# Patent

## Title
Pulse sequences for exciting nuclear quadrupole resonance

## Abstract

An apparatus and method for producing a multi-pulse sequence for irradiating a substance provided with quadrupole nuclei with either integer or half-integer spins to detect an NQR signal emitted therefrom. The apparatus has pulse sequence generating means adapted to produce multi-pulse sequences with pulse intervals exceeding T2* and containing a preparatory pulse or group of pulses for creating echo signals between the pulses. The pulse sequence is organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of pulses.

## Background
BACKGROUND OF THE INVENTION
This application claims priority to Australian Provisional Patent Application No. PS 3121 filed on Jun. 21, 2002 and to International Application No. PCT/AU03/00776 filed on Jun. 20, 2003.
FIELD OF THE INVENTION
The present invention relates to the practical use of the nuclear quadrupole resonance (NQR) phenomenon for identifying substances that contain quadrupole nuclei, particularly for identifying explosive or narcotic substances.
The invention has particular utility in multi-pulse radio frequency (RF) excitation of quadrupole nuclei and to the subsequent measurement of the NQR signal emitted therefrom where changes in temperature can effect measurement.
Throughout the specification, unless the context requires otherwise, the word “comprise” or variations such as “comprises” or “comprising”, will be understood to imply the inclusion of a stated integer or group of integers but not the exclusion of any other integer or group of integers.
The term “preparatory pulse” means both a separate preparatory pulse and a group of preparatory pulses.
Furthermore, “a group of preparatory pulses” means a group of pulses that precede a multi-pulse sequence distributed within a time interval <3T2 (T2 being the time of dipole-dipole relaxation), during which the NQR signal, as a rule, is not measured.
BACKGROUND ART
The following discussion of the background art is intended to facilitate an understanding of the present invention only. It should be appreciated that the discussion is not an acknowledgement or admission that any of the material referred to is or was part of the common general knowledge as at the priority date of the application.
To detect explosive and narcotic substances by means of multi-pulse NQR, steady state free precession (SSFP) type pulse sequences are usually preferred. The principal reasons for this preference are as follows:

    1. It is possible to receive a continuous chain of signals if the requirement






  
    
      
        
          n
          ·
          
            ω
            eff
          
        
        ≠
        
          m
          ·
          
            π
            τ
          
        
      
    
    
      
        (
        1
        )
      
    
  




     is met. Here, τ is the pulse spacing of the sequence, n and m are whole numbers, ωeff represents the effective field which substitutes the effect of the RF pulses and the resonance offset, ensuring unlimited time for signal accumulation.
    2. It is possible to receive an NQR signal phase that is different from the phase of the irradiating pulses, which can be used for cancelling intensity anomalies, or for subtracting spurious signals.
    3. Comparatively little RF power is required for detecting samples in large volumes


When requirement (1) is met, the SSFP sequences allow achievement of a greater signal-to-noise ratio per unit of time than any other multi-pulse sequences used for exciting the quadrupole spin system.
However, complying with condition (1) cannot be guaranteed in practice because the exact value of the resonance offset in most cases is unknown due to the fact that the exact temperature of the sample is not known either.
The dependence of the signal intensity on the resonance offset when using the SSFP sequences is characterised by the existence of intensity anomalies.
In the solid state when irradiating sequence parameters approach the resonance conditions, intensity anomalies are manifested specifically by the the amplitude reducing and damping of the signal accelerating as indicated by the equation:




  
    
      
        
          
            n
            ·
            
              ω
              eff
            
          
          =
          
            m
            ·
            
              
                π
                τ
              
              .
            
          
        
        ,
      
    
    
      
        (
        2
        )
      
    
  




which can result in a sharp decline in the signal intensity or even in the complete loss of information about the presence (or absence) of the sample in the examined volume.

The temperature dependence of the resonance frequencies of quadrupole nuclei in a number of substances is quite considerable. For example, for RDX at frequency v+=5.192 MHz and at temperatures close to room temperature, the change in 14N resonance frequency is −520 Hz/° K. For PETN at 14N frequency v+=890 kHz, it is −160 Hz/° K; and for KNO3 at nitrogen line v+=567 kHz, it is −140 Hz/° K.
The 14N NQR intensity anomalies were first demonstrated using a basic SSFP sequence of identical coherent RF pulses.
For the purpose of detecting explosive and narcotic substances, the SSFP type pulse sequences are followed by echo sequences. Multi-pulse versions of echo sequences refer to the class of saturating sequences, which exist because of the condition of the inequality T2*<τ<<T1.
Saturating properties of this type of sequence cause total decay of the NQR signals at any frequency offset and flip angle. The time of the signal decay is determined by the time constant T1e (“effective relaxation time”) and has a value within the limits T2≦T1e<T1 (or, to be more exact, T2≦T1e<T1ρ, where T1ρ is the relaxation time in a rotating frame of axis and where T1ρ should always be less than T1 (T1ρ<T1)).
In a number of cases, using echo sequences is preferable to using the SSFP-type sequences. This is true when working with high Q detectors, where the “dead time” of the receive system of the spectrometer is comparable in value with T2* time, and there exists the possibility to generate echo-signals which exceed the “dead time”. Hence echo sequences considerably increase the detection capability while reducing the requirements on the Q-switch system.
However, up to this point, intensity anomalies of echo signals after irradiation with multi-pulse sequences for the purpose of obtaining NQR have not been studied.
DISCLOSURE OF THE INVENTION
The principal object of the present invention is to increase the accuracy of detection of certain prescribed substances in specimens, compared with the previously known methods of detecting same using NQR, by reducing temperature effects. Such prescribed substances may include, but are not limited to, certain explosives and narcotics.
As temperature effects are based on the intensity anomalies effect, a preferred object of the invention is to increase the accuracy of detecting certain prescribed substances such as explosives and narcotics by reducing the intensity anomalies effect.
The purpose of the invention is generally achieved by using multi-pulse sequences with pulse intervals exceeding T2* and containing a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses, organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses.
The invention arises from the discovery, both theoretically and experimentally, that echo-sequences under certain conditions practically do not create intensity anomalies, and this permits detecting substances of interest in less time and with greater reliability.
In practical terms, the invention has great utility in the detection of substances containing nitrogen nuclei 14N with long spin-lattice relaxation times T1, although the invention is not limited to this area.
In accordance with a first aspect of the present invention, there is provided a nuclear quadrupole resonance apparatus for detecting a target substance containing quadrupole nuclei, comprising:

    
    
        a pulse sequence generator to generate a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2*, and to generate a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences;
        a detector to detect nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
        a combiner to combine the detected nuclear quadrupole resonance signals generated under conditions of unknown temperature of the target substance into a resulting signal such that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses, thereby mitigating the effect of intensity variations.
    
    


In accordance with a second aspect of the present invention, there is provided a method of nuclear quadrupole resonance for detecting a target substance containing quadrupolar nuclei, the method comprising the steps of:

    
    
        generating a combination of multi-pulse sequences, at least one of which contains a preparatory pulse or group of preparatory pulses, having intervals between pulses longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2*;
        irradiating the target substance with the combination of multi-pulse sequences;
        detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
        combining the detected nuclear quadrupole resonance signals generated under conditions of unknown temperature of the target substance into a resulting signal where echo generated by the preparatory pulse or group of preparatory pulses is a prevailing component, thereby mitigating the effect of intensity variations.
    
    


Preferably, the combination contains at least one sequence.
Preferably, the flip angles of pulses of sequences of the combination range between 60° and 345°
Preferably, at least one preparatory pulse is composite.
Preferably, all pulse sequences of the combination contain a preparatory pulse or a group of preparatory pulses.
Preferably, the carrier frequency of the pulse sequences is near to one of NQR frequencies of the target substance to be detected.
Preferably, the target substance to be detected comprises an explosive containing quadrupole nuclei.
Alternatively, it is preferred that the target substance to be detected comprises a narcotic containing quadrupole nuclei.
In accordance with a third aspect of the present invention, there is provided an apparatus for producing a multi-pulse sequence for irradiating a target substance containing quadrupole nuclei, comprising:

    
    
        a generator to generate a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2*, and containing a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses; and
        a detector to detect nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences.
    
    


In accordance with a fourth aspect of the present invention, there is provided a nuclear quadrupole resonance apparatus for detecting a target substance containing quadrupole nuclei, the apparatus comprising:

    
    
        means for generating a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2*, and for generating a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences;
        means for detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
        means for combining the detected nuclear quadrupole resonance signals into a resulting signal such that the resulting signal contains a prevailing echo component produced by the
        preparatory pulse or group of preparatory pulses, thereby mitigating the effect of intensity variations.
    
    


In accordance with a fifth aspect of the present invention, there is provided an apparatus for producing a multi-pulse sequence for irradiating a target substance provided with quadrupole nuclei, comprising:

    
    
        generator means for generating a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2 and containing a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses; and
        detector means for detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences.
    
    


In accordance with a sixth aspect of the present invention, there is provided a pulse sequence signal for irradiating a target substance provided with quadrupole nuclei, the pulse sequence signal comprising:

    
    
        a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2; and
        a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences, the pulse sequence signal being arranged for irradiating a target substance under conditions of unknown temperature of the target substance, and operable to mitigate the effect of intensity variations.
    
    


In accordance with a seventh aspect of the present invention, there is provided a pulse sequence for irradiating a target substance provided with quadrupole nuclei, the pulse sequence comprising:

    
    
        a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2; and
        a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences, the pulse sequence being arranged for irradiating a target substance under conditions of unknown temperature of the target substance, and operable to mitigate the effect of intensity variations.
    
    





BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be better understood in relation to the following description of several embodiments thereof. The description is made with reference to the following drawings, wherein:
FIG. 1 is a timing diagram showing the spin locking of the spin echo pulse sequence (SLSE) with all of the relevant parameters designated;
FIG. 2a is a graph showing the dependence of the calculated amplitude of the NQR signal plotted with respect to resonance offset in kHz, after a Fourier transform of the difference between the carrier frequency of the RF pulses of the SLSE sequence and the frequency of the resonance transition, if the sequence duration is  ̃T2; and FIG. 2b shows a similar graph of the same dependence, but the situation if the duration of the sequence is >>T2;
FIGS. 3a and 3b show the experimental dependence of the amplitudes of the NQR Fourier transform of the difference between the carrier frequency of the RF pulses of the SLSE sequence and the frequency of the resonance transition received at room temperature and the same parameters of the SLSE as the theoretical curves in FIGS. 2a and 2b respectively, wherein FIG. 3a depicts a received signal from an HMX sample and FIG. 3b depicts a received signal from NaNO2 powder;
FIG. 4 is a timing diagram showing an example of determining the acquisition window for an echo pulse sequence; and
FIGS. 5a, 5b and 5c show echo signals received after using sequences SLSE(φ0)0°−τ−(φ90°−2τ[90°])n1, MW-2 (φ0)0°−τ−(φ0°−2τ−φ180°−2τ[180°])n2 and WAHUHA-4(φ0)0°−τ−(φ180°−τ−φ90°−2τ−φ270°−τ−φ0°−2τ[0°])n3, respectively, for NaNO2 powder, wherein the acquisition time is set at the end of the sequence cycles at equal intervals from both the preceding and the posterior pulses for the purposes of isolating the echo part of the signal.



BEST MODE(S) FOR CARRYING OUT THE INVENTION
In order to justify the use of echo signals in multi-pulse sequences for eliminating intensity anomalies and thus provide for the best mode for carrying out the invention, a simplified theoretical analysis will be undertaken of the dependence of the echo signal on the frequency offset in the spin locking spin echo (SLSE) sequence represented by:

φ0x−(τ−φy−τ)n,

where the bottom index at the flip angle sign φ designates the phase of the carrier frequency for the RF pulse, n is the number of cycles of the sequence, and τ is the time interval between the preparatory pulse and the first pulse of the sequence. The results of this analysis will then be compared with experimental data to verify the correlation.

Thus the analysis is carried out in accordance with the following procedure.
Firstly, the echo signal in the sequence (φ0)0°−(τ−φ90°−τ)n is determined for the times t≦T2. The “spin locking” effect of the echo signal is then considered in terms of the concept of the effective field of a multi-pulse sequence. The essence of the effective field concept is to replace the effects of the real RF pulses and the resonance offset by a certain fictitious uninterrupted field. Using this concept, the echo signal in the interval between the n and n+1 pulses is determined, after which the Fourier transform of the signal is found and powder averaging is performed.
After that the echo signal is determined for the times t>T2 and N signal averages are carried out (N being the number of pulses in a pulse sequence) taking into account the spin-lattice relaxation. The dependence of the resulting signal against the resonance offset is plotted and the obtained dependence is compared with experimental results.
Now having regard to a crystalline sample containing spin-1 nuclei. It is assumed that the quadrupole principal axes for each of the spins have the same orientation. The quadrupole Hamiltonian in the quadrupole principal axes frame is given as:




  
    
      
        
          
            H
            q
          
          =
          
            
              
                
                  e
                  2
                
                ⁢
                q
                ⁢
                
                  
                
                ⁢
                Q
              
              
                4
                ⁢
                
                  I
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        I
                      
                      -
                      1
                    
                    )
                  
                
              
            
            ⁡
            
              [
              
                
                  3
                  ⁢
                  
                    I
                    z
                    2
                  
                
                -
                
                  I
                  2
                
                +
                
                  η
                  ⁡
                  
                    (
                    
                      
                        I
                        x
                        2
                      
                      -
                      
                        I
                        y
                        2
                      
                    
                    )
                  
                
              
              ]
            
          
        
        ,
      
    
    
      
        (
        1
        )
      
    
  




where e2qQ is the nuclear quadrupole coupling constant, and η is the asymmetry parameter of the electric field gradient tensor.

In terms of the fictitious spin-1⁄2 operators Hq takes the form:




  
    
      
        
          
            H
            q
          
          =
          
            
              
                
                  
                    e
                    2
                  
                  ⁢
                  q
                  ⁢
                  
                    
                  
                  ⁢
                  Q
                
                4
              
              ⁢
              
                (
                
                  
                    I
                    
                      x
                      ,
                      3
                    
                  
                  -
                  
                    I
                    
                      y
                      ,
                      3
                    
                  
                  -
                  
                    I
                    
                      z
                      ,
                      3
                    
                  
                
                )
              
            
            =
            
              
                
                  ω
                  p
                
                ⁢
                
                  I
                  
                    p
                    ,
                    3
                  
                
              
              +
              
                
                  ω
                  p
                  ′
                
                ⁢
                
                  I
                  
                    p
                    ,
                    4
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        with
      
    
    
      
        (
        2
        )
      
    
  
  
    
      
        
          
            ω
            x
          
          =
          
            
              
                
                  e
                  2
                
                ⁢
                q
                ⁢
                
                  
                
                ⁢
                Q
              
              4
            
            ⁢
            
              (
              
                η
                +
                3
              
              )
            
          
        
        ,
        
          
            ω
            y
          
          =
          
            
              
                
                  e
                  2
                
                ⁢
                q
                ⁢
                
                  
                
                ⁢
                Q
              
              4
            
            ⁢
            
              (
              
                η
                -
                3
              
              )
            
          
        
        ,
        
          
            
              ω
              z
            
            =
            
              
                -
                
                  
                    
                      e
                      2
                    
                    ⁢
                    q
                    ⁢
                    
                      
                    
                    ⁢
                    Q
                  
                  2
                
              
              ⁢
              η
            
          
          ;
        
      
    
    
      
        (
        3
        )
      
    
  
  
    
      
        
          
            ω
            x
            ′
          
          =
          
            
              
                
                  e
                  2
                
                ⁢
                q
                ⁢
                
                  
                
                ⁢
                Q
              
              4
            
            ⁢
            
              (
              
                η
                -
                1
              
              )
            
          
        
        ,
        
          
            ω
            y
            ′
          
          =
          
            
              
                
                  e
                  2
                
                ⁢
                q
                ⁢
                
                  
                
                ⁢
                Q
              
              4
            
            ⁢
            
              (
              
                η
                +
                1
              
              )
            
          
        
        ,
        
          
            ω
            z
            ′
          
          =
          
            
              -
              
                
                  
                    e
                    2
                  
                  ⁢
                  q
                  ⁢
                  
                    
                  
                  ⁢
                  Q
                
                2
              
            
            ⁢
            η
          
        
        ,
        
          

        
        ⁢
        or
      
    
    
      
        (
        4
        )
      
    
  
  
    
      
        
          ω
          p
          ′
        
        =
        
          
            1
            3
          
          ⁢
          
            
              (
              
                
                  ω
                  q
                
                -
                
                  ω
                  r
                
              
              )
            
            .
          
        
      
    
    
      
        (
        5
        )
      
    
  




Here the Hamiltonian Hq is written in frequency units (h=1), p,q,r=x,y,z or cyclic permutation.

The three parameters ωx, ωy and ωz, which have the dimension of angular frequency, are linked with the three transition frequencies ω+, ω− and ω0 between the quadrupolar energy level equalities ωx=ω+, ωy=−ω−, ωz=−ω0. The parameters ωx, ωy, ωz are treated as being equivalent to the resonant frequencies ω+, ω−, ω0, which will be described in more detail later, where one of these three transitions is considered with an arbitrary frequency ωp.
In this simplified one-particle analysis, all kinds of multi-particle interactions are neglected, as is the spin-lattice relaxation.
The interaction of quadrupolar nuclei with the radio frequency field is described with an RF Hamiltonian as follows:




  
    
      
        
          
            H
            1
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              -
              2
            
            ⁢
            
              γ
              ·
              
                
                  H
                  1
                
                ⁡
                
                  (
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        X
                      
                      ⁢
                      
                        I
                        
                          x
                          ,
                          1
                        
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        Y
                      
                      ⁢
                      
                        I
                        
                          y
                          ,
                          1
                        
                      
                    
                    +
                    
                      

                    
                    ⁢
                    
                      
                        +
                        cos
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        θ
                        Z
                      
                      ⁢
                      
                        I
                        
                          z
                          ,
                          1
                        
                      
                    
                  
                  )
                
              
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  φ
                
                )
              
            
          
          =
          
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  φ
                
                )
              
            
            ⁢
            
              
                ∑
                
                  
                    m
                    =
                    x
                  
                  ,
                  y
                  ,
                  z
                
              
              ⁢
              
                
                  a
                  m
                
                ⁢
                
                  I
                  
                    m
                    ,
                    1
                  
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




where ax=γH1·sin ΘL·cos φL, ay=γH1·sin ΘL·sin φL, γ is the gyromagnetic ratio, H1 is the amplitude of the RF field, φ is its initial phase, ΘL and φL are the polar and azimuthal angles of the field vector in the principal axes frame, and ω—is the carrier frequency of the RF pulses.

Now considering a transformation into the interaction representation defined by the Hamiltonian:




  
    
      
        
          H
          0
        
        =
        
          
            
              ω
              ·
              
                I
                p
              
            
            +
            
              
                W
                p
              
              ⁢
              
                I
                
                  p
                  ,
                  4
                
              
            
          
          =
        
      
    
    
      
        (
        7
        )
      
    
  
  
    
      
        =
        
          
            
              
                ω
                q
              
              ⁢
              
                I
                
                  q
                  ,
                  3
                
              
            
            +
            
              
                W
                q
              
              ⁢
              
                I
                
                  q
                  ,
                  4
                
              
            
          
          =
        
      
    
    
      
        (
        8
        )
      
    
  
  
    
      
        =
        
          
            
              ω
              r
            
            ⁢
            
              I
              
                r
                ,
                3
              
            
          
          +
          
            
              W
              r
            
            ⁢
            
              
                I
                
                  r
                  ,
                  4
                
              
              .
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




Here

Wp=(ωq−ωr)/3; Wq=(ωr−ω)/3; Wq=(ω−ωq)/3.  (10)

The secular independent of the time part of the RF Hamiltonian in the H0 representation looks as follows:

{tilde over (H)}1=exp(−iH0t)H1(t)exp(iH0t)=−ω1[cos φ·Ip,1+sin φ·Ip,2],  (11)

where (ω1=ap).

In the H0 representation the evolution of the density matrix {tilde over (p)} ({tilde over (p)}(t)=exp(−iH0t)ρ(t)exp(iH0t)) during the impact of the RF pulses is described by the equation




  
    
      
        
          
            i
            ⁢
            
              
                d
                
                  ρ
                  ~
                
              
              
                d
                t
              
            
          
          =
          
            
              [
            
            ⁢
            
              
                H
                ~
              
              1
            
          
        
        ,
        
          
            ρ
            ~
          
          ⁢
          
            ]
          
        
      
    
    
      
        (
        12
        )
      
    
  




and in the intervals between the pulses it is described by





  
    
      
        
          i
          ⁢
          
            
              d
              
                ρ
                ~
              
            
            
              d
              t
            
          
        
        =
        
          
            [
            
              
                Δ
                ⁢
                
                  
                
                ⁢
                ω
                ⁢
                
                  
                
                ⁢
                
                  I
                  
                    p
                    ,
                    3
                  
                
              
              ,
              
                ρ
                ~
              
            
            ]
          
          .
        
      
    
    
      
        (
        13
        )
      
    
  




Here and later on, all calculations and results will be done in H0 representation, so the tilde notation will be dropped.
Parameters of the SLSE are shown in FIG. 1.
It should be noted that in this case, tω<<τ.
The interaction with the k-pulse of the pulsed sequence is:

Hrf=−f(t)(Ip,1 cos φk+Ip,2 sin φk)  (14)

where φk is the phase of the k-pulse of the sequence; and f(t) is the pulsed function:





  
    
      
        
          f
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            ω
            1
          
          ⁢
          
            
              ∑
              
                k
                =
                0
              
              ω
            
            ⁢
            
              
                δ
                ⁡
                
                  (
                  
                    τ
                    +
                    
                      2
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      τ
                    
                    -
                    t
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




The equation (12) may be rewritten in the form:





  
    
      
        
          i
          ⁢
          
            
              d
              ρ
            
            
              d
              t
            
          
        
        =
        
          [
          
            
              
                ω
                e
              
              ⁢
              
                I
                p
              
            
            ,
            ρ
          
          ]
        
      
    
    
      
        (
        16
        )
      
    
  




Here ω=ωee, e=(ex,ey,ez) is the unit vector:




  
    
      e
      x
    
    =
    
      -
      
        
          
            
              ω
              1
            
            ·
            cos
          
          ⁢
          
            
          
          ⁢
          
            φ
            k
          
        
        
          ω
          e
        
      
    
  
  ,
  
    
      e
      y
    
    =
    
      -
      
        
          
            
              ω
              1
            
            ·
            sin
          
          ⁢
          
            
          
          ⁢
          
            φ
            k
          
        
        
          ω
          e
        
      
    
  
  ,
  
    
      e
      z
    
    =
    
      
        Δ
        ⁢
        
          
        
        ⁢
        ω
      
      
        ω
        e
      
    
  
  ,
  
    
      
        ω
        e
      
      =
      
        
          (
          
            
              ω
              1
              2
            
            +
            
              Δω
              2
            
          
          )
        
        
          1
          2
        
      
    
    ;
  




and Ip is the vector in Ip,i-operator space.

Before the impact of the RF pulses, the spin system is in equilibrium and the density matrix is described by the expression:




  
    
      ρ
      0
    
    =
    
      
        N
        
          
            2
            ⁢
            I
          
          +
          1
        
      
      ⁢
      
        exp
        ⁡
        
          (
          
            -
            
              
                H
                q
              
              kT
            
          
          )
        
      
    
  
  ,




where N is the number of units in the system; k is the Boltzmann's constant; and I is the spin of the nuclei.

In the high-temperature approximation for nitrogen-14, I=1, and consequently:




  
    
      
        
          ρ
          0
        
        =
        
          
            
              N
              3
            
            ⁢
            
              (
              
                E
                -
                
                  
                    H
                    q
                  
                  kT
                
              
              )
            
          
          =
          
            
              N
              3
            
            ⁢
            
              (
              
                E
                -
                
                  
                    α
                    0
                  
                  ⁢
                  
                    H
                    q
                  
                
              
              )
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




where





  
    α
    0
  
  =
  
    1
    kT
  




is a reverse spin temperature.

The solution for equation (16) during the impact of the preparatory pulse tw0 (φ=0) looks as follows:




  
    
      ρ
      ⁡
      
        (
        
          t
          
            w
            ⁢
            
              
            
            ⁢
            0
          
        
        )
      
    
    =
    
      
        N
        3
      
      ⁡
      
        [
        
          E
          -
          
            
              α
              0
            
            ⁢
            
              ω
              p
            
            ⁢
            
              n
              0
            
            ⁢
            
              I
              p
            
          
          -
          
            
              α
              0
            
            ⁢
            
              W
              p
            
            ⁢
            
              I
              
                p
                ,
                4
              
            
          
        
        ]
      
    
  
  ,




where n0=(n01, n02, n03) is a unit vector with components:





  
    
      n
      01
    
    =
    
      
        -
        
          
            Δω
            ·
            
              ω
              1
            
          
          
            ω
            e
            2
          
        
      
      ⁢
      
        (
        
          1
          -
          
            cos
            ⁢
            
              
            
            ⁢
            
              φ
              0
            
          
        
        )
      
    
  
  ,
  
    
      n
      02
    
    =
    
      
        
          ω
          1
        
        
          ω
          e
        
      
      ⁢
      sin
      ⁢
      
        
      
      ⁢
      
        φ
        0
      
    
  
  ,
  
    
      n
      03
    
    =
    
      
        
          Δω
          2
        
        +
        
          
            ω
            1
            2
          
          ⁢
          cos
          ⁢
          
            
          
          ⁢
          
            φ
            0
          
        
      
      
        ω
        e
        2
      
    
  
  ,




where φ0=ωetw0.

In the case when ω1>>Δω it can be assumed that:

n01≈0, n02≈sin φ0, n03≈cos φ0.  (18)

All the sequence pulses that follow the preparatory pulse have equal duration tw<<τ and carrier frequency phases




  
    φ
    k
  
  =
  
    
      π
      2
    
    .
  




An effective field ωeff will now be introduced, which substitutes the effect of the RF pulses and the resonance offset, by dependences:




  
    
      
        
          
            
              
                exp
                ⁡
                
                  (
                  
                    
                      -
                      
                        iω
                        eff
                      
                    
                    ⁢
                    
                      I
                      p
                    
                    ⁢
                    2
                    ⁢
                    τ
                  
                  )
                
              
              =
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    
                      -
                      
                        iω
                        eff
                      
                    
                    ⁢
                    
                      k
                      ·
                      
                        I
                        p
                      
                    
                    ⁢
                    2
                    ⁢
                    τ
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        iΔωτ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        
                          p
                          ,
                          3
                        
                      
                    
                    )
                  
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        
                          iω
                          e
                        
                      
                      ⁢
                      
                        I
                        p
                      
                      ⁢
                      
                        t
                        w
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    
                      -
                      iΔωτ
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        3
                      
                    
                  
                  )
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                        
                          
                        
                        ⁢
                        
                          Δω
                          ·
                          τ
                        
                      
                      2
                    
                  
                  -
                  
                    i2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Δω
                        ·
                        τ
                      
                      2
                    
                  
                
                )
              
            
          
        
        
          
            
              
              ⁢
              
                
                  (
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        2
                      
                    
                    -
                    
                      i2e
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        p
                      
                      ⁢
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        2
                      
                    
                  
                  )
                
                ×
              
            
          
        
        
          
            
              
              ⁢
              
                (
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Δω
                        ·
                        τ
                      
                      2
                    
                  
                  -
                  
                    i2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        3
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Δω
                        ·
                        τ
                      
                      2
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




Here φ=ωetw; at ω1>>Δω, so it can be accepted that φ≈ω1tw.
A unit vector is then defined as follows:




  
    k
    =
    
      (
      
        
          k
          1
        
        ,
        
          k
          2
        
        ,
        
          k
          3
        
      
      )
    
  
  ,
  
    
  
  ⁢
  
    k
    =
    
      
        ω
        eff
      
      
        
        
          ω
          eff
        
        
      
    
  
  ,
  
    
  
  ⁢
  
    
      
      
        ω
        eff
      
      
    
    =
    
      
        ω
        eff
      
      .
    
  




Assuming ω1>>Δω, then from (19):




  
    
      
        
          
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    eff
                  
                  ⁢
                  τ
                
                )
              
            
            =
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  φ
                  2
                
                ·
                cos
              
              ⁢
              
                
              
              ⁢
              Δωτ
            
          
          ;
          
            
          
          ⁢
          
            
              k
              1
            
            =
            0
          
          ;
          
            
          
          ⁢
          
            
              k
              2
            
            =
            
              -
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    φ
                    2
                  
                
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ω
                    eff
                  
                  ⁢
                  τ
                
              
            
          
          ;
        
        ⁢
        
          

        
        ⁢
        
          
            k
            3
          
          =
          
            
              1
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  ω
                  eff
                
                ⁢
                τ
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            
              φ
              2
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            
              Δωτ
              .
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




Here for the sake of convenience, an effective time-independent Hamiltonian is defined as:

Heff=ωeffIp.  (21)

From the moment t=0, when the induction signal reaches its maximum, the evolution of the spin system starts to be determined by the Hamiltonian (21) as:

ρ(2nτ)=exp(−iHeff·2nτ)·ρ(2τ)·exp(iHeff·2nτ)  (22)

The evolution of the system set by expression (22), is equivalent to the rotation of the operator of the angular momentum described by expression:




  
    
      
        
          
            
              
                I
                p
                ′
              
              =
              
              ⁢
              
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        iθ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        p
                      
                    
                    )
                  
                
                ⁢
                
                  I
                  p
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        +
                        i
                      
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                      ⁢
                      
                        
                      
                      ⁢
                      k
                      ⁢
                      
                        
                      
                      ⁢
                      
                        I
                        p
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  k
                  ⁡
                  
                    (
                    
                      k
                      ·
                      
                        I
                        p
                      
                    
                    )
                  
                
                +
                
                  
                    (
                    
                      
                        I
                        p
                      
                      -
                      
                        k
                        ⁡
                        
                          (
                          
                            
                              k
                              ·
                            
                            ⁣
                            
                              I
                              p
                            
                          
                          )
                        
                      
                    
                    )
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
                -
                
                  
                    (
                    
                      k
                      ×
                      
                        I
                        p
                      
                    
                    )
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    θ
                    .
                  
                
              
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




Thus, assuming θ=2nτωeff and bearing in mind equations (17) and (23), the following arises:




  
    
      
        
          
            
              
                ρ
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    τ
                  
                  )
                
              
              =
              
              ⁢
              
                
                  -
                  
                    
                      
                        α
                        0
                      
                      ⁢
                      
                        ω
                        p
                      
                      ⁢
                      N
                    
                    3
                  
                
                ⁢
                
                  n
                  0
                
                ⁢
                
                  I
                  p
                  ′
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                -
                
                  
                    
                      
                        α
                        0
                      
                      ⁢
                      
                        ω
                        p
                      
                      ⁢
                      N
                    
                    3
                  
                  [
                  
                    
                      I
                      
                        p
                        ,
                        1
                      
                    
                    ⁢
                    
                      {
                      
                        
                          
                            -
                            
                              n
                              02
                            
                          
                          ⁢
                          
                            k
                            3
                          
                          ⁢
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  ω
                                  eff
                                
                                ⁢
                                2
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                τ
                              
                              )
                            
                          
                        
                        +
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                ⁢
                
                  
                    n
                    03
                  
                  ⁢
                  
                    k
                    2
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          eff
                        
                        ⁢
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                
                }
              
              +
              
                
                  I
                  
                    p
                    ,
                    2
                  
                
                ⁢
                
                  {
                  
                    
                      
                        n
                        02
                      
                      ⁢
                      
                        k
                        2
                        2
                      
                    
                    +
                    
                      
                        n
                        02
                      
                      (
                      
                        1
                        -
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      
                      ⁢
                      
                        k
                        2
                        2
                      
                      )
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            ω
                            eff
                          
                          ⁢
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        )
                      
                    
                  
                  +
                  
                    
                      n
                      03
                    
                    ⁢
                    
                      k
                      2
                    
                    ⁢
                    
                      
                        k
                        3
                      
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            cos
                            ⁢
                            
                              (
                              
                                
                                  ω
                                  eff
                                
                                ⁢
                                2
                                ⁢
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                τ
                              
                              )
                            
                          
                        
                        )
                      
                    
                  
                
                }
              
              +
            
          
        
        
          
            
              
              ⁢
              
                
                  I
                  
                    p
                    ,
                    3
                  
                
                ⁢
                
                  {
                  
                    
                      
                        n
                        02
                      
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        
                          k
                          3
                        
                        ⁡
                        
                          (
                          
                            1
                            -
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    ω
                                    eff
                                  
                                  ⁢
                                  2
                                  ⁢
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  τ
                                
                                )
                              
                            
                          
                          )
                        
                      
                    
                    +
                    
                      
                        n
                        03
                      
                      ⁢
                      
                        k
                        3
                        2
                      
                    
                    +
                  
                
              
            
          
        
        
          
            
              
                
                  
                  ⁢
                  
                    
                      
                        n
                        03
                      
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            k
                            3
                            2
                          
                        
                        )
                      
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            ω
                            eff
                          
                          ⁢
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        )
                      
                    
                  
                  }
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




The echo is always proportional to n02, but not all parts of the density matrix which are proportional to n02 describe the echo. To separate the echo from the received expression, it is necessary to determine the density matrix ρ′(2nτ) for sequence (φ0)0°−(τ−φ0°−τ)n, which contains identical expressions for all components of the density matrix proportional to n02, with the exception of the fact that the echo components of the operator ρ′(2nτ) have the opposite sign.
Repeating the above procedure results in:




  
    
      
        
          
            ρ
            ′
          
          ⁡
          
            (
            
              2
              ⁢
              n
              ⁢
              
                
              
              ⁢
              τ
            
            )
          
        
        =
        
          -
          
            
              
                
                  
                    α
                    0
                  
                  ⁢
                  
                    ω
                    p
                  
                  ⁢
                  N
                
                3
              
              [
              
                
              
              ⁢
              
                
                  
                    I
                    
                      p
                      ,
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          -
                          
                            k
                            
                              2
                              ⁢
                              echo
                            
                          
                        
                        ⁢
                        
                          k
                          3
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              2
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              
                                τω
                                eff
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          k
                          
                            3
                            ⁢
                            echo
                          
                        
                        ⁢
                        
                          k
                          2
                        
                        ⁢
                        
                          
                            k
                            3
                          
                          ⁡
                          
                            (
                            
                              1
                              -
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    2
                                    ⁢
                                    n
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      τω
                                      eff
                                    
                                  
                                  )
                                
                              
                            
                            )
                          
                        
                      
                    
                    }
                  
                
                +
                
                  
                    I
                    
                      p
                      ,
                      2
                    
                  
                  (
                  
                    
                      
                        k
                        
                          2
                          ⁢
                          echo
                        
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            
                              τω
                              eff
                            
                          
                          )
                        
                      
                    
                    -
                    
                      
                        k
                        
                          3
                          ⁢
                          echo
                        
                      
                      ⁢
                      
                        k
                        2
                      
                      ⁢
                      
                        sin
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            
                              τω
                              eff
                            
                          
                          )
                        
                      
                    
                  
                  }
                
                +
                
                  
                    I
                    
                      p
                      ,
                      3
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          k
                          
                            2
                            ⁢
                            echo
                          
                        
                        ⁢
                        
                          k
                          2
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              2
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              
                                τω
                                eff
                              
                            
                            )
                          
                        
                      
                      +
                      
                        
                          k
                          
                            3
                            ⁢
                            echo
                          
                        
                        ⁢
                        
                          k
                          3
                          2
                        
                      
                      +
                      
                        
                          
                            k
                            
                              3
                              ⁢
                              echo
                            
                          
                          ⁡
                          
                            (
                            
                              1
                              -
                              
                                k
                                3
                                2
                              
                            
                            )
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              2
                              ⁢
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              
                                τω
                                eff
                              
                            
                            )
                          
                        
                      
                    
                    }
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        25
        )
      
    
  




Then, leaving in expressions (24) and (25) only parts proportional to n02, and subtracting them one from the other, the echo part of the density matrix is derived as:




  
    
      
        
          
            ρ
            echo
          
          ⁡
          
            (
            
              2
              ⁢
              n
              ⁢
              
                
              
              ⁢
              τ
            
            )
          
        
        =
        
          
            
              -
              
                
                  
                    α
                    0
                  
                  ⁢
                  
                    ω
                    p
                  
                  ⁢
                  N
                
                3
              
            
            ·
            2
          
          ⁢
          
            n
            02
          
          ⁢
          
            k
            2
            2
          
          ⁢
          
            I
            
              p
              ,
              2
            
          
          ⁢
          
            
              
                sin
                2
              
              ⁡
              
                (
                
                  
                    ω
                    eff
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  τ
                
                )
              
            
            .
          
        
      
    
    
      
        (
        26
        )
      
    
  




Both the in-phase and quadrature components of the echo after the n-th pulse of the sequence are:




  
    
      
        
          
            
              M
              1
            
            ⁡
            
              (
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                τ
              
              )
            
          
          =
          
            
              2
              ⁢
              
                Tr
                ⁡
                
                  (
                  
                    
                      
                        ρ
                        echo
                      
                      ⁡
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        )
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        1
                      
                    
                  
                  )
                
              
            
            =
            0
          
        
        ,
        
          

        
        ⁢
        
          
            
              M
              2
            
            ⁡
            
              (
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                τ
              
              )
            
          
          =
          
            2
            ⁢
            
              Tr
              ⁡
              
                (
                
                  
                    
                      ρ
                      echo
                    
                    ⁡
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                  ⁢
                  
                    I
                    
                      p
                      ,
                      2
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        27
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                -
                
                  
                    
                      α
                      0
                    
                    ⁢
                    
                      ω
                      p
                    
                    ⁢
                    N
                  
                  3
                
              
              ·
              
                
                  2
                  ⁢
                  
                    
                      sin
                      ⁢
                      
                        
                      
                    
                    2
                  
                  ⁢
                  
                    
                      (
                      
                        
                          ω
                          eff
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                      )
                    
                    ·
                    
                      sin
                      2
                    
                  
                  ⁢
                  
                    φ
                    2
                  
                
                
                  
                    sin
                    2
                  
                  ⁡
                  
                    (
                    
                      
                        ω
                        eff
                      
                      ⁢
                      τ
                    
                    )
                  
                
              
              ·
              sin
            
            ⁢
            
              
            
            ⁢
            
              
                φ
                0
              
              .
            
          
        
      
    
    
      
        
      
    
  




The density matrix at arbitrary point t, within the pulse interval (FIG. 1) equals:

ρecho([2nτ−τ]+t′)=exp(iHeff·(t′−τ))ρecho(2nτ)exp(−iHeff·(t′−τ)).  (28)

From (28) finally received is:




  
    
      
        
          
            
              
                
                  M
                  1
                
                ⁡
                
                  (
                  
                    
                      [
                      
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        -
                        τ
                      
                      ]
                    
                    +
                    
                      t
                      ′
                    
                  
                  )
                
              
              =
              
              ⁢
              
                2
                ⁢
                
                  Tr
                  ⁡
                  
                    (
                    
                      
                        
                          ρ
                          echo
                        
                        ⁡
                        
                          (
                          
                            
                              [
                              
                                
                                  2
                                  ⁢
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  τ
                                
                                -
                                τ
                              
                              ]
                            
                            +
                            
                              t
                              ′
                            
                          
                          )
                        
                      
                      ⁢
                      
                        I
                        
                          p
                          ,
                          1
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    -
                    
                      
                        
                          α
                          0
                        
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        N
                      
                      3
                    
                  
                  ·
                  
                    n
                    02
                  
                
                ⁢
                
                  
                    k
                    2
                    2
                  
                  ·
                  
                    
                      sin
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          ω
                          eff
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                  ·
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      Δω
                      ⁡
                      
                        (
                        
                          
                            t
                            ′
                          
                          -
                          τ
                        
                        )
                      
                    
                    )
                  
                
                ,
              
            
          
        
        
          
            
              
                
                  M
                  2
                
                ⁡
                
                  (
                  
                    
                      [
                      
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          τ
                        
                        -
                        τ
                      
                      ]
                    
                    +
                    
                      t
                      ′
                    
                  
                  )
                
              
              =
              
              ⁢
              
                2
                ⁢
                
                  Tr
                  ⁡
                  
                    (
                    
                      
                        
                          ρ
                          echo
                        
                        ⁡
                        
                          (
                          
                            
                              [
                              
                                
                                  2
                                  ⁢
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  τ
                                
                                -
                                τ
                              
                              ]
                            
                            +
                            
                              t
                              ′
                            
                          
                          )
                        
                      
                      ⁢
                      
                        I
                        
                          p
                          ,
                          2
                        
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    -
                    
                      
                        
                          α
                          0
                        
                        ⁢
                        
                          ω
                          p
                        
                        ⁢
                        N
                      
                      3
                    
                  
                  ·
                  
                    n
                    02
                  
                
                ⁢
                
                  
                    k
                    2
                    2
                  
                  ·
                  
                    
                      sin
                      2
                    
                    ⁡
                    
                      (
                      
                        
                          ω
                          eff
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                  ·
                
              
            
          
        
        
          
            
              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      Δω
                      ⁡
                      
                        (
                        
                          
                            t
                            ′
                          
                          -
                          τ
                        
                        )
                      
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        29
        )
      
    
  




The expression (35) is only true for times <T2.
After a time of  ̃3T2 the density matrix ρst is determined totally by the effective Hamiltonian (21). Non-commuting with Heff parts of the density matrix (27) decay and the density matrix ρst equals:




  
    ρ
    st
  
  =
  
    
      
        H
        eff
      
      ⁢
      
        
      
      ⁢
      
        
          Tr
          ⁢
          
            {
            
              
                H
                eff
              
              ⁢
              
                ρ
                ⁡
                
                  (
                  
                    t
                    
                      w
                      ⁢
                      
                        
                      
                      ⁢
                      0
                    
                  
                  )
                
              
            
            }
          
        
        
          Tr
          ⁢
          
            
              {
              
                H
                eff
              
              }
            
            2
          
        
      
    
    =
    
      
        -
        
          
            
              α
              0
            
            ⁢
            
              ω
              p
            
            ⁢
            N
          
          3
        
      
      ·
      
        (
        
          
            
              n
              02
            
            ⁢
            
              k
              2
            
          
          +
          
            
              n
              03
            
            ⁢
            
              k
              3
            
          
        
        )
      
      ·
      
        
          (
          
            k
            ·
            
              I
              p
            
          
          )
        
        .
      
    
  




The echo creating part of the density matrix ρecho-st is proportional to n02:




  
    
      
        
          ρ
          
            echo
            ⁢
            
              -
            
            ⁢
            st
          
        
        =
        
          
            
              -
              
                
                  
                    α
                    0
                  
                  ⁢
                  
                    ω
                    p
                  
                  ⁢
                  N
                
                3
              
            
            ·
            
              n
              02
            
          
          ⁢
          
            
              k
              2
            
            ·
            
              
                (
                
                  k
                  ·
                  
                    I
                    p
                  
                
                )
              
              .
            
          
        
      
    
    
      
        (
        30
        )
      
    
  




For the arbitrary point t′:

ρecho-st(t′)=exp(iHeff·(t′−τ))ρecho-stexp(−iHeff·(t′−τ)).

Consequently:




  
    
      
        
          
            M
            
              1
              ⁢
              
                -
              
              ⁢
              st
            
          
          =
          
            
              2
              ⁢
              
                Tr
                ⁡
                
                  (
                  
                    
                      ρ
                      
                        echo
                        ⁢
                        
                          -
                        
                        ⁢
                        st
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        1
                      
                    
                  
                  )
                
              
            
            =
            
              
                
                  -
                  
                    
                      
                        α
                        0
                      
                      ⁢
                      
                        ω
                        p
                      
                      ⁢
                      N
                    
                    3
                  
                
                ·
                
                  n
                  02
                
              
              ⁢
              
                
                  k
                  2
                  2
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      Δω
                      ⁡
                      
                        (
                        
                          
                            t
                            ′
                          
                          -
                          τ
                        
                        )
                      
                    
                    )
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            M
            
              2
              ⁢
              
                -
              
              ⁢
              st
            
          
          =
          
            
              2
              ⁢
              
                Tr
                ⁡
                
                  (
                  
                    
                      r
                      
                        echo
                        ⁢
                        
                          -
                        
                        ⁢
                        st
                      
                    
                    ⁢
                    
                      I
                      
                        p
                        ,
                        2
                      
                    
                  
                  )
                
              
            
            =
            
              
                
                  -
                  
                    
                      
                        α
                        0
                      
                      ⁢
                      
                        ω
                        p
                      
                      ⁢
                      N
                    
                    3
                  
                
                ·
                
                  n
                  02
                
              
              ⁢
              
                
                  k
                  2
                  2
                
                ·
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δω
                        ⁡
                        
                          (
                          
                            
                              t
                              ′
                            
                            -
                            τ
                          
                          )
                        
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        31
        )
      
    
  




Thus, the maximum amplitude that the echo signal will reach in the sequence window which is removed from the preparatory pulse by the time>T2, equals:




  
    
      
        
          M
          st
        
        =
        
          
            
            
              M
              
                2
                ⁢
                
                  -
                
                ⁢
                st
              
            
            
          
          =
          
            
              
                
                  α
                  0
                
                ⁢
                
                  ω
                  p
                
                ⁢
                N
              
              3
            
            ·
            
              
                
                  
                    sin
                    2
                  
                  ⁢
                  
                    
                      φ
                      2
                    
                    ·
                    
                      
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          0
                        
                      
                      
                    
                  
                
                
                  1
                  -
                  
                    
                      cos
                      2
                    
                    ⁢
                    
                      φ
                      2
                    
                    ⁢
                    
                      cos
                      2
                    
                    ⁢
                    Δωτ
                  
                
              
              .
            
          
        
      
    
    
      
        (
        32
        )
      
    
  




It should be noted that the solution (32) is arrived at with neglecting the influence of the effective relaxation time T2e and the echo signal shape.
Keeping in mind that for most explosive and narcotic substances the NQR signal line has the Lorentz line shape, the real echo-signal after the n-th pulse can be presented as follows:




  
    
      
        
          
            
              
                
                  M
                  echo
                
                ⁡
                
                  (
                  t
                  )
                
              
              =
              
              ⁢
              
                
                  M
                  echo
                
                ⁡
                
                  (
                  
                    
                      
                        [
                        
                          
                            2
                            ⁢
                            n
                          
                          -
                          1
                        
                        ]
                      
                      ⁢
                      τ
                    
                    +
                    
                      t
                      ′
                    
                  
                  )
                
              
            
          
        
        
          
            
              
                =
                
                ⁢
                
                  
                    
                      M
                      st
                    
                    ⁡
                    
                      (
                      
                        t
                        ′
                      
                      )
                    
                  
                  ⁢
                  
                    
                      exp
                      ⁡
                      
                        (
                        
                          -
                          
                            
                              
                                
                                  [
                                  
                                    
                                      2
                                      ⁢
                                      n
                                    
                                    -
                                    1
                                  
                                  ]
                                
                                ⁢
                                τ
                              
                              +
                              
                                t
                                ′
                              
                            
                            
                              T
                              
                                2
                                ⁢
                                e
                              
                            
                          
                        
                        )
                      
                    
                    ·
                    
                      exp
                      ⁡
                      
                        (
                        
                          
                            -
                            
                              1
                              
                                T
                                2
                                *
                              
                            
                          
                          ·
                          
                            
                            
                              
                                t
                                ′
                              
                              -
                              τ
                            
                            
                          
                        
                        )
                      
                    
                  
                
              
              ,
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




where M=M1+iM2 if t≦T2; and M=M1-st+iM2-st if t>T2.

Averaging for powder is performed in accordance with the following equation:




  
    
      
        
          M
          
            echo
            -
            powder
          
        
        =
        
          
            1
            2
          
          ⁢
          
            
              ∫
              
                -
                1
              
              1
            
            ⁢
            
              
                M
                echo
              
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              
                θ
                P
              
              ⁢
              
                d
                
                  (
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      θ
                      P
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  




In FIG. 2a the dependence of the amplitude of the Fourier-transform Mecho-powder on the difference between the carrier frequency of the RF pulses of the SLSE sequence and the resonance transition frequency when the duration of the sequence is  ̃T2 is shown, and FIG. 2b shows the same dependence when the duration of the sequence is >>T2.
The parameters chosen for the theoretical calculation in FIG. 2a are as follows: γH1tw=119°, tw0=80 μs, tw=80 μs, τ=1 ms, T2=260 ms, T2*=250 μs, n=200, which correspond to line v+=5301 kHz of HMX at a temperature of 300° K. The parameters chosen for the theoretical calculation in FIG. 2b are as follows: γH1tw=119°, tw0=80 μs, tw=80 μs, τ=2 ms, T2=3.3 ms, T2*=2.6 ms, n=200. These parameters correspond to line v−=3600 kHz of NaNO2 at a temperature of 300° K.
The comparison of the theoretical calculation with experimental results was done using two powder samples: HMX (line v+=5301 kHz) and NaNO2 (line v−=3600 kHz) at room temperature. The mass of the sample was 50 g for HMX and 40 g for NaNO2. The volume of the detector coil was approximately 1 litre. The peak power of the transmitter was 280 W.
The parameters of the sequence SLSE φ0x−(τ−φy−τ)n with which the sample was irradiated corresponded to those used for the calculation. The result of the dependence of the amplitude of the NQR echo signal on the frequency offset is presented in FIG. 3a for HMX and FIG. 3b for NaNO2.
The theoretical and experimental results demonstrate it is possible to use the echo sequences to eliminate the intensity anomalies.
Now describing specific embodiments of the best mode for carrying out the invention, the best mode for carrying out the invention is concerned with using multi-pulse RF sequences to excite an NQR signal in a substance containing quadrupole nuclei with either integer or half-integer spins for the purposes of detecting such a signal.
The particular apparatus for producing pulse sequences of this kind comprises a pulse generator, the hardware design of which is known, and described in the applicant's corresponding International Patent Application PCT/AU00/01214 (WO 01/25809), which is incorporated herein by reference.
In order to generate a pulse sequence, firstly a pulse programmer is used to create a low voltage level pulse sequence. Such programmer is capable of generating a continuous sine wave of a desired frequency (eg; 0.89 or 5.2 MHz) and of any phase by using a Direct Digital Synthesizer (DDS) or any RF source. To create a pulse sequence, a gate is used to divide the continuous sine wave into small pulses. For example, the gate switches on for  ̃300 μs and off for  ̃300 μs, repeatedly thereby creating a sequence of pulses. The user of the pulse generator generates the pulse sequence via a computer program in the controlling computer. The computer program enables the user to input the frequency, phase, duration and separation of any pulses and allows the user to repeat any parts of the pulse sequence in a loop. The entire pulse sequence is contained in the program and then converted into binary and sent to the pulse programmer and stored in memory. The CPU of the pulse programmer then takes the machine code stored in memory and creates the pulse sequence by changing the frequency and phase of the DDS and providing instructions to the gate as to when to switch, thereby creating the pulses.
A simplified example of the program used to create a pulse sequence is outlined below:

    Set Transmit Frequency: 0.89 MHz
    Set Phase: 0 degrees
    Gate Open
    Wait 300 μs
    Gate Closed (thus first pulse is created 300 μs long of phase 0 degrees)
    Wait 300 μs
    Set Transmit Frequency: 0.89 MHz
    Set Phase: 90 degrees
    For 1000 loops
    
        Gate Open
        Wait 300 μs
        Gate Closed
        Wait 300 μs
    
    
    End of Loop

(thus 1000 additional pulses are created each 300 μs long and spaced 300 μs of a phase 90 degrees).



Secondly, each pulse sequence is transmitted to the coil via a high power amplifier (1→5 kW), which amplifies the low voltage signal created by the pulse programmer to a higher voltage level which is sufficient to stimulate the nitrogen 14 nuclei.
The first embodiment of the invention is directed towards improving the cancelling of temperature effects by using an echo sequence without any additional measures for reducing the induction signals or echo signals appearing at the end of the window and which are not connected with the preparatory pulse. To separate the echo component from the preparatory pulse, the acquisition time is set at an interval of Δt from the preceding pulse and at the same interval from the posterior pulse, as shown in FIG. 4.
The following echo sequences are examples of the use of the first preferred embodiment:

    
    
        SLSE (φ0)0°−τ−(φ90°−τ)n,
        MW-2(φ0)0°−τ−(φ0°−2τ−φ180°−2τ)n,
        WAHUHA-4 (φ0)0°−τ−(φ180°−τ−φ90°−2τ−φ270°−τ−φ0°−2τ)n,
        φ0(45°)−τ−(φ0°−2τ−φ90°−2τ−180°−2τ−φ270°−2τ)n.
    
    


The difference between the above sequences and their steady-state analogs is that pulse separation in these sequences is at least several times longer than the T2* time.
The types of echo signals that occur when using sequences SLSE (φ0)0°−τ−(φ90°−2τ[90°])n1, MW-2(φ0)0°−τ−(φ0°−2τ−180°−2τ[180°])n2 and WAHUHA-4(φ0)0°−τ−(φ180°−τ−φ90°−2τ−φ270°−τ−φ0°−2τ[0°])n3 for NaNO2 powder are presented in FIGS. 5a, 5b and 5c respectively (square brackets in the sequence formulas contain the receiver phase).
Sequence parameters are as follows: τ=1980 μs, tw0=80 μs, tw=172 μs, φ0=γH1tw0≈119°, φ=γH1tw>257°, n=30.
The echo signal was measured only after the last pulse of the cycle. The length of the offset between the last pulse of the offset and acquisition time, and also between acquisition time and the first pulse of the following cycle (see FIG. 4) for all sequences was chosen to be the same Δt=870 μs. The duration of the acquisition time was also the same and equalled 3072 μs. All measurements were carried out at a temperature of 21° C.
The second embodiment of the invention is directed towards improving the diminishing effects of temperature by using a combination of sequences organised so as to cancel all signals, except for the echo signals generated by the preparatory pulse.
The third embodiment is directed towards a variant of the second embodiment, where the combination contains two sequences, the first sequence containing a preparatory pulse and the second sequence being different from the first by at least one parameter of the preparatory pulse, or the second sequence not having a preparatory pulse at all.
The receiver phases during the action of the second sequence follow the phases of the first sequence with an additional phase shift of 180°.
Examples of use of the third embodiment are:

(φ0)0°−τ−(φ90°−2τ[0°])n and (φ90°−2τ[180°])n,

(φ0)0°−τ−(φ90°−2τ[0°])n and (φ0)180°−τ−(φ90°−2τ[180°])n.

Here the square brackets contain the phase of the receiver.
The fourth embodiment is directed towards a second variant of the second embodiment, where the combination consists of two sequences that differ by the phases of all pulses, except the preparatory ones, by 180°. The preparatory pulse in the second sequence can differ from the preparatory pulse of the first sequence by any number of parameters or can coincide with it completely. The receiver phases for both sequences are identical.
Some examples of the third embodiment are:

(φ0)0°−τ−(φ90°−2τ[0°])n and (φ270°−2τ[0°])n,

(φ0)0°−τ−(φ90°−2τ[0°])n and (φ0)0°−τ−(φ270°−2τ[0°])n.

The third and the fourth embodiments are basically applicable to those sequences where the duration T lies within the limits T2<<T<T2e.
The fifth embodiment is directed towards a third variant of the second embodiment, where the combinations used consist of four sequences which are differentiated by the pulse phases as shown in the following Table.











TABLE 1








Phase of the n-th




Phase of the
pulse of the
Receiver



preparatory pulse
sequence
phase



















1st sequence




  φ
  0








  φ
  n








  φ
  reciever






 


2nd sequence




  φ
  0








  
    φ
    n
  
  +
  π








  φ
  reciever






 


3rd sequence




  φ
  0








  
    φ
    n
  
  +
  
    π
    2
  








  
    φ
    reciever
  
  +
  π






 


4th sequence




  φ
  0








  
    φ
    n
  
  -
  
    π
    2
  








  
    φ
    reciever
  
  +
  π













The fifth embodiment is intended mainly for sequences which have a duration T comparable with the relaxation time T2.
An example of the fifth embodiment of the SLSE sequences is:

(φ0)0°−τ−(φ90°−2τ[0°])n, (φ0)0°−τ−(φ270°−2τ[0°])n,

(φ0)0°−τ−(φ180°−2τ[180°])n and (φ0)0°τ−(φ0°−2τ[180°])n.

It is preferable to set the time interval between the sequences  ̃T1.
It should be appreciated that the scope of the present invention is not limited by the specific embodiments described herein.

## Claims

1. A nuclear quadrupole resonance apparatus for detecting a target substance containing quadrupole nuclei, comprising:
a pulse sequence generator to generate a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2, and to generate a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences;
a detector to detect nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
a combiner to combine the detected nuclear quadrupole resonance signals generated under conditions of unknown temperature of the target substance into a resulting signal such that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses, thereby mitigating the effect of intensity variations.

2. A method of nuclear quadrupole resonance for detecting a target substance containing quadrupole nuclei, the method comprising the steps of:
generating a combination of multi-pulse sequences, at least one of which contains a preparatory pulse or group of preparatory pulses, having intervals between pulses longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2;
irradiating the target substance with the combination of multi-pulse sequences;
detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
combining the detected nuclear quadrupole resonance signals generated under conditions of unknown temperature of the target substance into a resulting signal where echo generated by the preparatory pulse or group of preparatory pulses is a prevailing components, thereby mitigating the effect of intensity variations.

3. A method as claimed in claim 2, wherein the combination contains at least one sequence.

4. A method as claimed in claim 2, wherein the flip angles of pulses of sequences of the combination range between 60° and 345°.

5. A method as claimed in claim 2, wherein at least one preparatory pulse is composite.

6. A method as claimed in claim 2, wherein all pulse sequences of the combination contain a preparatory pulse or a group of preparatory pulses.

7. A method as claimed in claim 2, wherein the carrier frequency of the pulse sequences is near to one of NQR frequencies of the target substance to be detected.

8. A method as claimed in claim 2, wherein the target substance to be detected comprises an explosive containing quadrupole nuclei.

9. A method as claimed in claim 2, wherein the target substance to be detected comprises a narcotic containing quadrupole nuclei.

10. An apparatus for producing a multi-pulse sequence for irradiating a target substance containing quadrupole nuclei, comprising:
a generator to generate a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2, and containing a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses; and
a detector to detect nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences.

11. A nuclear quadrupole resonance apparatus for detecting a target substance containing quadrupole nuclei, comprising:
means for generating a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2, and for generating a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences;
means for detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences; and
means for combining the detected nuclear quadrupole resonance signals into a resulting signal such that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses, thereby mitigating the effect of intensity variations.

12. An apparatus for producing a multi-pulse sequence for irradiating a target substance provided with quadrupole nuclei, comprising:
generator means for generating a combination of multi-pulse sequences with pulse intervals longer than the time constant of free induction decay T2*, but shorter than the time of spin-spin relaxation T2, and containing a preparatory pulse or group of preparatory pulses for creating echo signals between the pulses of the multi-pulse sequences organised so that the resulting signal contains a prevailing echo component produced by the preparatory pulse or group of preparatory pulses; and
detector means for detecting nuclear quadrupole resonance signals generated from the target substance under conditions of unknown temperature of the target substance in response to the combination of multi-pulse sequences.