saffron: identical vs. no orientations of A and g

[JuliaL]

Hello!

I have one electron, hyperfine coupled to one deuteron, i.e. g tensor, A tensor and their respective orientations gpa and Apa. I want to simulate HYSCORE powder spectra at different field positions.

As far as I know, if g and A have the same orientation (Apa = gpa), I should get the same results for Apa = gpa = 0 as for a full rotation matrix like Apa = gpa = [1 0 0; 0 0 -1; 0 1 0]. The correlation between g and A is fixed, while "powder" should erase the correlation between molecule axes and g axes (/ A axes) due to averaging.

I tested this with saffron and I wonder why I see a difference between the two cases. Did I mix up something in my head?

Here is a (quite) minimal working example of both cases. The figure compares Apa = gpa = 0 (left panels) with Apa = gpa != 0 (right panels). Same contour levels etc.

Code: Select all

clear all;

Exp.Sequence = 'HYSCORE';
Exp.dt = [0.024, 0.024];
Exp.Flip = [1 1 2 1];
Exp.Inc = [0 1 2 0];
Exp.tau = 0.2;
Exp.t1 = 0.2;
Exp.t2 = 0.3;
Exp.nPoints = 256;
Exp.mwFreq = 9.72;
Exp.ExciteWidth = 1000/24;

Exp_gx = Exp; Exp_gy = Exp; Exp_gz = Exp;
Exp_gx.Field = 317.0; %gx
Exp_gy.Field = 325.0; %gy
Exp_gz.Field = 343.5; %gz

Sys.S = 1/2;
Sys.g = [2.197 2.139 2.015];
Sys.Nucs = '2H';
Sys.A = [18.4 -10.8 -18.0]/6.514;

Opt.nKnots = 50;
Opt.ZeroFillFactor = 2;

[xa1, xa2, ya, pa] = saffron(Sys,Exp_gx,Opt);
[xb1, xb2, yb, pb] = saffron(Sys,Exp_gy,Opt);
[xc1, xc2, yc, pc] = saffron(Sys,Exp_gz,Opt);

Sys2=Sys;
Sys2.gpa = eulang([1 0 0; 0 0 -1; 0 1 0]);
Sys2.Apa = Sys2.gpa;

[xd1, xd2, yd, pd] = saffron(Sys2,Exp_gx,Opt);
[xe1, xe2, ye, pe] = saffron(Sys2,Exp_gy,Opt);
[xf1, xf2, yf, pf] = saffron(Sys2,Exp_gz,Opt);


figure
hold on
colormap(jet(512))
vs=15:5:800; %contour level

% simulations, no orientations
% gx
subplot(3,2,1,'Position',[0.16 0.72 0.37 0.25])
[sa,hsa]=contour(pa.f1, pa.f2, pa.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_x'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTickLabel','',....
    'XTick',[0 2 4 6],'linewidth',1.5,'TickLength',[0.025 0.025]);

% gy
subplot(3,2,3,'Position',[0.16 0.46 0.37 0.25])
[sb,hsb]=contour(pb.f1, pb.f2, pb.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_y'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTickLabel','',...
    'XTick',[0 2 4 6],'linewidth',1.5,'TickLength',[0.025 0.025]);
ylabel('\nu_2\rm / MHz','Units', 'Normalized', 'Position',...
    [-0.2, 0.5, 0]);

% gz
subplot(3,2,5, 'Position',[0.16 0.2 0.37 0.25])
[sc,hsc]=contour(pc.f1, pc.f2, pc.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_z'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTick',[0 2 4 6],...
    'linewidth',1.5,'TickLength',[0.025 0.025]);


% simulations, same orientations
% gx
subplot(3,2,2, 'Position',[0.55 0.72 0.37 0.25])
[sd,hsd]=contour(pd.f1, pd.f2, pd.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_x'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTick',[0 2 4 6],...
    'linewidth',1.5,'YTickLabel','','TickLength',[0.025 0.025],...
    'XTickLabel','');

% gy
subplot(3,2,4, 'Position',[0.55 0.46 0.37 0.25])
[se,hse]=contour(pe.f1, pe.f2, pe.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_y'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTick',[0 2 4 6],...
    'linewidth',1.5,'YTickLabel','','TickLength',[0.025 0.025],...
    'XTickLabel','');

% gz
subplot(3,2,6, 'Position',[0.55 0.2 0.37 0.25])
[sf,hsf]=contour(pf.f1, pf.f2, pf.fd, vs,'linewidth',2);
text(6,6.5,sprintf('\\bf g_z'))
set(gca,'XLim',[0 7.5],'XMinorTick','off','Box','on','XAxisLocation',...
    'bottom','FontName','Helvetica','FontSize',24,'YAxisLocation',...
    'left','YMinorTick','off','YLim',[0 7.5],'XTick',[0 2 4 6],...
    'linewidth',1.5,'YTickLabel','','TickLength',[0.025 0.025]);

xlabel('\nu_1\rm / MHz','Units', 'Normalized', 'Position',...
    [-0.15, -0.25, 0]);

hold off

System Mac OS X 10.8.5, Easyspin 4.5.5 in Matlab 2012b

Thanks,
Julia

[Matt Krzyaniak]

Lets see if I can say this correctly.

When using orientation selection, the orientation of the g-tensor relative to the molecular axis is arbitrary. We initially assume a powder, all orientations of the g-axis relative to the lab frame are present, however taking into account g-anisotropy(and sometimes hyperfine such as with copper) only a limited number tensor orientations relative to the lab frame are resonant. And for hyperfine techniques, such as HYSCORE, determining the resonant orientations is more or less the extent of the influence of the g-tensor on the simulation. So your rotation of the hyperfine tensor is actually acting as a rotation away from being co-linear with the g-tensor.

A simple way to see that is to turn off Apa and just move around gpa, nothing happens within the simulation. on the otherhand if you were to fix gpa and rotate Apa you'll observe the features tracing out the powder contour.

[JuliaL]

Hi Matt, thanks for the fast answer!

...don't know if we are talking about the same thing, here.*hm* I don't want to shift Apa vs. gpa at this point.

Quoting the EasySpin homepage, Apa and gpa both refer to the molecular frame, so setting gpa=Apa means g and A should be co-linear. And if I change Apa and gpa in the same way, g and A would still be co-linear, right?
Example: I do a HYSCORE simulation on gz and see Az, due to co-linearity. Then I change Apa and gpa in the same way and still should see Az when doing the simulation for gz.

[Matt Krzyaniak]

We are talking about the same things, I think its just a slight confusion with regard to the documentation, since its written in a very general fashion.

I guess I should have asked first, are you working with a frozen powder sample or a single crystal/oriented sample?

In the former case, like I had said before, the g tensor euler angles should have no effect on the simulation. When you include anisotropy(g or zfs) in the electron spin system ES uses angle selection, and the electron spin interaction is used only to calculate the resonant orientations. The rotation on A is transforming the tensor relative to those resonant orientations.

In the latter case such as with a single crystal is when the g euler angle relative to the lab axis starts to become important since you'll shift in and out of resonance.

Does that make sense?

I guess the other thing you can think about is: what does a rotation on the g-tensor mean to the resonant field position? Alternatively at those resonant field position what does a rotation on the hyperfine tensor mean?

[ThomasMCasey]

In other words:

This is because of orientation selection. Your spectrum exceeds your excite width. When you rotate your g tensor you have a different subset of resonant transitions. Transitions are non-uniform with respect to your position in the spectrum when you have anisotropic tensors. Set your excite width excessively high and you'll notice the difference go away, set it lower and the difference will be exaggerated.

[JuliaL]

My sample is a frozen solution.

By "very general fashion" do you mean, that the documentation only describes the single crystal case and, in contrast, for a powder spectrum/sample Apa is given in the g frame?
Or is Apa always given in the g frame and the documentation is wrong?

That's what I'm referring to:

gpa - Each row of this array contains the three Euler angles (in radians) for the passive rotation which transforms the g matrix of the associated electron spin from its eigenframe to the molecular frame (see also the function erot). If not specified, gpa is assumed to be all-zero, that is, all tensors are aligned with the molecular frame.

and

Apa - Array of Euler angles giving the orientations of the various A matrices in the molecular frame, as described above for gpa. If Apa is not specified, it is assumed to be all-zero, that is, all tensors are aligned with the molecular frame. See also erot. Apa has a layout analogous to A. Each row contains the three Euler angles for one nucleus.

[ThomasMCasey]

For what you're doing, you really should just leave gpa out. This will let it set the molecular frame, then apa will relate the hyperfine tensor to it. Rotating them both is necessary for a single crystal where you actually have three frames (lab, g, A). With powders, since the lab frame is arbitrary it is easiest to just pick a tensor (in this case g) and rotate the others into it. The logic is: "the electron zeeman term is the dominant term in the hamiltonian, all others are referenced to it." Also, it is not entirely correct to call each spectrum simply x, y or z. When you have a powder situation, within each of your excite widths your sets of transitions are mixtures. That is, the three are not completely isolated from each other when you have excitation bandwidth.

Where is your Q and qpa? 2H is I>1/2

[Matt Krzyaniak]

My sample is a frozen solution.

By "very general fashion" do you mean, that the documentation only describes the single crystal case and, in contrast, for a powder spectrum/sample Apa is given in the g frame?
Or is Apa always given in the g frame and the documentation is wrong?

By general fashion I mean that just because the parameter is there in the documentation and you can include it without a warning or error doesn't mean its actually doing anything in this particular simulation.