1 

Absract  

 

 

In this work we wanted to underline the importance of Thermal Desorption Spectroscopy and 

its applications to several branches of Physics. 

Temperature-programmed desorption techniques (TPD) are important to determinate kinetic 

and thermodynamic parameters of desorption processes and decomposition reactions. 

Knowledge of the nature of the desorption process is fundamental to understand the nature of 

the elementary chemical processes of adsorbates, as the energetics of bonding, the 

specification of the chemical nature of the bound species and the nature and magnitude of 

interactional effect between adsorbed species. 

We focused our attention on the applications of Thermal Desorption Spectroscopy (TDS) to 

High-Energies Physics, Astrophysics and Geophysics; in fact this technique was used, 

respectively,  to investigate the molecular hydrogen adsorption on carbon nanotubes, the 

effects of electron bombardment on ammonia and methane ices and changes of zoisite mineral 

after heating. 

The molecular hydrogen adsorption on carbon nanotubes was studied to find a possible 

solution to vacuum system problems of Large Hadron Collider (LHC); in fact, the circular 

path of photon beams produces synchrotron radiation which deteriorates LHC vacuum 

desorbing gas molecules  from the ring walls. Among the desorbed species the most 

problematic to pump out is H2. Since LHC elements operate at low temperatures, a possible 

solution to vacuum problem is the installation of cryosorbent materials on the LHC walls. In 

this work we study the possibility to use carbon nanotubes as criosorbers in future 

accelerators. Our sample, furnished by Prof. Nagy group of Chemical Engineering 

Department of Calabria University, is constituted by MWNTs synthesized by chemical vapor 

deposition using C2H4 and subsequently purified. Our investigations confirm that the carbon 

nanotubes have a great adsorption capacity also at low temperatures both for H2 and noble 

gases as Kr; then we observed that H2 adsorption on CNT is described by a first kinetic-order, 

while Kr adsorption is characterized by a zero kinetic-order. By means of TDS we calculate 

the activation energy for H2 adsorption on carbon nanotubes and we found a value of about 

3KJ/mol, perfectly coherent with theoretic one. Moreover, from a comparison between 

nanotubes and other carbon-based material (as charcoal), we noted that adsorption efficiency 

for CNT is almost an order of magnitude higher then charcoal. So carbon nanotubes are good 

candidates to cryosorbers in future accelerators. 



 2 

As Thermal Desorption Spectroscopy application to Astrophysics we studied the effect of 

electron bombardment on ammonia and methane ices. The interstellar medium is composed 

for 99% by gas; molecules, atoms and radicals at gas state condense on dust grains surface of 

molecular clouds (at 10 K) creating an icy mantle with a thickness of  0.1 μm. The presence 

of ices is confirmed by IR spectroscopy of obscured stellar sources and in interstellar grains 

are localized solid mixture containing H2O, CO, CH4 and NH3. In these environments ices are 

subjected to chemical and physical processes, specifically to bombardment of photons and 

cosmic rays, with the consequent synthesis of new organic species 

In this work we conducted an investigation of the chemical processing of ammonia and 

methane ices subjected to energetic electrons. By Thermal Desorption Spectroscopy we verify 

the production of new organic species, after energetic irradiation in interstellar ices, as 

diazene (N2H2),  ethane (C2H6) and acetylene (C2H2).  

Finally, in Geophysics and Petrology Thermal Desorption Spectroscopy can be used to study 

minerals chemical composition. Our interest was focused on  zoisite and the sample 

investigated was furnished by prof. Ajò from “Institute of  Inorganic Chemistry and Surfaces” 

of CNR, in Padova. In this work we used TDS to investigate zoisite behaviour during heating 

form room temperature to 650
o
C and to understand if its modification into tanzanite variety 

after heating is due to structural changes or to a dehydration mechanism. 

 

 

 

 

 



 3 

Chapter 1 
 

 

Thermal desorption spectroscopy 

 

 
1.1 Adsorption and Desorption fundamental principles 

 

Temperature-programmed desorption techniques (TPD) are important to determinate kinetic 

and thermodynamic parameters of desorption processes and decomposition reactions. 

Knowledge of the distinctive features is fundamental to understand the nature of the 

elementary chemical processes of adsorbates, as the energetics of bonding, the specification 

of the chemical nature of the bound species and the nature and magnitude of interactional 

effect between adsorbed species[1].  

Adsorption takes place when attractive interaction  between a particle and a surface is strong 

enough to overcome thermal disorder effects. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1.1.  Adsorption potential diagram for a diatomic molecule X 2 interacting with a substrate surface along the 

direction z. Physisorption potential has been described through the  Lennard-Jones potential, while 

chemisorption has been represented by a Morse potential 

 



 4 

The weakest form of adsorption to a solid surface is called “physisorption”. It takes place if 

the interaction is mainly due to the ubiquitous van der Waals interaction and typical binding 

energies are lower than 50kJ/mol. On the contrary, if gas particles that strike the surface bind 

to it through the formation of a surface chemical bond “chemisorption” occurs and typical 

binding energies are higher than 50 kJ/mol and comparable to a fraction of the substrate 

sublimation energy. Moreover we should remark that chemisorption is an activated process, 

i.e. the formation of a chemisorptive bond requires the overcoming of an activation barrier. 

 

 

 

1.2 Adsorption and desorption kinetics 

1.2.1 Adsorption isotherms 

 

Theories from which arise studies about adsorption and desorption phenomena are essentially 

based on the following principles: 

 Adsorption is a localized phenomenon, i.e. absorbed particles are motionless 

 The substrate surface is saturated when all its adsorption sites are occupied, so a 

monolayer is formed : Θ = 1ML,  

with 

                                                               surfads NN ,                                                (1.1) 

where Nads is the adsorption particles number , while Nsurf  represents the number of 

adsorption sites on surface. 

 There is not interaction between adsorbed particles. 

 

Adsorption is usually described through isotherms, that is, the amount of adsorbate on the 

adsorbent as a function of its pressure at constant temperature. The most important adsorption 

isotherm in the analysis of adsorption/desorption phenomena is the Langmuir isotherm, 

described through the following set of equations: 

                                                                

   

  n

ndes

n

nad

Br

pAr



 1
,                                            (1.2) 

 

with An and Bn  constants, n=1,2 and rad/des  the adsorption/desorption rate. 

From the dynamic equilibrium condition, desad rr  , it follows that: 



 5 

                                                                  
 

  n
n

n
n

n

pb

pb
1

1

1
 .                                                (1.3) 

 

For n=1 it is possible to observe an adsorption/desorption first order which corresponds to a 

sort of condensation process of adsorbed molecules on the surface, while for n=2 there is a 

second order process, which corresponds to a dissociative adsorption and to a recombining 

desorption for diatomic molecules. Moreover molecules dissociation can be reduced 

decreasing surface temperature. 

Other common desorption isotherms are the Freundlich (4) and Brunauer, Emmett e Teller 

isotherms (BET) 

                                                      p n  ,    with 0.2<β<1.0                                    (1.4) 

 

                                                   
  0maxmax0

11

p

p

C

C

Cpp

p





 
,                                (1.5) 

where 

p0 = vapour pressure of purely liquid phase at fixed  temperature 

 

C = system specific constant 

 

      γ = adsorption molality,                        LAads NmN                                              (1.6) 

 

γmax = the maximum molality that can be achieved whit a monoatomic layer 

 

mA = mass of the adsorbed particle. 

 

The Bet isotherm extends the assumptions of Langmuir isotherm to multiple layers. It is 

hypothesized that only the first monoatomic layer is bonded to substrate surface and that, 

consequently, all the adsorbed layers are characterized by the same bonds that correspond to 

adsorbed molecules in liquid phase. This model describes chains of not interacting adsorbed 

molecules that grow perpendicularly to surface, so it is assumed that each molecule of the 

chain has coordination number of 2. For multiple coverages results achieved from BET 

isotherm are very different from experimental isotherms [2]. 

 

 



 6 

1.3 Polanyi-Wigner equation 

 

Adsorption of gaseous molecules on a solid surface, with pressure and temperature fixed, is 

governed by the equality of the chemical potential of the adsorbed layers and of the gas 

μad(N) = μgas(P,T), 

where N is the number of molecules adsorbed on surface, characterized by a total number of 

adsorption sites of N0. 

The chemical potential of the adsorbed species is usually calculated through different 

techniques (ex. density functional method), while μgas(P,T) can be obtained by thermal 

dynamics consideration. 

For a gas the Gibbs free Energy is defined as 

 

                                                               G= H- TS = U + PV-TS,                                        (1.7) 

 

where H ,S and U are respectively the enthalpy, the entropy and the energy of the system. 

For isothermal adsorption at a temperature Tad  dG= VdP, so 

 

                                                                     

G

G

P

P

VdPdG
0 0

 ,                                                  (1.8) 

from which 

                                                                    

P

P

VdPGG
0

0 .                                                (1.9) 

The chemical potential is defined as the molar Gibbs free energy μ=dG/dn, so 

                                                                    0

0

P

P

V
dP

n
    .                                             (1.10) 

For an ideal gas 

                                                                   
0

0 ln
P

P
RTad  ,                                        (1.11) 

 

where μ0 is the chemical potential at the reference pressure P0. 

If the gas is not ideal is necessary to calculate the integral in equation (1.10) using the 

equation of the state. For example, for  the van der Waals interaction, V can be expressed as a 

series of P: 



 7 

                                                             )
1

( cPb
P

nRTV  ,                                            (1.12) 

where the virial coefficients b and c are temperature dependent and the values can be found in 

literature. 

The desorption  rate is usually expressed through a law of n
th
 (with n = 0,1,2): 

                                                              n

ndes k
dt

d
r 


                                               (1.13) 

 

If  the rate constant kn is described by Arrhenius equation 

 

                                                              











 


RT

E
k

PW

des
nn exp            ,                             (1.14)                                                             

with νn = desorption frequency, equation (1.13) becomes: 

                                                        n
PW

des
ndes

RT

E

dt

d
r 












 



 exp  ,                         (1.15)    

that define the activation Energy of desorption PW

desE  and it is important in the interpretation 

of the experimental spectra. 

It is useful to observe the differences between the three kinetic orders: 

     

   i)Zero kinetic order 0 (n=0)  

 Desorption rate is not dependent on coverage. 

  Desorption exponentially increases with T. 

 When dose increases the desorption peak moves to higher temperatures (fig.1.2). 

 

 

Fig. 1.2. Representation of a zero-order kinetic 



 8 

       ii) First kinetic order (n=1)  

 Desorption rate is directly proportional to coverage. 

 When coverage increases the temperature at which the maximum of the peak 

occurs does not vary (fig.1.3). 

 The peak is asymmetrical.  

 

 

Fig. 1.3. First-order kinetic 

 

       iii) Second kinetic order (n=2) 

 Desorption rate varies with the square of the coverage. 

 The temperature at which maximum of the peak occurs increases with decreasing 

of the coverage (fig.1.4).  

 The peak is symmetrical. 

 

 

Fig. 1.4. Second-order kinetic 



 9 

Zero-order kinetics are often indicative of desorption from a multilayer where the rate of  

desorption is independent on surface coverage; first-order kinetics may be indicative of the 

presence of a single surface species; second-order kinetics are an indication of adsorbate atom 

recombination process  leading to the production of a diatomic molecules. 

A fundamental condition to determinate a thorough set of data is that the observed desorption 

signal is proportional to desorption rate; each step of the desorption process has to take place 

more slowly than any secondary reaction involved (rate-limiting). 

In equation (1.15) it is possible to note that the activation parameters depend generally from 

the coverage, but often it is possible a dependence from temperature, in particular a 

dependence from the heating rate  β, an experimental parameter. So it is reasonable to think 

that thermal desorption technique gives information about nature of adsorbed molecules in 

conditions very distant from  equilibrium and at temperatures lower than desorption 

temperature; as a consequence the Polany-Wigner equation can result insufficient for analysis 

and it is possible to couple with it equations based on simulation of desorption spectra  with 

complex statistic models to describe the interactions of adsorbates [2]. 

 

 

 

1.3.1 Material balance equation for Thermal Desorption in a vacuum system  

 

The basic equations for the measurement of thermal desorption spectra are given below [1]. 

The desorption rate (Rd) of a species from a surface should be a function of  the surface 

concentration N and of the temperature T of the species. The Arrhenius form is 

                                                            
 kTEnn

d
deNR

dt

dN 
 )(

0 ,                                  (1.16) 

where )(

0

n  is the pre-exponential factor, n the desorption order (usually n=0,1,2) and Ed the 

activation energy for desorption. It is often assumed as a first approximation that )(

0

n  and Ed 

are constant, but it has been shown that in many cases these kinetic parameters vary with  

coverage N. 

If the pumping speed of the system (s) is constant, if the pressure rise P above the base 

pressure P0 is used and if no adsorption/desorption processes occur on extraneous surface, the 

Redhead equation is [6] 

                                                             
 

V

Ps

dt

dN
A

V

kT

dt

Pd 












                                (1.17) 



 10 

= (rate of the gas evolution) –(incremental pumping rate above steady state background rate). 

 

Thus, 

                                                              
AkT

Ps

dt

Pd

AkT

V
Rd







)(
,                                     (1.18) 

which may be rewritten as 

                                                              


P

dt

Pd

V

AkT
Rd







)(
,                                       (1.19) 

 

where η = V/s is the characteristic pumping time of the system. 

Defining a characteristic time δt  for the desorption of a binding state, two limits exist: 

 

 for η >> δt the Eq.1.19 becomes 

 

                                                                     
dt

Pd

AkT

V
Rd

)(
 ;                                          (1.20) 

 for η << δt 

                                                                        


P

AkT

V
Rd


 .                                            (1.21) 

 

For most situations involving vacuum system with fast pumping speed (η0.25s) and δt 2s, 

Eq. (1.21) is applied conveniently and the pressure rise P is a direct measure of the 

desorption rate Rd. 

For intermediate cases it is convenient to use both terms in Eq. (1.19). 

In a glass or metal ultrahigh-vacuum system it is possible to measure η by rapidly opening or 

closing the leak valve for the gas admission or by producing a pulse of gas by rapid 

desorption from a filament and, subsequently, following the decay of pressure with time. It is 

often found that η is not constant. This problem may be due to wall effect or to re-emission of 

gas from ion pumps or other element within the system. 

Redhead [7] proposed a method to determinate that wall effects are absent in 

adsorption/desorption experiments under flow conditions. The pumping speed of a gas under 

molecular flow conditions (with no wall effects) is directly proportional to the molecular 

mean velocity and, for a Boltzmann distribution, it is therefore proportional to (mT)
-1/2

, where 

m is the molecular mass.  

 



 11 

1.4 Theories of Thermal desorption  

 

1.4.1 The Mobile Precursor Model in Adsorption and Desorption 

 

Several studies about chemisorption on surfaces have shown that the sticking coefficient 

remains nearly constant (and often near unity) over a wide coverage range. This observation 

has led to the postulate that in chemisorption is present a precursor state , that may sample 

both filled and empty adsorption sites  as it migrate, until finally becoming chemisorbed [1]. 

King [3] was the first to propose the passage through a mobile precursor state also in thermal 

desorption and he computed the influence that this process might have on the shape of thermal 

desorption spectra. 

Gorte and Schmidt [4] have formulated a desorption kinetic model involving transition 

through a precursor state, illustrated below for first-order desorption kinetics. 

For an adsorbed molecule A(s), the elementary reaction describing desorption process is 

 

 

                                                                                                                              (1.22)  

 

where the k’s represent first-order rate constants for the elementary steps. 

If the precursor state A
*
 is in equilibrium with A(s) at coverage θs and it is at a low steady-

state concentration during desorption, is it possible to write 

 

                                            01 ***

* 
ASAasdA

kkkdtd                  (1.23) 

and 

                                           SaSdAS kkkkkdtd   1***
*  .             (1.24) 

 

For k
*
>>ka the first-order desorption kinetics is obtained 

 

                                                                  sdS kdtd   ;                                           (1.25) 

while for k
*
<<ka  we have 

 

                                                          SaSdS kkkdtd   1*
                              (1.26) 

kd 

A(g) A
*
+ site A(s) 

ka 

k
* 



 12 

Using the rate expression, the equation (1.20) becomes 

 

                                        RTEEE

SSadS
ade dtd




*

1*                (1.27) 

 

The activation energy term in the exponential factor is just the total barrier height for the 

process going from A(s) to A(g), i.e. the heat of adsorption.  

Fig. 1.5 show a comparison between the theoretically calculated first-order temperature-

programmed desorption behaviour for a mobile precursor model and the normal first order 

desorption. 

 

 

Fig. 1.5. (a)A normal first order desorption spectra, calculated assuming Ed=26 kcal mole-1, ν0
(1)=1013 s-1 and 

β=100 K/s; (b)limiting case for single precursor. 

  

 

Is it possible to note that the θ/(1-θ) factor in equation (1.27) causes a significant broadening 

of the desorption peak and a shift to lower temperature. 

Thus if a mobile precursor is involved in desorption, estimates of pre-exponential factors and 

desorption energies from the peak shapes will be inexact, because of the dependence of these 

parameters on peak shape. 

Dissociative adsorption, followed by  second-order desorption via a mobile precursor, will 

introduce a  θ
2
/(1-θ

2
) term into Eq. (1.27), with a consequent desorption peak broadening and 

shifting to lower temperature. 

So the mobility of the activated complex is an important factor to consider in some kinetic 

measurements of thermal desorption.   

The absence of a precursor in condensation at low temperature implies the absence of a 

precursor expression  for desorption and the presence of a precursor in condensation suggests 

that these expressions might be anticipated in desorption [4]. 



 13 

1.4.2 Statistical Thermodynamics of Adsorption and Desorption 

 

The Eyring theory of reaction rates [5] postulates that the reaction (adsorption or desorption) 

proceeds via an energetically activated state (or complex) that is intermediate in structure 

between the reactants and the products in the process under consideration. It exist at the top of 

a potential energy barrier whose height is the activation energy  for the reaction process. 

Passage over the energy barrier occurs by motion along a path called the reaction coordinate, 

that describe the molecular configuration of the reactants and the products. The activated 

complex at the top of the barrier exists in low concentration in equilibrium with the reactants. 

For adsorption on a uniform surface, let N
*
 be the equilibrium concentration of activated 

complexes (molecules cm
-2

), NS the number of adsorption sites per square centimeter and Ng 

the number of gas-phase molecules per cubic centimeter. 

In an adsorption process in which a molecule is adsorbed without dissociation on the surface, 

the equilibrium constant is: 

                                                   sgsg fffNNNeqK *'** )(   ,                    (1.28) 

 

 where the three term f above are the complete partition functions for the species or for the site 

(
 





i

kT

i
iegf


 and gi is the degeneracy of the quantum state of energy εi). 

From Eq.(1.28) the activated complexes concentration is 

 

                                                            sgsg ffNNfN *'*  .                                (1.29) 

 

The rate of the reaction is equal to the product of the concentration of activated complexes N
*
 

and the frequency of crossing of the barrier. Let us assume that the activated complex exists in 

a region of length δ along the reaction coordinate at the top of the barrier. Using the Maxwell-

Boltzmann statistics is it possible to determinate the average velocity for translation over the 

barrier ( v ): 

                                                                  2
1

*2 m kTv  ,                                        (1.30) 

 

where m* is the effective mass of the complex. Thus the average time of crossing the barrier 

is 

 



 14 

                                                          2
1

*2


 m kTv  .                                    (1.31) 

 

Moreover it follows that the rate of transmission over the barrier is K’N
*
/η , where K’ is a 

transmission coefficient reflecting the probability of the activate complex of passing over the 

potential barrier. Thus, 

                                   )( *''*'

sgsg

g
ffNNfKNK

dt

dN
  .                  (1.32) 

 

It is convenient to re-express the complete partition function f
*’

 as f
*’

=f
*
f
*
trans,(1-dim), where   

f
*
trans,(1-dim) is the one-dimensional translation partition function corresponding to motion 

along the reaction coordinate, over the barrier. So, from mechanical consideration of a particle 

in a box of length δ, it follow that 

                                                         hkTm f trans  2
1

**

dim)1(, 2 .                            (1.33) 

So the Eq. (1.32) becomes 

                                              )( *'*'

sgsg

g
ffNNfhkTKNK

dt

dN
  .              (1.34) 

 

It is convenient to make the zero-point energy of the initial state of the system the arbitrary 

zero reference energy and to redefine f
*
. Thus, 

 

                                                  
 kT

sg

sg

g
eNN

ff

f

h

kT
K

dt

dN
1

*
' 

 ,                  (1.35) 

 

where ε1 is the activation energy for the process. 

This theory has been treated for the cases of immobile and mobile adsorption by Glasstone 

[5]. 

So the rate of the first-order desorption via an activated complex is given by 

 

                                                         
 kT

a

a

g
eN

f

f

h

kT
K

dt

dN
2

*
' 

 ,                         (1.36) 

 



 15 

where Na and fa refer to the adsorbed species and ε2 is the activation energy for desorption of a 

single molecule, referred to the zero-point energy of the adsorbed species. 

For the first-order kinetics of desorption in which both the adsorbate species and the activated 

complex are considered to be immobile,  f
* 

/fa  1. Many experiments have shown that first-

order desorption process exhibits pre-exponential factor of 
h

kT
K '  magnitude. 

For thermodynamics consideration is important to remember that 

 

                                                                                                

)(*

)(

*
0 RTG

eq eK


 ,                                                      
(1.37)

 

 

where *

0

*

0

*

0 STHG   and *

0H  and *

0S  are respectively the activation enthalpy and 

entropy for formation of the activated complex from the reactants. 

For the first-order desorption, with 

                                                                
 kT

aa

eq e
f

f

N

N
K 2

**
*

)(


  ,                              (1.38) 

the desorption rate is given by 

 

                                                        
 

a

RTHRSg
Nee

h

kT
K

dt

dN *
0

*
0' 

 .                         (1.39) 

For a n-order desorption the equilibrium constant becomes 

 

                                                               
n

a

n

seq NNNK )1(**

)(

 ,                          (1.40) 

 

where Ns is the number of sites per square centimeter. 

So the general equation for the nth order desorption rate is: 

 

                                                   
  n

a

RTHRS

n

s

g
Nee

h

kT

N

K

dt

dN *
0

*
0

)1(

'



 .                     (1.41) 

 

Ns is about 10
15

cm
-2

 for most adsorbents. For little or no activation entropy ( 0*

0 S ) and for 

K
’1, the pre-exponential factor ν0

(n)
 is  



 16 

                                                            
)1()(

0

 n

s

n hNkT .                                          (1.42) 

 

In the temperature range (50-2000)K, for first order kinetics, 113)1(

0

12 104)(10  shkT , 

while for the second order kinetics 122)2(

0

3 104))((10   scmhNkT s . 

When the activated complex is characterized by two degrees of translational freedom, the 

two-dimensional translational partition function for a single particle is 

 

                                                              A
h

mkT
f trD 2

*

),2(

2
 ,                                        (1.43) 

 

where A is the total surface area available to the translating complex. 

For N indistinguishable and independent particles , the canonical ensemble partition function 

is  

 

                                                               NtrDtrD f
N

F *

),2(

*

),2(
!

1
 .                               (1.44) 

 

The Helmholtz free energy is given by 

 

                                          









N

Ae

h

mkT
NkTFkTTSEA

2

**** 2
lnln


.             (1.45) 

 

So for a two-dimensional gas E
*
=NkT,  

 

                                               












N

A

h

mkT
NkNk

T

AE
S trD 2

**
*

),2(

2
ln2


,                   (1.46) 

 

with an increasing of the pre-exponential factor. Additional enhancement of the pre-

exponential factor can occur if other degrees of freedom are permitted for the activated 

complex in desorption.  

 

 

 



 17 

1.5 Treatment of experimental Desorption data 

 

Experimental data can be treated with several methods. In this work we used the Heating Rate 

Variation Method, through which we obtain a value of activation energy for H2 adsorption on 

carbon nanotubes absolutely coherent with the theoretic value. 

 

 

1.5.1 Heating Rate Variation Method 

 

Desorption spectra can be treated with several models. In this work we used the Heating Rate 

Variation Method, that is based on the collection of a series of spectra characterized by the 

same adsorbate coverage but different heating rates β = dT/dt = const. 

From each spectra it is possible to obtain the temperature corresponding to the maximum 

desorption rate (Tmax)  and then the expressions related to first and second kinetic order.   

From dt = dT/β and substituting it in Polany Wigner equation (1.15): 

 

                                                 ndes
ndes

RT

E

dT

d
r 












 



 exp

1



          ,                (1.47) 

where r is the desorption rate. 

Imposing at T=Tmax  the maximum condition  

                                                                     0
max


T

des

dT

dr
                                                 (1.48) 

in equation (1.14) we obtain that 

                                             
2

max

1

max

2

2

0
RT

E

dT

d
n

dT

d desnn

T








                               (1.49) 

 

So, from the expression for dΘ/dT in (1.13) 

 

                                                











 


 

RT

E
n

RT

E desn

n
des exp

1 1

2

max




.                               (1.50) 

 

The first kinetic order (n=1) is independent on the coverage, but dependent on the heating rate 

β: 



 18 

                                                     











 




RT

E

RT

E desdes exp
1

12

max




.                                        (1.51) 

 

Taking the logarithm of both parts of equation (1.51) , rearrangement yields: 

                                                         
R

E

RT

ET desdes









1max

2

max lnln


                                         (1.52) 

 

Plotting , )ln( 2

max T as a function of 1/Tmax for  a series of heating rate values, it is possible to 

obtain ΔEdes and ν1 respectively from the slope and the intercept with the ordinate of the line. 

For the first kinetic order desorption depends on the heating rate β, but not on the coverage Θ; 

instead the last dependence is present in equation for the second kinetic order.  

For the second kinetic order (n=2) instead, from (1.51) we obtain 

 

                                                  











 




RT

E
T

RT

E desdes exp)(
2

max22

max




                             (1.53) 

 

The second order curves are asymmetrical with respect to Tmax, so Θ0 = 2Θ(Tmax), where Θ0 is 

the coverage value before desorption. So 

 

                                                     
02max

2

max lnln








R

E

RT

ET desdes


                                      (1.54) 

 

Plotting )ln( 2

max T  vs 1/Tmax and if Θ0 is known, it is possible to obtain ΔEdes from the slope 

and ν2 from the intercept. Through this method we calculated an experimental value of the 

activation energy coherent with the theoretic value. 

In figure 1.6 is showed the Polany-Wigner equation plot at the first kinetic order, for a tipical 

value of desorption activation energy. As previously written, the peak is clearly asymmetrical.  



 19 

 

Fig. 1.6.  Polanyi-Wigner plot: emission intensity as function of temperature for different activation 

energies, with β and Edes fixed. 

 

 

 

 

 

1.5.2 Gas Evolution Method (constant Heating Rate) 

 

Let consider a surface desorbing gas into a pumped ultrahigh-vacuum system, in which the 

average residence time for a gas molecule is η. Equation (1.18) may be written, in a 

dimensionless, for as: 

                                                     Tg
dt

dN

N

P

dt

dP


0

** 1


,                                        (1.55) 

where P
*
 is a dimensionless quantity equal to P/Pmax and Pmax is the maximum pressure 

observed for desorption of N0 molecules cm
-2

 into the same vacuum system with zero 

pumping speed (η = ); N0 is the initial coverage on the surface. 

For β = dT/dt it is possible to obtain 

 

                        



































 
 

T

d

d

d dT
RT

E
T

E

R

RT

ETT
TP

0

2

)1(

0

)1(

0 expexpexp









,         (1.56) 

In
te

n
si

ty
 (

a.
u
.)

 



 20 

where )1(

0 is the pre-exponential factor at the first-order (s
-1

), Ed is the activation energy for 

desorption and R is the gas constant. P(T) describes the normalized profile of a desorption 

spectrum characterized by β, η, )1(

0  and Ed. 

To determinate the kinetic parameters )1(

0 and Ed it is convenient to utilize the following 

relationship, derived by Redhead: 

                                                            
p

dp

p

d

RT

ET

RT

E
lnln

)1(

0





,                                    (1.57) 

 

where Tp is the temperature at which the rate of desorption is maximum. 

Since the experimental desorption peak maximum temperature Tm Tp it is useful to 

determinate pairs of the parameters )1(

0 and Ed which satisfy Eq.(1.57). So the desorption peak 

shape may be generated from Eq.(1.56), using each pair of values which satisfy Eq.(1.57). By 

matching the theoretical peak to the experimental peak shape, it is possible to deduce the best 

values of )1(

0 and Ed  that fit data. 

  

 

 

1.5.3 Coverage measurement made during Programmed Desorption 

 

Pfnur et al. [8] have used the vibrating capacitor method for measuring the instantaneous 

surface coverage of CO on Ru(001) during TPD. This method is based on the observation that 

the work function change  was directly proportional to coverage of CO and independent on 

temperature. 

Therefore, 

                                                              
dt

dN
k

dt

d





)( 
                                                 (1.58) 

and it is possible to use Eq. (1.41) to determine )1(

0 and Ed as a function of coverage for the 

system plotting ln()/ versus 1/T for various constant coverages. 

 

 

 

 

 



 21 

1.5.4 Leading Edge Analysis 

 

This method [1] was introduced by Habenschaden and Kuppers and allows the determination 

of the activation parameters dependent on coverage and temperature. To keep temperature T 

and coverage Θ a small section of the spectrum on its leading edge is selected. This part of the 

spectrum is then plotted as ln rdes vs 1/T, according to the logarithmic Polany-Wigner 

equation (1.15): 

 

                                                      


 lnlnln n
RT

E
r n

des
des  .                                   (1.59) 

 

From the slope of this Arrhenius plot desE  can be determined, while from the intercept the 

frequency factor νn can be determined if the reaction order and coverage are known. 

Closer examination of the method reveals that an exact determination of νn is possible under 

certain condition. The first step is to differentiate Eq.(1.15) with respect to T for n=1 and 

T=Tmax , then taking account of the fact that the activation parameters are coverage dependent:  

 

                                                                      0

max

2

2




T
dt

d
,                                                 (1.60) 

so 

 

 







 



































 


dt

d

dt

d

d

Ed

RTRT

E

dt

d

d

d

RT

E

dt

d desdesdes
1

1

21
11 exp 









.  (1.61) 

 

Assuming n=1, using Eq.(1.47)  and imposing condition described by Eq.(1.60) yields: 

 

                         
 

0exp
1

max

2

1
1

1 






 
















 



















T

desdesdes

RT

E

RT

E

d

Ed

RTd

d







.  (1.62) 

Then it is possible to separate all terms containing ν1 : 

 

                                    
 

















 




d

Ed

RTd

d

RT

E

RT

E desdesdes 11
12

exp





.                 (1.63) 

 



 22 

The frequency factor ν1 can be determined directly in two cases: 

 when desE  and ν1  are independent of Θ so, from Eq.(1.63) 

                                           






 





RT

E

RT

E desdes exp
21


 ;                                       (1.64) 

 

 when the last two terms on the right of Eq.(1.63) compensate each other, i.e. when 

 

                                              
 

011 








 d

Ed

RTd

d des
                                    (1.65) 

 

In both cases ν1 may be easily determined. 

It is useful to note that Eq.(1.65) can be rearranged to 

 

                                                                    
 
RT

Ed
d des

1ln  ,                                        (1.66) 

which states that the logarithmic frequency factor scales linearly with the desorption energy. 

This result is an expression of the so-called compensation effect, that consists in the frequent 

observation that an increase in the exponential term in the Arrhenius equation is linked to a 

decrease of the frequency factor. 

 

 



 23 

Chapter 2 
 

 

Thermal Desorption Spectroscopy applications 

 

 
Thermal desorption spectroscopy is useful to determinate kinetic and thermodynamic 

parameters of desorption processes and decomposition reactions. In this chapter the 

application of TDS to astrophysical processes is explained. We used this technique to 

investigate recombination and decomposition mechanisms occurred in ammonia and methane 

ices subjected to electron bombardment. We obtained that new species are produced when 

solid methane or solid ammonia are irradiated with electrons. In the first case we observed the 

production of acetylene (C2H2) and ethane (C2H6) while in the second case we detected 

diazene (N2H2). 

 

 

 

 

 

 
2.1 Thermal desorption spectroscopy application to Astrophysics 

 

2.1.1 The interstellar medium 

 

 

The term “Interstellar Medium” (ISM) describes the matter diffused among the stars, that 

mostly thickens in wide regions named interstellar clouds (fig.2.1). In our galaxy, the Via 

Lattea, the interstellar clouds concentrate mainly along  the spiral arms, but also in the regions 

between the spiral arms and the galactic halo, on and under the galactic plan.  

The star light that crosses the interstellar medium is partially attenuated. This fact represented 

an important problem to the astronomers interested in the observation of stars and galaxies, 

until they understood the importance of studying the interstellar medium itself. The 

interstellar clouds, in fact, play an important role in the stars cycle of life: the explosions of 

supernovae enrich of heavier elements (synthesized into new stars) the interstellar clouds, 

which contribute to the rising of new stars because of  gravitational collapse. 

The interstellar medium represents the 10-15% of our galaxy mass and it is made largely of 

gas (99%) and for a small part (1%) of dust grains with dimensions in the order of micron. 

The dust grains are mainly the cause of the stellar extinction phenomenon, consisting in the 

partial stellar light attenuation due to the scattering and the adsorption physical processes. The 



 24 

interstellar clouds are composed by gases and dusts and their interaction produces a rich 

interstellar chemistry.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.1. The Horse’s head nebula belongs to the molecular Orione’s cloud. The nebula is made of dusts that 

block the light of the incandescent gas on background. 

 

 

 

2.1.2 The interstellar gas 

 

The interstellar matter chemical composition is similar to the cosmic composition, describing 

the abundance of several elements of our Sun. Thus the interstellar gas is made mainly of 

hydrogen (90%) and helium (10%), while oxygen, carbon and nitrogen represent only the 

0.1% of the entire mass; other heavier elements are present in very small amounts. The 

physical and chemical properties of interstellar clouds are mainly determined by interstellar 

gas because of its great abundance contrarily to dusts. Furthermore the interstellar clouds are 

classified according to hydrogen state. Many chemical species have been identified by the 

adsorption spectra of the stellar light that arrives on hearth passing through an interstellar 

cloud (Fig.2.2). 



 25 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.2. The interaction between the interstellar light and the interstellar medium, placed along the observation 

direction, gives information about the chemical elements constituting the interstellar clouds. 

 

The information about interstellar gas mostly come from radio observations. At the beginning 

of 1950 Purcell and his collaborators pointed out the line of 21cm of interstellar atomic 

hydrogen. Consequently to this observation the analysis of the 21cm line became a powerful 

instrument for the interstellar gas investigation; this technique gives information only about 

atomic hydrogen. Although the molecular hydrogen is the most abundant molecule in nature, 

its individuation is very difficult; in fact H2 is a diatomic homo-nuclear molecule with no 

permanent electric dipole, so it can emit radiation only through quadrupole transitions. For 

this cause the abundance of H2 in the interstellar medium is indirectly estimated by the 

observation of other molecules as CO, that is the second more abundant species (about 10
-4

-

10
-5

 of hydrogen concentration). 

When a CO molecule collides with a H2 molecule (which is the most probable partner in 

collisions because of its abundance) the former may be excited in a higher roto-vibrational 

state; the consequent CO de-excitation  through a photon emission gives information about 

the presence of H2. The ratio of H2 molecular abundance with H2 atomic abundance varies 

with the position of our Galaxy. Near the Sun the gas density is about 0.04 solar mass per 

(parsec)
3
, where a solar mass is equal to 1.9910

30
 kg and 1 parsec=3.0.8610

16
 m;  the entire 

mass is made of hydrogen for 77%, molecules for 17% and ions for 6%. 

The interstellar medium regions which are better screened from the stellar radiations are 

characterized by a great presence of H2, while regions with high radiation fields  are 

dominated by the H presence.  



 26 

Although H2 is the simpler molecule, its formation in the interstellar medium has generated 

many discussions. In fact the radioactive association of two hydrogen atoms is a less efficient 

process of formation because it involves prohibited roto-vibrational transitions; moreover the 

reaction of three bodies in gaseous phase in the interstellar medium is rare.  

Hollenbach and Salpeter suggested that H2 formation may occur efficaciously on the dust 

grains surface, which may operate as catalysts [9]. Many other studies both theoretical  

[10,11] and experimental [12, 13, 14] have been analyzed this topic. 

 

 

 

2.1.3 The interstellar dust 

 

Although the dust grains represent only  the 1% of the interstellar medium mass, they play a 

fundamental role in the chemistry and evolution of the interstellar clouds. In fact they operate 

as catalysts permitting to the chemical species adsorbed on their surface to interact and to 

form new species. The formation of new species on the grain surface can be described 

according to the following steps: 1) arrival of species in gaseous phase which are adsorbed on 

the grain surface; 2) scansion of the grain surface by species with more mobility; 3) meeting 

and chemical reaction between two chemical species on grain surface; 4) release of exceeding 

energy by the new species created and consequent molecular stabilization (as in the case of 

H2); 5) probable desorption of the new species formed (fig. 2.3). 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.2.3. Schematic illustration of interactions on grains surface. 

 



 27 

The presence of dust grains into interstellar medium is confirmed by some evidences, among 

which the stellar light extinction  curve at wavelengths of visible and ultraviolet (fig. 2.4). The 

stellar light extinction is due to scattering and adsorption processes involving the dust grains. 

The extinction is a function of wavelength; a wavelength decrease corresponds to an 

extinction increase. As a consequence of this observation the stellar light is more red than it is 

in reality. Another evidence of the interstellar dust grains existence is the weak linear 

polarization of stellar light; this implies also that the grains cannot have a spherical shape. The 

dust grains in fact are elongated particles made of paramagnetic materials and so aligned to 

interstellar magnetic fields. 

Although the evidence of these observations, the dust grains models (for example utilized for 

understanding the interstellar extinction curve) often consider a spherical shape for the grains. 

The typical dimensions of the dust grains are in the range of 1nm-3μm. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.4. Stellar extinction curve for several stars plotted as function of 1/λ 

 

 

The analysis of adsorption spectra in the infrared (for example spectra acquired by the 

satellite ISO) shows that the dust grains are mainly made of silicates and carbon materials. 

The spectral characteristics at 10 m and 12 m are due to vibrational transitions of silicates 



 28 

(SiO e SiO2) and water ice, while the interstellar extinction in ultraviolet may be explained by 

the presence of small graphitic grains. Another observation about the grains chemical 

composition is that some elements as N and S in gaseous phase have abundances similar to 

solar ones, while several elements (as Mg, Al, Si, Ti, Ca, Fe, Ni, Cr) are less abundant (figure 

2.5). It is note that the interstellar abundances are similar to solar ones [15], so the elements 

which are absent in gaseous  phase are probably located in the dust grains [16]. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.5. Reduction of elements (respect to cosmic abundances) in the interstellar cloud  Oph as a function of 

the condensation temperature of the elements. The ordinate is represented by the difference between the 

logarithm of measured abundances and the logarithm of cosmic abundances. For Tc  1200 K the elements are 

able to form refractory solids. For example Ca has an abundance 104 lower then solar one. 

 

 

It is thought that the dust grains are ejected by hot stars; in fact the dust grains cannot form in 

situ in interstellar cloud because of its very low density. So the dust grains formation 

mechanism is very probably the following. At the end of their life, the so-called Asymptotic 

Giant Branch stars (AGB) emit a large part of their envelope into space, so they are 

surrounded by very large atmospheres which are sufficiently cold and dense to permit atoms 

and molecules nucleation leading to the formation of small solid particles. The flow of stellar 



 29 

materials may contain mainly carbon or oxygen, characterizing the composition of the dust 

produced: if carbon is more abundant then oxygen, the latter forms mainly CO and the former 

produces several carbon molecules, so dust grains contain carbon; if oxygen is more 

abundant, the oxygen excess produces metallic oxides which for nucleation permit the 

formation of solid oxides and silicates [17]. 

 

 

 

2.1.4 Ices in interstellar medium 

 

In the very dense and cold interstellar clouds (the minimum density is about 10
4
 atoms for 

cubic centimetre) the observations have revealed the existence of frozen mantles on the dust 

grains surface [18]. These ices are mainly made of water, identified by the features at 6.0 m 

and 3.1 m, respectively assigned to the bending mode of the amorphous water ice and to the 

stretching mode of O-H in the water ice. Other molecules which contribute to formation of the 

grains ice mantel are CO, CO2, CH3OH, CH4, NH3, OCS (see table1 and figure 2.6). 

 

 

 

 

 

 

 

 

 

 

 

Table 2.1. Molecules observed in the interstellar ices, with abundances relative to water ice. 

 

In general these molecules condense on grains in gaseous phase, but the grains mantle 

composition does not exactly reflect the composition and the abundances of the gaseous 

phase. For example it is known that CO2 is present in the grains, but it is not observed in 

gaseous phase yet [19]. 



 30 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.6. Inventory of interstellar ices pointed out in the protostar W33A (Gibb et al., 2000).  The spectra shows 

the adsorption characteristics due both to the dust grains (silicates) and to the dirty ice mantles. 

 

 

So it is reasonable to think that when reactive chemical species condense on the dust grains 

may form new molecules through chemical reaction. New molecules form also after 

irradiation by UV photons and cosmic rays at which the interstellar ices are subjected, as 

noticed through experiments conducted in laboratories (see [20] for the ion bombardment and 

[21] for the photons irradiation) . Also the water ice forms by chemical reaction on the dust 

grains surface. Instead the direct increase of complex molecules from gaseous phase 

influences  little the chemical composition of the mantles. However mantles composition 

mainly reflects the local conditions (figure 2.7). Because hydrogen is the more abundant 

species in the Universe the chemistry on the grains surface is determined by the ratio H/H2. 

When this ratio is high the chemistry is dominated by hydrogenation reactions and the more 

abundant chemical species are H2O, NH3 and CH4. Instead if the ratio H/H2 is lower then 1, 

some very  reactive species as O and N may react between them and they may form CO, CO2, 

O2 and N2. Consequently it is expected that two different mantle types are produced by 

reactions on the grains surfaces: the first type is dominated by polar molecules linked by 

hydrogen bond, while the second one is dominated by unsaturated non-polar molecules (or 

weakly polar) [22].  In fact, from the analysis of carbon monoxide profiles in solid phase, 

generated by young stellar objects, it is possible to observe two types of ice mixtures (polar 



 31 

and non-polar) and a third type whose nature is uncertain. The profile attributed to CO 

stretching (2140 cm
-1

) is formed by the superimposition  of three components, describing 

three different ice mixtures. In general, a component prevails over the two others according to 

the physical condition of each interstellar cloud [23]. 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2.7. Schematic representation  of the different mantle types. The interstellar ice composition in interstellar 

clouds depends on local conditions as the ratio H/H2 , the intensity of cosmic rays and the radiation field. 

 

 

The main constituent of ice mantles is water, whose chemical origin is not jet understood. 

Some theoretical works [24, 25] suggest three possible reactions for the formation of H2O:  

hydrogenation of O, O2 and O3. Actually, because of the few experiments simulating these 

reactions, it is not possible to determinate all their contributes. However the water formation 

in gaseous phase and its consequent increase on grains does not explain the H2O abundances 

observed [26]. Both experimental [27] and theoretical [28] evidences confirm that the water 

ice in the interstellar medium is mainly amorphous, but there are few indications about its 

morphology. Some experiments have showed that the ice porosity considerably influences the 

H2 formation efficiency [29] and the H2 energetic content when, after its formation, it is 

released by grain in gaseous phase [30]. Moreover it is observed that a porous ice film has an 

adsorption efficiency 20-50 times larger then a compact ice film [31]. 



 32 

The ice porosity may be identified by the weak  adsorption features in the infrared ( 2.7 m) 

due to vibration of the O-H dangling bond on the pore surface [32]. This feature is not jet 

observed yet for the interstellar ices, so it is reasonable to think that  they have a compact 

nature [33]. 

 

 

2.1.5 Ices and energetic agents 

 

When inside an interstellar cloud bright stars form, they may induce both physical and 

chemical changes into surrounding material irradiating energy into their local environment. In 

particular the frozen mantles of the dust grains may be heated producing (partially or totally) 

the ices sublimation or crystallization. Moreover the mantles composition may be modified 

through chemical reactions inducted by the radiation field. The ices spectroscopy gives three 

methods to analyze these changes: 1) the abundance variations of the more volatile species, 

attributable to sublimation; 2) the profile evolution as function of temperature; 3) the 

individuation of features connected to photo-processes products. Figure 3.8 shows 

schematically the sublimation zones around a young stellar object (YSO). Among the 

molecules commonly observed in the ice mantles, CO is the most volatile, H2O is the least 

volatile and CO2 represents an intermediate case. 

 

 

 

 

 

 

 

 

 

 

Fig. 2.8. Schematic representation of chemical environment  of a massive young star inside a dense molecular 

cloud. The presence of several molecules into mantles is indicated, respect to the grains temperature (Td) and as 

a function of the star radial distance. 

 

So the heating may remove the non-polar layer (dominated by CO) faster then the polar layer 

(dominated by H2O); particularly the non-polar ice may vaporize at about 20 K, while the 



 33 

polar one may survive at least until 100 K. This assumption is confirmed by observations. 

Analyzing the differences in mantles sublimations observed along several observation lines, 

Chiar et al. proposed the existence of three general classes of objects, subdivided in 

accordance with an evolutionary sequence: 1) objects with few or with no sublimation of non-

polar mantles, with a ratio of the two ice type CO/H2O about (25-60)%; 2) objects subjected 

to a moderate or high sublimation of non-polar mantles, with a ratio CO/H2O about (1-20)%; 

3) objects characterized by a total CO sublimation.  

As already written, at the beginning of their formation the ices are amorphous, because 

accretion happens at temperature very lower then fusion or sublimation ones of remarkable 

molecules.  

The spectral profiles of amorphous solids are very large Gaussians. If material is heated the 

spectral lines become narrower and more similar to the crystalline solids spectral profiles, 

because of molecules rearrangement in energetically more favourable configurations. 

Through these differences it has been possible to verify that in the so-called quiescent dark 

clouds the ices are at a temperature between 10 K and 20 K, while in many YSO the ices are 

at a temperature in the range of (70-100)K [27]. It is important to note that although is 

favourable to characterize ices in accordance with distinct polar and non-polar components 

(deposited sequentially), it is possible to observe an intermediate stadium of mantle growth  in 

which H2O and CO accumulate in comparable rates. 

Oxidation and hydrogenation reactions, involving CO, may lead to CO2 and CH3OH 

production and so to formation of an amorphous mixture containing CO2, CH3OH e H2O with 

similar proportions. 

If this mixture is heated, the consequent structural changes lead to the formation of complexes 

in which the oxygen atoms of CH3OH bond with the carbon atoms of CO2 creating strong 

intermolecular bonds. 

The ices thermal evolution is mainly determined by the adsorption of infrared radiation 

emitted from stars inside gas and dusts entanglement. When the circumstellar material 

gradually disperses, more energetic radiation and particles flow may permeate the 

surrounding medium. Photons with energies in the order of eV may break the chemical bonds 

and they may convert saturated molecules as H2O, NH3 e CH4 in radicals. If irradiation is 

followed by heating, the radicals may migrate through the mantles and they may react with 

other chemical species. Moreover when the grains come back from the molecular clouds to 

the diffuse interstellar medium, they maintain their mantles which are consequently and 



 34 

remarkably modified by photochemical reactions; these reactions convert the volatile ices in 

refractory organic material, as pointed out in laboratory simulations (see [34], [ 35] and [36]). 

Into dense molecular clouds the grains mantles are also subjected to ion bombardment. 

Comparing the adsorption spectra obtained by observations with the spectra of ices 

bombarded in laboratory, it is possible to obtain information about the ion bombardment 

effects to which are subjected the interstellar ices. For example figure 2.9 ([20]) shows a 

comparison between the band profile at 2165 cm
-1

 observed in the direction of the protostellar 

object W33A and the band formed after ion bombardment of an ices mixture containing H2O, 

CH4 e N2. The good comparison confirms the hypothesis that the energetic agents play an 

important role in the grains mantels evolution  and that their chemical composition may be 

strongly influenced by interaction with cosmic rays [20]. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.2.9. Comparison between the band profile at a 2165 cm-1 observed in the direction of W33A and the spectra 

obtained in laboratory  from a mixture of H2O:CH4:N2 bombarded at 12 K and heated up to 120 K ([20]). 

 

 

 

The electronic bombardment effect on interstellar ices with laboratory experiments has been 

recently investigated [37]. 



 35 

Solid ammonia (NH3) was irradiated in the temperature range of (10-60) K with high-energy 

electrons to simulate the processing of ammonia-bearing ices in the interstellar medium and in 

the solar system.  

Ammonia and ammonia hydrate have been found in the atmosphere of Jupiter, Saturn, Uranus 

and on the surfaces of their icy satellites such as Miranda, Mimas, Enceladus, Tethys and 

Rhea. Ammonia has been identified in the dust grains in cold molecular clouds with an 

abundance of 1%-10% with respect to water. The ammonia abundance in the ices of comets 

and protostars is lower than water (H2O), carbon monoxide (CO) and carbon dioxide (CO2), 

but higher than methane (CH4). The ices in these cold molecular clouds and on solar system 

bodies are subjected to the irradiation by cosmic-rays, UV photons, the solar wind or by 

energetic particles trapped in planetary magnetospheres. 

Ammonia is hardly a pure ice, but mixed with species such as water, so  it is essential to study 

the mixture of ammonia with several molecules under realistic conditions. The solid ammonia 

properties are very interesting from both astrophysical and chemical viewpoint. 

As a result of the experiments of Zheng et al. the synthesis of hydrazine (N2H4), diazene 

(N2H2 isomers), hydrogen azide (HN3), the amino radical (NH2), molecular hydrogen (H2), 

and molecular nitrogen (N2) has been confirmed. So ammonia may be the main source of 

nitrogen in comets and the interaction of ammonia with ionizing radiation can produce a 

greater amount of molecular nitrogen. As consequence it is expected , on various solar system 

objects, an ammonia decreasing corresponding to an increasing of molecular nitrogen 

fraction. 

The results show that the production rates of hydrazine, diazene, hydrogen azide, molecular 

hydrogen, and molecular nitrogen are higher in amorphous ammonia than those in crystalline 

ammonia; this behaviour is similar to the production of molecular hydrogen, molecular 

oxygen, and hydrogen peroxide found in electron-irradiated water ices. 

These studies have been analyzed also in this work and they will be described particularly in 

chapter 4.  

 



 36 

2.2 Thermal desorption spectroscopy application to Geophysics 

 

In Geophysics and Petrology Thermal Desorption Spectroscopy can be used to study minerals 

chemical composition. Our interest was focused on  zoisite. In fact, in this work we used TPD 

to  investigate zoisite behaviour and to understand if its modification into tanzanite variety 

after heating is due to structural changes or to a dehydration mechanism. 

 2.2.1 Zoisite 

 

Zoisite (Ca2Al3[Si2O7][SiO4]O(OH) ) is a sorosilicate, i.e. a silicate in which are present 

isolate groups made of tetrahedral couples (SiO4) sharing an apical oxygen. The zoisite 

empirical formula is Ca2Al3Si3O12(OH). 

The crystal structure consists of one type of endless octahedral chains  parallel to b with two 

distinct octahedral sites M1,2 and M3. These chains are cross by isolated tetrahedral SiO4 

(T3) and Si2O7 groups (T1 and T2) in the a and c directions(fig.3.10).  

 

 

Fig.2.10. The elementary crystal structure of zoisite. 

 

Zoisite is an allocromatic mineral, because its milling produces a white dust. So the colour is 

not due to its chemical composition, but it can vary because of the presence of cromophores 

ions or structural defects of crystal lattice. 

Zoisite chemical composition (in %) is mainly described in the following table [38]: 

 

a

c

b
a

b

c



 37 

 

 

Calcium (Ca) 17.64% CaO 24.68 

Aluminum (Al) 17.82% Al2O3 33.66 

Silicon (Si) 18.52% SiO2 39.67 

Hydrogen (H) 0.22% H2O 1.98 

Oxygen (O) 45.78%   
 

Table 2.2 

 

Among the typical impurities of zoisite the most common is water. 

The optical adsoprtion spectra of natural yellow zoisite shows three spectral bands, 

respectively at 13.400 cm
-1

, 22.500 cm
-1

 and 27.000 cm
-1

. 

Tanzanite is the blue variety of zoisite, which was discovered in the Merelani Hills of 

Northern Tanzania in 1967. When yellow zoisite is heated between 370°C-650°C, it becomes 

an intense violet-blue colour (variety tanzanite) [39]; this behaviour was explained in terms of 

change of the oxidation state of transition metal ions, such as V [40, 41]. 

Rodeghero et al. [42] studied the dependence of the colour changes in yellow and blue zoisite 

after heating not only on the concentration of metal ions and their oxidation states, but also on 

possible changes in bond lengths and bond angles or on distortion of coordination polyhedron. 

Yellow and blue crystal of zoisite were characterized by XRF analysis, that reveal that the 

same minor elements (particularly V and Sr) are present in both samples but in different 

amounts; this aspect can play a fundamental role in the change colour. 

The samples were heated in air flux from room temperature to 520°C (heating rate 5°C/min). 

PXRD data were collected on a diffractometer before and after heating; XRD diffraction 

patterns were analysed by the Rietveld method. Finally, the samples were optically 

characterized by UV-VIS technique, using a Xenon lamp and an integrating sphere. 

UV-VIS spectra show that yellow zoisite becomes blue during heating both in air and 

nitrogen. The spectral changes induced by heating indicates that the colour change is 

connected the disappearance of the 22500 cm
-1

 band (fig.3.11).  



 38 

 

Fig.2.11. UV-VIS spectra for (a) yellow zoisite, (b) blue zosite and (c) yellow zoisite after heating. 

 

 

Differences in the spectrum between the originally blue-coloured zoisite and the heat-treated 

zoisite  could be explained on the basis of the structural refinements. 

The analysis of PXRD patterns shows, for  structural refinement of natural yellow (a) and 

blue (b) samples at room temperature, differences in the unit cell parameters and also in the 

orientation and dimension of coordination polyhedron. In both samples, M3 polyhedron is 

larger than M1 one (fig. 2.12). After heating, structural refinement of yellow and blue samples 

reveals that the orientation and dimension of  coordination polyhedron becomes similar. The 

rotation of M3 parallel to a is compensated by a rotation of the Si2O7 group, which is reflected 

by the shortening of the O7-O8 distance and the narrowing of T1-O9-T2 angle. 

 

 

Yellow zoisite (a) 

Blue zoisite (b) 

Yellow zoisite after heating (c) 



 39 

 

 

Fig. 2.12. Structural refinement of natural yellow (a) and blue (b) zoisite at room temperature 

 

The essential elements of zoisite (Ca2Al3[Si2O7][SiO4]O(OH)) are Al, Ca, H, O, Si, but water 

is a common impurity. When zoisite is heated its structure is subject to a change and it is 

important to verify if the structural change is due to the loss of an essential element as H or to 

a H2O release. In the following chapter, a TPD experiment on zoisite heating from room 

temperature to 650°C is analysed to evaluate a possible H2O release and to explain changes in 

structure. 

In fact, the products of zoisite heating are H, O, OH and H2O; by analysing thermal 

desorption  spectra for these species it is clear that intensity and shape are the same for all 

elements. So it is reasonable to think that H, O, OH are due to cracking fraction of water 

(released by zoisite during heating) operated by quadrupole filament. So the main mechanism 

of zoisite change in colour after heating  is the dehydration.  



 40 

2.3 Thermal desorption spectroscopy application to high-energies Physics 

 

In this work we use Thermal Desorption Spectroscopy to evaluate molecular hydrogen 

adsorption properties of carbon nanotubes to find a possible solution for vacuum system 

problems in future accelerators. In fact molecular hydrogen  represents the critical species for 

desorption in Large Hadron Collider. We verified that carbon nanotubes have a great H2 

adsorption capacity also at very low temperatures (12K-20K) so they may be good candidates 

to criosorbers in the future accelerators. 

 

 

 

2.3.1 Large Hadron Collider: problems to vacuum system 

 

The Large Hadron Collider (LHC) is made of a couple of two semi-conducting rings with a 

circumference about 26.7 Km; inside these rings two protons beams with energy of 7 TeV 

circulate [43]-[46]. 

 

2.13. Schematic representation of LHC 



 41 

There are also two magnets in a direct contact with liquid helium at 1.9 K. The inner walls of 

semi-conducting magnets (“cold bore”) work at very low temperatures and they are efficient 

cryo-pumps. The core of these magnets is made of super-conducting cables, wrapped in coils, 

which generate a magnetic field at current passage. A couple of coils (super-conducting 

dipole) provides the magnetic field necessary to bend the beams forcing them on a circular 

trajectory. So is produced an intense centripetal acceleration. 

 

 

2.14. LHC section 

 

As consequence, inside the magnetic rings every beam emits a synchrotron radiation with a 

flow of about 10
7
 photons s

-1
m

-1
, a critical energy of 45.4 eV and a power of 0.22 Wm

-1
. The 

radiation emitted by relativistic protons deteriorates the vacuum system; in fact, because of  

photons scattering with residual gas, the desorption of gas and electrons from the ring walls is 

produced [47]. 

It is important to take into account the possibility that desorbed particles settle on super-

conducting magnetic coils, stopping the machine operations. To avoid a decrease of cryogenic 

power, the heat load (due to synchrotron radiation) and the dissipation of resistive power (due 

to beam currents) must be adsorbed on a beam screen, inserting inside the vacuum chamber 

and operating at temperature between 5 K and 20 K (fig.2.15). 



 42 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

2.15. Schematically representation of LHC section 

 

The cryo-pumping on cold surfaces gives a good vacuum in chamber, but it is important to 

guarantee that the vapour pressure of cryo-adsorbed molecules (whose critical species are H2 

and He) remains in acceptable limits. This is a serious problem because it may compromise 

the LHC functioning. 

The possibility to use Carbon Nanotubes as cryo-pumps to solve vacuum system problems in 

LHC and their adsorption power are analysed in detail in the following chapter. In fact using 

CNTs to cover the vacuum chamber walls it may promote the adsorption of residual gases 

(particularly for H2  [58]) at temperatures on the range of (5÷20)K. 

 

 

 

 

 

 

 



 43 

2.3.2 The LHC vacuum system 

 

The LHC vacuum system is made of two parts, according to the different vacuum conditions 

at which the apparatus is subjected: 

 the cryogenic insulation vacuum, which required pressures in  the order of 10
-6 

mbar; 

 the dynamic vacuum, necessary to guarantee a good beam lifetime; pressures required 

are about 10
-10

 mbar. 

 

The dynamic vacuum is a specific vacuum for the beam. 

The “beam pipe” is localized in the centre of super-conducting magnetic coil (figure 3.15) and 

it is  in direct contact with helium at 1.9K(“cold bore beam pipe”). 

At these temperatures the vacuum chamber walls become an efficient cryo-pump with an 

infinite adsorption power, with the exception of helium and hydrogen. 

It is possible to individuate four distinct heat sources inside the system, whose temperature is 

maintained at 1.9K (fig.2.16): 

 

2.16. Losses induced by beam in LHC 

 

 Synchrotron radiation  

As already written, because of the centripetal acceleration into magnetic rings, the protons 

beam emits a synchrotron radiation  with a flow of about 10
7
 photons s

-1
m

-1
, a critical 

energy of 45.4 eV and a power of 0.22 Wm
-1

. The radiation flux depends on beam current 

and energy and on the system bending radius. The flux of photo-desorbed gas is 

proportional to photons flux and to the molecular desorption yield, defined as the number 

of emitted particles after collisions per incident particles. 



 44 

Among the adsorbed gas from walls, the more critical species for vacuum are H2 ,CO2, 

CO, H2O and CH4. The following table shows the molecular desorption yield values of 

several gases for temperature in the range of (4.2K300K). 

 

 

Table 2.3. Molecular desorption yield 

 

The previous results show a decrease of molecular desorption yield with the increase of 

temperature, with significant values also at very low temperatures. In the figure 2.17 it is 

possible to observe the desorption yields trend as function of dose, after 3.6 h of LHC 

functioning at 7 TeV. 

 

 

2.17. Molecular desorption yield as function of dose 

 

 

 

The H2 molecular yield from a cupper surface at 10 K is typically in the order of 510
4
 

molecules per photon; this value is considerably grater then values of other species. So 

molecular hydrogen  represents the critical species for desorption. 

 



 45 

 Beam current 

 

The interior of beam pipe dissipates power in the order of 0.05Wm
-1

, directly depending 

on resistivity ρ of the vacuum chamber material. For limiting this heat loss and for 

avoiding the beam instability, it is necessary to maintain low the resistivity of chamber 

wall. This effect can be obtained covering the beam pipe (made of stainless steel) with a 

thin cupper layer. In fact, at low temperature the resistivity for pure cupper is strongly 

reduced and it is possible to observe a gain factor   10
3
 times greater than the factor of  

only stainless steel. 

 

 Photoelectrons 

 

The synchrotron radiation determinates the electrons production through the 

photoemission and the Auger effect. The produced electrons may create an electronic 

cloud which may led to the system instability. 

The electronic cloud can be caused by the following mechanisms and by their 

combination: 

1) residual gas ionization; 

2) photoemission due to synchrotron radiation; 

3) secondary emission or multipacting induced by the beam. 

 

When LHC operates at 7 TeV, the main source of primary electrons is photoemission, due 

to synchrotron radiation. Primary electrons hit the chamber producing secondary electrons 

and a consequent increase of electronic density during successive crossing of protons 

beam (fig.2.18). 

 

Fig.2.18. Schematically representation of creation mechanism of electronic cloud  

 



 46 

The secondary electrons are defined, conventionally, through three contributes: the real 

secondary electrons (produced from primary electrons), the elastically reflected electrons 

and re-diffuse electrons. 

The electron beam is characterized by a strong electric field, so these electrons may be 

accelerated and they may transfer their energy to cold walls, creating an intense heat load 

[47]. 

 

 Nuclear scattering 

 

The nuclear scattering of high-energy protons on residual gas generates a secondary 

particles fall; particles with a sufficient energy escape from  the vacuum chamber and 

deposit their energy on cryo-magnets then they are adsorbed into the system at 1.9K. The 

loss due to nuclear scattering on residual gas generates a heat load not negligible. So the 

heat provided to cryogenic system is directly proportional to gas density. 

Two limits for residual gas density exist; the first is due to a local heat load which can 

deteriorates the magnet, while the second is connected to the total heat load in the 

cryogenic system. 

The first limit regards the high pressure regions of system, while the second one can be 

expressed through the beam lifetime in nuclear scattering (η).  

Considering for protons with an energy of 7 TeV , on hydrogen atoms, a cross section of ζ 

~ 5 10
-30

 m
2
, the beam lifetime is given by 

                                                                    n  c



1
 ,                                                   (2.1) 

where c is the lightning speed. 

The linear power P(W/m) is proportional to the beam current I and to the beam energy E 

and it can be expressed as a function of lifetime η:  

                                                     
)(

)()(
93.0

h

AITeVE

c

IE
P


  .                                     (2.2) 

So for H2 a lifetime in the order of 100h implies a relative pressure -9101  torr; heavier 

gases obviously require lower densities, as showed in table 2.4. 



 47 

 

Table 2.4.Cross-sections and maximum density for several gas species  

 

 

The heat loads due to synchrotron radiation and photoelectrons can be limited at about 20 

K, inserting a thermally isolated screen (beam screen), cooled by gaseous helium, into the 

cold bore of vacuum chamber (fig. 2.19). 

 

Fig. 2.19. Beam screen section with active cooling into magnet cold bore 

 

 

The beam screen is equipped for all its length of a series of small holes (“pumping slots”) 

which permit to gas molecules to leave the beam pipe for being adsorbed by cold bore 

walls. 

The pumping holes represent about 4% of the entire beam screen surface.  

The beam screen is partially transparent to permit to H2 molecules desorbed by 

synchrotron radiation to be permanently pumped by cold bore surface at 1.9K. 



 48 

Among the desorbed molecules hydrogen has the bigger molecular desorption yield and it 

represents a very critical species because of high value of its vapour pressure, contrary to 

beam screen pumping power. 

 

Fig. 2.20. Cold bore and beam screen section 

 

Without pumping holes the beam screen cryo-pumping could be little efficient, since the 

vapour density at equilibrium for a H2 monolayer (T~5K) overcomes with several orders 

of magnitude the acceptable limits; instead the presence of holes on surface induces into 

beam screen an approximately infinite pumping power. So hydrogen and other species 

may be pumped on a surface at 1.9K with a saturated vapour density negligible. Moreover 

pumping holes give a constant pumping speed which is proportional to the beam screen 

surface occupied from them. 

 

 

 

 

2.3.3 Gas recycling and instability pressure induced by ions 

 

Because of photons collision on the beam screen inner surface, the cryo-pumped gas 

molecules on screen may desorb again giving place to the secondary desorption phenomenon. 

Studies about the secondary adsorption coefficient (recycling coefficient) for H2 show an 

increase proportional to the amount of covered surface, until the value of one molecule per 

photon respect as a monolayer; for H2 the effect of secondary desorption is very bigger than 

primary desorption for great screen coverages. 

 consequence of Thermal Desorption and Recycling, only a limited amount of gas 

stay on beam screen surface; beyond this limit the gas load due to primary 

desorption migrates to vacuum chamber wall (at 1.9K) through pumping holes, with a 

pumping speed determined by the total area occupied by holes. 

As a 



 49 

It has been observed that  heavier gases as CH4,CO and CO2 have very low recycling 

coefficients for thin physisorbed layers (10-20 monolayers) and these secondary desorption 

coefficients are indistinguishable respect to primary desorption coefficients; instead for H2 the 

coefficients remains constant as is it possible to note in figure 2.21. 

 

Fig. 2.21. Recycling coefficients as function of coverage for several gas species 

 

The behaviour of the vacuum system is showed in figure 2.22, which describes the H2 density 

evolution during the photons exposition. 

 

 

Fig.2.22. H2 density as function of LHC operation time 

 

 



 50 

The ionization of gas molecules in LHC produces ions, accelerated toward the beam screen or 

the vacuum chamber, with a final energy higher than 300 eV. 

Also energetic ions (as photons) may desorb thin layers of gas molecules, so it is possible to 

define a molecular yield for ions (ηi); the desorbed molecules may be ionized and they may 

take part in the consequent desorption process.  

So there is an instability pressure due to ions, dependent on the local purity of surface, on the 

molecular desorption yield and on the effective pumping speed (Seff ). 

The condition which describes the stability limit is  

                                                                      effi S
e

I


  ,                                                    (2.3) 

where I is the beam current, e is the electron charge and ζ the crossing section of residual gas 

molecules for high energy protons. 

In the cold regions of LHC, the pumping inferior limit is determined by cryosorption on 

vacuum chamber surface at 1.9 K; the stable vacuum condition is expressed by 

                                                                  f
B

i A
e

m

TK
I




2
 ,                                             (2.4) 

where Af  is the pumping holes area per length unit, KB the Boltzmann constant and m the 

molecule mass. 

With LHC parameters, the calculated stability limit for hydrogen is ηiI=103 A. Since the 

primary desorption yield for H2 is unitary and a great recycling coefficient does not allow the 

formation of a H2 thin layer on screen, this limit is acceptable for the LHC cold sections. For 

sections operating at room temperature the limit is defined by the beam pipe conductance. 

 

 

 

2.3.4 Cryosorbent materials 

 

In recent years several materials have been analysed to investigate their cryogenic properties 

[48]. Charcoal is one of the most famous cryosorbent materials and it presents a sufficiently 

high adsorption power. However this material shows some disadvantages for the use in 

vacuum chambers: in fact it has a little free surface ad it is difficult to create bonds in surface. 

Recent studies about different cryosorbent materials   (at 4.2 K) show that the adsorption 

power of filters, anodized aluminium and other materials is lower than charcoal and only the 

latter may be utilized in LHC. 



 51 

Other cryosorbers, as new forms of anodized aluminium, porous cupper and charcoal 

materials, are analysed in collaboration with numerous research institutes in H2 cryo-pumping 

at temperatures in the range of 10 K20 K. 

Among the cryo-sorbent material which may be utilized in LCH there are also the carbon 

fibres. 

Figure 2.23 shows the pumping speed at different temperatures as function of surface density 

of adsorbed molecules (molecules/cm
2
), for several materials subjected to a fast cooling: 

anodized aluminium (AA3), porous cupper (Cu 1), charcoal with super-conducting ceramics 

(C+SCC) and carbon fibres covered by a thin layer of stainless steel. 

 

 

Fig. 2.23. Pumping speed trend for several cryo-sorbent materials as function of H2 adsorbed molecules density. 

 

 

It is possible to note that the H2 adsorption power overcomes the value of 10
17

 molecules/cm 

only for charcoal and carbon fibres samples, while for other materials the surface density of 

adsorbed molecules does not have appreciable values. So only charcoal and carbon fibres 

have the correct characteristics to be used as cryo-sorbers in LHC. 

The experimental results described in chapter 4 show that MWCNTs have an efficiency  

clearly higher than charcoal and they may be a possible solution to vacuum problems of LHC. 

S
cr

y
o
(T

R
T
/T

cr
y
o
)1

/2
, 

(m
3
/s

/c
m

2
) 

D (molecole/cm
2
) 



 52 

Chapter 3 

 

Carbon nanotubes: properties and applications 

 

In this chapter we analyse structure and properties of  carbon nanotubes. We focuse our 

attention on gas adsorption property, particularly on molecular hydrogen adsorption at low 

temperatures. Problems of Large Hadron Collider vacuum system are due mainly to H2 

desorption from chamber wall caused by synchrotron radiation; we studied gas adsorption 

properties to candidate carbon nanotubes as good criosorbers in future accelerators.  

 

 

3.1 Single-Wall  Carbon Nanotubes (SWCNT) 

 

Carbon exists in nature in several allotropic structures, depending on  the different regular 

atomic arrangement (fig. 3.1); the main carbon structures are: 

 

 Graphite: described by a planar geometry (hybridization sp
2
) 

 Diamond: characterized by a three-dimensional tetrahedral geometry (hybridization 

sp
3
) 

 Fullerene: formed by regular arrangement of hexagonal and pentagonal atomic 

structures. 

 

 

Fig.3.1.  Allotropic carbon structures. 

Graphite 

Diamond 
Fullerene C-60 



 53 

In 1991 the researcher Sumio Iijimaal observed that the product of an electric discharge 

between two carbon electrodes contained structures made of  concentric carbon tubes: the 

nanotubes [49, 50]. It is possible to distinguish two general groups of nanotubes: single-

walled (SWNT) and multi-walled (MWNT) nanotubes. 

An ideal SWNT can be described through a graphitic tube rolled up into itself  forming a 

cylinder, closed at the end by two hemispherical caps (fig.3.2). 

 

 

 

Fig. 3.2. Carbon SWNT structure 

 

The lateral wall of cylinder is made only of hexagons, while the two caps are constituted also 

by pentagons as fullerene(fig. 3.3). 

 

 

 

Fig. 3.3.  Ideal SWNT closed at the end by two fullerenes. 

 

SWCNT diameter is in the range of 0,7 nm (corresponding to twice the graphite interplanar 

distance) and 10 nm, but for their big ratio between diameter and length  SWCNT are 

considered as virtually monodimensional nanostructures. 



 54 

MWCNTs consist of many concentric SWCNT ; their diameter increases with walls number 

and it is about 10 nm (fig. 3.4). In MWCNT it is possible to observe  lip-lip interactions that 

stabilize the growth. 

 

Fig. 3.4.  MWCNT example  

 

The CNTs’ bonding mechanism is obviously similar to graphite’s one. Graphite is 

characterized by sp2 hybridization, which occurs when one s-orbital and two p-orbitals 

combine to form three hybrid sp2-orbitals (fig.3.5).  

 

 

Fig. 3.5. The bonding structure for one graphite plane: ζ-bonds connect carbon nuclei atoms (filled circles) in the 

plane and  π-bonds contribute to a delocalization. 

 

The strong covalent bond that binds the atoms in the plane (ζ-bond) produces the high 

stiffness and the high strength of CNts. The p-orbital, perpendicular to the plane, contributes 



 55 

to interlayer interaction forming a π-bond. This π-bond interacts with the π-bond in the 

neighboring layer ,but this interaction is weaker than a  ζ-bond. 

So the remaining p-electron of CNT contributes to a delocalized p-electron system [51]. 
 

Nanotubes often present structural defects as, for example, the presence of pentagonal and 

octagonal structure in the lateral wall of cylinder (fig. 3.6). 

 

 

Fig. 3.6.  Examples of defects on  SWNT nanostructure 

 

Figure 3.7 shows a generic graphitic plane rolled up; it is possible to observe the CNT axes 

(green line) and  1a , 2a   vectors of direct lattice. 

 

Fig. 3.7.  Graphite layer and possible rolling up directions. 

 



 56 

The main characteristics of SWCNT are due to chiral vector Ch, describing  the rolling up 

direction of graphite with respect to its axes and to nanotube length [59]. This vector links the 

two carbon atoms that superimpose themselves in nanotube formation and it is due to linear 

combination of 1a ed 2a : 

 

                                          21h a m   anC  , with m,n integers, 0≤ m ≤n                            (2.1) 

 

Based on simple geometry, the diameter d is given by L/π, where L is the length of carbon 

nanotube circumference: 

 

                                         22 mnmn
aCCCL

d
hhh

t 





 Å ,                      (3.2) 

 

 

where the lattice constant a=2.49 Å= 2211 aaaa   

The angle between chiral vector and the straight line defined by 1a  is the chiral angle θ and 

some nanotube properties depend on it. Because of the hexagonal symmetry and the 

honeycomb lattice 0≤θ≤30
o
 and it is described by the following equation: 

                                                     
22

1

1

2

2
cos

mnmn

mn

aC

aC

h

h







 .                                 (3.3) 

 

According to m and n there are several SWCNT typologies, particularly “zigzag” structures 

correspond to m = 0, while “armchair” one for m = n ; finally if m ≠ n (con m ≠ 0  and n ≠ 0) 

it is possible to observe “chiral ” structures (fig. 3.8). 

 

 

 

Fig. 2.8.  Examples of SWCNT structures: “chiral”, “armchair” and “zigzag”. 



 57 

In particular zigzag and armchair nanotubes correspond to θ=0
o
 and θ=30

o
, respectively.  

Among carbon nanotubes it is also possible to distinguish between achiral (symmorphic) or 

chiral (non-symmorphic). The former is a carbon  nanotubes whose mirror image has the 

same structure of the original one, while chiral nanotube mirror image cannot be 

superimposed on to the original one. The only two types of achiral carbon nanotubes are 

armchair and zigzag nanotubes. 

Another important vector characterizing carbon nanotubes is the translation vector T ,defined 

as the unit vector of a 1D carbon nanotube. The vector T is parallel to nanotube axis and it is 

normal to the vector Ch in the unrolled honeycomb lattice (fig. 3.7). 

The expression of vector T in terms of the basic vectors is [52]: 

 

                                                  2211 a t   atT  , with t1 and t2 integers.                             (3.4) 

 

From the condition  Ch · T = 0, is it possible to determinate the expression for t1 and t2: 

 

                                                                       

Rd

nm
t




2
1 , 

Rd

mn
t




2
2                                         (3.5) 

 

where dR is the greatest common divisor of (2m + n) and (2n + m). Defining d as the greatest 

common divisor of m and n, dR is given by: 

 








 3d of multiple a is m if 3d

3d of multiple a not is m- n if d
. 

 

The length of  the translation vector T is defined by 

                                                                         RdLT /3 ,                                                (3.6) 

 

where L is the circumferential nanotube length. 

The length of T is reduced when m and n have a common divisor or when (n-m) is a multiple 

of 3d.  

Furthermore, dividing the area of  nanotube unit cell TCh   by the area of a hexagon 

( 21 aa  ) it is possible to obtain the number of hexagons per unit cell (N) as a function of m 

and n: 

 



 58 

                                                    
 

RR

h

da

L

d

nmnm

aa

TC
N

2

222

21

22








 ,                         (3.7) 

 

where dR is given by equation (3.5). Each hexagon contains to carbon atoms, so there are N 

carbon atoms in each unit cell of the carbon nanotube. 

The coordinates of carbon atoms in a nanotubes unit cell are described by the vector R, 

expressed in terms of its projections on the orthogonal vectors Ch and T (fig. 3.9): 

 

                                                                      21 a    ap qR  ,                                               (3.8) 

where p and q are integers and they not have any common divisor except for unity. 

 

Fig. 3.9. The symmetry vector ψ denotes the angle of rotation around the nanotube axes  and η is the translation 

in the T direction. 

 

 

It is possible to note that  

                                                                     
T

RT

L

RCh





 ,                                             (3.9)  

where  

                                                                   2121 aaptqtRT                                  (3.10) 

  

and  ptqt 21   is an integer. 

It is important to choose p and q of R to form the smallest site vector (i = 1),such that 

                                                            

                                                               ptqt 21  =1,(0 < mp – nq ≤ N).                            (3.11) 

 



 59 

The vector R consists of a rotation around the nanotube axis, by an angle ψ, combined with a 

translation η in the direction of T, and reflect the basic space group symmetry operation of a 

chiral nanotube. 

The projection of R on Ch gives the angle ψ, while the projection of R in T represents the 

translation η of the basic symmetry operation of the 1D space group of the carbon nanotube.   

From definition of T, N, d and from the equation (3.11), it is possible to obtain the expression 

for the length of  η and the rotation angle ψ: 

 

                                               
   

N

Tnqmp

L

aanqmp

L

CR h 








21
                    (3.12) 

                                       

                                                  
 

N

a

L

ptqtd

LT

RT 


2

2

3

3

2 2

21 





 ,                      (3.13) 

 

where N is the number of hexagons in the 1D unit cell of the nanutube (eq. (3.7)). 

 

Fig. 3.10. The vector NR=(ψ│η)N on the cylindrical surface, in the case of Ch=(4,2) and M=6. 

 

 

The symmetry operator (ψ│η)
N
 brings the lattice point O to an equivalent lattice point C 

(fig.3.10); this mechanism is described by the equation below: 

 

                                                                           TMCRN h  ,                                       (3.14) 

 



 60 

where M = mp-nq is an integer that gives indication of the number of vector T that are 

necessary for reaching the distance from O to NR. 

After a rotation of 2π around the tube, the vector  NR reaches a lattice point C equivalent to 

O, but separated from it by the vector MT. 

The unit cell for a carbon nanotube in real space is represented by the rectangle generated by 

the chiral vector Ch and the translation vector T (fig. 3.11). 

 

 

 

Fig. 3.11. The unrolled honeycomb lattice of a nanotube. C h is the chiral vector, T the translational vector and R 

the symmetry vector of the nanotube. The rectangle OAB'B defines the unit cell for the nanotube. 

 

 

 

In the unit cell there are 2N carbon atoms, there are N pairs of π bonding and anti-bonding π
* 

electronic energy bands. 

Defining K2 as the reciprocal lattice vector and K1 as a vector that gives the discrete k values 

in the direction of Ch, their expression is obtained from the relation ijji 2 = K · R  , where Ri 

and Kj are, respectively, the lattice vectors in real and in reciprocal space.  

Using the definition of t1, t2 and N and the following relations 

                                                          21 KCh                  01 KT  

                                                                                                                   ,                         (3.15) 

                                                          02 KCh                    22 KT  

the expression for K1 and K2 are: 

                                                 21121
N

1
 = K btbt         212

N

1
 = K bnbm   ,                  (3.16) 

where  b1 and b2 are the reciprocal lattice vectors of two-dimensional graphite (fig. 3.12). 



 61 

 

Fig. 3.12. The Brillouin zone of a carbon nanotube, represented by the line segment WW’. 

 

The vector K1 and K2  are reciprocal lattice vectors corresponding to Ch and T, respectively. 

The first Brillouin zone of carbon nanotube is represented by the segment WW’. From 

equation (3.16), NK1 corresponds to a reciprocal lattice vector  of two dimensional graphite, 

so two wave vector which differ by NK1 are equivalent. The length of the one-dimensional 

first Brillouin zone , corresponding to the length of all the parallel lines in figure 2.12, is 2π/T. 

Because of the translational symmetry of T, there are continuous wave vectors in the K2 

direction for a nanotube of infinite length. It has been experimentally observed that for a 

nanotube of finite length L, the space between wave vector is 2π/L [52]. 

 

 

 

 

 

3.2 Electronic structure of Single-Wall Carbon Nanotubes 

 

The electronic structure of a single-wall carbon nanotube can be obtained simply from that of 

two-dimensional graphite. Using the periodic boundary condition for the chiral vector Ch, the 

wave vector associated to the Ch direction becomes quantized, while the wave vector 

associated to the direction of the translational vector T is continuous for a nanotube of infinite 

length. Thus the energy bands consist in a set of one-dimensional energy dispersion  relations. 

If Eg2D is the energy dispersion relations of two dimensional graphite, the 1D energy 

dispersion relations for single-wall carbon nanotubes are given by 

 

                         













 1

2

2
g2DE = kE K

K

K
k  ,   with µ=0,…..,N-1 and   –π/T < k <  π/T    (3.17) 



 62 

The N pairs of energy dispersion curves, described in the previous equation, correspond to the 

cross section of the two-dimensional energy dispersion surface, shown in figure 3.13. 

 

 

 

Fig. 3.13. The energy dispersion relation for 2D graphite in the whole region of the Brillouin zone 

 

 

For a particular (n,m) nanotube, if the cutting line passes through a K point of the 2D 

Brillouin zone, where the π and π
*
 energy band of 2D graphite are degenerated by symmetry, 

the 1D energy band has a zero energy gap (inset of figure 2.13). In this case the carbon 

nanotube are metallic and the density of state at the Fermi level  has a finite value, while if the 

cutting line does not pass through a K point it is expected that these nanotubes have a 

semiconducting behaviour with a finite energy gap between the valence and the conductance 

bands (fig. 3.14). 

 

 

 

Fig. 3.14. Map of chiral vectors (n,m): the carbon nanotube (n,m) metallic and semi-conductive are denoted , 

respectively, by open and solid circles. 



 63 

In particular  SWCNTs  can be characterized by electric properties of a semiconductor with an 

energetic gap in the range of (0,2 ÷ 1.2) eV in the band structure or of a conductor: nanotubes 

with armchair structure (n,n) have always metallic characteristic, while zigzag nanotubes 

(n,0) show a metallic behaviour if n is multiple of 3, otherwise they have semi-conductive 

properties. 

 

 

 

 

3.2.1 Energy dispersion relation of Armchair, Zigzag and Chiral nanotube 

  

To obtain explicit expression for the dispersion relations, it is useful to analyse the simplest 

case of nanotube having the highest symmetry. 

Starting from the energy dispersion relation from the two-dimensional graphite 

  

                            
2

1

2

g2D
2

cos4
2

cos 
2

3
cos41,E 


















































akakak
tkk

yyx

yx           (3.18) 

 

and using the discrete allowed values for the wave vector kx,q  

 

                                                            qakn qx 23 ,        (q=1,……,2n) ,                        (3.19) 

 

is it possible to obtain the energy dispersion relations  kE a

q for the armchair nanotube 

(Ch=(n,n): 

 

                                        
2

1

2a

q
2

cos4
2

cos cos41E 



































kaka

n

q
tk


,                (3.20) 

 

in which  -π< ka < π , q = 1,….2n, the superscript a indicates an armchair type and k is a one-

dimensional vector in the direction corresponding to the Г to K point vector in the two-

dimensional Brillouin zone of graphite (fig.3.15). 

 

 



 64 

 

 

Fig.3.15. One-dimensional energy dispersion relations for (a) armchair (5,5), (b) zigzag (9,0) and (c) zigzag 

(10,0) carbon nanotubes. The a-bands are no-degenerate ,while the e-bands are doubly degenerate at a general k-

point. x point for armchair and zigzag nanotubes correspond to ak   and  ak 3 , respectively. 

In figure a) it is possible to observe that armchair structure has semi-conductive characteristic, how confirmed by 

the absence of a gap; figure b) represents the properties of a semi-conductive zigzag nanotube (9,0), how showed 

by the small gap width(~ 0.2 eV); figure c) describes the semi-conductive behaviour of a zigzag nanotube (10,0) 

wit