Studying multi-nucleon transfer reaction in a recoil mass spectrometer

Abstract

We present simulation of a recoil mass spectrometer for measurement of multi-nucleon transfer reactions, based on Monte Carlo techniques. Target-like ions are generated event by event and transported to the focal plane of the spectrometer by the method of transfer matrices. Measured focal plane spectra for the elastic and six transfer channels of the reaction $^{28}$Si+$^{94}$Zr are reproduced at two projectile energies near the Coulomb barrier. Excellent reproduction is achieved even with first-order ion-optical calculations, indicating not so significant role of higher order aberrations in conventional recoil separators with smaller acceptance. This is in striking contrast with the large acceptance magnetic separators used for studying transfer reactions, for which complex trajectory reconstruction algorithms involving higher order aberrations are essential. Transmission efficiency of the spectrometer, calculated by the reported code, is crucial to convert measured transfer probabilities to differential and integral transfer cross sections for each channel. We show effectiveness of the proposed methodology by extracting differential elastic scattering cross sections from measured yields for the reaction $^{28}$Si+$^{94}$Zr. Besides being useful for other recoil mass spectrometers, our methodology can be employed for measurement of quasi-elastic reaction cross sections in velocity filters and gas-filled separators.

Formalism for extraction of differential quasi-elastic scattering cross sections from measured yields in an RMS

We consider measurement of differential quasi-elastic scattering cross section using a recoil mass spectrometer (RMS) like the HIRA. In the experiment, the target-like ions, moving in the forward direction (in the laboratory frame of reference) are detected in the focal plane (FP) of the RMS, instead of detecting the projectile-like ions in the backward direction. Two monitor detectors are used to detect elastically-scattered projectiles for absolute normalization of cross sections. We also assume that both projectile-like and target-like ions are recorded simultaneously for a duration t.

Yield of projectile-like ions recorded by the normalization detector (placed at an angle, say, $\theta _{\text {norm}}$ in the centre of mass (c.m.) frame of reference) is given by

$$\begin{aligned} Y^{\text {Ruth}}_{\text {norm}} = \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{\theta _{\text {norm}}}\ . \ \varOmega _{\text {norm}}\ .\ i\ .\ d.\ t \end{aligned}$$

(A.1)

where $\varOmega _{\text {norm}}$, i and d are acceptance of the normalization detector, beam intensity and thickness of the target, respectively. The differential Rutherford scattering cross section at an angle $\theta _{\text {c.m.}}$ (in units of mb/sr) is computed by the well-known relation

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{\theta _{\text {c.m.}}} = 1.296\ .\ \left( \frac{Z_{\text {p}}Z_{\text {t}}}{E_{\text {c.m.}}}\right) ^{2} \ .\ \frac{1}{\sin ^{4}\frac{\theta _{\text {c.m.}}}{2}}\ . \end{aligned}$$

(A.2)

Here, $E_{\text {c.m.}}$ is the energy available for collision in the c.m. frame of reference and $Z_{\text {p}}$ and $Z_{\text {t}}$ are the atomic numbers of projectile and target nuclei, respectively.

In the c.m. frame of reference, projectile-like and target-like ions fly $180^{\circ }$ apart from each other. We further assume here that the measurement is carried out at an incident energy sufficiently below the Coulomb barrier, so that all scattering events can be described by Rutherford scattering (i.e. by Eq. A.2). With this assumption, the yield of target-like ions, recorded at the FP of the RMS (operated at an angle, say, $\theta _{\text {RMS}}$ in the c.m. frame of reference) can be described by the relation

$$\begin{aligned} Y^{\text {Ruth}}_{\text {RMS}} = \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{180^{\circ } - \theta _{\text {RMS}}}\ .\ \varOmega _{\text {RMS}}^{\text {eff}}\ .\ \varepsilon \ .\ i\ .\ d\ .\ t \end{aligned}$$

(A.3)

where $\varOmega _{\text {RMS}}^{\text {eff}}$ and $\varepsilon$ are effectve solid angle and transmission efficiency of the RMS, respectively.

Eliminating the product $(i\ .\ d\ .\ t)$, between Eqs. A.1 and A.3, and rearranging, one can easily get

$$\begin{aligned} \varOmega ^{\text {eff}}_{\text {RMS}} = \frac{Y^{\text {Ruth}}_{\text {RMS}}}{Y^{\text {Ruth}}_{\text {norm}}}\ .\ \varOmega _{\text {norm}}\ .\ \frac{1}{\varepsilon }\ .\ \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{\theta _{\text {norm}}} \ /\ \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{180^{\circ } - \theta _{\text {RMS}}}\ . \end{aligned}$$

(A.4)

One may note here that the ratio of the two differential cross sections on the right hand side of Eq. A.4 boils down to $\frac{\sin ^{4}\left( \frac{180^{\circ } - \theta _{\text {RMS}}}{2}\right) }{\sin ^{4}\frac{\theta _{\text {norm}}}{2}}$.

To obtain the differential cross-section from measured yields for any other quasi-elastic (i.e. elastic, inelastic, transfer) channel, the following relation is used

$$\begin{aligned} \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {qel}}_{180^{\circ } - \theta _{\text {RMS}}} = \frac{Y_{\text {RMS}}}{Y_{\text {norm}}}\ .\ \frac{\varOmega _{\text {norm}}}{\varOmega ^{\text {eff}}_{\text {RMS}}}\ .\ \frac{1}{\varepsilon }\ .\ \left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}_{\theta _{\text {norm}}}\ .\nonumber \\ \end{aligned}$$

(A.5)

In deriving quation A.5, it has been assumed that the differential cross sections do not vary significantly within the respective acceptances. While the assumption is quite justified for $\left( \frac{d\sigma }{d\varOmega }\right) ^{\text {Ruth}}$ (as the acceptance of the normalization detector is usually very small), the same may not be strictly valid for $\left( \frac{d\sigma }{d\varOmega }\right) ^{\text {qel}}$ (because of finite acceptance of the RMS) in every situation.