Clean up, minor corrections to keep notation consistent.
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@ -29,9 +29,12 @@ Dy164IsotopicAbundance = 0.2826;
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DyMagneticMoment = 9.93*BohrMagneton;
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%% 2-D DDI Potential in k-space, with Gaussian ansatz width determined by constrained minimization
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% With user-defined values of interaction, density and tilt
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w0 = 2*pi*61.6316; % Angular frequency unit [s^-1]
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l0 = sqrt(PlanckConstantReduced/(Dy164Mass*w0)); % Defining a harmonic oscillator length
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% w0 = 2*pi*61.6316; % Angular frequency unit [s^-1]
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% l0 = sqrt(PlanckConstantReduced/(Dy164Mass*w0));
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% % Defining a harmonic oscillator length - heredue to the choice of w0, l0
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% is 1 micrometer
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wz = 2 * pi * 300; % Trap frequency in the tight confinement direction
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lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
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@ -39,14 +42,14 @@ lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz));
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% Number of grid points in each direction
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Params.Nx = 128;
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Params.Ny = 128;
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Params.Lx = 150*l0;
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Params.Ly = 150*l0;
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Params.Lx = 150*1e-6;
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Params.Ly = 150*1e-6;
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[Transf] = setupSpace(Params);
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nadd2s = 0.110;
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as_to_add = 0.782;
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Params.alpha = 10;
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Params.phi = 0;
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nadd2s = 0.110; % Number density * add^2
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as_to_add = 0.782; % 1/edd
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Params.theta = 57; % Polar angle of dipole moment
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Params.eta = 0; % Azimuthal angle of dipole moment
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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@ -61,15 +64,13 @@ ub = [10];
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nonlcon = [];
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fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
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AtomNumberDensity = nadd2s / add^2; % Areal density of atoms
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AtomNumberDensity = nadd2s / add^2; % Number density of atoms
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as = as_to_add * add; % Scattering length
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eps_dd = add/as; % Relative interaction strength
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
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sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
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%% Chosen values of interaction, density and tilt
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MeanWidth = sigma * lz;
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% == 2-D DDI Potential in k-space == %
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@ -79,13 +80,13 @@ VDk_fftshifted = fftshift(VDk);
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figure(11)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, VDk_fftshifted); % Specify x and y data for axes
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, VDk_fftshifted); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
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ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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title(['2-D DDI Potential: $\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'fontsize',16,'interpreter','latex')
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% == Quantum Fluctuations term == %
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
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@ -108,33 +109,33 @@ EpsilonK = double(imag(EpsilonK) ~= 0); % 'is
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figure(12)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, EpsilonK); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
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ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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title(['2-D Dispersion: $\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'fontsize',16,'interpreter','latex')
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%%
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%% Cycle through angles
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% Define values for alpha and phi
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alpha_values = 0:5:90; % Range of alpha values (you can modify this)
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phi_values = 0:2:90; % Range of phi values (you can modify this)
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% Define values for theta and eta
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theta_values = 0:10:90; % Range of theta values (you can modify this)
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eta_values = 0:10:90; % Range of eta values (you can modify this)
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% Set up VideoWriter object
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v = VideoWriter('potential_movie', 'MPEG-4'); % Create a video object
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v.FrameRate = 5; % Frame rate of the video
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open(v); % Open the video file
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% Set up VideoWriter object to produce a movie
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% v = VideoWriter('potential_movie', 'MPEG-4'); % Create a video object
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% v.FrameRate = 5; % Frame rate of the video
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% open(v); % Open the video file
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% Loop over alpha and phi values
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for alpha = alpha_values
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for phi = phi_values
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% Update Params with current alpha and phi
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Params.alpha = alpha;
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Params.phi = phi;
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% Loop over theta and eta values
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for theta = theta_values
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for eta = eta_values
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% Update Params with current theta and eta
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Params.theta = theta;
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Params.eta = eta;
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% Compute the potential for the current alpha and phi
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% Compute the potential for the current theta and eta
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% == 2-D DDI Potential in k-space == %
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VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
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VDk_fftshifted = fftshift(VDk);
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@ -158,25 +159,25 @@ for alpha = alpha_values
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EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
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% Plot the result
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figure(12)
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figure(13)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, EpsilonK); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x$','fontsize',16,'interpreter','latex');
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ylabel('$k_y$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
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ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'fontsize',16,'interpreter','latex')
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% Capture the frame and write to video
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frame = getframe(gcf); % Capture the current figure
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writeVideo(v, frame); % Write the frame to the video
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% frame = getframe(gcf); % Capture the current figure
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% writeVideo(v, frame); % Write the frame to the video
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end
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end
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% Close the video file
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close(v);
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% close(v);
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%%
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function [Transf] = setupSpace(Params)
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@ -216,14 +217,14 @@ end
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function VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth)
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% == Calculating the DDI potential in Fourier space with appropriate cutoff == %
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% Angles of the dipole moment are defined in and away from the X-Z plane
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% Interaction in K space
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QX = Transf.KX*MeanWidth/sqrt(2);
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QY = Transf.KY*MeanWidth/sqrt(2);
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Qsq = QX.^2 + QY.^2;
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absQ = sqrt(Qsq);
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QDsq = QX.^2*cos(Params.phi)^2 + QY.^2*sin(Params.phi)^2;
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QDsq = QX.^2*cos(Params.eta)^2 + QY.^2*sin(Params.eta)^2; % eta here is the azimuthal angle of the dipole moment (angle from the x-axis)
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% Bare interaction
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Fpar = -1 + 3*sqrt(pi)*QDsq.*erfcx(absQ)./absQ; % Scaled complementary error function erfcx(x) = e^(x^2) * erfc(x)
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@ -231,7 +232,7 @@ function VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth)
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Fpar(absQ == 0) = -1;
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% Full DDI
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VDk = (Fpar*sin(Params.alpha)^2 + Fperp*cos(Params.alpha)^2);
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VDk = (Fpar*sin(Params.theta)^2 + Fperp*cos(Params.theta)^2); % theta here is the polar angle of the dipole moment (angle from the z-axis)
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end
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function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
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@ -66,7 +66,7 @@ end
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% ====================================================================================================================================================== %
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alpha = 0; % Polar angle of dipole moment
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theta = 0; % Polar angle of dipole moment
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phi = 0; % Azimuthal angle of momentum vector
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k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
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instability_boundary = zeros(length(as_to_add), length(nadd2s));
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@ -88,7 +88,7 @@ for idx = 1:length(nadd2s)
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
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% == Quantum Fluctuations term == %
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
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@ -154,4 +154,79 @@ plot(xvals', yvals,LineWidth=2.0)
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xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
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ylabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
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title('$k_{roton}$ at the instability boundary','fontsize',16,'interpreter','latex')
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grid on
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grid on
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%%
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function [Transf] = setupSpace(Params)
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Transf.Xmax = 0.5*Params.Lx;
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Transf.Ymax = 0.5*Params.Ly;
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Nx = Params.Nx;
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Ny = Params.Ny;
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% Fourier grids
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x = linspace(-0.5*Params.Lx,0.5*Params.Lx-Params.Lx/Params.Nx,Params.Nx);
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Kmax = pi*Params.Nx/Params.Lx;
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kx = linspace(-Kmax,Kmax,Nx+1);
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kx = kx(1:end-1);
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dkx = kx(2)-kx(1);
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kx = fftshift(kx);
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y = linspace(-0.5*Params.Ly,0.5*Params.Ly-Params.Ly/Params.Ny,Params.Ny);
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Kmax = pi*Params.Ny/Params.Ly;
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ky = linspace(-Kmax,Kmax,Ny+1);
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ky = ky(1:end-1);
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dky = ky(2)-ky(1);
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ky = fftshift(ky);
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[Transf.X,Transf.Y] = ndgrid(x,y);
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[Transf.KX,Transf.KY] = ndgrid(kx,ky);
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Transf.x = x;
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Transf.y = y;
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Transf.kx = kx;
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Transf.ky = ky;
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Transf.dx = x(2)-x(1);
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Transf.dy = y(2)-y(1);
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Transf.dkx = dkx;
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Transf.dky = dky;
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end
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function VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth)
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% == Calculating the DDI potential in Fourier space with appropriate cutoff == %
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% Interaction in K space
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QX = Transf.KX*MeanWidth/sqrt(2);
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QY = Transf.KY*MeanWidth/sqrt(2);
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Qsq = QX.^2 + QY.^2;
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absQ = sqrt(Qsq);
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QDsq = QX.^2*cos(Params.phi)^2 + QY.^2*sin(Params.phi)^2;
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% Bare interaction
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Fpar = -1 + 3*sqrt(pi)*QDsq.*erfcx(absQ)./absQ; % Scaled complementary error function erfcx(x) = e^(x^2) * erfc(x)
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Fperp = 2 - 3*sqrt(pi).*absQ.*erfcx(absQ);
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Fpar(absQ == 0) = -1;
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% Full DDI
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VDk = (Fpar*sin(Params.alpha)^2 + Fperp*cos(Params.alpha)^2);
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end
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
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Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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Fka = (3 * cos(deg2rad(theta))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(theta))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(theta))^2));
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Ukk = (gs + (gdd * Fka)) * gamma4;
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end
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function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
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eps_dd = add/as; % Relative interaction strength
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MeanWidth = x * lz;
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
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gQF = gamma5 * gammaQF;
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Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
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Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
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ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
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end
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@ -62,7 +62,7 @@ end
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% ====================================================================================================================================================== %
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alpha = 0; % Polar angle of dipole moment
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theta = 0; % Polar angle of dipole moment
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phi = 0; % Azimuthal angle of momentum vector
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k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
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instability_boundary = zeros(length(as_to_add), length(nadd2s));
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@ -83,7 +83,7 @@ for idx = 1:length(nadd2s)
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
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% == Quantum Fluctuations term == %
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
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@ -126,10 +126,10 @@ ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
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title('Roton instability boundary','fontsize',16,'interpreter','latex')
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%%
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi)
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
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Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2));
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Fka = (3 * cos(deg2rad(theta))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(theta))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(theta))^2));
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Ukk = (gs + (gdd * Fka)) * gamma4;
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end
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@ -143,7 +143,4 @@ function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, g
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Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
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Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
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ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
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end
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end
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@ -0,0 +1,318 @@
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%% Physical constants
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PlanckConstant = 6.62607015E-34;
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PlanckConstantReduced = 6.62607015E-34/(2*pi);
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FineStructureConstant = 7.2973525698E-3;
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ElectronMass = 9.10938291E-31;
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GravitationalConstant = 6.67384E-11;
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ProtonMass = 1.672621777E-27;
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AtomicMassUnit = 1.660539066E-27;
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BohrRadius = 5.2917721067E-11;
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BohrMagneton = 9.274009994E-24;
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BoltzmannConstant = 1.38064852E-23;
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StandardGravityAcceleration = 9.80665;
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SpeedOfLight = 299792458;
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StefanBoltzmannConstant = 5.670373E-8;
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ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% Roton instability boundary for tilted dipoles
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
nadd2s = 0.005:0.0025:0.5;
|
||||
as_to_add = 0.3:0.025:0.95;
|
||||
var_widths = zeros(length(as_to_add), length(nadd2s));
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
var_widths(jdx, idx) = sigma;
|
||||
end
|
||||
end
|
||||
|
||||
%% ====================================================================================================================================================== %
|
||||
|
||||
figure(6)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 1850 750])
|
||||
|
||||
theta = 66; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
subplot(1, 2, 1); % 1 row, 2 columns, first subplot
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
caxis([0 1])
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
title(['Along Y: $\theta = ',num2str(theta), '; \phi = ', num2str(phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
theta = 66; % Polar angle of dipole moment
|
||||
phi = 90; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
% set(gcf,'Position',[50 50 950 750])
|
||||
subplot(1, 2, 2); % 1 row, 2 columns, first subplot
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
caxis([0 1])
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
title(['Along X: $\theta = ',num2str(theta), '; \phi = ', num2str(phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
%{
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
caxis([0 1])
|
||||
% ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
%}
|
||||
|
||||
sgtitle('Mean-field instability boundary','fontsize',16,'interpreter','latex')
|
||||
|
||||
%% Cycle through angles
|
||||
|
||||
% Define values for theta and phi
|
||||
theta_values = 0:2:90; % Range of theta values (you can modify this)
|
||||
|
||||
% Set up VideoWriter object to produce a movie
|
||||
v = VideoWriter('rib_movie', 'MPEG-4'); % Create a video object
|
||||
v.FrameRate = 5; % Frame rate of the video
|
||||
open(v); % Open the video file
|
||||
|
||||
for theta = theta_values
|
||||
figure(6)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 1850 750])
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
subplot(1, 2, 1); % 1 row, 2 columns, first subplot
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
caxis([0 1])
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
title(['Along Y: $\theta = ',num2str(theta), '; \phi = ', num2str(phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
phi = 90; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
% set(gcf,'Position',[50 50 950 750])
|
||||
subplot(1, 2, 2); % 1 row, 2 columns, first subplot
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
caxis([0 1])
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
title(['Along X: $\theta = ',num2str(theta), '; \phi = ', num2str(phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
%{
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
caxis([0 1])
|
||||
% ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
%}
|
||||
|
||||
% Capture the frame and write to video
|
||||
frame = getframe(gcf); % Capture the current figure
|
||||
writeVideo(v, frame); % Write the frame to the video
|
||||
|
||||
% sgtitle('Mean-field instability boundary','fontsize',16,'interpreter','latex')
|
||||
end
|
||||
|
||||
% Close the video file
|
||||
close(v);
|
||||
|
||||
%%
|
||||
function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
|
||||
Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
Fka = (3 * cos(deg2rad(theta))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(theta))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(theta))^2));
|
||||
Ukk = (gs + (gdd * Fka)) * gamma4;
|
||||
end
|
||||
|
||||
function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
MeanWidth = x * lz;
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
|
||||
Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
|
||||
ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
|
||||
end
|
||||
|
||||
|
||||
|
||||
@ -34,7 +34,7 @@ wz = 2 * pi * 72.4;
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
as = 102.515 * BohrRadius; % Scattering length
|
||||
Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
theta = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
|
||||
k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
|
||||
@ -47,7 +47,7 @@ eps_dd = add/as;
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
@ -74,7 +74,7 @@ AtomNumber = 1E5;
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
theta = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
|
||||
k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
|
||||
@ -92,7 +92,7 @@ for idx = 1:length(ScatteringLengths)
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
@ -128,7 +128,7 @@ legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
theta = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
|
||||
@ -154,7 +154,7 @@ for idx = 1:length(nadd2s)
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
@ -249,7 +249,7 @@ ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
% ====================================================================================================================================================== %
|
||||
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
theta = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
@ -270,7 +270,7 @@ for idx = 1:length(nadd2s)
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
@ -312,10 +312,10 @@ title('Roton instability boundary','fontsize',16,'interpreter','latex')
|
||||
%}
|
||||
|
||||
%%
|
||||
function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi)
|
||||
function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
|
||||
Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2));
|
||||
Fka = (3 * cos(deg2rad(theta))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(theta))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(theta))^2));
|
||||
Ukk = (gs + (gdd * Fka)) * gamma4;
|
||||
end
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user