deerlab.dipolarbackground¶

dipolarbackground(t, pathways, basis)[source]¶

Computes the (multi-pathway) background function.

The function computes the two-spin interaction dipolar background which accounts for the structure and shape of the echo modulation due to intermolecular dipolar interactions over a sample (background contribution). The dipolar background is constructed from the contributions arising from different dipolar pathways [1] during a pulse sequence. The two-spin dipolar background reads:

$B(\mathbf{t}) = \prod_k B_\mathrm{basis} \Big( \lambda_k, \mathbf{\delta}_k (\mathbf{t} - \mathbf{t}_{\mathrm{ref},k}) \Big)$

where $\mathbf{t}=(t_1,\dots,t_D)$ are the experimental time incrementation vectors of a $D$ -dimensional experiment. The dipolar pathways are characterized by their amplitudes $\lambda_k$ , their refocusing times $\mathbf{t}_{\mathrm{ref},k}$ , and their (sub)harmonic factors $\mathbf{\delta}_k$ . The function $B_\mathrm{basis}(\lambda,t)$ represents the background basis function that describes the shape of the intermolecular contributions for a single dipolar pathway.

The function can also compute three-spin interaction dipolar background functions that account for the additional the structure and shape of the echo modulation in multi-spin systems [2]. The three-spin dipolar background is constructed from the contributions arising from the combinations of different pair dipolar pathways [2] during a pulse sequence. The three-spin dipolar background function reads:

$B^{(3)}(\mathbf{t}) = \prod_k\prod_q^3 B_\mathrm{basis}\Big(\lambda_k, \mathbf{\delta}_{k,q} (\mathbf{t} - \mathbf{t}_{\mathrm{ref},k,q}) \Big)$

The three-spin dipolar pathways are now characterized by an amplitude $\lambda_k$ , and by a set of three refocusing times: $\mathbf{t}_{\mathrm{ref},k,q}$ and three (sub)harmonic factors $\mathbf{\delta}_{k,q}$ .

Parameters:

tndarray or list thereof

Experimental time incrementation vectors (time coordinates) $t$ , in microseconds. For multi-dimensional experiments, specify a list [t1,...,tD] of the time coordinates $\mathbf{t}$ along each of the D different experimental dimensions.

pathwayslist of dictionaries or None, optional

List of dipolar pathways.

For two-spin dipolar interactions, each pathway is defined as a dictionary containing the pathway’s amplitude $\lambda_k$ ('amp'), refocusing time $t_{\mathrm{ref},k}$ ('reftime') in microseconds, and (sub)harmonic $\delta_k$ ('harmonic'). For example, for a one-dimensional experiment {'amp:'lambda, 'reftime':tref, 'harmonic':delta} or {'amp:'lambda, 'reftime':tref}. If 'harmonic' is not specified, it is assumed to be 1 (modulated). For an unmodulated pathway, specify only the amplitude, i.e. {'amp':Lambda0}.

For multi-dimensional experiments, the pathway refocusing times mathbf{t}_{mathrm{ref},k}` and harmonics $\mathbf{\delta}_k$ along the different dimensions must be specified as a list, e.g. for a pathway in a two-dimensional experiment {'amp:'lambda, 'reftime':[tref1,tref2], 'harmonic':[delta1,delta2]} ` For three-spin interactions, each pathway is defined as a dictionary containing the pathway’s amplitude $\lambda_k$ ('amp'), a tuple of refocusing times $t_{\mathrm{ref},k,q}$ ('reftime'), and a tuple of (sub)harmonics $\delta_{k,q}$ ('harmonic'). For example, for a one-dimensional experiment {'amp:'lambda, 'reftime':(tref1,tref2,tref3), 'harmonic':(delta1,delta2,delta3)}.

If neither pathways or mod are specified (or set to None), pathways=[{'amp':1,'reftime':0}] is used as the default.

basiscallable or array_like or None, optional

Dipolar background basis function. Must be a callable function, accepting a time-vector array as first input and a pathway amplitude as a second, i.e. bg = lambda t,lam: bg_model(t,par,lam). If set to None, no background function is included. For a single-pathway model, the numerical background decay can be passed as an array.

Returns:

Bndarray: Background decay.

References

[1]

L. Fábregas Ibáñez, G. Jeschke, and S. Stoll. DeerLab: A comprehensive toolbox for analyzing dipolar EPR spectroscopy data, Magn. Reson., 1, 209–224, 2020

[2] (1,2)

L. Fábregas Ibáñez, M. H. Tessmer, G. Jeschke, and S. Stoll. Dipolar pathways in multi-spin and multi-dimensional dipolar EPR spectroscopy, Phys. Chem. Chem. Phys., 24 2022, 22645-22660

Examples

To specify single-pathway background based on a homogenous 3D distribution of spins that refocuses at time $t=0$ , use:

lam = 0.4  # modulation depth
pathways = [
    {'amp': lam, 'reftime': 0}
    ]

def basis(t,lam):
    return dl.bg_hom3d(t,conc,lam)

B = dl.dipolarbackground(t, pathways, basis)

To specify a three-pathway kernel of the 4-pulse DEER experiment that, e.g that includes the 2+1 contribution, use:

tau1 = 0.5  # First interpulse delay (µs) 
tau2 = 4.0  # Second interpulse delay (µs)

lam1 = 0.40  # Pathway #1 amplitude
lam2 = 0.10  # Pathway #2 amplitude
lam3 = 0.10  # Pathway #3 amplitude

tref1 = tau1       # Pathway #1 refocusing time
tref2 = tau1+tau2  # Pathway #2 refocusing time
tref3 = 0          # Pathway #3 refocusing time

# Define the list of dipolar pathways
pathways = [
    {'amp': lam1, 'reftime': tref1},
    {'amp': lam2, 'reftime': tref2},
    {'amp': lam3, 'reftime': tref3},
]

def basis(t,lam):
    return dl.bg_hom3d(t,conc,lam)

B = dl.dipolarbackground(t, pathways, basis)

To specify a dipolar background for an experiment on a three-spin system based on a single-pathway model use:

tau1 = 0.5  # First interpulse delay (µs)         
lam = 0.40  # Pathway #1 amplitude

# Three dipolar pathways needed, one for each dipolar interaction in the three-spin system 
pathways = [
    {'amp': lam, 'reftime': (tau1,None,None), 'harmonic': (1,0,0)},
    {'amp': lam, 'reftime': (None,tau1,None), 'harmonic': (0,1,0)},
    {'amp': lam, 'reftime': (None,None,tau1), 'harmonic': (0,0,1)},
]

def basis(t,lam):
    return dl.bg_hom3d(t,conc,lam)

B = dl.dipolarbackground(t, pathways, basis)

To construct a background function for a three-spin system studied with a two-dimensional experiment, with a single dipolar pathway refocusing at, e.g., $t=(\tau_1,\tau_2)$ use:

# Three dipolar pathways needed, one for each dipolar interaction in the three-spin system 
pathways = [
    {'amp': 1-3*lam},
    {'amp': lam, 'reftime': ([tau1,tau2],None,None), 'harmonic': ([1,1],0,0)},
    {'amp': lam, 'reftime': (None,[tau1,tau2],None), 'harmonic': (0,[1,1],0)},
    {'amp': lam, 'reftime': (None,None,[tau1,tau2]), 'harmonic': (0,0,[1,1])},
]

def basis(t,lam):
    return dl.bg_hom3d(t,conc,lam)

B = dl.dipolarbackground(t, pathways, basis)