a)))./(p.*exp(-2*i.*alpha.*a)-q);disp('displaying value of b');disp(b);disp('displaying the value of Reflection coefficient R:');r=(abs(b)).^2;

disp(r');f=(q-p)./((exp(i.*(k+alpha).*a).*(q))-(exp(i.*(k-alpha).*a).*(p)));disp('displaying value of f');disp(f);disp('displaying the value of Transmission coefficient T:');t=(abs(f)).^2;disp(t');disp('displaying the value of R+T:');disp((r+t)');plot(e,t,'r');hold on;plot(e,r,'b');hold on;A=1-r-t;

disp(A');plot(e,A,'g');hold on;legend('T','R','A');xlabel('E[fm^-2]');ylabel('T/,R/,A');title('Variation of R,T,A with variation of energy');end


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Figure 2: Plot of T, R and A as a function of Energy (E) for fixed value of W0=0.5 and V0=0.0.


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[Figure 3: Plot of T, R and A as a function of Energy (E) for fixed value of W0=1.0 and V0=0.0]


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Figure 4: Plot of T, R and A as a function of Energy (E) for fixed value of W0=0.5 and V0=2.0


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Figure 5: Plot of T, R and A as a function of Energy (E) for fixed value of W0=1.0 and V0=2.0


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MATLAB SCRIPT FOR VARIATION OF RC, Tc, A WITH W0, E AND V0 FOR COMPLEX

BARRIER

(a)

%VARIATION OF R,T ,A WITH ABSORPTION STRENGTHfunction absorptionstrength_varitionm=1;h=1;%we have choosen constant parameters as 1.a=2;v0=0;e=3.8;w0=[0:0.01:100];alpha=sqrt(e-v0+i.*w0);k=sqrt(e);p=(1+k./alpha).^2;q=(1-k./alpha).^2;b=((1-e./alpha.^2).*(1-exp(-2*i.*alpha.*a)))./(p.*exp(-2*i.*alpha.*a)-q);disp('displaying value of b');disp(b);disp('displaying the value of Reflection coefficient R:');r=(abs(b)).^2;disp(r');f=(q-p)./((exp(i.*(k+alpha).*a).*(q))-(exp(i.*(k-alpha).*a).*(p)));disp('displaying value of f');disp(f);disp('displaying the value of Transmission coefficient T:');t=(abs(f)).^2;disp(t');disp('displaying the value of R+T:');

disp((r+t)');plot(w0,t,'r');hold on;plot(w0,r,'b');hold on;disp('displaying the value of A:');A=1-r-t;disp(A');plot(w0,A,'g');hold on;legend('T','R','A');xlabel('W0[fm^-2]');ylabel('T/,R/,A');title('R,T,A with variation of absorption strength wo');end


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Figure 6: Plot of T, R and A as a function of W0 for fixed value of E=3.8 and V0=0.0


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Figure 7: Plot of T, R and A as a function of W0 for fixed value of E=3.8 and V0=2.0


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(b)

%VARIATION OF R,T,A WITH ENERGY Efunction barrierenergy_varitionm=1;h=1;%we have choosen constant parameters as 1.a=2;v0=2;w0=0.5;%we can choose our required absorption strength.e=[0:0.005:10];alpha=sqrt(e-v0+i*w0);k=sqrt(e);p=(1+k./alpha).^2;q=(1-k./alpha).^2;b=((1-e./alpha.^2).*(1-exp(-2*i.*alpha.*a)))./(p.*exp(-2*i.*alpha.*a)-q);disp('displaying value of b');disp(b);disp('displaying the value of Reflection coefficient R:');r=(abs(b)).^2;disp(r');f=(q-p)./((exp(i.*(k+alpha).*a).*(q))-(exp(i.*(k-alpha).*a).*(p)));disp('displaying value of f');disp(f);disp('displaying the value of Transmission coefficient T:');t=(abs(f)).^2;disp(t');disp('displaying the value of R+T:');

disp((r+t)');plot(e,t,'r');hold on;plot(e,r,'b');hold on;A=1-r-t;disp(A');plot(e,A,'g');hold on;legend('T','R','A');xlabel('E[fm^-2]');ylabel('T/,R/,A');title('Variation of R,T,A with variation of energy');end

(c)%VARIATION OF R,T,A WITH POTENTIAL V0function potential_varitionm=1;h=1;%we have choosen constant parameters as 1.a=2;w0=0.05;e=3.8;v0=[0:1:100];


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alpha=sqrt(e-v0+i.*w0);k=sqrt(e);p=(1+k./alpha).^2;q=(1-k./alpha).^2;b=((1-e./alpha.^2).*(1-exp(-2*i.*alpha.*a)))./(p.*exp(-2*i.*alpha.*a)-q);%disp('displaying value of b');%disp(b);disp('displaying the value of Reflection coefficient R:');r=(abs(b)).^2;disp(r');f=(q-p)./((exp(i.*(k+alpha).*a).*(q))-(exp(i.*(k-alpha).*a).*(p)));%disp('displaying value of f');%disp(f);disp('displaying the value of Transmission coefficient T:');t=(abs(f)).^2;disp(t');disp('displaying the value of R+T:');disp((r+t)');plot(v0,t,'r');hold on;

plot(v0,r,'b');hold on;disp('displaying the value of A:');A=1-r-t;disp(A');plot(v0,A,'g');hold on;legend('T','R','A');xlabel('v0[fm^-2]');ylabel('T/,R/,A');title('R,T,A with variation of barrier potential Vo');end

SECTION-III:

TABLE-II:

DATA FOR POTENTIAL VARIATION (Absorptive Barrier)

a=2; w0=0.05; e=3.8; v0= [0:1:100]Reflect ion coeff ic ient

Rc:

Transmission

coeff ic ient Tc:

Rc + Tc Absorp t ion

coeff ic ient(A r )Rc + T c + A r

0.0000 0.9500 0.9500 0.0500 1.0000

0.0009 0.9410 0.9419 0.0581 1.00000.0258 0.8964 0.9222 0.0778 1.0000

0.3778 0.5349 0.9127 0.0873 1.0000

0.7984 0.1461 0.9445 0.0555 1.0000

0.9326 0.0349 0.9675 0.0325 1.0000

0.9690 0.0096 0.9786 0.0214 1.0000

0.9816 0.0030 0.9846 0.0154 1.0000

0.9871 0.0011 0.9882 0.0118 1.0000


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0.9901 0.0004 0.9905 0.0095 1.0000

0.9920 0.0002 0.9922 0.0078 1.0000

0.9933 0.0001 0.9934 0.0066 1.0000

0.9943 0.0000 0.9943 0.0057 1.0000

0.9951 0.0000 0.9951 0.0049 1.0000

0.9956 0.0000 0.9956 0.0044 1.00000.9961 0.0000 0.9961 0.0039 1.0000

0.9965 0.0000 0.9965 0.0035 1.0000

0.9968 0.0000 0.9968 0.0032 1.0000

0.9971 0.0000 0.9971 0.0029 1.0000

0.9974 0.0000 0.9974 0.0026 1.0000

0.9976 0.0000 0.9976 0.0024 1.0000

0.9978 0.0000 0.9978 0.0022 1.0000

0.9979 0.0000 0.9979 0.0021 1.0000

0.9981 0.0000 0.9981 0.0019 1.0000

0.9982 0.0000 0.9982 0.0018 1.0000

0.9983 0.0000 0.9983 0.0017 1.00000.9984 0.0000 0.9984 0.0016 1.0000

0.9985 0.0000 0.9985 0.0015 1.0000

0.9986 0.0000 0.9986 0.0014 1.0000

0.9987 0.0000 0.9987 0.0013 1.0000

0.9987 0.0000 0.9987 0.0013 1.0000

0.9988 0.0000 0.9988 0.0012 1.0000

0.9989 0.0000 0.9989 0.0011 1.0000

0.9989 0.0000 0.9989 0.0011 1.0000

0.9990 0.0000 0.9990 0.0010 1.0000

0.9990 0.0000 0.9990 0.0010 1.0000

0.9990 0.0000 0.9990 0.0010 1.00000.9991 0.0000 0.9991 0.0009 1.0000

0.9991 0.0000 0.9991 0.0009 1.0000

0.9992 0.0000 0.9992 0.0008 1.0000

0.9992 0.0000 0.9992 0.0008 1.0000

0.9992 0.0000 0.9992 0.0008 1.0000

0.9992 0.0000 0.9992 0.0008 1.0000

0.9993 0.0000 0.9993 0.0007 1.0000

0.9993 0.0000 0.9993 0.0007 1.0000

0.9993 0.0000 0.9993 0.0007 1.0000

0.9993 0.0000 0.9993 0.0007 1.0000

0.9994 0.0000 0.9994 0.0006 1.00000.9994 0.0000 0.9994 0.0006 1.0000

0.9994 0.0000 0.9994 0.0006 1.0000

0.9994 0.0000 0.9994 0.0006 1.0000

0.9994 0.0000 0.9994 0.0006 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000


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0.9995 0.0000 0.9995 0.0005 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000

0.9995 0.0000 0.9995 0.0005 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.00000.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9996 0.0000 0.9996 0.0004 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.00000.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.00000.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9997 0.0000 0.9997 0.0003 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.00000.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000

0.9998 0.0000 0.9998 0.0002 1.0000


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Figure 8: Plot of T, R and A as a function of V0 for fixed value of E=3.8 and W0=0.0


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Results and discussion:

In figure-1, the transmission co-efficient (T) and Reflection co-efficient (R) are plotted against the

energy of the particle (E) for a real barrier with width a=1.5 fm and height V0=4 fm-2

. The variation

of T and R with E are as usual up to the barrier height when energy increases sharply above the

barrier then T=1.0 and R=0.0. That means particle is free from potential influence.

In figure-2, the transmission co-efficient (T), Reflection co-efficient (R) and Absorption co-

efficient (A) are plotted against the energy of the particle (E) for a complex barrier with width a=2.0

fm, height V0=0 fm-2

and absorption strength W0=0.5 fm-2

. At low energies the T, R and A are

effective. Reflection is almost zero for higher energies. Absorption co-efficient is peaking at a

particular energy for W0=0.5 fm-2

. Then A gradually decreases for higher energies. Transmission is

affected by W0 in the low energy region then it increases slowly from low energy to high energy.

In figure 3, we have plotted a similar plot to figure 2 but with W 0=1.0 fm-2. Here the nature of T, R

and A are similar to figure 2 but the Absorption is more and T and R both are less in comparison

with the figure 2.

Again, in figure-4, T, R and A are plotted as a function of E for a complex barrier with width a=2.0

fm, height V0=2.0 fm-2

and absorption strength W0=0.5 fm-2

. In comparison with figure-2, the peak

of A is shifted and Reflection is more in low energies. If W0 increases from0.5 fm-2

to 1.0 fm-2

then

it will affect the transmission and absorption more in comparison to reflection. This is shown in

figure-5.

In figure-6, T,R and A are plotted as a function of W0 for a complex barrier with width a=2.0 fm,

height V0=0.0 fm-2

and energy of the particle E=3.8 fm-2

. If W0 increases from zero then gradually

the maximum absorption occurs for a specific value of W0 further increase reduces the absorption in

a slow rate. Here, T decreases with increase in W0. In figure-7, we have plotted the same plot with

V0=2.0 fm-2

. T and A decrease with increase in barrier height but R increases with the same.


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In figure-8, T, R, A are plotted as a function of V0 for a complex barrier with width a=2.0 fm,

strength W0=0.0 fm-2

and energy of the particle E=3.8 fm-2

. So variation is visible in the plot from

0.0 to V0=E. After that R=1.0 and T=0.0. Absorption co-efficient A is zero. (W0=0.0). Now,

increase the W0 from 0.0 to 0.05 and 0.5 while all other parameters are fixed. In this case T

decreases but A increases in the low energy region. This is shown in figure-9 and figure-10.

References:

1. Quantum Mechanics, Joachain

2. Quantum Mechanics, Ghatak & Lokanathan

3. Introduction to Quantum Mechanics, D.J.Griffith

4. Quantum Mechanics, V.Devanathan

5. Mathematical Methods for Physicist, Arfken&Weber.

Date post:	05-Apr-2018
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Barrier Draft2

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