**3. Full‐field transmission soft X‐ray spectromicroscopy**

In this paragraph the full‐field transmission spectromicroscopy technique will be shortly explained. In Section 3.1, some basic concepts will be introduced following references [16, 17]. Then two possible experimental setups, the scanning and the full‐field X‐ray transmission microscope will be compared [16]. Finally, a brief paragraph will describe the full‐field transmission microscope installed at the Mistral beamline (Alba Light Source, Spain) [18, 19].

#### **3.1. The interaction of X‐rays with matter**

Usually for X‐rays, the refraction index is written as:

$$n(\omega \mid = 1 - \delta(\omega \mid -i\beta(\omega \mid \tag{1}))) \tag{1}$$

while E = ℏ*ω* is the energy of the incident radiation. It is interesting to consider the asymptotic trend of *δ* and *β* for high energy (*E* far from absorption edges):

$$
\delta(E) \propto \frac{1}{E^2} \quad \beta(E) \propto \frac{1}{E^4} \tag{2}
$$

Let us consider now the propagation of a plane wave in the sample. Assuming **kr** = kr and using the *dispersion relation k*<sup>2</sup> = *ω*<sup>2</sup> *n*<sup>2</sup> /*c*<sup>2</sup> , one has:

$$\mathbf{E}\_0 e^{i(\omega t - k\gamma)} = \mathbf{E}\_0 e^{i[\omega t - \frac{\omega}{\gamma}(1 - \delta - \beta)\nu]} = \mathbf{E}\_0 e^{i(\omega t - k\gamma)} e^{i k\beta\nu} e^{-k\beta\nu} \tag{3}$$

where *k*<sup>0</sup> *= ω/c = 2π/λ* is the wave vector in vacuum. In Eq. (3), the first factor is the phase advance had the wave been propagating in vacuum; the second factor containing *δ* represents the modified phase shift due to the interaction with the medium; and the last factor containing *β* represents the decay of the wave amplitude in the medium due essentially to photoelectron absorption for X‐rays. Hence, the phase shift due to the interaction with the sample is determined by *δ*, while the attenuation by *β*. The *linear absorption coefficient μ* is defined as the inverse of the distance into the material for which the intensity related to the wave amplitude, Eq. (3), is diminished by a factor 1/*e*. Using Eq. (3):

$$
\mu(E) = \frac{4\pi}{\lambda} \beta(E) \tag{4}
$$

It is usually measured in μm‐1 and the corresponding characteristic distance is called the *atten‐ uation length*. *μ* is a rapidly increasing function of the atomic number and a rapidly decreasing function of the energy, taking into account Eq. (2):

$$
\mu(E) \propto \frac{1}{E^{\flat}} \tag{5}
$$

and at some particular energies called "absorption edges" it has peaks that correspond to the energies required to eject an electron from an internal core level to a final available electronic state. It is useful to introduce the *mass absorption coefficient*, defined as

$$
\mu\_w = \frac{\mu}{\rho} \tag{6}
$$

with *ρ* being the mass density, *<sup>ρ</sup>* <sup>=</sup> *ma na* (*ma* is the atomic mass, *na* is the density of the atoms).

To properly account for the transitions corresponding to the absorption edges, the use of quantum mechanics is needed. Considering a well‐defined initial core state and using the *Fermi golden rule* in the *dipole approximation* one can write *μ*m as:

Studies of Lithium-Oxygen Battery Electrodes by Energy-Dependent Full-Field Transmission Soft X-Ray Microscopy http://dx.doi.org/10.5772/66978 99

$$
\mu\_n(E\_\cdot) \propto \left| \left< \psi\_i \middle| \middle| \hat{\varepsilon} \cdot r \middle| \psi\_j \right> \right|^2 \rho\_f(E\_\cdot) \tag{7}
$$

where *ψ<sup>i</sup>* and *ψ<sup>f</sup>* are the initial and final single electron states, *E*<sup>i</sup> and *E*<sup>f</sup> are the corresponding energies and *E = E*<sup>f</sup>  *– E*<sup>i</sup> is the energy of the incident photons, *ρ<sup>f</sup>* the density of final state, and ^ *ε* the polarization of the electric field. The interaction between the atom and the electromagnetic field, classically pictured by the wave in Eq. (3), removes an X‐ray photon whose energy is used to promote an electron from the initial core state *ψ<sup>i</sup>* to the final state *ψ<sup>f</sup>* . The absorption is modulated by the density of final state which has peaks in correspondence of the absorption edges. The dipole matrix element, almost flat with the energy, contains the angular momentum selection rules for dipole transitions (l = orbital angular momentum, *s* = spin, *j* = total angular momentum, *m* = *z*‐component of the total angular momentum):

$$
\Delta 1 = \pm 1, \quad \Delta \text{s} = 0, \quad \Delta \text{j} = \pm 1, 0, \quad \Delta m = 0 \tag{8}
$$

and the dependence on the direction of the photon polarization **^***ε* **.**

**3.1. The interaction of X‐rays with matter**

*<sup>δ</sup>*(*<sup>E</sup>* ) <sup>∝</sup> \_\_<sup>1</sup>

**E**<sup>0</sup> *ei*(*ωt*−*kr*) = **E**<sup>0</sup> *ei*[*ωt*−\_\_

Eq. (3), is diminished by a factor 1/*e*. Using Eq. (3):

*μ*(*E* ) = \_\_\_ <sup>4</sup>*<sup>π</sup>*

function of the energy, taking into account Eq. (2):

*<sup>μ</sup>*(*<sup>E</sup>* ) <sup>∝</sup> \_\_<sup>1</sup>

*<sup>μ</sup><sup>m</sup>* <sup>=</sup> *<sup>μ</sup>*\_\_

with *ρ* being the mass density, *<sup>ρ</sup>* <sup>=</sup> *ma na*

state. It is useful to introduce the *mass absorption coefficient*, defined as

*Fermi golden rule* in the *dipole approximation* one can write *μ*m as:

(*ma*

using the *dispersion relation k*<sup>2</sup>

It is usually measured in μm‐1

Usually for X‐rays, the refraction index is written as:

trend of *δ* and *β* for high energy (*E* far from absorption edges):

98 X-ray Characterization of Nanostructured Energy Materials by Synchrotron Radiation

 = *ω*<sup>2</sup> *n*<sup>2</sup> /*c*<sup>2</sup>

*n*(*ω* ) = 1 − *δ*(*ω* ) −*iβ*(*ω* ) (1)

while E = ℏ*ω* is the energy of the incident radiation. It is interesting to consider the asymptotic

Let us consider now the propagation of a plane wave in the sample. Assuming **kr** = kr and

where *k*<sup>0</sup> *= ω/c = 2π/λ* is the wave vector in vacuum. In Eq. (3), the first factor is the phase advance had the wave been propagating in vacuum; the second factor containing *δ* represents the modified phase shift due to the interaction with the medium; and the last factor containing *β* represents the decay of the wave amplitude in the medium due essentially to photoelectron absorption for X‐rays. Hence, the phase shift due to the interaction with the sample is determined by *δ*, while the attenuation by *β*. The *linear absorption coefficient μ* is defined as the inverse of the distance into the material for which the intensity related to the wave amplitude,

*uation length*. *μ* is a rapidly increasing function of the atomic number and a rapidly decreasing

and at some particular energies called "absorption edges" it has peaks that correspond to the energies required to eject an electron from an internal core level to a final available electronic

To properly account for the transitions corresponding to the absorption edges, the use of quantum mechanics is needed. Considering a well‐defined initial core state and using the

is the atomic mass, *na*

, one has:

*ω*

*<sup>E</sup>*<sup>2</sup> *<sup>β</sup>*(*<sup>E</sup>* ) <sup>∝</sup> \_\_<sup>1</sup>

*<sup>c</sup>* (1−*δ*−*iβ*)*r*] = **E**<sup>0</sup> *ei*(*ωt*−*k*<sup>0</sup>

*<sup>r</sup>*) *eik*<sup>0</sup> *<sup>δ</sup><sup>r</sup> e* <sup>−</sup>*k*<sup>0</sup>

and the corresponding characteristic distance is called the *atten‐*

*<sup>E</sup>*<sup>4</sup> (2)

*<sup>λ</sup> β*(*E* ) (4)

*<sup>E</sup>*<sup>3</sup> (5)

*<sup>ρ</sup>* (6)

is the density of the atoms).

*<sup>β</sup><sup>r</sup>* (3)

X‐ray absorption imaging technique consists in detecting the photons transmitted through the observed object. Experimentally, the number of photons *N* after the transmission through the sample, along *z*, obeys to the *Beer‐Lambert's law*:

$$\mathcal{N}(\mathbf{x}, y) = \mathcal{N}\_0(\mathbf{x}, y) \exp\left[-\int\_{\text{sample}} \mu(\mathbf{x}, y, z) \, dz\right] \tag{9}$$

where *N* and *N*<sup>0</sup> are the emerging and incident number of photons, respectively, and the integral is extended through all the sample thickness. The measured transmission *T = I/I*<sup>0</sup> (with *I* and *I* 0 proportional to *N* and *N*<sup>0</sup> , respectively) depends exponentially on the linear absorption coefficient *μ*, integrated along the X‐ray path in the sample (**Figure 1**).

In a transmission microscope based on X‐ray photons, the contrast will therefore depend on the sample thickness, the elements by which it is composed, their density, and the energy and polarization of the incident radiation. Assuming not oriented samples (so that we neglect the polarization) and *μ* constant along *z* in a thickness *t,* we can write:

$$
\mu \langle \mathbf{x}, y, \mathbf{E} \rangle \, t = \mu\_n \langle \mathbf{x}, y, \mathbf{E} \rangle \, \rho t = -\ln \left( \frac{I}{I\_0} \right) \tag{10}
$$

This product is proportional to the *absorbance A*, which is defined as *A* = *‐*log10*(I/I*<sup>0</sup> ), and is usually considered in the place of the transmission because of its additivity:

$$
\mu t = \sum \mu\_i t\_i \tag{11}
$$

**Figure 1.** Optical absorption and the Beer‐Lambert's law.

which, for the particular case of *i* chemical species in a thickness *t*, becomes:

$$
\mu t = t \sum\_{\mathbb{T}} \mu\_{u,t} \rho\_i \tag{12}
$$

As a function of the photon energy parts of the images will then suddenly become darker (or brighter if we use the absorbance) when the radiation is triggering some electronic transitions allowed by Eqs. (7) and (8). As the exact energy is also dependent on the atom environment, it will also be possible to detect chemical states of the same element. This chemical information is available with spatial resolution down to few tens of nanometers in a synchrotron‐based transmission X‐ray microscope and the corresponding technique is called transmission X‐ray spectromicroscopy. By using X‐rays of the "soft" energy region (<3 KeV), it is possible to access transitions from core levels of light elements, among them the K‐edge of nitrogen, oxygen, fluorine, as well as L – and M ‐ edges of other elements (**Figure 2**).


**Figure 2.** Accessible absorption edges with a soft X‐ray radiation of 300–800 eV in the periodic table.

#### **3.2. Zone plate‐based X‐ray transmission microscopes**

Using the "radiography setup" depicted in **Figure 1**, the spatial resolution would be limited by the detector pixel size to few microns. In lens‐based microscopy, this limit is imposed by the lenses. In the following we will briefly describe two examples of lens‐based microscope geometries used at synchrotron radiation sources: the full‐field transmission X‐ray microscope (TXM) and the scanning transmission X‐ray microscope (STXM), both represented in **Figure 3**.

**Figure 3.** Scheme of the two common transmission X‐ray microscopes: (a) Full‐field microscope, in which a full sample image is formed on the detector; (b) Scanning microscope, in which the sample is scanned in the focal spot of the incoming beam.

Both of them are lens microscopes based on the application of Fresnel Zone Plate diffractive lenses (ZP). A ZP lens works like a circular diffraction grating whose period radially decreases from the center in such a way that all the waves of the same diffracted order are redirected to the same point (**Figure 4**). The expression for the focal length can be calculated as:

**Figure 4.** Conceptual scheme of a Fresnel Zone Plate lens.

which, for the particular case of *i* chemical species in a thickness *t*, becomes:

100 X-ray Characterization of Nanostructured Energy Materials by Synchrotron Radiation

*i*

As a function of the photon energy parts of the images will then suddenly become darker (or brighter if we use the absorbance) when the radiation is triggering some electronic transitions allowed by Eqs. (7) and (8). As the exact energy is also dependent on the atom environment, it will also be possible to detect chemical states of the same element. This chemical information is available with spatial resolution down to few tens of nanometers in a synchrotron‐based transmission X‐ray microscope and the corresponding technique is called transmission X‐ray spectromicroscopy. By using X‐rays of the "soft" energy region (<3 KeV), it is possible to access transitions from core levels of light elements, among them the K‐edge of nitrogen, oxygen, fluorine, as well as L – and M ‐ edges of other elements

Using the "radiography setup" depicted in **Figure 1**, the spatial resolution would be limited by the detector pixel size to few microns. In lens‐based microscopy, this limit is imposed by the lenses. In the following we will briefly describe two examples of lens‐based microscope geometries used at synchrotron radiation sources: the full‐field transmission X‐ray microscope (TXM) and the scanning transmission X‐ray microscope (STXM), both represented in

**Figure 2.** Accessible absorption edges with a soft X‐ray radiation of 300–800 eV in the periodic table.

*μ<sup>m</sup>*,*<sup>i</sup> ρ<sup>i</sup>* (12)

*μt* = *t*∑

**3.2. Zone plate‐based X‐ray transmission microscopes**

(**Figure 2**).

**Figure 3**.

Present technology allows the fabrication of ZP capable to produce focal spots down to few tens of nanometers (≥15 nm) [20]. Spatial resolution (*r*<sup>s</sup> ) and depth of focus (DoF) both depend on *λ* and the *numerical aperture* (NA) of the lens which is, in the case of ZP lenses, related to the dimension of the outermost zone width ∆*r*:

$$r\_s = \frac{0.61\lambda}{NA} = 1.22\lambda r \quad DoF = \pm \frac{1}{2} \frac{\lambda}{(NA)^2} = \pm \frac{2\left(\lambda r\right)^2}{\lambda} \tag{14}$$

In the first equation of (14) the spatial resolution is defined following the *Rayleigh* criterion. It is interesting to note that a better spatial resolution will imply also a smaller DoF.

The nanofabrication of a ZP is typically realized on a thin silicon nitride membrane and involves high technology processes such as electron beam lithography. In general, the choice of the material for the ZP's opaque zones determines the efficiency depending on the energy range. For the TXM case we have the typical scheme of a common visible light microscope: a sample is placed at the focal plane of two lenses. The first lens is usually called "condenser" and it focalizes the beam on to the sample plane, while the second lens is called "objective lens" and it produces a full magnified image of the sample on to the detector plane. The objective lens is a ZP lens which works as a *thin lens* between the sample and the detector plane so that the magnification *M* is:

$$M = \frac{p}{q} \tag{15}$$

With *p* being the distance ZP‐detector and *q* being the distance ZP‐sample as indicated in **Figure 5**.

**Figure 5.** Mistral beamline layout. In the inset, a scheme of the TXM optics setup is reported.

The spatial resolution is limited by the NA of the objective lens, i.e., by the external zone width of the ZP lens as indicated in Eq. (14). This limit is reached if the microscope operates at *M* big enough to neglect the dimension of the pixels size of the detector (typically a CCD‐ based detector) and if the NA of the objective lens is filled by the beam emerging from the sample. Also, the ZP lens is strongly chromatic as indicated by Eq. (13), so that also a minimal requirement on the energy resolution of the incident beam has to be satisfied: ∆*λ*/*λ* ≤ 1/*N*, with *N* being the number of zones.

In a STXM system, the sample is placed in the focal spot of a ZP lenses. If illuminated with the proper coherent beam the ZP will produce a focal spot (more precisely an *airy pattern*) with resolution set again by the first equation of (14). In this case, higher order diffracted beams will be stopped using an order sorting aperture (OSA) as indicated in **Figure 3**. The sample is scanned through the ZP focal spot and the full image of the sample is reconstructed electronically, step by step. The spatial resolution will be limited by the dimension of the focal spot produced by the ZP on the sample. **Table 1** summarizes the main properties of the two microscopes.


**Table 1.** Comparison between the main properties of full‐field and scanning transmission electron microscopy.

The main advantage of the STXM over the TXM system is in terms of radiation dose: the lens is before the sample and then all the photons emerging from the sample are used to produce the image. Instead in a TXM system, the ZP is after the sample: due to its limited efficiency, most of the photons arising from the sample will not contribute to the formation of the image, therefore to obtain the same image quality of a STXM, the TXM will require more radiation dose absorbed by the sample. The main advantage of the TXM over the STXM system is in terms of the exposure time: in one "snapshot" the full image of the sample is obtained.

#### **3.3. The MISTRAL microscope**

**Figure 5.** Mistral beamline layout. In the inset, a scheme of the TXM optics setup is reported.

on *λ* and the *numerical aperture* (NA) of the lens which is, in the case of ZP lenses, related to

In the first equation of (14) the spatial resolution is defined following the *Rayleigh* criterion.

The nanofabrication of a ZP is typically realized on a thin silicon nitride membrane and involves high technology processes such as electron beam lithography. In general, the choice of the material for the ZP's opaque zones determines the efficiency depending on the energy range. For the TXM case we have the typical scheme of a common visible light microscope: a sample is placed at the focal plane of two lenses. The first lens is usually called "condenser" and it focalizes the beam on to the sample plane, while the second lens is called "objective lens" and it produces a full magnified image of the sample on to the detector plane. The objective lens is a ZP lens which works as a *thin lens* between the sample and the detector plane so

With *p* being the distance ZP‐detector and *q* being the distance ZP‐sample as indicated in

The spatial resolution is limited by the NA of the objective lens, i.e., by the external zone width of the ZP lens as indicated in Eq. (14). This limit is reached if the microscope operates at *M* big enough to neglect the dimension of the pixels size of the detector (typically a CCD‐ based detector) and if the NA of the objective lens is filled by the beam emerging from the sample. Also, the ZP lens is strongly chromatic as indicated by Eq. (13), so that also a minimal requirement on the energy resolution of the incident beam has to be satisfied: ∆*λ*/*λ* ≤ 1/*N*, with

In a STXM system, the sample is placed in the focal spot of a ZP lenses. If illuminated with the proper coherent beam the ZP will produce a focal spot (more precisely an *airy pattern*) with

2 \_\_\_\_\_ *λ* (*NA* )2 <sup>=</sup> <sup>±</sup>

2 (*Δr* )2 \_\_\_\_\_

*<sup>q</sup>* (15)

*<sup>λ</sup>* (14)

*NA* <sup>=</sup> 1.22*Δ<sup>r</sup> DoF* <sup>=</sup> ±\_\_<sup>1</sup>

It is interesting to note that a better spatial resolution will imply also a smaller DoF.

the dimension of the outermost zone width ∆*r*:

102 X-ray Characterization of Nanostructured Energy Materials by Synchrotron Radiation

*<sup>M</sup>* <sup>=</sup> *<sup>p</sup>*\_\_

*rs* <sup>=</sup> \_\_\_\_\_ 0.61*<sup>λ</sup>*

that the magnification *M* is:

*N* being the number of zones.

**Figure 5**.

An example of a state‐of‐the‐art soft X‐ray full‐field transmission microscope is the one installed at the Mistral beamline of the ALBA Light source. A picture of the beamline (BL) is reported in **Figure 5**. The BL is devoted to transmission cryotomography of biological cells in the water window energy range (284.2–543.1 eV [21]). It is useful to distinguish between the BL optical elements and the TXM which is just the end station of the BL, as indicated in the picture. Presently the TXM can works in the energy range (ER) between the C–K edge and the Ni L edge excluded, i.e., 290–850 eV. All the optical elements before the TXM work to prepare the beam for the condenser lens of the microscope and are in ultra‐high vacuum chambers (working *p* ~1 × 10‐9 mbar) to minimize absorption and scattering of the X‐ray beam. Moreover, the full beamline from the source to the CCD detector chip is windowless. In the case of Mistral the condenser lens is a glass capillary which work like a single reflection elliptical mirror. It is characterized by a good efficiency (50–75% in the ER) and achromaticity: its focal length of about 1 cm is energy independent. The 2 × 2 μm2 condenser focal spot is wobbled to cover a field of view of typically 10 × 10–16 × 16 μm2 on the sample plane. Two ZP are available as objective lens: 25 and 40 nm ZP. The first one is used typically for experiments in which one wants to maximize the 2D spatial resolution and the depth of focus is not a critical parameter. More technical details on the MISTRAL beamline are reported in references [18, 19].
