2022 NIST Center for Neutron Research School

on Small Angle Neutron Scattering and Neutron Reflectometry

NG7 SANS

Self-assembly in binary lipid mixtures

Abstract

We will use small angle neutron scattering (SANS) to study the structure and composition of 

assemblies formed by binary mixtures of long- and short- tail lipids. The measurements will 

illustrate the advantage of using contrast matching in neutron scattering as well as introduce data 

treatment and analysis. The effects of mixing different amphiphilic molecules on the structure of 

the assemblies as well as the design of the SANS experiment are introduced here. References are 

given for more in-depth information.

1. Introduction

Lipid assemblies are found throughout cells in nature and in numerous products we encounter in 

our everyday lives from food to personal care products and even drug products and vaccines. 

Lipids are a type of amphiphilic molecule, with a hydrophilic – or water-loving – headgroup and 

hydrophobic – or water-hating – tails. In water, lipids form a wide variety of nanoscale structures 

where the hydrophilic headgroups are exposed to the aqueous environment and shield the inner 

hydrophobic core.

This self-assembly is the basis of biomembranes that compartmentalize cells, where the 

hydrophobic lipid tails form a low dielectric oil that hosts transmembrane proteins and prevents 

free passage of polar and charged species into and out of the cell (Fig. 1). The formation of welldefined hydrophobic pockets in an otherwise aqueous environment is also why lipid assemblies

are used as containers for membrane proteins for protein structure determination measurements 

and for hydrophobic guest molecules in personal care products and drug delivery vehicles.

Figure 1. Cartoon illustration 

of a cell encapsulated and 

compartmentalized by 

membranes formed by selfassembled lipids
One type of self-assembled lipid structure that is widely used in biochemical and biophysical 

studies are “bilayed micelles” or bicelles for short. Bicelles are easily aligned in a magnetic field, 

making them a useful tool for the nuclear magnetic resonance (NMR) based protein structure 

determination and studies of membrane associated peptides and 

proteins.(1, 2) More recently it was also demonstrated that the 

proton pump transmembrane protein bacteriorhodopsin could 

be crystallized from suspension in these lipid mixtures(3, 4) and 

their use has even expanded to topical drug delivery.(5)

One of the most widely used bicellar systems is a mixture of 

long tail 14-C dimyristoyl-phosphatidylcholine (DMPC) and

shorter 6-C tail dihexanolyl-phosphatidylcholine (DHPC)

lipids (see Fig. 2 for corresponding structures).(6) DMPC and 

DHPC have identical phosphocholine (PC) headgroups and 

double chain carbon tails of 14 and 6 carbons per tail, 

respectively. However, variations in lipid ratio, lipid 

concentration, and even temperature can have profound 

effects on the resulting structures.

1.1. Molecular shapes and the assemblies they form

In this experiment we will study the study the structures formed by mixing a long-tail lipid, DMPC, 

and a short-tail lipid, DHPC. At certain conditions, these lipid mixtures are reported to form 

bicelles that are widely used as model membrane systems.

So what determines the structure of self-assembled systems? While there are different ways to 

think about this, including thermodynamic frameworks,(7, 8) the simplest - yet very powerful -

approach is a geometric one.(9, 10) Fundamentally, the structures formed are determined by the 

mean packing shape of the molecule (Fig. 3).(9, 10) Depending on the relative sizes of the 

hydrophobic and hydrophilic portions of the lipids, they can form a wide variety of structures from 

spherical micelles to rodlike micelles, to bilayers and vesicles, and even more complex inverted 

micellar structures that are important for example in some of the more complex membrane 

structures found in cells. These differences in molecular shape can be quantified by the critical 

Figure 2. Molecular structures 

of DMPC and DHPC 

Figure 3. Amphiphilic 

molecules come in all sorts 

of shapes and sizes that selfassembly into a wide variety 

of structures depending on 

their critical packing 

parameter (CPP)
packing parameter (CPP, sometimes known as the surfactant packing parameter or SPP). CPP = 

v/a0lc where v is the molecular volume, a0 is the area per molecule, and lc is the length of the 

hydrocarbon tails.(10) A CPP ≤ 1⁄3 leads to spheres, 1⁄3 < CPP < 1⁄2 to rodlike elongations, CPP ~ 1 

to bilayers (including vesicles) and CPP > 1 to “inverted” structures. Thus the CPP tells us 

something about the intrinsic preferred curvature in the structures formed.

Comparing the molecular shapes of DMPC and 

DHPC reveals that they have a very different 

CPP and therefore prefer to form different 

structures (Fig. 4). With a long tail relative to 

the size of the PC headgroup, DMPC has a large 

CPP close to 1 and forms flexible bilayers like 

those found in the cell membrane.(11)

Meanwhile, because of the short tails, DHPC 

self-assembles into small spheroidal micelles in 

solution.(12)

The role of thermodynamics. Keep in mind that 

while the CPP tells us about the intrinsic 

curvature and what shape structures to expect, 

the driving force to actually create these 

aggregates, rather than the molecules being 

dissolved in solution or aggregating 

uncontrollably and precipitating out of 

solution, is the details of the system energetics: thermodynamics. The shape of the self-assembled 

structure is thus the minimum energy state, but without covalent bonds the individual amphiphile 

molecules are free to move about within kT of the minimum energy structure. The consequence is 

that the self-assembled systems are not “frozen” shapes but actually undergo constant 

deformations from the nominal, idealized shape. Furthermore, the self-assembled structures can 

grow and shrink by insertion and ejection of individual amphiphilic molecules, leading to a 

polydispersity in the size of the structures formed. It is thus important to remember when 

measuring such systems that there is actually a distribution of sizes and shapes, not a single frozen 

one. This is where a bulk technique like scattering (and luckily this is a scattering school) that 

directly measures the statistically relevant average of that ensemble becomes a critically important 

tool.

What happens when we mix DMPC and DHPC? As one might guess, typically when mixing 

amphiphiles with very different CPP, the structures formed are based on an average CPP (strictly 

speaking the CPP of a stoichiometric ratio of the different molecules). Once again, the dynamic 

nature of the assembly has consequences. Indeed, it can induce transient local enrichment in one 

or the other amphiphile, leading to enhanced local curvatures which gives rise to much more 

dynamic self-assembled structures, e.g. sponge phases.(13, 14) However, there is a large body of 

literature that suggests at least certain DMPC/DHPC mixtures under certain conditions form 

relatively monodisperse discotic micelles, termed bicelle in the lipid literature (Fig 5).(15, 16) The 

Figure 4. Mean shapes of lipids with different tail 

lengths and the structures they form
explanation is that the high CPP DMPC molecules segregate into a planar bilayer portion of the 

bicelle while the high intrinsic curvature DHCP molecules (small CPP) segregate into the curved 

rim, thereby decreasing the edge energy by protecting the hydrophobic tails from the surrounding 

solvent. Assuming that the minimum energy for these mixtures is the bicelle despite the high 

entropic cost of segregation, the question remains as to what extent to the lipids actually separate? 

Is the rim mainly purely of DHPC? Does some DHPC get trapped in the bilayer region? There is 

believed to be some degree of lipid mixing and a “mixed bicelle” model has therefore been 

proposed.(17-19) However, the degree of mixing has proven difficult to quantify. The precise 

quantification of mixing, or lack thereof, the key component to understanding the underlying 

physics and chemistry, and, in turn, the structures of bicelles, would allow for better control and 

very specific tunability. 

1.2. SANS from colloidal solutions: measuring the structure and 

composition of particles

The previous section suggested that mixing these two lipids, with different critical packing 

parameters and molecular shapes, leads to the assembly of new structures, but how do we confirm 

that the cartoons are accurate representations of the structures? And if they are, to what extent do 

the lipids segregate? How do we know which lipids go where? And finally, to the extent the lipids 

are segregating in some fashion, what is the physics/chemistry that drives that? As you might have 

guessed, being at a neutron scattering summer school – we can measure these structures and the 

degree of lipid mixing with small angle neutron scattering (SANS). But how does scattering from 

a solution tell us about the structure and composition of bicelles? 

The coherent scattering intensity, I(q), from a monodisperse colloidal solution can be expressed as

I(q) = φVΔρ

2P(q)S(q) [1]

Where φ is the particle volume fraction, V is the particle volume, Δρ is the difference in scattering 

length density between the particles and the solvent, also called the scattering contrast, P(q) is the 

form factor and S(q) is the structure factor. In all of these expression q is the scattering vector 

which is given by q = 4π⁄λ sin θ where 2θ is the scattering angle. The goal of every elastic 

scattering experiment is to measure the structures and interactions in the samples. To do this, we 

Figure 5. Cartoon illustration of a bicelle formed by mixtures of lipids with long- and short tails.
need to determine the contrast, form factor and structure factor for our sample. We will discuss 

each of these terms in the next 3 sections.

1.2.1. Scattering contrast: Δρ2

 

To have enough scattered intensity to characterize the sample, there must be scattering contrast 

(Δρ) between the sample and the surrounding solvent. In other words, the sample must ‘look’ 

different from the solvent as far as the neutrons are concerned. This is very similar - and in fact is 

related to - the index of refraction in light, where one needs a difference in index of refraction to 

“see” an object as separate from the matrix.

Neutrons interact with the nucleus of the of an atom and the probability that the neutron will 

interact with a given atomic nucleus is characterized by the so-called scattering length, bi. The 

scattering length varies by element and by isotopes within a given element, which is a unique 

advantage of neutron scattering. Scattering by X-rays will increase with the atomic number while 

the neutron scattering length varies seemingly randomly across the periodic table. The bi values 

for the elements and their isotopes are tabulated and can be found on the NCNR website: 

https://www.ncnr.nist.gov/resources/n-lengths/

That said, we are interested in length scales that are much larger than atomic distances in small 

angle scattering, so it is useful to define the scattering length density (SLD), ρ of a material,

ρ = ∑

bi

V

N

i

[2]

In which V is the volume containing the N atoms. Calculating the SLD averages the scattering 

length over the volume of the material, and we are most interested in the differences in materials 

properties ρ1 and ρ2 where 1 and 2 are for example our particles and solvent, respectively. We 

refer to the difference in SLDs as the scattering contrast, Δρ = ρ1 − ρ2. The larger the Δρ, the 

larger the intensity. The SLD’s can be calculated by hand from the tabulated values of bi or by 

using the NCNR online calculator: https://www.ncnr.nist.gov/resources/activation/

Shown in Table 1 are calculated values for the SLD of DMPC, deuterated DMPC (d-DMPC), 

DHPC, D2O and H2O, the materials we will be using in our experiment. Notice that the ρ values 

for H2O and D2O differ by an order of magnitude and by sign. The difference is because hydrogen 

and deuterium have very different scattering lengths despite being isotopes (remember bi varies 

with element and isotope). This means that we can vary the H and D content in our samples to 

change the scattering contrast for different components. In fact, it is often possible to find 

conditions where we match the SLD of the solvent to one of the components of the sample, a

technique referred to as contrast matching. Contrast matching is a unique advantage of neutron 

scattering compared to other scattering techniques and makes SANS a very powerful tool for 

studying multicomponent systems – such as the ones we are interested in studying in this 

experiment.
Material Chemical 

formula

Mass density @ 

10 °C

(g/cm3)

Scattering length density

(cm-2)

DMPC C36H72NO8P 1.08 0.30 x 1010

d-DHPC C36H5D67NO8P 1.19 7.00 x 1010

DHPC C20H40NO8P 1.16 0.67 x 1010

Heavy water D2O 1.11 6.34 x 1010

Light water H2O 1.00 -0.56 x 1010

Table 1. Calculated scattering length densities for the materials we will be using in our experiment. Note 

that the values for silica and PEG are based on densities from literature and may not be precise. In particular, 

the silica density can vary with particle preparation method.

To study the structures formed by mixtures of DMPC and DHPC, we need to make sure there is 

enough contrast with the surrounding solvent to be able to see them. Contrast between the 

deuterated lipid and light water ∆ρ

2 = (ρd−DMPC − ρH2O)

2

≈ (8.11 − (−0.561))

2 ≈ 75 is 

almost two times greater than the contrast between a protiated lipid and heavy water ∆ρ

2 =

(ρDMPC − ρD2O)

2

≈ (0.29 − 6.34)

2 ≈ 36. However, maximizing contrast is not the only thing 

we need to consider. We also need to consider the incoherent scattering (or background). 

Incoherent scattering contributes an isotropic (i.e. flat) background that can make it difficult to 

resolve the signal of interest at high q or short length scales. In the case of water, the incoherent 

scattering from H2O is about 30 times greater than that from D2O! Because hydrogen contributes 

an extremely large incoherent background compared to deuterium, the general rule of thumb in 

neutron scattering is to always make the majority component (usually the solvent in colloidal 

systems) deuterated and the minority component hydrogenated - at least in the absence of other 

considerations. Finally, it is important to note that deuterated versions of molecules are almost 

always more expensive than the protiated version (by a factor of ~ 1000 in the case of lipids used 

here) so cost can sometimes also be a consideration. Given that h-lipids are significantly cheaper 

than d-lipids, and that the majority component is water, we will do the experiments to determine 

the overall structure of the sample with h-lipids in D2O.

Now, how do we figure out which lipids go where in the bicelle? Comparing the scattering length 

densities of DMPC and DHPC suggest they look very similar as far as the neutrons are concerned. 

Fortunately, we can take advantage of the difference in scattering length densities between h- and 

d-lipids and be clever about designing the contrast in our experiment. As shown in Fig. 6, we 

should be able to distinguish between d-DMPC from DHPC and figure out if the lipids evenly mix 

or segregate in the bicelles.
1.2.2. Form Factor: P(q)

The form factor is scattering from interference from different parts of the same object and therefore 

provides information on the structure of the scattering object. P(q) is related to the Fourier 

transform of the real space density distribution,

P(q) = |

1

Vp

∫ e

−iqr⃗

Vp

dr⃗|

2

[3]

In the limit q → 0 , the above expression can be simplified for randomly oriented monodisperse 

particles, independent of their shape, by using a Taylor series expansion around 0. When truncated 

to the first term, this Taylor series expansion yields

I(q → 0) ∝ exp (

−q

2Rg2

3

) [4]

Where Rg is the radius of gyration of the particle. 

Rg

2 =

∑ ρi(ri − r0)

2

i

∑i ρi

Where the mass term in the classical mechanics definition has been replaced with the SLD and r0

is the SLD center of mass. The radius of gyration can thus be fairly easily calculated for various 

shapes. For example, for a homogenous sphere with radius R, Rg

2 = 3R2

/5, for a cylinder with 

Figure 6. Cartoon illustration of potential 

bicelles structures where d-DMPC and hDHPC are mixed or segregated to the rim and 

bilayer regions as expected as seen in 

contrast-variation experiments with SANS.
length L and radius R, Rg

2 = L2

/12+R

2

/2, and for an ellipsoid with axes lengths a, b, and c, Rg

2 = 

(a

2+b2+c2

)/5.

The above equation is an example of Guinier’s Law and, as we are truncating at the first term in 

the expansion, is valid only for very small qRg. A typical rule of thumb is that qRg < 1. More 

importantly, Eq. 4 can be used to estimate the particle size independent of any assumption of the 

shape and without needing a detailed form factor by simply plotting ln(I) versus q

2

at low q.

To extract more detailed information on the structure of the assemblies, an appropriate form factor 

model is needed. Fortunately, models for P(q) for common shapes such as spheres, cylinders, and 

ellipsoids have been developed. Here we will try fitting with these different models to see which 

best the describes the data. We can also double check the accuracy of assumptions and fit results 

by comparing the Rg calculated from the different P(q) fit results against the value determined 

from the Guinier analysis of the data.

1.2.3. Structure Factor: S(q)

The structure factor S(q) is due to interference from different objects in the sample. In a crystalline 

lattice, the structure factor describes the positions of different atoms in the lattice. In solution, S(q)

is a measure of the correlation function between the center of masses of the different particles. 

Because the relative positions are determined by their interaction potential, S(q) therefore contains 

information on the interactions between objects in solution. 

At low concentrations, there are no correlations between the particle positions and S(q) ≈ 1. The 

scattering pattern from dilute solutions is therefore determined only by the form factor as 

I(q) = AP(q). At higher concentrations, the interactions between the objects in solution need to be 

taken into consideration. 

Because S(q) arises from interference between different objects, we see the effects at larger length 

scales which corresponds to the scattering at low q. In the limit of q → 0, the scattered intensity is 

related to the osmotic compressibility of the system:

S(q → 0) = kBT (

∂n

∂Π

) [5]

Where n is the particle number density and is the Π osmotic pressure.(20) Therefore S(q → 0) >

1 for attractive interactions when the system is more compressible and S(q → 0) < 1 for repulsive 

interactions. Expressions for S(q) have been determined from statistical mechanical theories for 

the inter-particle potential. Details on how to calculate S(q) are beyond the scope of this write-up, 

but the calculated structure factors for sticky hard sphere attractive interaction as well as the 

repulsive hard sphere and Coulombic interactions are shown in Fig. 7. As can be seen in the graph,

S(q) increases at low q for the attractive interactions and is less than 1 for repulsive interactions.
Figure 7. Calculated structure factors for sticky hard spheres (red), hard spheres (green), and charged 

spheres with Coulombic repulsion (blue). The calculations are for 100 nm spheres with 10% polydispersity.

Measuring the structure factor is a very powerful method for characterizing the interactions between 

nanoscale objects in solution. Even if we’re not interested in these interactions per se, it is important to 

remember that the structure contributes to the measured intensity. If we are only interested in measuring 

the shape of our assemblies (P(q)), then we’ll want to double check that we are in the dilute limit and S(q) 

≈ 1 for our samples. But how can we do this? The simplest way is to measure the same sample at 

multiple concentrations and make sure the scaled data overlap. We will perform these checks here 

to make sure our measured intensity is not changing with concentration, but keep in mind – this 

experiment assumes that the structure does not change with concentration – which may or may not 

be true for all amphiphiles! 

2. Planning a SANS experiment

Now that we know what we are interested in measuring, how to we go about planning a successful 

experiment? To get good data we want to maximize the scattered intensity from our sample and 

minimize the background. We already introduced important things to consider for SANS sample 

preparation above such as making sure there is enough scattering contrast to get a good signal 

(Section 1.2.1) and if our samples are in a dilute limit where S(q) → 1 (Section 1.2.3), but there are 

a few more things to think about both in terms of sample preparation and measurements before 

starting a SANS experiment that we discuss below. 

2.1. Sample thickness

What sample thickness do we need for a SANS experiment? Choosing a sample path length not 

only determines the amount of sample needed for the experiment, but more importantly, the 

measured scattering intensity from the sample. 

The intensity that we measure is proportional to the sample thickness, ds, and sample transmission, 

T. Increasing ds increases the measured intensity – but at the same time, decreases the sample 
transmission. This tradeoff means that there should be an optimum ds to maximize the measured 

intensity.

To find the optimum ds, we first need to consider how increasing the thickness affects the sample 

transmission. The sample transmission is the fraction of neutrons that pass through the sample 

without being scattered or absorbed and is given by:

T = e

−Σtds [6]

Where Σtis the total cross section and equal to the sum of the coherent, incoherent, and absorption 

macroscopic cross sections: Σt = Σc + Σi + Σa. 

The absorption cross section (Σa) depends on the cross section of elements in the sample and can 

be calculated from tabulated values for the different elements if the mass density and chemical 

composition of the sample are known. Σa is wavelength dependent and increases linearly with λ 

for almost all wavelengths, meaning the sample transmission will also vary wavelength and must 

be measured for very wavelength used during the experiment (see Section 3). The incoherent cross 

section (Σi) can also be estimated from tabulated values for the different elements in the sample 

as well but will also depend on the atomic motions in the sample and therefore can vary slightly 

with temperature. Last but not least is the coherent cross section (Σc) which will depend on both 

the structure and dynamics of the material (which should not be a surprise since Σcis what we are 

aiming to measure!).

After estimating the sample transmission, the optimum sample length can be calculated because 

the measured intensity is proportional to

Imeas ∝ dse

−Σtds [6]

Which will have a maximum at ds = 1⁄Σt. The optimum transmission, Topt = 1⁄e ≈ 0.37. The 

sample thickness that gives Topt for a given sample is known as the “1/e-length”.

Estimating T for our samples is straightforward but can be time consuming. Fortunately, there are 

tools to help. The NCNR’s Web-based Neutron activation and scattering calculator

(https://www.ncnr.nist.gov/resources/activation/) not only computes the scattering length density, 

but also estimates the incoherent and absorption cross sections and the 1/e-length for the entered 

materials. It is good to estimate the optimum thickness before starting a SANS experiments. In 

some cases, we can increase the sample thickness to get better counting statistics in shorter times

or in other cases, we may need to decrease the sample thickness to prevent multiple scattering 

discussed next Section 2.2. Finally, it should be noted that sometimes the amount of sample 

available precludes optimizing the scattering volume.

2.2. Multiple scattering

It is possible to have too much scattering. Analyzing SANS data assumes that the neutron is only 

scattered once as it passes through the sample and that any measured scattering angle is therefore 

related to the structure of the sample. However, this assumption may not be true is samples that 
scatter very strongly, i.e. the neutron may scatter multiple times while passing through the sample. 

Multiple scattering distorts the shape of the measured SANS curves and makes data analysis 

almost impossible. (21)

If the sample scatters too strongly, we need to consider ways to reduce the scattered intensity such 

as decreasing the scattering contrast Δρ2or reducing the sample thickness. A good check before an 

experiment is to estimate Σi and Σa, because if Σc >> Σi + Σa the sample thickness should be 

reduced so that the transmission due to the coherent scattering remains higher than 0.9 rather than 

Topt = 0.37 to avoid multiple scattering effects. 

We will be using 1 mm pathlength sample cells for our experiments. 

2.3.Choosing an instrument

Here at the NCNR we have a total of 5 SANS instruments: 3 SANS instruments, a very small angle 

scattering (VSANS) instrument, and the ultra-small angle neutron scattering (USANS) instrument. 

Each instrument is designed to cover a specific q-range as illustrated below in Fig. 8. It is important 

that we know the length scales of interest the sample before picking an instrument. SANS is best 

suited for studying samples from ≈ 1 nm to ≈ 100 nm while USANS captures length scales from

≈ 100 nm to ≈ 20 μm. VSANS bridges these two extremes and allow us to measure samples with 

size scales ranging from ≈ 1 nm to ≈ 1 μm.

Figure 8. Comparison of the accessible q ranges for the BT-5 USANS instrument, NG-3 and NG-7 SANS 

instruments, and VSANS at the NCNR. The plot is reproduced from reference (22). 

The cartoons in Fig. 5 suggest that a bicelle is only two molecules thick, or approximately 5 nm, 

so we will need to collected data at high q. However, we do not know the overall size of the bicelle, 

so it would be best to measure the intensity over a wide q-range to make sure we can characterize 

the overall size of the structures. To get a better idea of the required q-range, we can estimate the 

scattered intensity for the simplest case of non-interacting randomly oriented spherical particles

using the spherical form factor in SasView (https://www.sasview.org/), a community-developed 

software package for the analysis of small angle scattering data.
Fig. 9 compares the estimated scattered intensity from spheres with radii of 5 nm and 50 nm. 

These calculations highlight the scattering from the smaller spheres are at higher q, and the 

scattering from the larger spheres is at lower q. Given the two length scales of interest in bicelles

that we are studying here, i.e., the bicelle thickness and radius, the scattering form these features 

will likely appear in different regions of q. Also note that the incoherent background (Iinc) present 

at high q will have a greater effect on the smaller features, which is another reason why we decided 

to prepare our samples in D2O.

Figure 9 Estimated scattering from non-interacting spheres with an SLD of 0.67 x 1010 cm-2in 

D2O with different levels of incoherent background (Iinc). Note that Iinc primarily affects the data 

at high q. Also note that the larger spheres have a greater I0 and scatter more strongly at low q.

Therefore, for this experiment we know we will need to measure the intensity over a wide q-range 

since the information we are looking for is likely distributed in the low and high q-regime. 

Information on the q-range of each instrument is available on the NCNR’s website. If you are not 

sure which instrument is best for your experiment, you can always contact your friendly instrument 

scientists to ask! As long as you have some idea of the approximate structures and their sizes in

your systems, we can help design the best possible SANS experiment. 

3. Running the Experiment

After all the planning, there are a few more things to take into consideration when we start our 

SANS experiment. We will need to run ‘backgrounds’ to subtract from our sample data. We also 

need to figure out how long to run the measurement to ensure that the data are not too noisy and 

that we have good statistic when it comes time to analyze the data. Each of these topics is 

discussed below.
3.1.‘Backgrounds’ to run

We are ultimately interested in the scattering from our sample, but there are additional 

measurements we need to make during a SANS experiment to correct for the “background.” The 

neutrons counted by the detector come from 3 places: [a] neutrons scattered by the sample itself 

(and what we want to measure), [b] neutrons scattered by everything that is surrounding the sample 

as the beam passed through the sample, and [c] everything else such as stray neutrons that reach 

the detector without going through the sample and the electronic noise of the detector. To separate 

these contributions, we need to make 3 measurements during our SANS experiments:

1) Scattering from our samples, which will contain contributions from everything listed above 

(a, b and c), referred to as Isam in the next section

2) Scattering from the empty cell, which includes scattering from everything around the 

sample but is not from the sample itself as well as the stray neutrons and detector sensitivity 

(b and c), referred to as Iemp in the next section

3) Counts measured with a neutron absorber in the sample position, which we call the blocked 

beam and will account for the stray neutrons that are measured as well as the electronic 

dark count (c), referred to as Ibgd

We also need to measure the transmission of our samples as well as the empty sample cell to 

correctly subtract the background measurements listed above. Remember from Section 2.1 that 

transmission is wavelength dependent, so we may need to measure multiple transmissions 

depending on how we select our instrument configurations!

3.2. How long do we need to count?

Now that we know all the measurements we need to make to be able to correctly subtract the 

“background” from our data – how long will we need to measure?

A SANS experiment is an example of a counting experiment where the uncertainty in the measured 

intensity (the standard deviation, σ, to be exact) scales with the (square root of the) total number 

of counts I(t): σ = √I(t) . The longer we measure, the more total counts we accumulate and the 

lower the uncertainty. A very basic rule of thumb would be to try to accumulate ≈ 200,000 from 

your sample above the sample background where the sample background may be the solvent or 

simply the empty cell. If the 200,000 are circularly averaged into 50 data points when plotting I(q)

vs. q, then we will have about 4000 counts per data point. This averaging would mean that the 

standard deviation on a given point would be √4000 ≈ 60, or about 1.5%, which is more than 

enough in most cases. In practice, most low q measurements will accumulate much fewer counts 

and the highest q measurements will often collect more. It is important to remember that σ scale 

with √t, which means that if we needed to decrease our uncertainty by 2x from 1.5% to 0.75%, 

then we would need to increase out counting time by a factor of 4. Likewise cutting our time in 

half (in order to run more samples for example) will only increase the uncertainty by √2 or ~ 

1.4x from 1.5% to ~ 2%
Along the same lines, how long should the background and empty cell be counted relative to the 

sample measurement? The σ = √I(t) relationships means the optimum counting times are 

approximately

tbackground

tsample

= √

background count rate

sample count rate

[7]

If the sample scattering is weak, then the background should be counted for as long as the sample 

(but not longer!). But if the sample scattering is strong, say 4x that of the background, then we 

would need to measure the background for half the time of the sample.

4. After the experiment

After we have completed our measurements, we will need to process the raw data into a form 

that can be analyzed. In the next two sections we quickly review SANS data reduction and 

analysis.

4.1. Data reduction

What we mean by data reduction is to correct the measured scattering from the sample from the 

sources of background discussed in Section 3 as well as taking into consideration things such as 

the counting time and sample thickness to put the measured data on an absolute scale. The final 

data will, hopefully, have removed all the instrumental artifacts and just be the scattering cross 

section per unit volume which contains only the sample information of interest that we can analyze 

to answer the questions we set at the beginning of the experiment. 

The background corrected intensity, Icor, is calculated according to 

Icor = (Isam − Ibgd) − (

Tsam+cell

Tcell

) (Iemp − Ibgd)

[8]

And the corrected intensity is related to the differential cross section of our sample, dΣ(Q)⁄dΩ by

(

dΣ(Q)

dΩ

)

sam

=

Icor

K ds Tsam+cell

[9]

Where Ix denotes a measured scattering intensity, Tx a measured transmission, and ds the sample 

thickness. We already talked about the different measured intensities and transmissions in 
Section 3. K is an instrumental scale factor that is specific to the instrument set up we use to 

measure our desired q range:

K = φ A ΔΩ ε t [10]

Where:

φ = neutron flux at the sample (neutrons/cm2/s)

A = area of incident beam on the sample

ΔΩ = solid angle subtended by one pixel of the detector

ε = detector efficiency

t = counting time

Also note that the equation above is per pixel. The data reduction is performed on the 2D detector 

images, and the data are radially integrated to get I(q) vs. q after the reduction. To do this there is 

one other correction, shown in Fig 8, which corrects for the variation of each pixel’s efficiency 

from the mean efficiency, ε, described above. The so-called sensitivity file is collected by the 

NCNR staff and not something you would worry about other than to make sure you have been 

given the file before you start reducing your data.

At the NCNR we use macros written by Steve Kline in IGOR Pro to perform the data 

reduction.(23) Details on the reduction are reduction are provided in reference (23) and written 

documentation and video tutorials on the reduction software are also available on the NCNR’s 

website: https://www.ncnr.nist.gov/programs/sans/data/red_anal.html

Snapshots of the representative scattering data at the different reduction steps are also shown in 

Fig. 10 on the next page
Figure 10. Overview of the data reduction process with corresponding SANS data

Figure 8: Overview of the SANS data reduction process

corrected intensity:

COR = (SAM − BGD) −

TSAM

TEMP

(EMP − BGD)

correct for detector sensitivity:

CAL =

COR

DIV

calculate absolute intensity:

ABS =

CAL

K ds TSAM

and remove masked pixels from data

circularly average data

→ output .ABS text file

measured scattering files:

SAM: sample

EMP: empty cell

BGD: blocked beam
4.2.Data analysis

At this point, we have reduced data on an absolute intensity that we can fit with models with 

different form factors to determine the shape of the assemblies formed by mixtures of DMPC and 

DHPC and where the different lipids are in these structures. Since these data no longer contain any 

instrumental artifacts, they can be analyzed with any available small angle scattering data analysis 

software package, of which there are many (see for example 

http://smallangle.org/content/Software). Here at the summer school we will be using SasView to 

model the data. SasView is an analysis software package that is developed and managed by an 

international collaboration of scattering facilities. More information on SasView can be found 

here: https://www.sasview.org/.

5. Summary and Objectives

We have covered the basics of planning a SANS experiment, performing the measurements, 

reducing the data, and analyzing the data. During the summer school we will analyze data from 

two series of experiments to answer our questions about mixtures of DMPC and DHPC:

1) Determine the form factor of the structures formed by different mixtures of DMPC and 

DHPC in the dilute limit where S(q) → 1

2) Fit the data from a contrast variation experiment using d-DMPC and DHPC to determine 

where the different lipids are in the structures. 

6. Links for helpful tools and resources

The SANS Toolbox, by Boualem Hammouda:

https://www.ncnr.nist.gov/staff/hammouda/the_SANS_toolbox.pdf

Table of scattering lengths:

https://www.ncnr.nist.gov/resources/n-lengths/

SLD Calculator:

https://www.ncnr.nist.gov/resources/activation/

Video documentation on the NCNR’s data reduction by Steve Kline:

https://www.ncnr.nist.gov/programs/sans/data/movies/reduction_analysis_movies.html

SASView:

https://www.sasview.org/
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