Kinetics of X-ray conductivity for an ideal wide-gap semiconductor irradiated by X-rays

1. Volodymyr Ya. Degoda1)
Andrii Sofiienko2)
1)Kyiv National Taras Shevchenko University,
Department of Physics
Acad. Glushkova ave., 4, Kyiv 03680,
e-mail: degoda@univ.kiev.ua
2)University of Bergen,
Department of Physics and Technology
Norway, 5020 Bergen, 55 Allegaten str.,
e-mail: asofienko@gmail.com
Kinetics of X-ray conductivity
for an ideal wide-gap semiconductor irradiated
by X-rays

2. 22
Why the creation of a general kinetic theory of X-ray conductivity (XRC)
of the semiconductors and dielectrics is necessary?
1. The analysis of all processes of X-ray conductivity will allow to determine the most
relevant aspects for the development of novel detectors of the ionizing
radiation with wide operating temperature range and higher irradiation
stability.
2. To explain the specific experimental results of X-ray conductivity (XRC):
1) An anomalous XRC of ZnSe crystals, when sometimes the decreasing of the
current is observed at higher X-ray excitation;
2) A non-linear current-voltage XRC dependences.
Which characteristics of the semiconductors the kinetic theory of X-
ray conductivity should describe?
1. Spectrometric (amplitude and shape of the current pulse).
2. Integral (voltage-ampere and lux-ampere characteristics at the excitation by X-
rays).

3. 3
3
First, we need to choose a geometric and physical model of the object. The
geometric model can be following as shown below:
In accordance with X-ray absorption law, the spatial distribution of the generated
free electron-hole pairs is following:
0( ) X y
XdF y F e dy 
     0( , )
3
X yX
G X
g
h
N y t F t e
E
 
    
 0
0 1
3
X YX
GG
g
h
N F e Hd
E

  
Also we believe that the electrical contacts of the detector are ohmic.

4. 44
The material model can be gradually complicated step by step by the incorporation
of the point defects until it becomes adequate to real crystals:
- ideal semiconductor;
- semiconductor with shallow traps;
- semiconductor with shallow and deep traps and recombination centers.
The logic scheme for the development of X-ray conductivity kinetic theory
can have several stages and the following succession can be proposed:
I. Kinetics of X-ray conductivity at the absorption of one photon
1. Free hot carriers are generated after the absorption of X-ray photon;
2. Thermalization of hot carriers during inelastic scattering;
3. Current impulse in an ideal semiconductor;
4. Current impulse in a semiconductor with one type of shallow traps;
5. Current impulse in a semiconductor with different types of shallow traps;
6. Current impulse in a semiconductor with a recombination centers;
7. Current impulse in a semiconductor with shallow and deep traps and
recombination centers.

5. 5
5
II. X-ray conductivity kinetics at the X-ray excitation of the
semiconductors
1. Ideal semiconductor at the low excitation level;
2. Influence of the Coulomb interaction on the concentration of free-carriers;
3. Ideal semiconductor at the high excitation level;
4. Recombination of free electrons and holes;
5. Influence of shallow traps on the kinetics of XRC at the low excitation level;
6. Semiconductor with shallow traps at the high excitation level;
7. Deep traps and recombination centers. The stationary state;
8. The near-contact volume of the charges of XRC;
9. Accumulation of charge carriers on the deep traps;
10. VAC and LAC of the real semiconductor at steady-state excitation;
11. Kinetics of the rise and decay of the current of X-ray conductivity.

6. 6
6
Ideal semiconductor at the low excitation level
A concept of an ideal semiconductor means that such semiconductor doesn't have
any local centers affecting on the charge transport.
For determining the integral characteristics of XRC it is necessary to appoint the
spatial distribution of the generated free electrons and holes and the value of the
additional electric field created by them.
On the first stage we believe that direct recombination of free electrons with free
holes is extremely small (but this assumption can be useful only if there is no
external electric field end carriers drift to the contacts).
The kinetic equations system for an ideal semiconductor simplifies to two
equations for the free electrons and holes and the Poisson equation as follows:
 
 
 
0
G
G
N
N D N E N
t
N
N D N E N
t
e N N
E


 

   

   
 

       


       

 
 


7. 7
7
For low excitation level, when the additional electric field of free charge carriers can
be neglected, the following interchange is acceptable: ∂(E·N±)/∂x on E0·∂N±/∂x and
∂E/∂x=0 (the Poisson equation is not used). A solution of the kinetic equations
system for the selected geometry with boundary conditions N±(x=0)=0 and
N±(x=d)=0 for the spatial distribution of the concentration of free electrons and
holes will be following:
The application of the Einstein relation (µ±/D±=e/kT) enables to simplify the
expressions in the exponents and we can see that they are equal for electrons and
holes and even at small values U0 = E0∙d become considerable.
 
 
0
0
00
0
0
00
1 exp
,
1 exp
exp 1
,
exp 1
G
G
E x
N D
N x E d x
E dE
D
E x
N D
N x E x d
E dE
D














     
           
              

    
          
            

8. 8
8
Spatial distributions of electrons (a) and holes (b) in an ideal semiconductor with parameters of
ZnSe when NG = 1010 cm-3∙s-1, d = 1.0 cm and at different external electric field : 0.0 V/cm (1);
0.01 V/cm (2), 0.1 V/cm (3), 1.0 V/cm (4), 10 V/cm (5).
As seen, even if the electric field strength is greater than 1 V/cm then diffusive
motion is weak in comparison with the drift and can be neglected (but the
charge drift velocity is much smaller than the thermal velocity). The unequal
maximum values of the carriers concentrations are determined by the different values
of the diffusion coefficients of free electrons and holes (D-≠D+) and by their effective
masses.

9. 9
9
It is possible to determine a statistically mean lifetime of free electrons and holes as
follows:
The lifetime of free carriers in the sample is inversely related to E0 and is
determined by the drift time of the charge carriers to the electrical contact.
The total current of XRC can be computed as the additive sum of the currents of all
layers of a semiconductor:
   
1
0 02
2
6D U
d
 
  
 
 
0
0 0
0 0 0 0
1 exp
2
1 exp
Y Z
РП GG
eU
kTkT
i dy dz j j eN
eU eU
kT
 
  
   
     
       
 
Hence, VAC of an ideal semiconductor is conditional by the temperature and total
amount of generated free charge carriers. The obtained dependence has entirely
lost the material parameters and will be the same for all ideal semiconductors.
For the integral characteristics of XRC we have got linear LAC . VAC has a rapid
saturation and when U0 > 10 V it remains practically unchanged at the level of
eNGG.

10. 10
10
The current-voltage characteristics of an ideal semiconductor when
NGG = 1010 s-1 (1,3) and NGG = 108 s-1 (2,4) at 295 К (1,2) and 80 К (3,4).

11. 11
If E0 > 1 V/cm and if the diffusion motion can be neglected in the kinetic
equation, then the equation Eeh(x) is simplified to the following form:
The value of the electric field of free charge carriers can be computed by integrating
of the Poisson equation using the stationary spatial distributions of electrons and
holes:
 
   
0 0 0
1
2
x d
eh
x
e N N dx e N N dx
E x
 
   
  
   
 
 
  22
2
0 0 0
1
1 2 1
2 2
eh G
E x eN d x x
E E d d
  
    
  
   
        
            
        
The maximum value of the electric field Eeh(x) will be at x = d·(µ+/µ++µ-). Next figure
illustrates the relations |Eeh|/E0 at different values of NG. We can define such
concentration of free charge carriers at which their electric field Eeh is equivalent to
the external field E0.
When NG > 1012 cm-3·s-1 then it is necessary to take into account the electrical
field of free electrons and holes.

12. 12
12
The computed ratios Eeh(x)/E0 for the ideal ZnSe (a) in which the electrons
and holes mobilities are different in 25 times and for the semiconductor with
equal mobilities µ- = µ+ = 100 cm2V-1s-1 (b) if NG = 108 cm-3s-1 , d = 1 cm,
U0 = 1 V (1,1’); 3 V (2,2’); 10 V (3,3’); 30 V (4,4’); 100 V (5,5’)

13. The dependences of max|Eeh|/E0 for ZnSe (full
lines) and for a semiconductor with µ- = µ+ =
100 cm2V-1s-1 (dashed lines) on the applied
voltage to the contacts U0 and NG = 108 cm-3s-1
(1,1’); 1010 сm-3s-1 (2,2’); 1012 сm-3s-1 (3,3’);
1014 сm-3s-1 (4,4’); d=1 cm 13
The value of the free carriers
electric field is directly
proportional to the excitation
intensity. At strong X-ray
irradiation the generated field of
charge carriers is comparable
with the external electric field.
Then the spatial distributions of
the electrons and holes can be
determined by the full system of
kinetic equations and the
Poisson equation can’t be
neglected.

14. 14
14
The influence of the Coulomb interaction on the
concentration of free charge carriers
The main feature of XRC of the semiconductors at high excitation levels is the
necessity to take into account the Coulomb interaction between free electrons and
holes. Free charge carriers create an additional electric field and move themselves in
the diffusion-drifting way in this self-consistent field.
First we consider the case when there is no external electric field (E0 = 0), and then
we will define the spatial distributions N-(x) and N+(x), we should take into account
the field of free charge carriers Eeh(x) and the Coulomb interaction of free electrons
and holes. It’s obvious that Eeh=0 at any excitation level if µ- = µ+. The electric field of
free harge carriers will be caused only by different concentrations of electrons and
holes.
The Coulomb interaction between electrons and holes will slow down of the drift
motion of the electrons and will speed up the drift of the holes to the contacts. That
is, the internal field Eeh will cause the decrease in the free holes concentration and
will increase the concentration of free electrons and as a result the value of the field
Eeh will slightly decrease until a dynamic balance between the value of the field and
the equilibrium carriers density occurs.

15. 15
15
We can try to find an approximate solutions of the kinetic equations as simple
analytical functions. The changing of the concentration will increase monotonically at
the excitation intensity increment due to the Coulomb interaction and this change will
be given in the form of the addend that has functional dependence as bell-shaped.
Also we use a small difference between following functions:
where 0 < x < d. This allows to use the approximate relation for the Poisson equation:
4 1
x x x
Sin
d d d
   
    
   
 
2
1 1
8
G
N d x
N N Sin
D D d
 
 
   
      
   
And for the value of the electric field
of free charge carriers we have:
 
3
0
1 1
8
G
eh
eN d x
E x Cos
D D d

  
   
      
   
Spatial distribution of the electric field strength of free charge carriers in
ZnSe at NG = 108 cm-3s-1 and d = 1 cm, U0 = 0 V; the exact calculation
is a dashed line and the approximate calculation is a full line

16. 16
16
For the third addend of the kinetic equation for free carriers we also use the function
the free holes concentration in the form
in the kinetic equation, we get a system with two algebraic equations for defining of
the constant A+ and B+
Hence, such a functional relationship for the spatial distribution of the concentration
of free holes gives the correct tendency for changing their concentration. The same
result can be obtained for the concentration of free electrons, considering that B- = -
B+. Dealing with such functional relationship for the spatial distributions of
concentrations N-(x) and N+(x) we can obtain the values of concentrations that takes
into account the Coulomb interaction of the free charge carriers.
 
2
.
8
G
N d x
N x Sin
D d


 
   
 
    2 x
N x A d x x B Sin
d
    
    
 
of the carriers concentration in the form Substituting
2 2 4
0
2
1 1
2
64
G
G
N A D
eN d
B
d D D D
 

 


  


    
     
   
2 6
2
0
2
1 1
128
G
G
N
A
D
eN d
B
D D D

 




  


 
     
  

17. 17
17
where the constant b is computed as follows:
   
 
2
2
0 0 0
2
2
0 0 0
4 1 1 ,
8
4 1 1 ,
8
G
G
x x x x N d
N x N bSin N b Sin N
d d d d D
x x D x D x N d
N x N b Sin N b Sin N
d d D d D d D
 
 
   

 
   
  
      
           
      
        
            
       
2 1
2 2 2
0 0 0
2 4 2 4 2 4
1 4 1 1 8 1 8
3 3 2 1 1 1
4 2 2 2G G G
D kT D D kT D D D kT D
b
D e d N D e d N D D e d N
     

      
   
          
                   
          
The computed relative changing
in the concentrations of free
charge carriers b(NG) by virtue of
their Coulomb interaction as the
function of the intensity of their
generation in ZnSe and Si at
room temperature.

18. 18
18
Evidently, due to the Coulomb
interaction between free electrons and
holes at NG > 1010 cm-3s-1 their
concentrations become almost equal,
and the value of their electric field
reaches of the maximum value less
than 1 V/cm. It should be noted that
the physical basis of this phenomenon
is the same as the Dember effect. The
Coulomb interaction of the generated
free carriers causes the equalization
of electrons and holes concentrations
in an ideal semiconductor at
increasing of the excitation intensity,
and the diffusion motion of electrons
and holes is determined by the same
diffusion coefficient
D-1 = ½ ·[(D+)-1 + (D-)-1].
It is important for the case of
screening of the external field by free-
carriers.
The maximum value of the electric field
of free carriers as function of the intensity
of their generation in ZnSe and Si at
room temperature

19. 19
19
The ideal semiconductor at the high excitation level
The key feature of XRC of a semiconductor at the high excitation level is the
necessity to take into account the additional electric field created by the free charge
carriers that move in the drift-diffusion way in the self-consistent field. The integrated
XRC is usually experimentally studied at the high values of the external fields
(E0 > 10 V/cm), and that is why for the determining of N-(x), N+(x) and Eeh(x) we can
neglect of the diffusion component in the kinetic equation system
      2
2
, G
E x N xN x
D N
x x


 


 
The kinetic equation system will be simplified:
 
 
 
0
G
G
N
N E N
t x
N
N E N
t x
e N NE
x


 

 

 
 
 
   
 
 
   
 
 


In an ideal semiconductor the
steady state is reached very
quickly, and that is what we are
going to consider.

20. 20
20
After of the integrating with boundary conditions N+(x=0)=N-(x=d)=0 we obtain next
relations:
These relations are confirmed by the change conservation law, by virtue of the fact
that the value of the current density at XRC will be constant for each dy layer along
the direction of the current flux (OX axis):
     
       
G
G
G
G
N
E N N E x N x x
x
N
E N N E x N x d x
x




  

  


    


      

         G
j j j e N x E x e N x E x eN d Const x      
     
From the physical standpoint it is logical that the current density is determined only by
the number of carriers generated in a unit time in the layer, if the field is big enough to
neglect the diffusion losses of the current at the across the contacts and in the case
there are no recombination channels of the electron-hole pairs in the semiconductor
sample. The obtained relation can be inserted into the third equation (Poisson
equation) and for E(x) we get:
 0 0
1 1 2 1 1
2G G
E eN d eN d
x EdE x dx
x E x            
      
                   
22
2 2
2 1
0
2 1 1 1 1
2
GeN d x x
E C C
d d     
     
         
     
After of the integration:

21. 21
21
The first constant of integration C1 arises from the fact that the average charge of
the free carriers in terms of the sample is non-zero at the different mobilities of the
electrons and holes in semiconductors and dielectrics. And the total field around the
electrical contacts should differ from E0 by the equal value of E(x=0)-E0=E0-E(x=d)
which is determined by the total charge of the free carriers generated in the
semiconductor.
The second constant C2 is determined by two conditions: at the small excitation level
the total electric field tends to the value of the created external field, and the second
condition is caused by the possibility of the significant change of the electric field in the
semiconductor due to the high concentrations of free carriers. The external field is
brought about by the potential difference applied to the electrical contacts,
1
1 1 1
4
C
  
 
  
 
that is a negative quantity when µ- > µ+.
0
0
( )
d
E x dx Uthat is why the constant C2 is defined from
Simple approximated relation can be proposed for the spatial distribution of
total electric field E(x) inside a semiconductor as a result of the solving of the
Poisson equation:
 
24 4
0
2 2
0 0 0 0
1
1 2 1 2 1
8 2
G GU eN d eN d x x
E x
d U U d d
   
       
   
     
          
                 
            

22. 22
22
Normalized spatial dependences of the electric field E(x)d/U0 for an ideal semiconductor with equal (a) µ- =
µ+ = 100 cm2V-1s-1, ε=10 and different (b) µ- = 700 cm2V-1s-1, µ+ = 28 cm2V-1s-1, ε=8.66 (ZnSe) mobilities of
the electrons and holes (d=1 cm, U0=10 V) at different levels of the excitation intensity NG = 1010 cm-3s-1
(1,1’), 1011 cm-3 s-1 (2), 4∙1011 cm-3 s-1 (3), 3∙1010 cm-3 s-1 (2’), 6∙1010 cm-3 s-1 (3’)
If E(x) is known, we can obtain the spatial distributions of free carriers in first
approximation as follows:
 
 
 
24 4
0 2 2
0 0 0 0
4 4
0 2 2
0 0 0 0
1
1 2 1 2 1
2
1 2 1
G
G G
G
G G
N d x
N x
eN d eN d x x
U
d dU U
N d d x
N x
eN d eN d x
U
dU U
   

       
 

     

   

     

 

   


          
                
              


    
       
     
2
1
2 1
2
x
d
 
 
 
 
    
      
     

23. 23
23
Spatial distributions of the concentrations of
electrons (1,1’) and holes (2,2’) in the ideal ZnSe
semiconductor when NG = 5∙1010 сm-3 s-1 (1,2)
and in the semiconductor with µ- = µ+ = 100
cm2V-1s-1, ε=10 when NG = 4∙1011 cm-3 s-1
(1’,2’); d=1 cm, U0=10 V
The spatial distributions of the
concentrations of the electron and holes
at high excitation level assume a bell-
shaped form and we cannot neglect the
diffusion motion of the carriers anymore,
because the condition
      2
2
, G
E x N xN x
D N
x x


 


 
is not used.
For the high-power
excitations
2
0 0
4
8
G
U
N
ed
 

it is necessary to consider the case of
screening of the external electric field by
the free charge carriers. And still should
be remembered that in the ideal
semiconductor at E=0 we can observe
the electrons and holes concentrations
leveling due to their Coulomb
interaction.

24. 24
24
The general kinetic equation system for electrons and holes remains unchanged. The
approximate solution at the high excitation level, when the electric field in the central
part of the sample reaches zero will be following:
where relations for the electrons and holes f1e(x)=f1h(x) will be equal and symmetric at
x=d/2 and also are bell-shaped, and the values of the concentrations will be
determined as the average diffusion coefficient. The functions f2e(x) and f2h(x) describe
the spatial screening distributions of the concentrations of electrons and holes near of
the electrodes. Since f1e(x)=f1h(x) than these spatial distributions of carriers do not
create the electric field and for the Poisson equation can be written the following:
           1 2 1 2
,h h e e
N x f x f d x N x f x f x 
    
 2
0
e
E e
f x
x 



 2
0
h
E e
f x
x 



0,E  0
E
x



Such simplifications allow to write the kinetic equation for the stationary distribution of
the electrons or holes as follows:
- in the central
part of the sample
- Close to the positive electrode,
- Close to the negative electrode
 
   
2
1 2
1 22
0G
f f
D E x f f N
x x
   
       

25. 25
25
that can be divided into two parts
and
2
1
2
0G
f
D N
x
 
 

 
2
2
1 22
0
f
D E f E f
x x
  
    
 
Due to the Coulomb interaction between the free charge carriers of the opposite
signs, in the first equation D+ and D- should be changed on the effective values.
 
2
1
1
2
G
N d x x
f x
d dD
 
   
 
1
0 0 0
2 2 2
1
0 0 0
2 2 2
2
exp
4
2
exp 1
4
e
h
E kT eU x
f
e x e d kT d
E kT eU x
f
e x e d kT d
 
 


   
        
     
             
and
The values of the free charge carriers concentrations in the screening layer are
determined only by the values of the applied potential difference and temperature.
And the total electric field in the sample of the ideal crystal at the high excitation level
will be:
 
2 1 1
e e
kT
E x
e d l x l x
 
  
   
where
0
0
exp
4
exp 1
4
e
d eU
l d
eU kT
kT
 
    
    
 

26. 26
26
The value of the electric field, that is close to the contacts, is much larger than the
average value of the electric field U0/d with the screening of the external field by the
free charge carriers at intensive excitation.
Total concentrations of free electrons and holes will be following:
Hence, at the high excitation level, the generated free charge carriers will shield of
the external electric field. As a result, the concentrations of the carriers in the central
part of the sample can be very greater in proportion to the excitation intensity. And the
diffusion kind of motion becomes dominating and also it is the same for electrons and
holes with effective diffusion coefficient D.
The integral current-voltage and lux-ampere characteristics of X-ray conductivity of
the ideal semiconductor will remain linear, because the number of the generated
carriers per unit of time will be equal to the number of carriers that reach the electric
contacts and recombine on them.
 
 
12
0 0
2 2
12
0 0
2 2
2
1 exp 1
42
2
1 exp
42
G
G
N d kT eUx x x
N x
d d e d kT dD
N d kT eUx x x
N x
d d e d kT dD






     
           
     
   
         
    

27. 27
27
Recombination of free electrons and holes
In an ideal semiconductor model, there is only one fundamental possibility for the
disappearance of the generated free charge carriers in the sample. It is the
recombination of free electrons and holes. The probability of this process (r0) is very
small and in the real semiconductors with very low concentration of recombination
centers (~1012 cm-3) such process can be neglected in contrast to the probability of
recombination on the recombination centers. Also, we consider the potential effect of
the direct recombination of electrons and holes in the spatial distribution of the free
carriers equilibrium concentrations. This process leads to the occurrence of the
summand r0N-N+, that basically changes kinetic equations for free carriers:
 
 
 
2
02
2
02
0
G
G
N N
N D E N r N N
t x x
N N
N D E N r N N
t x x
e N NE
x


 
 
    
 
    
 
  
      
  
  
      
  
 


Parameter r0 is the product of the recombination cross-section of free electrons on
their thermal velocity: r0 = σ0∙vT ≈ 3·10-14 cm3 s-1.

28. 28
28
Hence, the direct recombination of free electrons and holes has influence upon the
spatial distributions of carriers only at high excitation levels (NG > 1014 см-3 с-1), that
causes the reduction in current of XRC. The intensity of current is directly proportional
to N±
max and the relation of LAC for current XRC will follow the relation N±
max(NG).
This equation can not be solved by using of simple analytical functions, but we can
try to find the approximate solution in the following form:
Obviously, this process may be observed only at the high excitation levels, when the
concentrations of free carriers in the ideal semiconductor increase up to ~1014 сm-3.
At such concentrations of N+(x) and N-(x) the screening of the external electric field is
observed, so we should consider the kinetic equation:
2
21
0 12
0G
f
D r f N
x

  

    2
1
x
f x A d x x C Sin
d
 
    
 
and
2
~
4
Ad
C
As a result here is an approximate solution for the spatial distributions of the
concentration of free charge carriers:
 
2 2 6
2
30
2
1
2 256
G G
N d x x N d x
N x r Sin
d d dD D


    
     
   

29. 29
29
The calculated relations N±
max(NG) (a) and the spatial distributions of the
concentrations N±(x) (b) when NG > 1016 cm-3 s-1 without (1) and with (2) the direct
recombination of free electrons and holes (r0 = 3·10-14 cm3s-1) when d = 1 cm, D =
1.35 cm2s-1.

30. 30
30
Conclusions
Analysis of the X-ray conductivity for an ideal semiconductor shows the necessity
to consider thef X-ray excitation according to its intensity: low, medium and high
level.
At the low excitation level, when the time interval between the two successive
acts of absorption of X-ray quanta is greater in many times than the drift time of
the charge carriers to electric contacts of the sample, we can consider the
kinetics of X-ray conductivity as the additive sum of current pulses when one X-
ray photon is absorbed. In this case, the integral characteristics are determined
by the additive sum.
At the medium excitation level of an ideal semiconductor when the macroscopic
interaction between free charge carriers generated by the absorption of different
X-ray photons occurs, and this interaction is caused by change of the electric
field E(x) ≠ Const, the consideration of spectrometric characteristics makes no
sense. And the integral characteristics are determined by the spatial distributions
N+(x), N-(x) and E(x) provide for LAC the linear dependence on the excitation
intensity, and for VAC it’s the characteristic dependence on the current saturation
of the XRC.

31. 31
31
At the high excitation level there is a screening of external electric field by the
charge carriers in the middle of the sample and the two barriers around
electrical contacts occur. The barriers are transparent for the carriers of one
sign and nontransparent for the carriers of the opposite sign. At that the integral
characteristics are also determined by the spatial distributions N+(x), N-(x) and
E(x) and provide the linear dependence for LAC and the indicative VAC for the
ideal semiconductor.
At extremely high excitation levels (NG > 1015 см-3 с-1) the process of the direct
recombination of free electrons and holes can be observed that changes the
linear dependence of LAC on the excitation intensity into sublinear. At that, the
character of VAC remains unchanged but the level of current saturation of the
XRC decreases.

32. 3232