barrier draft2
TRANSCRIPT
-
7/31/2019 Barrier Draft2
1/26
1
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.
-
7/31/2019 Barrier Draft2
2/26
2
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.
-
7/31/2019 Barrier Draft2
3/26
3
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; =
-
7/31/2019 Barrier Draft2
4/26
4
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,
-
7/31/2019 Barrier Draft2
5/26
5
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
-
7/31/2019 Barrier Draft2
6/26
6
+
= 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
=
[
]
-
7/31/2019 Barrier Draft2
7/26
7
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.
-
7/31/2019 Barrier Draft2
8/26
8
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
-
7/31/2019 Barrier Draft2
9/26
9
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]
-
7/31/2019 Barrier Draft2
10/26
10
(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
-
7/31/2019 Barrier Draft2
11/26
11
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.
-
7/31/2019 Barrier Draft2
12/26
12
[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]
-
7/31/2019 Barrier Draft2
13/26
13
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
-
7/31/2019 Barrier Draft2
14/26
14
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
-
7/31/2019 Barrier Draft2
15/26
15
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
-
7/31/2019 Barrier Draft2
16/26
16
Figure 6: Plot of T, R and A as a function of W0 for fixed value of E=3.8 and V0=0.0
-
7/31/2019 Barrier Draft2
17/26
17
Figure 7: Plot of T, R and A as a function of W0 for fixed value of E=3.8 and V0=2.0
-
7/31/2019 Barrier Draft2
18/26
18
(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];
-
7/31/2019 Barrier Draft2
19/26
19
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
-
7/31/2019 Barrier Draft2
20/26
20
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
-
7/31/2019 Barrier Draft2
21/26
21
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
-
7/31/2019 Barrier Draft2
22/26
22
Figure 8: Plot of T, R and A as a function of V0 for fixed value of E=3.8 and W0=0.0
-
7/31/2019 Barrier Draft2
23/26
23
Figure 9: Plot of T, R and A as a function of V0 for fixed value of E=3.8 and W0=0.05
-
7/31/2019 Barrier Draft2
24/26
24
Figure 10: Plot of T, R and A as a function of V0 for fixed value of E=3.8 and W0=0.5
-
7/31/2019 Barrier Draft2
25/26
25
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.
-
7/31/2019 Barrier Draft2
26/26
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.