MATLAB: Numerical integral over the whole spherical volume of a crystallite

integrationnumerical integrationspherical volumetriple integration

I am new to matlab. I want to solve the above equation to get the total carrier concentration in the spherical grain volume;

below are the values of the parameters:

Total Volume of the sphere= 4/3*pi*r^3

r= 40*10^-9 %meters

Ld= 18*10^-9 %meters

phi= 20

Nd= 1e23

Best Answer

Nd = 1e23;
phi = 20;
Ld = 18e-9;
R = 40e-9;
fun = @(r) 4*pi*r.^2.*(1-exp(phi + 1/6*(r/Ld).^2));
value_integral = Nd*integral(fun,0,R)

Related Solutions

MATLAB: Plotting the different solution via fzero solver against a range of one of the variable

N = 100;
% define the parameters
Nd = 1e23;
e  = 12*1e-12;
%r  = 0
R  = 20*1e-9;
Et_Ef = -0.21*1.6*1e-19;
K  = 1.3807e-23;
T  = 600;
Nt = linspace(1e15,1e17,N);
q  = 1.6e-19;
phi0value = zeros(1,N);
for ii = 1:N
Ld = sqrt(e*K*T/(q^2*Nd))
funct= @(R,phi0) (4.*pi.*(Nd-Nd.*exp(-phi0)).*R.^2)
equation = @(phi0)integral(@(R)funct(R,phi0),0,R,'ArrayValued',true)-(4*pi.*R.^2.*Nt(ii)./(1+2*exp((Et_Ef + K*T .*(phi0 + (1/6).*(R/Ld).^2))/(K.*T))));
%X= [0 10];
phi0value(ii) = fzero(equation,10)
end
plot (Nt,phi0value)

MATLAB: Problem in integration in an array inside integrand

You are using ‘R’ in too many contexts. It is a vector, however it is also an argument to your anonymous functions, and anonymous functions pick up variables from the workspace. Changing the anonymous functions argument ‘R’ to ‘Rv’ allows your code to run:

neff = zeros(numel(R),N);
phi0val = zeros(numel(R),N);
for j = 1:numel(R)
    for i= 1:N
        funct = @(Rv,phi0) (4*pi*(Nd-Nd*exp(-phi0))*Rv.^2);
        equation = @(phi0)integral(@(Rv)funct(Rv,phi0),0,R(j))-(4*pi*R(j)^2*Nt(i)/(1+2*exp((Et_Ef + K.*T(i) .*(phi0 + (1/6)*(R(j)/Ld)^2))./(K.*T(i)))));
        %equation = @(phi0)integral(@(r)funct(r,phi0),0,R)-(4*pi*R^2*Nt(i)/(1+2*exp((Et_Ef + K*T *(phi0 + (1/6)*(R/Ld)^2))/(K*T))));
        phi0val(j,i) = fzero(equation,40);
        funneff= @(Rv)1./(4/3.*pi.*Rv.^3).*(Nd.*exp(-phi0val(j,i))).*Rv.^2;
        neff(j,i)= (1./(4./3.*pi.*R(j).^3)).*4.*pi.*integral(funneff,0,R(j),'Arrayvalued',true);
    end
end
figure
hold all
for k = 1:size(neff,1)
    plot(neff(k,:),phi0val(k,:))
end
hold off
expstr = @(x) [x(:).*10.^ceil(-log10(abs(x(:))))  floor(log10(abs(x(:))))];
Rexp = expstr(R)
hl = legend(sprintfc('$R\\ =\\ %2d\\times10^{%d}$',Rexp));
hl.Interpreter = 'latex';

Check to be certain this does what you want to to do.

The Plot —

Problem in integration in an array inside integrand - 2019 06 08.png

EDIT — Added plot image.

Best Answer

Related Solutions

MATLAB: Plotting the different solution via fzero solver against a range of one of the variable

MATLAB: Problem in integration in an array inside integrand

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